On Geometric-Analytic Aspects of Solvable Nonlinear Ordinary Differential Equations and Some Applications

Abstract

A geometric-analytic approach to studying invariants of solvable nonlinear ordinary differential equations is developed. In particular, there is described in detail a general scheme of constructing solvable nonlinear ordinary differential equations, based on a linear differential spectral problem and its related invariants. Examples of nonlinear differential equations applications are discussed, generalizing those previously studied in the literature. The analytical properties of the invariants and determining the Noether-Lax evolution equation, including its asymptotic properties, are analyzed in detail. Some interesting from a practical point examples of the second ordinary differential equations are analyzed in detail, including the classical Van der Pol and Painlevé equations. The backgrounds of the isolvability problem are also presented and applied to ordinary super-differential equations on the superaxis, which are of interest for research in the field of modern quantum physics.

1. Introduction

As is well known, the famous Newton’s secret message sent to the Royal Society of London sounds as “Data aeqeatione quotcunque fluents quantitatflue involvente flusiones invenire et vice versa” (in Latin), meaning “that it is very useful to solve differential equations”. The modern theory of differential equations presents a hardly observable set of a great number of ideas, approaches, methods, tricks and artificial speculations regarding solving the special classes of differential equations. For example, Lie symmetry analysis, Jacobi multiplier Prelle-Singer and Darboux transformations methods [1,2,3,4,5,6,7,8,9] are widely used in modern diverse applications. Having analyzed most of these existing methods subject to algorithmic solving the second order nonlinear ordinary differential equations with polynomial coefficients, we observed that, in fact, almost all of them do not make use of the existing modern techniques and analytical tools developed for investigating such as the important characteristic as invariants. Moreover, there are also almost neglected important subtle geometric properties of such equations as flows on the cotangent space $T^{*}\left(\mathbb{R}\right)$ to the base space $\mathbb{R},$ deeply related with the well-known [10,11] Legendre transformation from the tangent space $T\left(\mathbb{R}\right)$ to the cotangent space $T^{*}\left(\mathbb{R}\right)$ and so on. In particular, we demonstrated the importance of such a characteristic of invariants as their gradients, governed by the Noether–Lax-type linear evolution equation and allowing to retrieve them via the classical Volterra homotopy formula.

In our work, we also presented a systematic geometric–analytic approach, which takes its origin in classical works by S. Novikov and O. Bogoyavlensky [12,13] to describe nonlinear solvable ordinary differential equations, naturally generated by spectral invariants of generalized linear spectral problems on the axis. Taking into account great importance of invariants of nonlinear ordinary differential equations, we re-analyzed asymptotic solutions to the corresponding Noether–Lax-type equation, which proves to be very useful both for analytic studying the integrability properties of a given equation and for analytic construction its invariants. Keeping in mind all of them, we described two of these analytic approaches to constructing invariants to the nonlinear second order ordinary differential equations on the axis.

The interesting from practical point examples of the second ordinary differential equations are analyzed in detail, including the classical van der Pol equation. As a natural and interesting for modern mathematical physics generalization of the devised approach we presented a short introduction into the integrability theory of ordinary super-differential equations on the super-axis [14,15,16,17], in part initiated in [18,19], and having applications [17,20,21,22] for studying quantum mechanical fermionic models of quantum physics [18,19] and gravity [23] like instantons and solitons.

2. A Generalized Linear Spectral Problem, Its Differential Invariants and Related Solvable Nonlinear Ordinary Differential Equations

Consider a smooth $2\pi$-periodic functional manifold $X=C^r(\mathbb{R}/\left\{2\pi \mathbb{Z}\right\};{\mathbb{R}}^m),r\in \mathbb{N},$ and a vector fibre bundle $E(X,C^1(\mathbb{R}/\left\{2\pi \mathbb{Z}\right\};{\mathbb{C}}^n))$ over the manifold X with the complex valued space $C^1(\mathbb{R}/\left\{2\pi \mathbb{Z}\right\};{\mathbb{C}}^n)$ as its bundle jointly with a linear generalized rational in $\lambda \in \mathbb{C}$ differential spectral problem of the first order:

$$df/dt=l[x;\lambda ]f,$$

(1)

with respect to the temporal parameter $t\in \mathbb{R},$ where $f\in L_{\infty }(\mathbb{R};{\mathbb{C}}^k),$ a matrix $l[x;\lambda ]\in \mathrm{End}$$\left(L_{\infty }(\mathbb{R};{\mathbb{C}}^n)\right),$$x\in X,$ is the corresponding $2\pi$-periodic local matrix functional on the manifold X and $\lambda \in \mathbb{C}$ is a complex spectral parameter. Regarding the $2\pi$-periodic linear Equation (1) one can determine the fundamental matrix solution $F:=F(t,t_0;\lambda )$$\in \mathrm{End}\left(L_{\infty }(\mathbb{R};{\mathbb{C}}^n)\right)$ satisfying the matrix equation

$$dF/dt=l[x;\lambda ]F$$

(2)

and normalized by the unit matrix at an arbitrary point $t=t_0\in \mathbb{R},$ that is $F(t_0,t_0;\lambda )=I\in \mathrm{End}$${\mathbb{C}}^k$ for all $x_0\in \mathbb{R},$ $\lambda \in \mathbb{C}.$

Based on the linearity of (2), it is evident that any solution to the Equation (1) can be represented in the form

$$f=F(t,t_0;\lambda )f_0,$$

(3)

where $f_0\in {\mathbb{C}}^n$ is an initial vector value at the point $t_0\in \mathbb{R}.$ By virtue of the $2\pi$-periodicity in the variable $t\in \mathbb{R}$ of the matrix $l[x;\lambda ]\in \mathrm{End}{L}_{\infty }(\mathbb{R};{\mathbb{C}}^n),$ we easily obtain that for arbitrary integer $N\in \mathbb{Z}$ the following Floquet [24] properties:

$$f(t_0\pm 2\pi N,t_0;\lambda )=S^{\pm N}(t_0;\lambda )f_0,$$

(4)

hold, where $S(t_0;\lambda ):=F(t_0+2\pi ,t_0;\lambda )\in \mathrm{End}$${\mathbb{C}}^n,t_0\in \mathbb{R},$ is the so-called monodromy matrix of the matrix differential Equation (1), defining simultaneosuly [13,24,25,26] a smooth nonlinear functional on the functional manifold $X,$ meromorphically depending on the spectral parameter $\lambda \in \mathbb{C}.$

Let $\xi \left(\lambda \right)\in \sigma \left(S\right)\subset \mathbb{C}$ be an eigenvalue of the monodromy matrix $S(t_0;\lambda )\in \mathrm{End}$${\mathbb{C}}^n,t_0\in \mathbb{R}.$ Then from (4) it directly follows [25,27,28] that the solution $f(·,t_0;\lambda )\in C(\mathbb{R};{\mathbb{C}}^n)$ is bounded on the whole axis $\mathbb{R}$ if and only if any eigenvalue $\xi \left(\lambda \right)\in \sigma \left(S\right)$ of the monodromy matrix $S(t_0;\lambda )$ has the absolute value equal to one, that is $\left|\xi \right(\lambda \left)\right|=1.$

Lemma 1.

The spectrum $\sigma \left(S\right)\subset \mathbb{C}$ of the periodic problem (1) does not depend on a point $t_0\in \mathbb{R}$ and generates the fool system of invariants to the periodic linear problem (1).

Proof. 

To demonstrate that the eigenvalues $\xi \left(\lambda \right)\in \sigma \left(S\right)$ do not depend on a point $t_0\in \mathbb{R},$ we derive the corresponding differential equation for monodromy matrix $S(t_0;\lambda )\in \mathrm{End}$ ${\mathbb{C}}^n$:

$$\begin{array}{c}\partial S(t_0;\lambda )/\partial t_0=\partial F(t_0+2\pi ,t_0;\lambda )/\partial t_0=\\ =\partial F(t+2\pi ,t_0;\lambda )/\partial {t|}_{t=t_0}+\partial F(t+2\pi ,t_0;\lambda )/\partial {t_0|}_{t=t_0}.\end{array}$$

(5)

Taking into account that

$$F(t+2\pi ,t_0;\lambda )=I+{\int }_{t_0}^{t+2\pi }l[x\left(s\right);\lambda ]F(s,t_0;\lambda )ds$$

(6)

and

$$\partial (\partial F/\partial t_0)/\partial t=l[x;\lambda ](\partial F/\partial t_0),$$

(7)

for any points $t,t_0\in \mathbb{R},$ one easily derives from (6) that

$$\begin{array}{c}\partial F(t+2\pi ,t_0;\lambda ){/\partial t|}_{t=t_0}={\left.l[x(t+2\pi );\lambda ]\partial F(t+2\pi ,t_0;\lambda )/\partial t_{0}ds\right|}_{t=t_0}=\\ =l[x\left(t_0\right);\lambda ]F(t_0+2\pi ,t_0;\lambda )=l[x\left(t_0\right);\lambda ]S(t_0;\lambda ),\end{array}$$

(8)

and from (3), (6), and (7)

$$\begin{array}{ccc}\hfill \partial F(t,t_0;\lambda )/\partial t_0{|}_{t=t_0}& =& {\left.{\int }_{t_0}^{t}l[x\left(s\right);\lambda ]\partial F(s,t_0;\lambda )/\partial t_{0}ds\right|}_{t=t_0}-\hfill \\ \hfill -l[x\left(t_0\right);\lambda ]\partial F(t_0,t_0;\lambda )& =& -l[x\left(t_0\right);\lambda ],\hfill \\ \hfill \partial F(t+2\pi ,t_0;\lambda )/\partial t_0{|}_{t=t_0}& =& F(t_0+2\pi ,t_0;\lambda )\partial F(t,t_0;\lambda )/\partial t_0{|}_{t=t_0}=\hfill \\ & =& S(t_0;\lambda )l[x\left(t_0\right);\lambda ].\hfill \end{array}$$

(9)

As a result of expressions (8) and (9), we obtain the following Novikov–Marchenko commutator equation

$$\partial S(t_0;\lambda )/\partial t_0=[l[x\left(t_0\right);\lambda ],S(t_0;\lambda )]$$

(10)

where $[·,·]$ denotes here the usual matrix commutator in the space $\mathrm{End}$${\mathbb{C}}^n.$ From Equation (10), we conclude that the traces $\mathrm{tr}$$S^k(t_0;\lambda ),$$k\in \mathbb{Z},$ do not depend on a point $t_0\in \mathbb{R}.$ This means, equivalently, that the eigenvalue $\xi \left(\lambda \right)\in$$\sigma \left(S\right)$ of the monodromy matrix $S(t_0;\lambda )\in \mathrm{End}$${\mathbb{C}}^n$ does not depend on a point $t_0\in \mathbb{R},$ that is

$$\partial \xi \left(\lambda \right)/\partial t_0=0.$$

Taking, in addition, into account the invariance of the spectrum $\sigma \left(S\right)\subset \mathbb{C}$ on the related gauge type transformations of the linear problem (1), we obtain [29] the completeness of obtained invariants, thus proving the lemma. □

For more detail examination of the mondromy matrix spectrum $\sigma \left(S\right)\subset \mathbb{C}$ properties, it is necessary to concretize the form of the matrix $l[x;\lambda ]\in \mathrm{End}$${\mathbb{C}}^n$ for $x\in X$ and its dependence on the spectral parameter $\lambda \in \mathbb{C},$ as it is presented in the classical manuals [13,24,26,30,31].

Recall now that the monodromy matrix eigenvalue $\xi \left(\lambda \right)\in$$\sigma \left(S\right)$ is smooth and invariant with respect to the variable $t_0\in \mathbb{R}$ functional on the functional manifold $X.$ To study its variational properties with respect to the functional variable $x\in X,$ we will analyze the related variational properties of the fundamental matrix $F(t,t_0;\lambda )\in \mathrm{End}$${\mathbb{C}}^n,$ satisfying the linear differential matrix Equation (2). Since the fundamental matrix $F(t,t_0;\lambda )\in \mathrm{End}$${\mathbb{C}}^n$ is also a functional on the linear manifold $X,$ then from the Equation (2) one can easily obtain the corresponding equation for its variation $\delta F$$\in \mathrm{End}$${\mathbb{C}}^n,$ when the value $x\in X$ changes to $(x+\delta x)\in X.$ Hence, we have

$$\partial \left(\delta F\right)/\partial t=l[x;\lambda ]\delta F+\delta l[x;\lambda ]F$$

(11)

under the evident condition $\delta F(t_0,t_0;\lambda )=0$ for all $t_0\in \mathbb{R}.$ The solution to the Equation (11) allows the integral representation

$$\delta F(t,t_0;\lambda )=\underset{t_0}{\overset{t}{\int }}F(t,y;\lambda )\delta l[x\left(y\right);\lambda ]F(y,t_0;\lambda )dy,$$

(12)

which holds for all $t\in \mathbb{R}$ and $\lambda \in \mathbb{C}.$ Now taking into account that the monodromy matrix $S(t_0;\lambda )=F(t_0+2\pi ,t_0;\lambda ),$ from relation (12) we find (see [25,28]) that

$$\delta S(t_0;\lambda )=\underset{t_0}{\overset{t_0+2\pi }{\int }}F(t_0+2\pi ,t;\lambda )\delta l[x\left(t\right);\lambda ]F(t,t_0;\lambda )dt,$$

(13)

satisfied for all $t_0\in \mathbb{R}$ and $\lambda \in \mathbb{C}.$ Next, having defined the smooth functional

$$\gamma \left(\lambda \right):=\mathrm{tr}S(t_0;\lambda ),$$

(14)

which by virtue of Lemma 1 generates invariants of the spectral problem (1), and calculated the matrix trace $\mathrm{tr}S(t_0;\lambda ),$ from (13) and (14), we obtain the following governing variation expression

$$\delta \gamma \left(\lambda \right)=\underset{0}{\overset{2\pi }{\int }}\mathrm{tr}\left[F(t_0+2\pi ,t;\lambda )\delta l[x\left(t\right);\lambda ]F(t,t_0;\lambda )\right]dt,$$

(15)

which holds for an arbitrary matrix variation $\delta l[x;\lambda ]\in \mathrm{End}$${\mathbb{C}}^n$ at point $x\in X.$ Whence, making use of the Formula (15), we derive for the gradient covector $\mathsf{grad}\gamma \left(\lambda \right)\in T^{*}\left(X\right)$ at any $x\in X$ the following expression:

$$\mathsf{grad}\gamma \left(\lambda \right)\left[x\right]=\mathrm{tr}\left(l_x^{\prime ,*}S\right),$$

(16)

where the conjugation $“*”$ is taken with respect to the natural bi-linear form $(·|·):T^{*}\left(X\right)\times T\left(X\right)$$\to \mathbb{R}$ on the functional manifold $X.$ Morerover, from the integral relationship (15) one also obtains the important expression

$$\partial \gamma \left(\lambda \right)/\partial \lambda ={\int }_0^{2\pi }\mathrm{tr}(S\partial l/\partial \lambda )dt,$$

(17)

for all $\lambda \in \mathbb{C}.$ For example, if $l[x;\lambda ]=l(x,x_t;\lambda )\in \mathrm{End}$$L_{\infty }(\mathbb{R};{\mathbb{C}}^k),$ where $(x,x_t)\in$$J^1(\mathbb{R}/\left\{2\pi \mathbb{Z}\right\};{\mathbb{R}}^m)$—an arbitrary point of the corresponding [10,11] jet-manifold $J^1(\mathbb{R}/$$\left\{2\pi \mathbb{Z}\right\};{\mathbb{R}}^m)\simeq X,$ from the matrix Equation (10) and the representation (16) we can easily obtain that

$$\mathsf{grad}\gamma \left(\lambda \right)\left[x\right]=\mathrm{tr}\left(S([l,l_{x_t}]-\partial l_{x_t}/\partial t+l_x\right)),$$

(18)

where $l_x$ and $l_{x_t}$ are the usual partial derivatives with respect to the variables $(x,x_t)\in$$J^1(\mathbb{R}/\left\{2\pi \mathbb{Z}\right\};{\mathbb{R}}^m)$ of the matrix $l[x;\lambda ]\in \mathrm{End}$$L_{\infty }(\mathbb{R};{\mathbb{C}}^n)$ at $x\in X.$ Obviously, in general cases there exist such an m-dimensional matrix valued vector $\mathrm{a}=\{{\mathrm{a}}_{ij}[x;\lambda ]:i,j=\overline{1,k}\}$∈${\left(\mathrm{End}{\mathbb{C}}^n\right)}^m$ at $x\in X,$ being a local functional on $X,$ that the following representation

$$\mathsf{grad}\gamma \left(\lambda \right)\left[x\right]=\mathrm{tr}\left(S\mathrm{a}\right[x;\lambda \left]\right).$$

(19)

holds at any $x\in X.$ Having assumed now that the vector-matrix $\mathrm{a}$ ∈${\left(\mathrm{End}{\mathbb{C}}^n\right)}^m$ in (19) is non-trivial and making use both of the system of differential equations (10) for the monodromy matrix $S(t;\lambda )\in \mathrm{End}$${\mathbb{C}}^n,$$t\in \mathbb{R},$ and the functional relationship (19), one can derive the following useful recursion expression for the gradient vector $\mathsf{grad}\gamma \left(\lambda \right)\in T^{*}\left(X\right):$

$$\Lambda \left[x\right]\mathsf{grad}\gamma \left(\lambda \right)={\lambda }^{k\left(l\right)}\mathsf{grad}\gamma \left(\lambda \right),$$

(20)

satisfied for some integer $k\left(l\right)\in \mathbb{N},$ and where in general case the linear mapping $\Lambda \left[x\right]:T^{*}\left(X\right)\to$$T^{*}\left(X\right),$ called [25,26,30,32] the recursion operator, is an integro–differential operator, parametrically depending on $x\in X.$ If we force now the complex parameter $\lambda \in \mathbb{C}$ to tend to any singularity $\{\infty \}\in \Gamma ,$ we can construct the corresponding asymptotic expansions both of the monodromy matrix

$$S(t;\lambda )\sim \sum _{j\in {\mathbb{Z}}_{+}}{S}_j\left[x\right]{\lambda }^{-j}$$

(21)

and of the related gradient expression $\mathsf{grad}\gamma \left(\lambda \right)\sim {\sum }_{j\in {\mathbb{Z}}_{+}}{\lambda }^{-j}\mathsf{grad}{\gamma }_j,$ we will obtain from (21) and (20) the recursion relationship

$$\Lambda \left[x\right]\mathsf{grad}{\gamma }_j=\mathsf{grad}{\gamma }_{j+k\left(l\right)},$$

(22)

for the gradient covectors $\mathsf{grad}{\gamma }_j\in T^{*}\left(X\right),$$j\in {\mathbb{Z}}_{+},$ generated by the corresponding invariants ${\gamma }_j\in \mathcal{D}\left(X\right),j\in {\mathbb{Z}}_{+},$ defined by the expression (14).

The obtained above infinite hierarchy of invariants ${\gamma }_j\in \mathcal{D}\left(X\right),j\in {\mathbb{Z}}_{+},$ on the functional manifold X can be put into a Bogoyavlensy–Novikov type [13,25,32] geometric-analytic scheme for describing and constructing solvable nonlinear ordinary differential equations on axis. Namely, let us take a finite set of invariants ${\gamma }_j\in \mathcal{D}\left(X\right),j=\overline{0,N},$ and construct a finite-dimensional functional submanifold

$$X_c=\{x\in X:\mathsf{grad}{\mathcal{L}}_N\left[x\right]=0\},$$

(23)

where, by definition,

$${\mathcal{L}}_N:={\int }_0^{2\pi }{\mathcal{L}}_N\left[x\right]dt={\gamma }_N+\sum _{j=0}^{N-1}{c}_j{\gamma }_j$$

(24)

is an invariant, parameterized by means of some constants $c_j\in \mathbb{R},j=\overline{0,N-1}.$ It is also evident that the submanifold $X_c\subset X$ is also invariant with respect to the shifting vector field $d/dt:X\to T\left(X\right)$ on $X.$ Assume now, for convenience, that the smooth functional ${\mathcal{L}}_N\in \mathcal{D}\left(X\right)$ is nondegenerate, that is $det{\left({\displaystyle \frac{{\partial }_N^2\mathcal{L}\left[x\right]}{\partial x_j^{(k_j)}\partial x_s^{\left(k_s\right)}}}\right)}_{j,s=\overline{1,m}}\ne 0$ at $x\in X_c,$ where $k_j\in \mathbb{N}$ denotes the maximum order of the derivative of the variable $x_j\in C^{\infty }(\mathbb{R};\mathbb{R}),$ $j=\overline{1,m}.$ Then we can calculate the differential

$$d{\mathcal{L}}_N\left[x\right]=\langle \mathsf{grad}{\mathcal{L}}_N\left[x\right]|dx\rangle +d{\alpha }_N^{\left(1\right)}\left[x\right]/dt$$

(25)

and observe that at $x\in X_c$ the fifferential one-form ${\alpha }_N^{\left(1\right)}\left[x\right]\in {\Lambda }^1\left(X_c\right)$ is invariant with respect to the vector field $d/dt:X_c\to T\left(X_c\right).$ Moreover, one can check that the differential two-form ${\omega }^{\left(2\right)}\left(\tilde{x}\right):=d{\alpha }_N^{\left(1\right)}\left[x\right]\in {\Lambda }^2\left(X_c\right)$ is closed, nondegenerate and also invariant with respect to the vector field $d/dt:X_c\to$$T\left(X_c\right),$ where $\tilde{x}\in J(\mathbb{R}/\left\{2\pi \mathbb{Z}\right\};{\mathbb{C}}^m)\simeq X_c$ are the corresponding symplectic coordinates on the finite dimensional submanifold $X_c\subset X.$ The latter means that within the classical symplectic geometry theory [10,11,25,32] the vector field $d/dt$: $X_c\to$$T\left(X_c\right)$ is representable on the submanifold $X_c\subset X$ as the Hamiltonian flow

$$d\tilde{x}/dt=\mathrm{sgrad}{h}_N\left(\tilde{x}\right),$$

(26)

generated by the Hamiltonian function $h_N\in C^{\infty }(X_c;\mathbb{R}),$$d{h}_N\left(\tilde{x}\right)/dt=0,$ via the differential form relationship

$$d{h}_N\left(\tilde{x}\right)(·)=-{\omega }^{\left(2\right)}\left(\tilde{x}\right)(\mathrm{sgrad}{h}_N\left(\tilde{x}\right),·),$$

(27)

giving rise to the classical expression

$$h_N\left(\tilde{x}\right)={\alpha }^{\left(1\right)}\left(\tilde{x}\right)(d\tilde{x}/dt)-{\mathcal{L}}_N\left[\tilde{x}\right]{|}_{X_c}$$

(28)

for all $\tilde{x}\in X_c\subset X.$ Regarding the integrability of the flow (26) on the submanifold $X_c\subset X$ it is enough to show that it possesses additional invariants $h_j\in C^{\infty }(X_c;\mathbb{R}),$$j=\overline{0,N-1},$ for which the following conditions

$$d{h}_j\left(\tilde{x}\right)/dt=0,{\omega }^{\left(2\right)}\left(\tilde{x}\right)(\mathrm{sgrad}{h}_k\left(\tilde{x}\right),\mathrm{sgrad}{h}_j\left(\tilde{x}\right))=0$$

(29)

hold $j,k=\overline{0,N}.$ To show the existence of such invariants $h_j\in C^{\infty }(X_c;\mathbb{R}),$$j=\overline{0,N-1},$ we define them via the equalities

$$\langle \mathsf{grad}{\mathcal{L}}_N\left[\tilde{x}\right]|{\left({\Lambda }^{*}\right)}^j{\tilde{x}}_t\rangle =d{h}_j\left(\tilde{x}\right)/dt$$

(30)

for all $j=\overline{0,N-1},$ which easily follow from the recursion relationship (22) and the condition that $\mathsf{grad}{○}_0\left[\tilde{x}\right]={\Lambda }^{*}{\tilde{x}}_t\in T^{*}\left(X\right)$ for all $x\in X.$ As the functions $h_j\in C^{\infty }(X_c;\mathbb{R}),j=\overline{0,N-1},$ satisfy on the submanifold $X_c\subset X$ the invariance conditions $d{h}_j\left(\tilde{x}\right)/dt=0,j=\overline{0,N-1},$ we can easily observe that

$$\begin{array}{c}{\displaystyle \frac{d}{dt}}{\omega }^{\left(2\right)}\left(\tilde{x}\right)(\mathrm{sgrad}{h}_k\left(\tilde{x}\right),\mathrm{sgrad}{h}_j\left(\tilde{x}\right))=\\ =L_{{\displaystyle \frac{d}{dt}}}{i}_{\mathrm{sgrad}{h}_k\left(\tilde{x}\right)}{i}_{\mathrm{sgrad}{h}_j\left[x\right]}{\omega }^{\left(2\right)}\left(\tilde{x}\right)=\\ =i_{\mathrm{sgrad}{h}_k\left(\tilde{x}\right)}{i}_{\mathrm{sgrad}{h}_j\left(\tilde{x}\right)}{\displaystyle \frac{d}{dt}}{\omega }^{\left(2\right)}\left(\tilde{x}\right)=0,\end{array}$$

(31)

owing to vanishing both the commutators $[L_{{\displaystyle \frac{d}{dt}}},i_{\mathrm{sgrad}{h}_j\left(\tilde{x}\right)}]=L_{{\displaystyle \frac{d}{dt}},\mathrm{sgrad}{h}_j\left(\tilde{x}\right)}=0$ for $j=\overline{0,N-1},$ and the Lie derivative $L_{{\displaystyle \frac{d}{dt}}}{\omega }^{\left(2\right)}\left(\tilde{x}\right)=0$ for all $\tilde{x}\in X_c\subset X.$ The obtained results can be reformulated as the following theorem.

Theorem 1.

The Hamiltonian flow (26) is a completely solvable system of nonlinear ordinary differential equations on the finite dimensional submanifold $X_c\subset X,$ possessing a complete set of differential invariants, naturally related with a linear generalized differential spectral problem (1).

The statement above makes it possible to generate a wide number interesting from application point of view solvable systems of nonlinear ordinary differential equations on finite dimensional submanifolds, simultaneously related with generalized linear spectral problems on the axis.

Example 1.

As an example, we consider the following generalized second order linear pseudo-differential spectral problem

$$-f_{tt}+2yf-2ix{(\partial /\partial t)}^{-1}{x}^{*}f-4{\lambda }^{2}f=0$$

(32)

on the axis $\mathbb{R},$ where, by definition,

$${\left(\partial /\partial t\right)}^{-1}:=1/2\left[{\int }_{t_0}^t(\dots )ds-{\int }_t^{t_0+2\pi }(\dots )ds\right],$$

(33)

$f\in L_{\infty }(\mathbb{R};\mathbb{C})\cap C^3(\mathbb{R};\mathbb{C}),$ the spectral parameter $\lambda \in \mathbb{C}$ and the coefficients $(x,y,x^{*})\in$$X\simeq$$C^{\infty }(\mathbb{R}/2\pi \mathbb{Z};\mathbb{C}\times \mathbb{R}\times \mathbb{C})$ are $2\pi$-periodic. The problem (32) can be equivalently rewritten in the following first order differential matrix form:

$$df/dt=l[x,y,x^{*};\lambda ]f,$$

(34)

where $f\in L_{\infty }(\mathbb{R};{\mathbb{C}}^3)\cap C^1(\mathbb{R};{\mathbb{C}}^3)$ and the coefficient matrix

$$l[x,y,x^{*};\lambda ]=\left(\begin{array}{ccc}0& 2i& 0\\ i(2{\lambda }^2-y)& 0& -2x^{*}\\ x/2& 0& 0\end{array}\right).$$

(35)

By means of simple enough yet slightly cumbersome calculations [25,28] one can get convinced that the corresponding mondoromy matrix $S(t;\lambda )\in \mathrm{End}$${\mathbb{C}}^3$ is with respect to the parameter $\lambda \in \mathbb{C}$ a meromorphic function on a hyper-elliptic Riemann surface Γ with two infinite points ${\infty }^{\pm }\in \Gamma ,$ and whose trace-functional $\gamma \left(\lambda \right)=\mathrm{tr}$$S(t;\lambda )$ is invariant in $t\in \mathbb{R}.$ Moreover, its gradient $\mathit{grad}\gamma \left(\lambda \right)[x,y,x^{*}]\in T^{*}\left(X\right)$ satisfies the recursion relationship

$$\Lambda \mathsf{grad}\gamma \left(\lambda \right)[x,y,x^{*}]={\lambda }^2\mathsf{grad}\gamma \left(\lambda \right)[x,y,x^{*}]$$

(36)

for all $(x,y,x^{*})\in X$ and $\lambda \in \Gamma ,$ where the recursion operator $\Lambda :T^{*}\left(X\right)\to T^{*}\left(X\right)$ is given by the following matrix operator:

$$\begin{array}{ccc}\hfill \Lambda & =& \left(\begin{array}{ccc}0& 0& i\\ 0& {\left(\partial /\partial t\right)}^{-1}& 0\\ -i& 0& 0\end{array}\right)\times \left(\begin{array}{c}-3/4x{\left(\partial /\partial t\right)}^{-1}x\\ 1/4x\partial +1/8\partial /\partial tx\\ 3/4x{\left(\partial /\partial t\right)}^{-1}x+4i{\partial }^2-8iy\end{array}\right.\hfill \\ & & \left.\begin{array}{cc}1/4\partial /\partial tx+1/8x\partial /\partial t& 3/4x{\left(\partial /\partial t\right)}^{-1}{x}^{*}-4i{\partial }^2/\partial t^2+8iy\\ -1/16{\partial }^3/\partial t^3+1/4y\partial /\partial t+1/4\partial /\partial ty& 1/4x^{*}\partial /\partial t+1/8\partial /\partial t{x}^{*}\\ 1/4\partial /\partial t{x}^{*}+1/8x^{*}\partial /\partial t& -3/4x{\left(\partial /\partial t\right)}^{-1}{x}^{*}\end{array}\right).\hfill \end{array}$$

(37)

The recurrent relationship (36) makes it possible to construct the next invariants

$$\begin{array}{ccc}\hfill {\gamma }_0& =& {\displaystyle \frac{1}{2}}{\int }_0^{2\pi }ydt,{\gamma }_1={\int }_0^{2\pi }{x}^{*}xdt,\hfill \\ \hfill {\gamma }_2& =& {\displaystyle \frac{1}{2}}{\int }_0^{2\pi }[y^2-i(x^{*}{x}_t-x_t^{*}x)]dt,\hfill \\ \hfill {\gamma }_3& =& {\int }_0^{2\pi }[x_t^{*}{x}_t+2y{x}^{*}x)]dt,{\gamma }_4={\int }_0^{2\pi }[1/2y^3-6{\left(x^{*}x\right)}^2-\hfill \\ & & -1/2y_t^2+6iy(x^{*}{x}_t-x_t^{*}x)-2i(x^{*}{x}_{3t}-x_{3t}^{*}x)]dt,\dots ,\hfill \end{array}$$

(38)

and their gradients:

$$\begin{array}{ccc}\hfill \mathit{grad}{\gamma }_0& =& {(0,1/2,0)}^{⊺},\mathit{grad}{\gamma }_1={(x^{*},0,x)}^{⊺},\hfill \\ \hfill \mathit{grad}{\gamma }_2& =& {(1/2i{x}_t^{*},y,x)}^{⊺},\hfill \\ \hfill \mathit{grad}{\gamma }_3& =& {(-x_{tt}^{*}+4y{x}^{*},4y{x}^{*}x,-x_{tt}+4yx)}^{⊺},\hfill \\ \hfill \mathit{grad}{\gamma }_4& =& (-12{\left(x^{*}\right)}^{2}x-6i{\left(y{x}^{*}\right)}_t-6iy{x}_t^{*}+4i{x}_{ttt}^{*},\hfill \\ & & y_{tt}++6i(x^{*}{x}_t-x_t^{*}x),-12x^2{x}^{*}+6i{\left(yx\right)}_t+6iy{x}_t-4i{x}_{3t}{)}^{⊺},\dots \hfill \end{array}$$

(39)

Now let us take the nondegenerate invariant ${\mathcal{L}}_3={\gamma }_3+{\sum }_{j=\overline{0,2}}{c}_j{\gamma }_j\in \mathcal{D}\left(X\right)$ and construct the related finite dimensional submanifold

$$X_c=\{(x,y,x^{*})\in X:\mathit{grad}{\mathcal{L}}_3=\mathit{grad}{\gamma }_3+\sum _{j=\overline{0,2}}{c}_j\mathit{grad}{\gamma }_j=0\},$$

(40)

which is geometrically described, owing to the determining differential identity:

$$d{\mathcal{L}}_3[x,y,x^{*}]=\langle \mathit{grad}{\mathcal{L}}_3[x,y,x^{*}]|{(dx,dy,d{x}^{*})}^{⊺}\rangle +{\displaystyle \frac{d}{dt}}{\alpha }^{\left(1\right)}[x,y,x^{*}],$$

(41)

where

$$\begin{array}{c}\mathit{grad}{\mathcal{L}}_3[x,y,x^{*}]=(c_1{x}^{*}+c_{2}i{x}_t^{*}-x_{tt}^{*}+2y{x}^{*},\\ \\ c_0/2+c_{2}y+2x^{*}x,c_{1}x-c_{2}i{x}_t-x_{tt}{+2yx)}^{⊺},\end{array}$$

(42)

by means of the following [10,11,25] geometric structures: the Liouville type differential 1-form

$$\begin{array}{c}{\alpha }^{\left(1\right)}[x,y,x^{*}]=d{x}^{*}{x}_t+x_t^{*}dx+i{c}_2/2(d{x}^{*}x-x^{*}dx)\end{array}$$

(43)

and the symplectic differential 2-form

$${\omega }^{\left(2\right)}\left(\tilde{x}\right):=d{\alpha }^{\left(1\right)}[x,y,x^{*}]=d{x}_t^{*}\wedge dx-d{x}^{*}\wedge d{x}_t+i{c}_{2}dx\wedge d{x}^{*},$$

(44)

defined on the finite dimensional functional submanifold $X_c\subset X.$ As follows from (43), the finite dimensional submanifold $X_c\subset X$ is four-dimensional with the corresponding independent symplectic geometry coordinates $\tilde{x}=(x,x^{*},x_t,x_t^{*})\in J^1(\mathbb{R}/2\pi \mathbb{Z};{\mathbb{C}}^2)\simeq X_c.$ Regarding the Hamiltonian function $h_3$$\in \mathcal{D}\left(X_c\right),$ governing the evolution of the above jet-variables along the manifold $X_c,$ we can easily obtain from (28) and (44) that

$$h_3\left(\tilde{x}\right)=x_t^{*}{x}_t-c_1{x}^{*}x-c_2^{-1}{(c_0+4x^{*}x)}^2/8$$

(45)

for all $\tilde{x}\in X_c,$$c_2\ne 0.$ Moreover, taking into account the definition (40) and the gradient expression (42), we obtain the following system

$$\begin{array}{c}{x}_{tt}=c_{1}x-i{c}_2{x}_t/2-c_2^{-1}(c_0+4x^{*}x)x=0,\\ x_{tt}^{*}=c_1{x}^{*}+i{c}_2{x}_t^{*}/2+c_2^{-1}(c_0+4x^{*}x)x^{*}=0,\end{array}$$

(46)

of nonlinear solvable second order ordinary differential equations, generated by the Hamiltonian function (45) on the four-dimentional submanifold $X_c\subset X.$

It is worth remarking here that having chosen another matrix linear spectral problem, we would have obtained other infinite hierarchy of solvable ordinary differential equations on a suitably reduced finite-dimensional manifold $X_c\subset X,$ many of which can prove to be important for practical applications in studying diverse problems of mechanics, mathematical biology and economics as well as in other applied fields.

3. Geometric–Analytic Approach Setting

Consider the following nonlinear second order differential equation

$$\ddot{x}=K\left(t;x,\dot{x}\right)$$

(47)

on the real axis $\mathbb{R},$ where $x\in C^2(\mathbb{R};\mathbb{R}),$ dotted variables $x,\dot{x}$ denote the first and second time derivatives, $K\in \mathbb{R}\left((t;x,p)\right)$ is assumed to be an element of the Laurent series field, and represent it as a simple nonautonomous vector field

$${(\dot{x},\dot{p})}^{⊺}=\mathcal{K}(t;x,p)$$

(48)

on the cotangent [10,11] space $T^{*}\left(\mathbb{R}\right)$ to the axis $\mathbb{R}$ with coordinates $(x,p=\dot{x})\in T^{*}\left(\mathbb{R}\right)$ subject to the evolution parameter $t\in \mathbb{R}.$ It is worth to mention that, in general, the Equation (47) can be represented in the form (48) in many but equivalent ways.

Taking into account a great importance of invariants to the vector field (48) regarding a construction of solutions to the differential Equation (47), we will proceed to studying their properties from the geometric points [10,11,25,33] of view.

Definition 1.

A smooth scalar element $H\in \mathcal{D}\left(T^{*}\left(\mathbb{R}\right)\right)$ is called the invariant of the vector field (48), if it is constant along the admissible vector field (48) orbits, that is

$$dH/dt=0\sim \partial H/\partial t+\langle \mathit{grad}H|\mathcal{K}\rangle =0$$

(49)

for all $(t;x,p)\in \mathbb{R}\times T^{*}\left(\mathbb{R}\right).$

The set of all invariants to (48) will be denoted by $I\left(K\right).$ Let us now denote the gradient covector

$$\phi :=\mathsf{grad}H={\left(\partial H/\partial x,\partial H/\partial p\right)}^{⊺}$$

(50)

and formulate the next important Noether-Lax [25,33,34] proposition.

Proposition 1

(Noether-Lax). For any invariant $H\in I\left(K\right)$ its gradient $\phi =\mathit{grad}H\in T^{*}\left(T^{*}\left(\mathbb{R}\right)\right)$ satisfies the linear differential-functional relationship

$$d\phi /dt+{\mathcal{K}}^{\prime *}\phi =0,$$

(51)

where ${\mathcal{K}}^{\prime *};T^{*}(T^{*}\left(\mathbb{R}\right)\to T^{*}\left(T^{*}\left(\mathbb{R}\right)\right)$ is the adjoint mapping to the Frechet derivative ${\mathcal{K}}^{\prime }:T\left(T^{*}\left(\mathbb{R}\right)\right)\to T\left(T^{*}\left(\mathbb{R}\right)\right)$ of the mapping vector field $\mathcal{K}:T^{*}\left(\mathbb{R}\right)\to T\left(T^{*}\left(\mathbb{R}\right)\right)$ with respect to the natural bi-linear form $\langle ·|·\rangle :T^{*}\left(T^{*}\left(\mathbb{R}\right)\right)\times T\left(T^{*}\left(\mathbb{R}\right)\right)\to \mathbb{R}.$

Proof. 

Let $H\in I\left(K\right)$ be an invariant of the vector filed (48) and ${\alpha }_0:T^{*}\left(\mathbb{R}\right)\to T\left(T^{*}\left(\mathbb{R}\right)\right)$ be an arbitrary vector field on $T^{*}\left(\mathbb{R}\right)$ and construct a supplementary vector field $\alpha :T^{*}\left(\mathbb{R}\right)\to T\left(T^{*}\left(\mathbb{R}\right)\right),$ satisfying the linear evolution equation

$$d\alpha /dt={\mathcal{K}}^{\prime }{\alpha ,\alpha |}_{t=\tau }={\alpha }_0.$$

(52)

Then one easily checks that the related evolution flow

$${\left(dx/d\tau ,dp/d\tau \right)}^{⊺}=\alpha (t,\tau ;x,p)$$

with respect to the evolution parameter $\tau \in \mathbb{R}$ under the Cauchy data

$${(x\left(\tau \right),p\left(\tau \right))}^{⊺}{|}_{\tau =t}={(x\left(t\right),p\left(t\right))}^{⊺},$$

(53)

for $t\in \mathbb{R},$ taken to be a solution to the evolution Equation (48), generates a new solution ${(x(t;\tau ),p(t;\tau ))}^{⊺}\in T^{*}\left(\mathbb{R}\right),$ to this equation, parameterized by means of the variable $\tau \in \mathbb{R}.$ As a simple consequence from (53), one easily ensures that the function $H\in I\left(K\right)$ will persist to be invariant along this constructed above solution too, depending only on the supplementing evolution parameter $\tau \in \mathbb{R}$ through the solution ${(x(t;\tau ),p(t;\tau ))}^{⊺}\in T^{*}\left(\mathbb{R}\right)$ at each fixed temporal parameter $t\in \mathbb{R}.$ Thus, having differentiated this invariant with respect the parameter $\tau \in \mathbb{R},$ we obtain the following quantity:

$$dH/d\tau =\langle \mathsf{grad}H|\alpha \rangle =\langle \phi |\alpha \rangle ,$$

(54)

being, evidently, invariant with respect to the evolution parameter $t\in \mathbb{R}$ too, that is the condition $d\langle \phi |\alpha \rangle /dt=0$ for all $t\in \mathbb{R}$ holds. As a result of this condition, one obtains from (54) the following differential relationship:

$$〈d\phi /dt|\alpha 〉+\langle {\mathcal{K}}^{\prime *}\phi |\alpha \rangle =0$$

(55)

for any fixed $t\in \mathbb{R}$ and all $\tau \in \mathbb{R}.$ Substituting $t=\tau \in \mathbb{R},$ one derives from (55) that

$${\langle d\phi /d\tau +{\mathcal{K}}^{\prime *}\phi |\alpha \rangle }_{t=\tau }=〈d\phi /d\tau +{\mathcal{K}}^{\prime *}\phi |{\alpha }_0〉=0$$

(56)

for all $\tau \in \mathbb{R},$ or, equivalently, owing to the arbitrariness of the vector field ${\alpha }_0:T^{*}\left(\mathbb{R}\right)\to T\left(T^{*}\left(\mathbb{R}\right)\right),$ after changing the variables $\mathbb{R}\ni t\rightleftarrows \tau \in \mathbb{R},$ one obtains

$$d\phi /dt+{\mathcal{K}}^{\prime *}\phi =0$$

(57)

for all $t\in \mathbb{R},$ finishing the proof. □

The statement above means, in particular, that for constructing an invariant $H\in I\left(K\right)$ for the evolution Equation (48) it is enough to calculate the gradient co-vector $\phi =\mathsf{grad}H\in T^{*}\left(T^{*}\left(\mathbb{R}\right)\right)$ and to retrieve it via the classical Volterra homotopy formula

$$H={\int }_0^{1}d\lambda \langle \mathsf{grad}H(t;\lambda x,\lambda p)|{(x,p)}^{⊺}\rangle .$$

(58)

To continue our analysis of the set of invariant $I\left(K\right),$ remark that the gradient co-vector $\phi \in T^{*}\left(T^{*}\left(\mathbb{R}\right)\right)$ satisfies the symmetry condition

$${\phi }^{\prime }(t;x,p)={\phi }^{\prime *}(t;x,p)$$

(59)

for all $t\in \mathbb{R}$ and $(x,p)\in T^{*}\left(\mathbb{R}\right).$ Moreover, if to take into account the expressions for matices

$$K^{\prime }=\left(\begin{array}{cc}0& 1\\ {\displaystyle \frac{\partial K}{\partial x}}& {\displaystyle \frac{\partial K}{\partial p}}\end{array}\right),K^{\prime *}=\left(\begin{array}{cc}0& {\displaystyle \frac{\partial K}{\partial x}}\\ 1& {\displaystyle \frac{\partial K}{\partial p}}\end{array}\right)$$

(60)

and

$${\phi }^{\prime }=\left(\begin{array}{cc}{\displaystyle \frac{\partial {\phi }_1}{\partial x}}& {\displaystyle \frac{\partial {\phi }_1}{\partial p}}\\ {\displaystyle \frac{\partial {\phi }_2}{\partial x}}& {\displaystyle \frac{\partial {\phi }_2}{\partial p}}\end{array}\right),{\phi }^{\prime *}=\left(\begin{array}{cc}{\displaystyle \frac{\partial {\phi }_1}{\partial x}}& {\displaystyle \frac{\partial {\phi }_2}{\partial x}}\\ {\displaystyle \frac{\partial {\phi }_1}{\partial p}}& {\displaystyle \frac{\partial {\phi }_2}{\partial p}}\end{array}\right),$$

(61)

where, by definition, $\phi ={({\phi }_1,{\phi }_2)}^{⊺}\in T^{*}\left(T^{*}\left(\mathbb{R}\right)\right),$ the determining Noether–Lax condition can be equivalently rewritten as the differential relationships

$$\begin{array}{cc} & {\displaystyle \frac{\partial {\phi }_1}{\partial t}}+{\displaystyle \frac{\partial }{\partial x}}({\phi }_{1}p+{\phi }_{2}K)=0,\hfill \\ & {\displaystyle \frac{\partial {\phi }_2}{\partial t}}+{\displaystyle \frac{\partial }{\partial p}}({\phi }_{1}p+{\phi }_{2}K)=0\hfill \end{array}$$

(62)

for all $(t;x,p)\in \mathbb{R}\times T^{*}\left(\mathbb{R}\right).$ The result (62) we now will consider as determining the co-vector $\phi \in T^{*}\left(T^{*}\left(\mathbb{R}\right)\right)$ for calculating via the Formula (58) the corresponding conservation law $H\in I\left(K\right).$

The construction, devised above, can be naturally generalized by means of introducing the coadjoint space $T^{*}\left(\mathbb{R}\right)$ and defining on it the induced vector field

$$\left(\begin{array}{c}\dot{x}\\ \dot{p}\end{array}\right)=\left(\begin{array}{c}\xi (t;x,p)\\ \eta (t;x,p)\end{array}\right):=\mathcal{K}(t;x,p)$$

(63)

subject to a suitably specified mapping $(\xi ,\eta ):T^{*}\left(\mathbb{R}\right)\to T\left(T^{*}\left(\mathbb{R}\right)\right).$ The relationship (63), in particular, means that if there exists some smooth mapping $\mathcal{L}:$$\mathbb{R}\times T\left(\mathbb{R}\right)\to \mathbb{R},$ for which $\partial \mathcal{L}(t;x,\dot{x})/\partial \dot{x}{|}_{\dot{x}=\xi (t;x,p)}=$p for all $(x,p)\in T^{*}\left(\mathbb{R}\right),$ then the scalar quantity

$$H(t;x,p):=\langle p\left|\xi \right(t;x,p)\rangle -\mathcal{L}(t;x,\xi (t;x,p)\left)\right|$$

(64)

will present a priori an invariant of the nonlinear second order differential Equation (47), that is $dH(t;x,p)/dt=0$ for arbitrary $t\in \mathbb{R}.$ Regarding the given nonlinear second order differential Equation (47), the $\eta$-component of the vector field $\mathcal{K}:\mathbb{R}\times T^{*}\left(\mathbb{R}\right)\to T\left(T^{*}\left(\mathbb{R}\right)\right)$ is easily calculated as

$$\eta (t;x,p)={\left(\partial \xi (t;x,p)/\partial p\right)}^{-1}\left(K(t;x,\xi (x,p\left)\right)-\xi (t;x,p)\partial \xi (t;x,p)/\partial x\right)).$$

(65)

Moreover, the vector field (63) can be equivalently rewritten as

$$\mathcal{K}(t;x,p)=(\xi (t;x,p),\partial \mathcal{L}{(t;x,\xi (x,p)/\partial x)}^{⊺}$$

(66)

for all $(t;x,p)\in \mathbb{R}\times T^{*}\left(\mathbb{R}\right).$ Based on the representation (66), under the autonomous condition $\partial K(t;x,p)/\partial t=0=$$\partial \mathcal{L}(t;x,\xi (x,p\left)\right)/\partial t$ for all $(t;x,p)\in \mathbb{R}\times T^{*}\left(\mathbb{R}\right)$ the system (63) can be reduced to the Pfaff form

$$\xi (x,p)dp-\partial \mathcal{L}(x,\xi (x,p\left)\right)dx=0$$

(67)

on the cotangent space $T^{*}\left(\mathbb{R}\right),$ always solvable subject to some invariant of the nonlinear second order differential Equation (47).

Under the condition ${\displaystyle \frac{\partial \xi }{\partial p}}\ne 0$ on $T^{*}\left(\mathbb{R}\right)$ with regard to the Noether-Lax equation (51), any smooth invariant $H\in I\left(K\right)$ of the flow (63) is characterized by its gradient co-vector $\phi ={({\phi }_1,{\phi }_2)}^{⊺}$∈ $T^{*}\left(T^{*}\left(\mathbb{R}\right)\right),$ whose components satisfy the following diffusion-like linear differential relationship:

$$\partial {({\phi }_1,{\phi }_2)}^{⊺}/\partial t+\mathsf{grad}({\phi }_1\xi +{\phi }_2\eta )=0$$

(68)

on the whole space $T^{*}\left(\mathbb{R}\right).$ Assuming, as before, the meromorphic temporal dependence of the co-vector ${({\phi }_1,{\phi }_2)}^{⊺}\in T^{*}\left({\mathbb{R}}^2\right)$ as

$${({\phi }_1,{\phi }_2)}^{⊺}=\sum _{j\in {\mathbb{Z}}_{+}}{\left({\phi }_1^{\left(j\right)},{\phi }_2^{\left(j\right)}\right)}^{⊺}{t}^{q-j}$$

(69)

for some $q\in {\mathbb{Z}}_{+},$ one can reduce the differential relationship (68) to an equivalent set of differential-algebraic equations on the components ${\left({\phi }_1^{\left(j\right)},{\phi }_2^{\left(j\right)}\right)}^{⊺}\in T^{*}\left(T^{*}\left(\mathbb{R}\right)\right),$$j\in {\mathbb{Z}}_{+},$ which are often truncated at some finite $j\in \mathbb{N}.$ It is worth remarking here that the mapping $\phi :T^{*}\left(\mathbb{R}\right)\to T\left(T^{*}\left(\mathbb{R}\right)\right)$ is assumed to be chosen as a one depending on functional parameter. In addition, it is necessary to take care of the solution (69) to Equation (68) to satisfy the necessary compatibility symmetry condition ${\phi }^{\prime }(x,p)-$${\phi }^{\prime *}(x,p)=0$ on the whole space $T^{*}\left(\mathbb{R}\right).$ This way, the obtained co-vector $\phi (x,p)\in T^{*}\left(T^{*}\left(\mathbb{R}\right)\right)$ at $(x,p)\in$$T^{*}\left(\mathbb{R}\right)$ makes it possible to retrieve the corresponding invariant $H\in I\left(K\right)$ via the Volterra homotopy formulae (58).

4. The Geometric-Analytic Approach Applications and Its Generalization

Below we first consider some typical applications of the devised geometric-analytic approach to second-order differential equations with meromorphic coefficients.

Example 2.

Let a differential Equation (47) be given in the following form:

$$\ddot{x}={\displaystyle \frac{3}{x}}{\left(\dot{x}\right)}^2+{\displaystyle \frac{1}{t}}\dot{x}=K\left(t;x,\dot{x}\right),$$

(70)

having some interesting applications in gravity theory. The corresponding vector field

$${(dx/dt,dp/dt)}^{⊺}={\left(p,3p^2/x+p/t\right)}^{⊺}:=\mathcal{K}$$

(71)

on the cotangent $T^{*}\left(\mathbb{R}\right)$ is defined via the simple relationship $(x,\dot{x}=p)\in T^{*}\left(\mathbb{R}\right).$ Since the invariant $H\in I\left(K\right)$ depends, in general, on the temporal variable $t\in \mathbb{R},$ it is natural to assume the following meromorphic expansions:

$${\phi }_1=\sum _{k\in {\mathbb{Z}}_{+}}{\phi }_1^{\left(k\right)}{t}^{-k+q},{\phi }_2=\sum _{k\in {\mathbb{Z}}_{+}}{\phi }_2^{\left(k\right)}{t}^{-k+q},$$

(72)

for some integer $q\in {\mathbb{Z}}_{+}$ and coefficients ${\phi }_1^{\left(k\right)},{\phi }_2^{\left(k\right)}\in T^{*}\left(T^{*}\left(\mathbb{R}\right)\right),k\in {\mathbb{Z}}_{+}$ depending only on variables $(x,p)\in T^{*}\left(\mathbb{R}\right).$ Concerning the vector field (71), it is easy to see that $q=1,$ $k=\overline{0,1}$ or $q=0,k=1.$ For the first case $q=1,k=\overline{0,1},$ we have the expansions

$${\phi }_1={\phi }_1^{\left(0\right)}t+{\phi }_1^{\left(1\right)},{\phi }_2={\phi }_2^{\left(0\right)}t+{\phi }_2^{\left(1\right)},$$

(73)

where substitution into the determining relationships (62) gives rise to the following conditions:

$$\begin{array}{cc} & {\phi }_1^{\left(0\right)}+{\displaystyle \frac{\partial }{\partial x}}\left[{\phi }_1^{\left(0\right)}tp+{\phi }_1^{\left(1\right)}p+{\phi }_2^{\left(0\right)}t\left({\displaystyle \frac{3p^2}{x}}+{\displaystyle \frac{p}{t}}\right)+{\phi }_2^{\left(1\right)}\left({\displaystyle \frac{3p^2}{x}}+{\displaystyle \frac{p}{t}}\right)\right]=0,\hfill \\ & {\phi }_2^{\left(0\right)}+{\displaystyle \frac{\partial }{\partial p}}\left[{\phi }_1^{\left(0\right)}tp+{\phi }_1^{\left(1\right)}p+{\phi }_2^{\left(0\right)}t\left({\displaystyle \frac{3p^2}{x}}+{\displaystyle \frac{p}{t}}\right)+{\phi }_2^{\left(1\right)}\left({\displaystyle \frac{3p^2}{x}}+{\displaystyle \frac{p}{t}}\right)\right]=0,\hfill \end{array}$$

(74)

simply reducing to the set of linear equations

$$\begin{array}{cc} & {\displaystyle \frac{\partial }{\partial x}}\left[{\phi }_1^{\left(0\right)}p+{\phi }_2^{\left(0\right)}{\displaystyle \frac{3p^2}{x}}\right]=0,{\displaystyle \frac{\partial }{\partial x}}\left({\phi }_2^{\left(1\right)}p\right)=0,{\displaystyle \frac{\partial }{\partial p}}\left[{\phi }_1^{\left(0\right)}p+{\phi }_2^{\left(0\right)}{\displaystyle \frac{3p^2}{x}}\right]=0,\hfill \\ & {\phi }_1^{\left(0\right)}+{\displaystyle \frac{\partial }{\partial x}}\left[{\phi }_1^{\left(1\right)}p+{\phi }_2^{\left(0\right)}p+{\phi }_2^{\left(1\right)}{\displaystyle \frac{3p^2}{x}}\right]=0,{\displaystyle \frac{\partial }{\partial p}}\left({\phi }_2^{\left(1\right)}p\right)=0,\hfill \\ & {\displaystyle \frac{\partial }{\partial x}}\left[{\phi }_2^{\left(0\right)}p+{\phi }_2^{\left(0\right)}{\displaystyle \frac{3p^2}{x}}\right]=0,{\displaystyle \frac{\partial }{\partial x}}\left({\phi }_2^{\left(1\right)}p\right)=0,{\phi }_2^{\left(0\right)}+{\displaystyle \frac{\partial }{\partial p}}\left[{\phi }_1^{\left(1\right)}p+{\phi }_2^{\left(0\right)}p+{\phi }_2^{\left(1\right)}{\displaystyle \frac{3p^2}{x}}\right]=0,\hfill \end{array},$$

(75)

whose solution is

$${\phi }_1^{\left(0\right)}=-{\displaystyle \frac{3p}{x^4}},{\phi }_1^{\left(1\right)}=-{\displaystyle \frac{2}{x^3}},{\phi }_2^{\left(0\right)}={\displaystyle \frac{1}{x^3}},{\phi }_2^{\left(1\right)}=0.$$

(76)

Applying the Formula (58) to (73) and (76), we get that

$$H_1={\displaystyle \frac{pt}{x^3}}+{\displaystyle \frac{1}{x^2}}\Rightarrow {\displaystyle \frac{t}{x^3}}\dot{x}+{\displaystyle \frac{1}{x^2}}$$

(77)

is an invariant of the nonlinear ordinary second order differential Equation (70). In a similar way to case 2, when $q=0,k=1,$ we obtain

$${\phi }_1^{\left(0\right)}=-{\displaystyle \frac{3p}{x^4}},{\phi }_2^{\left(0\right)}=-{\displaystyle \frac{1}{x^3}},$$

from which one obtains the second invariant

$$H_2={\displaystyle \frac{p}{t{x}^3}}\Rightarrow {\displaystyle \frac{\dot{x}}{t{x}^3}}$$

(78)

of the differential Equation (70).

Example 3.

Consider the following second order nonlinear ordinary Mathews–Lakshmanan differential equation

$$\ddot{x}={\displaystyle \frac{kx}{1+k{x}^2}}\left({\dot{x}}^2-{\omega }^2{x}^2\right):=K(x,\dot{x}),$$

(79)

where $k,\omega \in \mathbb{R}$ are arbitrary parameters. As the Equation (79) is autonomous, the co-vector $\phi \in T^{*}\left(T^{*}\left(\mathbb{R}\right)\right)$ does not depend on the temporal variable $t\in \mathbb{R}.$ Having imbeded the Equation (79) into the flow on the related cotangent space with the phase variables $(x,p=\dot{x}{(1+k{x}^2)}^{-1/2})\in T^{*}(\mathbb{R}),$ one obtains the flow

$$\left.\begin{array}{c}\dot{x}=p{(1+k{x}^2)}^{1/2}\\ \dot{p}=-{\omega }^2{x}^{3}k{(1+k{x}^2)}^{-3/2}\end{array}\right\}=\mathcal{K}(x,p),$$

(80)

which reduces to the following differential relationship:

$$p{p}^{\prime }\left(x\right)=-{\omega }^2{x}^{3}k{(1+k{x}^2)}^{-2},$$

(81)

equivalent to the Pfaff form (67), derived before. The latter is easily integrated as

$$\begin{array}{ccc}\hfill H_1& =& k{p}^2+{\omega }^2{(1+k{x}^2)}^{-1}+{\omega }^{2}ln(1+k{x}^2)\Rightarrow \hfill \\ & \Rightarrow & {\displaystyle \frac{k{\dot{x}}^2+{\omega }^2}{(1+k{x}^2)}}+{\omega }^{2}ln(1+k{x}^2),\hfill \end{array}$$

(82)

presenting an invariant of the Equation (79). Yet, if we assume that there exists an invariant $H_2\in I\left(K\right),$ depending on the temporal variable $t\in \mathbb{R},$ then the simplest case is given by the expansion (72) of the gradient co-vector ${({\phi }_1,{\phi }_2)}^{⊺}=\mathsf{grad}{H}_2$ with $p=0,k=1:$

$${\phi }_1={\phi }_1^{\left(0\right)}t+{\phi }_1^{\left(1\right)},{\phi }_2={\phi }_2^{\left(0\right)}t+{\phi }_2^{\left(1\right)},$$

(83)

from (62) one derives the simplest solutions

$$\begin{array}{ccc}\hfill {\phi }_1^{\left(0\right)}& =& {\displaystyle \frac{kx[p^2+x^2(k{p}^2-{\omega }^2)]}{{(1+k{x}^2)}^{3/2}\sqrt{H_1}}},{\phi }_1^{\left(1\right)}={\displaystyle \frac{p\sqrt{H_1}}{{(1+k{x}^2)}^{3/2}}},\hfill \\ \hfill {\phi }_2^{\left(0\right)}& =& -{\displaystyle \frac{k^2}{2{(1+k{x}^2)}^{1/2}\sqrt{H_1}}},{\phi }_2^{\left(1\right)}=-{\displaystyle \frac{x{(1+k{x}^2)}^{1/2}\sqrt{H_1}}{p^2(1+k{x}^2)+H_1{x}^2}}.\hfill \end{array}$$

(84)

Having applied to the co-vector components (83) and (84) the Volterra homotopy formula (58), one obtains the second non-autonomous invariant

$$\begin{array}{ccc}\hfill H_2& =& arctan\left({\displaystyle \frac{x\sqrt{H_1}}{p{(1+k{x}^2)}^{1/2}}}\right)-t\sqrt{H_1}\Rightarrow \hfill \\ & \Rightarrow & arctan\left({\displaystyle \frac{x\sqrt{H_1}}{\dot{x}}}\right)-t\sqrt{H_1}\hfill \end{array}$$

(85)

to the differential Equation (79).

Example 4.

Let a nonlinear second order differential equation be in the form

$$\ddot{x}={\displaystyle \frac{k{\dot{x}}^2}{x}}+C_0\left(x\right)\dot{x}+C_1\left(x\right),$$

(86)

where $C_0\left(x\right),C_1\left(x\right)\in \mathbb{R}\left[x\right]$ and parameter $k\in \mathbb{R}\setminus \left\{0\right\}.$ It is natural to imbed the Equation (86) into the flow on the cotangent space, parameterized by the variables $(x,p)\in T^{*}\left(\mathbb{R}\right),$ where

$$\dot{x}=x^{k}p.$$

(87)

Whence upon substituting (87) into (86) one obtains the second vector field component

$$\dot{p}=C_0\left(x\right)p+C_1\left(x\right)x^{-k}$$

(88)

on the phase space $T^{*}\left(\mathbb{R}\right).$ Now, if a function $H\in \mathbb{R}\left(\right(x,p\left)\right)$ is invariant with respect to the vector field

$${(\dot{x},\dot{p})}^{⊺}=K(x,p),$$

(89)

then its gradient $\phi ={({\phi }_1,{\phi }_2)}^{⊺}\in T^{*}\left(T^{*}\left(\mathbb{R}\right)\right)$ satisfies the determining equation $\langle \phi |K\rangle =0,$ or equivalently

$${\phi }_1{x}^{k}p+{\phi }_2\left[C_0\left(x\right)p+C_1\left(x\right)x^{-k}\right]=0.$$

(90)

Taking into account that, by definition ${\phi }^{\prime }={\phi }^{\prime *},$ the latter gives rise to the existing of a function $\alpha \in \mathbb{R}\left(\right(x,p\left)\right),$ satisfying the conditions

$${\phi }_1={\displaystyle \frac{\partial \alpha }{\partial x}},{\phi }_2={\displaystyle \frac{\partial \alpha }{\partial p}},$$

(91)

for all $(x,p)\in T^{*}\left(\mathbb{R}\right).$ As a result of relationships (90) and (91), we can write down the following differential form expression

$${\displaystyle \frac{dp}{p{C}_0\left(x\right)+C_1\left(x\right)x^{-k}}}={\displaystyle \frac{dx}{x^{k}p}},$$

(92)

reducing to the first-order ordinary differential equation:

$$p^{\prime }\left(x\right)=C_0\left(x\right)x^{-k}+C_1\left(x\right)x^{-2k}{p}^{-1}.$$

(93)

The obtained above Equation (93) can be simplified by means of the substitution $p=1/\beta ;$

$${\beta }^{\prime }\left(x\right)+C_0\left(x\right)x^{-k}{\beta }^2+C_1\left(x\right)x^{-2k}{\beta }^3=0,$$

(94)

presenting, in general, a non-solvable equation of the Abel type. Nevertheless, for some constraints on the coefficients $C_0\left(x\right)$ and $C_1\left(x\right)\in \mathbb{R}$ the Equation (94) can be solved. Namely, assume that $\beta =x+\omega ,$ where the function $\omega \in \mathbb{R}\left(\right(x\left)\right)$ is chosen from the condition that the resulting equation on the function $x\in \mathbb{R}\left(\right(x\left)\right)$ possesses no $x^2$-component:

$$\begin{array}{cc} & C_0{x}^k+C_{1}3\omega =0,{\omega }^{\prime }+2C_0{x}^{-k}{\displaystyle \frac{{\omega }^2}{3}}=0,\hfill \\ & u^{\prime }+u{C}_0{x}^{-k}\omega -{\displaystyle \frac{C_0{x}^{-k}}{3\omega }}{u}^3=0.\hfill \end{array}$$

(95)

The latter equation of (95) belongs to the Bernoulli type and can be easily integrated in quadratures. Having integrated it, we obtain that

$$\omega ={\displaystyle \frac{1}{H_1+\frac{2}{3}{\partial }^{-1}{x}^{-k}{C}_0}},$$

(96)

where $H_1\in \mathbb{R}\left((x,\dot{x})\right)$ is the first invariant of the Equation (86). Taking into account that

$$\begin{array}{ccc}\hfill u& =& (H_{2}exp\left({\partial }^{-1}2C_0{x}^{-k}\omega \right)+\frac{1}{3}exp\left({\partial }^{-1}\left(2C_0{x}^{-k}\omega \right)\right)\times \hfill \\ & & \times {\partial }^{-1}{C}_0{x}^{-k}{\omega }^{-1}exp{\left(-{\partial }^{-1}2C_0{x}^{-k}\omega \right)}^{-1/2},\hfill \end{array}$$

(97)

where $H_2\in \mathbb{R}\left((x,\dot{x})\right)$ is the second invariant of the differential Equation (86), one can write down the determining analytic relationship for the both invariants $H_1,H_2\in \mathbb{R}\left((x,\dot{x})\right)$ of the Equation (86) under regard:

$$x^k=\dot{x}(u+\omega ),$$

(98)

where functions u and $\omega \in \mathbb{R}\left(\right(x\left)\right)$ are given by the expressions (96) and (97), thus solving the posed problem.

Example 5.

As an instructive application of the reasonings above, we consider the following second order ordinary differential equation

$$\ddot{x}={\displaystyle \frac{a\left(\dot{x}\right)\dot{x}}{a^{\prime }\left(\dot{x}\right)}}\left[b\left(x\right)+{\displaystyle \frac{c_1\left(x\right)}{a\left(\dot{x}\right)}}+c_0\left(x\right)\right],$$

(99)

where $a,b,c_0$ and $c_1\in \mathbb{R}\left((x,\dot{x})\right)$ are arbitrary and $a^{\prime }\left(\dot{x}\right)\ne 0$ for any $\dot{x}\in T\left(\mathbb{R}\right).$ To analyze invariants to the differential Equation (99), we need to introduce the supplementary coadjoint phase space $T^{*}\left(\mathbb{R}\right)$ with coordinates $(x,p)\in T^{*}\left(\mathbb{R}\right)$ and to determine a suitable vector field (63) on $T^{*}\left(\mathbb{R}\right)$ via the substitution $\dot{x}=\xi (x,p),$ which we take in the following functional form:

$$a\left(\dot{x}\right)=pexp\left({\partial }^{-1}b\right)\rightleftarrows \dot{x}=\xi (x,p).$$

(100)

The expression (100) makes it possible to calculate the temporal dependence

$$\dot{p}=c_1\left(x\right)\dot{x}exp\left({\partial }^{-1}b\right)+{\displaystyle \frac{c_0\left(x\right)exp\left({\partial }^{-1}b\right)\dot{x}}{a\left(\dot{x}\right)}},$$

(101)

that can be reduced to the following form:

$$p^{\prime }\left(x\right)=c_1\left(x\right)exp\left({\partial }^{-1}b\right)+{\displaystyle \frac{c_0\left(x\right)exp(-2{\partial }^{-1}b)}{p}}.$$

(102)

It is well known that the nonlinear differential Equation (102) is of the Riccati–Abel type and is not solvable in general case. Nonetheless, we can analyze its integrability for specially chosen functions $c_0\left(x\right)$ and $c_1\left(x\right)\in \mathbb{R}\left(\left(x\right)\right).$ For instance, if to make the substitution

$$p=\alpha {\left(x\right)}^{-1}exp(-{\partial }^{-1}b),$$

(103)

into (102), we get that the function $\alpha \left(x\right)\in \mathbb{R}\left(\right(x\left)\right)$ satisfies the Abel type equation

$${\alpha }^{\prime }+b\left(x\right)\alpha +c_1\left(x\right){\alpha }^2+c_0\left(x\right){\alpha }^3=0.$$

(104)

Having eliminated the ${\alpha }^2$-component from the equation above, we arrive at an already solvable equation of the Bernoulli type [7] via the following substitutions:

$$\begin{array}{cc} & c_1\left(x\right)=-3c_0\left(x\right)\omega \left(x\right),\alpha \left(x\right)=u\left(x\right)+\omega \left(x\right),\hfill \\ & {\omega }^{\prime }+\omega b\left(x\right)-2c_0\left(x\right){\omega }^3=0,\hfill \end{array},$$

(105)

reducing (104) to the next Lagrange form

$$u^{\prime }+u\left[b\left(x\right)-3\omega {\left(x\right)}^2{c}_0\left(x\right)\right]+u^3{c}_0\left(x\right).$$

(106)

Thus, having solved the first Bernoulli equation of (105) and substituted its solution into the next Lagrange Equation (106), we successfully obtain a general solution to the Equation (104) in the functional form

$$\alpha =\alpha (x;H_1,H_2),$$

(107)

where $H_1,H_2\in \mathbb{R}\left(\left(x\right)\right)$ are the corresponding invariants to our second order differential Equation (99). Moreover, taking into account the relationships (100) and (103), we finally obtain the equality

$$\alpha (x;H_1,H_2)a\left(\dot{x}\right)=1,$$

(108)

allowing us to construct two invariants $H_1$ and $H_2$$\in \mathbb{R}\left(\right(x\left)\right),$ explicitly using the determining expression from the Supplement.

5. Determining Equations and Its Asymptotical Properties

Consider a general nonlinear second order differential Equation (47) and its invariants determining Noether-Lax type equation

$$d\phi /\partial t+{\mathcal{K}}^{\prime ,*}\phi =0,$$

(109)

on their gradients $\phi \in T^{*}\left(T^{*}\left(\mathbb{R}\right)\right),$ written down for the simplest vector field form

$$\left(\begin{array}{c}dx/dt\\ dp/dt\end{array}\right)=\left(\begin{array}{c}p\\ K(t;x,p)\end{array}\right)=\mathcal{K}(t;x,p)$$

(110)

on the cotangent space with coordinates $(x,p=\dot{x})\in T^{*}\left(\mathbb{R}\right).$ Taking into account that matrices

$${\mathcal{K}}^{\prime }=\left(\begin{array}{cc}0& 1\\ K_x& K_p\end{array}\right),{\mathcal{K}}^{\prime ,*}=\left(\begin{array}{cc}0& K_x\\ 1& K_p\end{array}\right),$$

(111)

the component-wise form of the Equation (109) for the co-vector $\phi ={({\phi }_1,{\phi }_2)}^{⊺}\in T^{*}\left(T^{*}\left(\mathbb{R}\right)\right)$ looks as

$$\partial {\phi }_1/\partial t+K_x{\phi }_2=0,\partial {\phi }_2/\partial t+{\phi }_1+K_p{\phi }_2=0.$$

(112)

Recall now, that the general solution to the vector field (110) depends on two arbitrary parameters $\lambda ,\mu \in \mathbb{R},$ thus implying the same dependence on them of the co-vector components ${({\phi }_1,{\phi }_2)}^{⊺}\in T^{*}\left(T^{*}\left(\mathbb{R}\right)\right).$ In particular, fixing the second parameter $\mu \in \mathbb{R},$ we can assume, adapting reasoning from [35], that the solution to (112) possesses as $\left|\lambda \right|\to \infty ,$ the following regularized asymptotic representation:

$${\phi }_1=a(t;\lambda )exp\left[\sigma (t;\lambda )\right],{\phi }_2=exp\left[\sigma (t;\lambda )\right],$$

(113)

where

$$a(t;\lambda )\sim \sum _{j\in {\mathbb{Z}}_{+}}{a}_j(t;x,p){\lambda }^{-j},\sigma (t;\lambda )\sim \sum _{j\in {\mathbb{Z}}_{+}}{\sigma }_j(t;x,p){\lambda }^{-j}.$$

(114)

Having substituted the representation (113) into (112), one obtains the following nonlinear Riccati type reduced equation:

$$\partial a/\partial t-a(a+K_p)+K_x=0.$$

(115)

The Equation (115) can be analyzed analytically, if to assume that elements $K_x,K_p\in$$\mathbb{R}\left(\right(x,p\left)\right),$ allowing an expansion

$$a(t;x,p)=p^k\sum _{s\in {\mathbb{Z}}_{+}}{a}^{\left(s\right)}\left(x\right)p^{-s}$$

(116)

for some non-negative integer $k\in {\mathbb{Z}}_{+}.$ Moreover, taking now into account the expansions (114), the following recurrent set of equations

$$\partial a_j/\partial t-\sum _{k\in {\mathbb{Z}}_{+}}{a}_{j-k}{a}_k-a_j{K}_p+{\delta }_{j,0}{K}_x=0$$

(117)

holds for all $j\in {\mathbb{Z}}_{+}.$ The obtained recurrent set of Equation (117) can be similarly analyzed analytically, if to assume that elements $a_j\in \mathbb{R}\left((x,p)\right),j\in {\mathbb{Z}}_{+},$ therefore allowing the expansions

$$a_j(t;x,p)=p^{k_j}\sum _{s\in {\mathbb{Z}}_{+}}{a}_j^{\left(s\right)}\left(x\right)p^{-s},$$

(118)

where $k_j\in {\mathbb{Z}}_{+},j\in {\mathbb{Z}}_{+},$ are some non-negative integers. In particular, taking into account the functional relationships ${\phi }_1=a{\phi }_2$ and the compatibility condition $\partial {\phi }_1/\partial p=$$\partial {\phi }_2/\partial x$ for all $(x,p)\in T^{*}\left(\mathbb{R}\right),$ one can easily retrieve the asymptotic representation (113) for the gradient element ${({\phi }_1,{\phi }_2)}^{⊺}\in$$T^{*}\left(T^{*}\left(\mathbb{R}\right)\right).$

Example 6.

As a first example, let us take the nonlinear second order differential Equation (70), for which the determining nonlinear Riccati type reduced Equation (115) looks as $\partial a/\partial t-a(a+t{x}^{-3})-(3pt{x}^{-4}+2x^{-3})=0.$

Having used an expansion (116), one easily ensues that $k=1$ and

$$a^{\left(0\right)}=-3/x,a^{\left(1\right)}=-2/t,a^{\left(2\right)}=0=a^{\left(3\right)}=\dots a^{\left(s\right)}$$

(119)

for all $s\in {\mathbb{Z}}_{+},$ giving rise to the exact expression $a(t;x,p)=-(3p{x}^{-1}+2t^{-1})$ for all $t\in \mathbb{R}\setminus \left\{0\right\}$ and $(x,p)\in$$T_x^{*}{\left(\mathbb{R}\right)|}_{x\ne 0}.$ Making use now of the compatibility conditions ${\phi }_1=a{\phi }_2$ and $\partial {\phi }_1/\partial p=$$\partial {\phi }_2/\partial x,$ one easily obtains the corresponding covector

$$\left(\begin{array}{c}{\phi }_1\\ {\phi }_2\end{array}\right)=\left(\begin{array}{c}-(3p{x}^{-1}+2t^{-1})t{x}^{-3}\\ t{x}^{-3}\end{array}\right)$$

(120)

for all $(x,p)\in$$T_x^{*}{\left(\mathbb{R}\right)|}_{x\ne 0}$, which generates via the Volterra homotopy formula (58) the invariant $H_1=pt{x}^{-3}+x^{-2},$ coinciding with that, obtained before in (77) for the nonlinear second order differential Equation (70).

Example 7.

As a second instructive example we consider the classical nonlinear Van der Pol equation

$$\ddot{x}=\mu (1-x^2)\dot{x}-x$$

(121)

with $\mu \in {\mathbb{R}}_{+}$ as a parameter.

The Equation (121) can be equivalently rewritten as the vector field

$$\left(\begin{array}{c}dx/dt\\ dp/dt\end{array}\right)=\left(\begin{array}{c}p+\mu (x-x^3/3)\\ -x\end{array}\right)=\mathcal{K}(t;x,p)$$

(122)

on the cotangent space $T^{*}\left(\mathbb{R}\right).$ The problem of studying analytical solutions to the Van der Pol equation (121) for either large enough Cauchy data or the parameter $\mu \to \infty$ is of most important ones [11,36,37,38,39] amongst the “Open problems in mathematics” and still presents an open problem today. The determining equation for invariants to the vector field (122) looks as

$$\partial {\phi }_1/\partial t+\mu (1-x^2){\phi }_1-{\phi }_2=0,\partial {\phi }_2/\partial t+{\phi }_1=0,$$

(123)

and can be rewritten as the second order differential equation

$${\ddot{\phi }}_2+{\phi }_2+\mu (1-x^2){\dot{\phi }}_2=0$$

(124)

on the smooth component ${\phi }_2\in C^{\infty }(\mathbb{R};\mathbb{R}).$ We can now make the functional substitution

$${\dot{\phi }}_2=a{\phi }_2,$$

(125)

where the coefficient $a\in C^{\infty }(\mathbb{R};\mathbb{R})$ satisfies the Riccati type differential equation

$$a_x^{\prime }[p+\mu (x-x^3/3)]-a_p^{\prime }x+\mu (1-x^2)a+a^2+1=0,$$

(126)

which possesses as $\mu \to \infty$ the following asymptotical solution:

$$a(x,p;\mu )\sim a_{-1}(x,p)\mu +a_0(x,p)+a_1(x,p){\mu }^{-1}+a_2(x,p){\mu }^{-2}+\dots ,$$

(127)

whose coefficients $a_j(x,p),j\in \mathbb{N}\cup \{-1,0\}$ can be determined from the next set of recurrent relationships:

$$a_{j,x}^{\prime }p+a_{j+1,x}^{\prime }(x-x^3/3)]-a_{j,p}^{\prime }x+(1-x^2)a_{j+1}+\sum _{k\ge -2}{a}_{j-k}{a}_k+{\delta }_{j,0}=0.$$

(128)

The corresponding solutions to the set (128) look as follows:

$$\begin{array}{ccc}\hfill a_{-1}(x,p)& =& 1-x^2,a_0(x,p)=[c_0\left(p\right)+I_0(x,p;\lambda )](x-x^3/3)],\hfill \\ \hfill a_1(x,p)& =& [c_1\left(p\right)+I_1(x,p;\lambda )](x-x^3/3)],\hfill \\ \hfill a_2(x,p)& =& [c_2\left(p\right)+I_2(x,p;\lambda )](x-x^3/3)],\dots ,\hfill \\ \hfill a_n(x,p)& =& [c_n\left(p\right)+I_n(x,p;\lambda )](x-x^3/3)],\hfill \end{array}$$

(129)

where the integration constants $c_j\left(p\right),j\in {\mathbb{Z}}_{+},$ should be determined from the convergence condition of the series (127), and we denoted the integral expressions

$$\begin{array}{ccc}\hfill I_0(x,p;\lambda )& =& {\int }_{\lambda }^x{\displaystyle \frac{[a_{-1,p}^{\prime }(x,p)s-a_{-1,x}^{\prime }(x,p)]ds}{{(s-s^3/3)}^2}},\hfill \\ \hfill I_1(x,p;\lambda )& =& {\int }_{\lambda }^x{\displaystyle \frac{[a_{0,p}^{\prime }(x,p)s-a_{0,x}^{\prime }(x,p)+a_0{(x,p)}^2-1]ds}{{(s-s^3/3)}^2}},\hfill \\ \hfill I_2(x,p;\lambda )& =& {\int }_{\lambda }^x{\displaystyle \frac{[a_{1,p}^{\prime }(x,p)s-a_{1,p}^{\prime }(x,p)-2a_1(x,p)a_0(x,p)]ds}{{(s-s^3/3)}^2}},\hfill \\ \hfill I_3(x,p;\lambda )& =& {\int }_{\lambda }^x{\displaystyle \frac{[a_{2,p}^{\prime }(x,p)s-a_{2,p}^{\prime }(x,p)-a_1{(x,p)}^2)]ds}{{(s-s^3/3)}^2}},\hfill \\ \hfill I_4(x,p;\lambda )& =& {\int }_{\lambda }^x{\displaystyle \frac{[a_{3,p}^{\prime }(x,p)s-a_{3,p}^{\prime }(x,p)-2a_2(x,p)a_1(x,p)-2a_3(x,p)a_0(x,p)]ds}{{(s-s^3/3)}^2}},\dots ,\hfill \end{array}$$

(130)

for all $n\in {\mathbb{Z}}_{+}$ and some fixed $\lambda \in \mathbb{R}\setminus \left\{0\right\}.$ Here we need to remark, that owing to the expressions (129), the functional coefficients $a_j(x,p),j\in {\mathbb{Z}}_{+},$ are regular at all points $x\in \mathbb{R}.$ Taking into account that the relationships (123), (125) and (127), one can obtain uniquely the gradient vector $\phi :=\mathsf{grad}H={({\phi }_1,{\phi }_2)}^{⊺}\in T^{*}\left(T^{*}\left(\mathbb{R}\right)\right),$ where the asymptotic as $\mu \to \infty$ expansions

$${\phi }_1\sim \sum _{j\in {\mathbb{Z}}_{+}}{\phi }_{1,j}{\mu }^{-j},{\phi }_2\sim \sum _{j\in {\mathbb{Z}}_{+}}{\phi }_{1,j}{\mu }^{-j}$$

(131)

are easily calcualted from the following relationships:

$$\begin{array}{c}{\phi }_1(x,p;\mu )=-{\phi }_{2,x}^{\prime }[p+\mu (x-x^3/3)]+{\phi }_{2,p}^{\prime }x,\\ {\phi }_{2,x}^{\prime }[p+\mu (x-x^3/3)]-{\phi }_{2,p}^{\prime }x=a(x,p;\mu ){\phi }_2,\end{array}$$

(132)

if to take additionally into account that its Frechet derivative is symmetric: ${\phi }^{\prime }(x,p)={\phi }^{\prime ,*}(x,p)$ for all $(x,p)\in T^{*}\left(\mathbb{R}\right).$ Thus, we have constructed the analytic in variables $(x,p)\in T^{*}\left(\mathbb{R}\right)$ and asymptotic as $\mu \to \infty$ gradient vector $\mathsf{grad}H={({\phi }_1,{\phi }_2)}^{⊺}\in T^{*}\left(T^{*}\left(\mathbb{R}\right)\right),$ generating via the Volterra homotopy formula (58) an asymptotic as $\mu \to \infty$ expression for the corresponding invariant $H\in \mathcal{D}\left(T^{*}\left(\mathbb{R}\right)\right)$ to the Van der Pol equation (121). In general, one cane make use directly of the linear determining Equation (124) and construct its asymptotic as $\mu \to \infty$ solution, generating the corresponding invariant $H\in \mathcal{D}\left(T^{*}\left(\mathbb{R}\right)\right)$ to the Van der Pol equation. Its analytical structure for both cases is of special interest and is postponed for another research.

Example 8.

As a final example, the classical [40] Painleve-3 second order ordinary differential equation

$$\ddot{x}={\dot{x}}^2{x}^{-1}-\dot{x}{t}^{-1}+t^{-1}(\alpha x^2+\beta )+\gamma x^3+\delta x^{-1}$$

(133)

with $\alpha ,\beta ,\gamma$ and $\delta \in \mathbb{R}$ as parameters.

Observe first that the Painlevé-3 Equation (133) can be equivalently rewritten as the vector field

$$\left(\begin{array}{c}dx/dt\\ dp/dt\end{array}\right)=\left(\begin{array}{c}px{t}^{-1}\\ (\alpha x+\beta x^{-1}+\gamma x^{2}t+\delta x^{-2}t)\end{array}\right)=\mathcal{K}(t;x,p)$$

(134)

on the cotangent vector space $T^{*}\left(\mathbb{R}\right).$ Then the corresponding determining Noether-Lax equation (109) with respect to the invariants to (134) reduces to the following system of linear partial differential equations:

$$\begin{array}{c}d{\phi }_1/dt+p{t}^{-1}{\phi }_1+(\alpha -\beta x^{-2}+2\gamma xt-2\delta x^{-3}t)=0,\\ d{\phi }_2/dt+x{t}^{-1}{\phi }_1=0\end{array},$$

(135)

where $\phi :=\mathsf{grad}H\in T^{*}\left(T^{*}\left(\mathbb{R}\right)\right)$ and is generated by the adjoint matrix

$${\mathcal{K}}^{\prime ,*}=\left(\begin{array}{cc}p{t}^{-1}& \alpha -\beta x^{-2}+2\gamma xt-2\delta x^{-3}t\\ x{t}^{-1}& 0\end{array}\right),$$

(136)

to the Frechet derivative ${\mathcal{K}}^{\prime }:T\left(T^{*}\left(\mathbb{R}\right)\right)\to T\left(T^{*}\left(\mathbb{R}\right)\right)$ and acting on the cotangent space $T^{*}\left(T^{*}\left(\mathbb{R}\right)\right).$ The system (135) can be simplified, if to define a mapping $\xi :T^{*}\left(\mathbb{R}\right)\times \mathbb{R}\to \mathbb{R}$ via the substitution ${\phi }_1=-\xi t{x}^{-1}{\phi }_2:$

$$td\xi /dt+\xi +t{\xi }^2-x{a}_x^{\prime }=0,$$

(137)

where we have denoted $a:=\alpha x+\beta x^{-1}+\gamma x^{2}t+\delta x^{-2}t.$ Moreover, if to introduce the dependent variable $w:=\xi t:T^{*}\left(\mathbb{R}\right)\times \mathbb{R}\to \mathbb{R},$ the Riccati type Equation (137) transforms into the simpler one

$$w_t^{\prime }+w_x^{\prime }px{t}^{-1}+w_p^{\prime }a+w^2{t}^{-1}-x{a}_x^{\prime }=0,$$

(138)

which possesses two different asymptotic as $\left|p\right|\to 0\vee \infty$ solutions

$$w(x,t;p)\sim w_{-1}(x,t)p^{-1}+\sum _{j\in {\mathbb{Z}}_{+}}{w}_j(x,t)p^j$$

(139)

and

$$w(x,t;p)\sim w_{-1}(x,t)p+\sum _{j\in {\mathbb{Z}}_{+}}{w}_j(x,t)p^{-j},$$

(140)

respectively. For instance, if $\left|p\right|\to 0,$ we obtain from (138) the recurrent sequence

$$\partial w_j/\partial t+x{t}^{-1}\partial w_{j-1}/\partial x=(j+1)a{w}_{j+1}+t^{-1}\sum _{k\ge -2}{w}_{j-k}{w}_k-x\partial a/\partial x{\delta }_{j,0}=0$$

(141)

on the coefficients $w_j:\mathbb{R}\setminus \left\{0\right\}\times \mathbb{R}\setminus \left\{0\right\}$ for all $j\ge -1,$ whose solutions are given by the expressions

$$\begin{array}{ccc}\hfill w_{-1}& =& at,w_0=-{\displaystyle \frac{t}{2}}{\displaystyle \frac{\partial }{\partial t}}ln\left(at\right),w_1=-{\displaystyle \frac{\partial w_0/\partial t+t^{-1}{w}_0^2}{3a}},\hfill \\ \hfill w_2& =& -{\displaystyle \frac{\partial w_1/\partial t+x{t}^{-1}\partial w_0/\partial x+2t^{-1}{w}_0{w}_1}{4a}},\dots ,\hfill \end{array}$$

(142)

and so on, allowing the general representation

$$w_j={\displaystyle \frac{k_j(x,t)}{(j+2)!a{(x,t)}^2}}$$

(143)

for some smooth mappings $k_j:\mathbb{R}\setminus \left\{0\right\}\times \mathbb{R}\setminus \left\{0\right\}\to \mathbb{R},j\ge -1,$ which can be easily derived from the sequence (141). Moreover, on an arbitrary compact subset $W_a\subset \mathbb{R}\setminus \left\{0\right\}\times \mathbb{R}\setminus \{a=0\vee \infty \}$ the following estimations

$$|w_j|\le \left|{\displaystyle \frac{k_j(x,t)}{a{(x,t)}^{j+1}}}\right|{\displaystyle \frac{1}{(j+2)!}},\underset{(x,t)\in W_a,j\ge -1}{sup}\left|k_j(x,t)\right|=K_a<\infty ,$$

(144)

for some value $K_a>0$ and all $j\ge -1$ holds. As the series (139) can be summed up and estimated as

$$\begin{array}{ccc}\hfill |w(x,t;p)|& \le & |K_a{p}^{-1}|+K_a\left|a^{-1}\sum _{j\in {\mathbb{Z}}_{+}}{\displaystyle \frac{{\left(p{a}^{-1}\right)}^{j+2}}{(j+2)!}}\right|\le \hfill \\ & \le & K_a\left(\right|p^{-1}|+|a^{-1}+p^2{a}^{-3}|+|aexp\left(p{a}^{-1}\right)\left|\right)\hfill \end{array}$$

(145)

for all $0<$$\left|p\right|$≤${inf}_{(x,t)\in W_a}\left|a^{-1}\right|,$ we can obtain the multiplier $\xi =wt,$ the corresponding covector $\phi \in T^{*}\left(T^{*}\left(\mathbb{R}\right)\right)$ and, suitably, the invariant $H\in \mathcal{D}\left(T^{*}\left(\mathbb{R}\right)\right)$ via the Volterra homotopy formula (58). These reasonings make it possible to formulate the following proposition.

Proposition 2.

The Painleve-3 differential Equation (133), considered as the vector field (134) on the cotangent space $T^{*}\left(\mathbb{R}\right),$ possesses a global invariant $H\in \mathcal{D}\left(T^{*}\left(\mathbb{R}\right)\right)$ on a very compact subset ${\tilde{W}}_a=W_a\times \{0<|p|$$\le {inf}_{(x,t)\in W_a}|a^{-1}\left|\right\}\subset T^{*}\left(\mathbb{R}\right)\times \mathbb{R}/\left\{0\right\}.$

The obtained above result look very interesting from practical points of view, in particular, for studying both periodical solutions and the corresponding resonances, as well as the related critical phase structures, allowed by dynamics of the the Van der Pol and Painlevé equations.

6. Supersymmetric Generalization-Solvable Super-Differential Equations on the Super-Axis

Nowadays, superanalysis is widely accepted as important both from physical point of view, making a contact with reality, related to the phenomenology of the simplest potentially realistic supersymmetric field theory [17,20,21,22] like the minimal Supersymmetric Standard Model, and from its rich mathematical properties, which often allow us to shed a new light on many aspects of modern differential-geometric and topological structures on smooth manifolds and Riemannian metric spaces. From a mathematical physics point of view, there is an interesting generalization of the devised above approach for studying integrability properties of super-differential equations on the super-axis [14,15,17], in part initiated in [18,19], and having diverse applications for analysing fermionic models of quantum physics.

To introduce these super-differential objects, let ${\mathbb{R}}^{1|1}\simeq \mathbb{R}\times {\Lambda }_1^{\left(1\right|1)}$ denote the superized [14,15,17] real axis $\mathbb{R}$ by means of the one-dimensional ${\mathbb{Z}}_2$-graded Grassmann algebra ${\Lambda }^{\left(1\right|1)}={\Lambda }_0^{\left(1\right|1)}\oplus {\Lambda }_1^{\left(1\right|1)}$ with coordinates $(x,\theta )\in \mathbb{R}\times {\Lambda }_1^{\left(1\right|1)}$ and $D_{\theta }:=\partial /\partial \theta +\theta \partial /\partial x$ be the corresponding super-derivation:

$$D_{\theta }\left(a^{\left(p\right)}{b}^{\left(q\right)}\right)=D_{\theta }\left(a^{\left(p\right)}\right)b^{\left(q\right)}+{(-1)}^{pq}{a}^{\left(p\right)}{D}_{\theta }\left(b^{\left(q\right)}\right)$$

(146)

acting on uniform elements $a^{\left(p\right)}$ and $b^{\left(q\right)}$ with parities $\left(p\right)$ and $\left(q\right)\in {\mathbb{Z}}_2,$ respectively, from the space $C^{\infty }({\mathbb{R}}^{1|1};{\Lambda }^{\left(1\right|1)})$ of smooth functions on ${\mathbb{R}}^{1|1}.$

The derivation $D_{\theta }:C^{\infty }({\mathbb{R}}^{1|1};{\Lambda }^{\left(1\right|1)})\to$$C^{\infty }({\mathbb{R}}^{1|1};{\Lambda }^{\left(1\right|1)})$ satisfies also the following important operator relationships: $D_{\theta }^2=\partial /\partial x$ at any point $(x,\theta )\in \mathbb{R}\times$${\Lambda }_1^{\left(1\right|1)}.$ Moreover, for an arbitrary smooth function $u^{\left(p\right)}\in C^{\infty }({\mathbb{R}}^{1|1};{\Lambda }^{\left(1\right|1)})$ of parity $\left(p\right)\in {\mathbb{Z}}_2$ there exists the expansion

$$u^{\left(p\right)}(x,\theta )=u_{\left(p\right)}\left(x\right)+\theta u_{(p+1)}\left(x\right)$$

(147)

at any point $(x,\theta )\in {\mathbb{R}}^{1|1}$ with uniquely defined smooth mappings $u_{\left(k\right)}\in C^{\infty }({\mathbb{R}}^{1|1};{\Lambda }_{\left(k\right)}^{\left(1\right|1)}),$$k=\overline{p,p+1}.$ For the function (147) there is defined the super-integral $\int u^{\left(p\right)}(x,\theta )d\theta ,\left(p\right)\in {\mathbb{Z}}_2,$ over the super-variable $\theta \in {\Lambda }_1^{\left(1\right|1)}$ via the rules

$$\int 1d\theta =0,\int \theta d\theta =1.$$

(148)

As a simplest example of supersymmetric ordinary differential equations, we can consider a general linear first-order non-uniform super-differential equation

$$D_{\theta }u+a^{\left(1\right)}(x,\theta )u+g^{\left(0\right)}(x,\theta )=0,$$

(149)

where, for instance, an odd function $u\in C^2({\mathbb{R}}^{1|1};{\Lambda }_1^{\left(1\right)})$ is unknown, the coefficients $a^{\left(1\right)}\in C^{\infty }({\mathbb{R}}^{1|1};{\Lambda }_1^{\left(1\right|1)}),$$g^{\left(0\right)}\in C^{\infty }({\mathbb{R}}^{1|1};{\Lambda }_0^{\left(1\right|1)})$ are assumed to be given at all points $(x,\theta )\in {\mathbb{R}}^{1|1}.$ To solve the Equation (149), we apply to it the derivation $D_{\theta }$ and take into account that $D_{\theta }^2=\partial /\partial x$:

$$\begin{array}{c}\partial u/\partial x+\left(D_{\theta }{a}^{\left(1\right)}\right)u-a^{\left(1\right)}{D}_{\theta }u+\left(D_{\theta }{g}^{\left(0\right)}\right)=\\ =\partial u/\partial x+\left(D_{\theta }{a}^{\left(1\right)}\right)u+a^{\left(1\right)}[a^{\left(1\right)}u+g^{\left(0\right)})]+\left(D_{\theta }{g}^{\left(0\right)}\right)=\\ =\partial u/\partial x+\left(D_{\theta }{a}^{\left(1\right)}\right)u+\left(a^{\left(1\right)}{a}^{\left(1\right)}\right)u+a^{\left(1\right)}{g}^{\left(0\right)}+\left(D_{\theta }{g}^{\left(0\right)}\right)=\\ =\partial u/\partial x+\left(D_{\theta }{a}^{\left(1\right)}\right)u+a^{\left(1\right)}{g}^{\left(0\right)}+\left(D_{\theta }{g}^{\left(0\right)}\right)=0.\end{array}$$

(150)

This way, we obtained the following equivalent linear ordinary differential equation

$$\partial u/\partial x+\left(D_{\theta }{a}^{\left(1\right)}\right)u+a^{\left(1\right)}{g}^{\left(0\right)}+\left(D_{\theta }{g}^{\left(0\right)}\right)=0,$$

(151)

with respect to the real variable $x\in \mathbb{R},$ whose solution is easily given by means the following classical expression:

$$\begin{array}{c}u(x,\theta )=exp\left(-{\int }_{x_0}^x{D}_{\theta }{a}^{\left(1\right)}(s,\theta )ds\right)c_0+\\ +{\int }_{x_0}^x\left[a^{\left(1\right)}(s,\theta )g^{\left(0\right)}(s,\theta )+D_{\theta }{g}^{\left(0\right)}(s,\theta )\right]exp\left({\int }_x^{s}ds{D}_{\theta }{a}^{\left(1\right)}(s,\theta )\right)ds,\end{array}$$

(152)

where $c_0{:=u(x,\theta )|}_{x=x_0}\in {\Lambda }_1^{\left(1\right)}-$ an arbitrary constant.

Consider now a supersymmetric ordinary linear non-uniform differential equation of second order

$$D_{\theta }^{2}u+b^{\left(1\right)}(x,\theta )D_{\theta }u+a^{\left(0\right)}(x,\theta )u+g^{\left(1\right)}(x,\theta )=0,$$

(153)

where an odd function $u\in C^2({\mathbb{R}}^{1|1};{\Lambda }_1^{\left(1\right|1)})$ is unknown, the coefficients $a^{\left(0\right)},b^{\left(1\right)}$ and $g^{\left(1\right)}\in C^{\infty }({\mathbb{R}}^{1|1};{\Lambda }^{\left(1\right|1)})$ are assumed to be given at all points $(x,\theta )\in {\mathbb{R}}^{1|1}.$ The Equation (153) can be equivalently rewritten as the following vector equation:

$$D_{\theta }\left(\begin{array}{c}{v}_0\\ v_1\end{array}\right)=\left(\begin{array}{cc}0& 1\\ -a^{\left(0\right)}& -b^{\left(1\right)}\end{array}\right)\left(\begin{array}{c}{v}_1\\ v_1\end{array}\right)+\left(\begin{array}{c}0\\ -g^{\left(1\right)}\end{array}\right),$$

(154)

we where we put, by definition, $v_0:=u,v_1:=D_{\theta }u.$ Taking into account that $D_{\theta }^2=\partial /\partial x,$ we obtain from Equation (154) the following ordinary differential equation with respect to the axis variable $x\in \mathbb{R}:$

$$\begin{array}{c}{\displaystyle \frac{\partial }{\partial x}}\left(\begin{array}{c}{v}_0\\ v_1\end{array}\right)=\left(\begin{array}{cc}0& 1\\ -\left(D_{\theta }{a}^{\left(0\right)}\right)& -\left(D_{\theta }{b}^{\left(1\right)}\right)\end{array}\right)\left(\begin{array}{c}{v}_1\\ v_1\end{array}\right)+\\ +\left(\begin{array}{cc}0& 1\\ -a^{\left(0\right)}& b^{\left(1\right)}\end{array}\right)\left[\left(\begin{array}{cc}0& 1\\ -a^{\left(0\right)}& -b^{\left(1\right)}\end{array}\right)\left(\begin{array}{c}{v}_1\\ v_1\end{array}\right)+\left(\begin{array}{c}0\\ -g^{\left(1\right)}\end{array}\right)\right]+\\ +\left(\begin{array}{c}0\\ -\left(D_{\theta }{g}^{\left(1\right)}\right)\end{array}\right)=\left[\left(\begin{array}{cc}0& 1\\ -\left(D_{\theta }{a}^{\left(0\right)}\right)& -\left(D_{\theta }{b}^{\left(1\right)}\right)\end{array}\right)+\right.\\ +\left.\left(\begin{array}{cc}0& 1\\ -a^{\left(0\right)}& b^{\left(1\right)}\end{array}\right)\left(\begin{array}{cc}0& 1\\ -a^{\left(0\right)}& -b^{\left(1\right)}\end{array}\right)\right]\left(\begin{array}{c}{v}_1\\ v_1\end{array}\right)+\\ +\left(\begin{array}{cc}0& 1\\ -a^{\left(0\right)}& b^{\left(1\right)}\end{array}\right)\left(\begin{array}{c}0\\ -g^{\left(1\right)}\end{array}\right)+\left(\begin{array}{c}0\\ -\left(D_{\theta }{g}^{\left(1\right)}\right)\end{array}\right).\end{array}$$

(155)

The obtained vector differential equation

$$\begin{array}{c}{\displaystyle \frac{\partial }{\partial x}}\left(\begin{array}{c}{v}_0\\ v_1\end{array}\right)=\left[\left(\begin{array}{cc}0& 1\\ -\left(D_{\theta }{a}^{\left(0\right)}\right)& -\left(D_{\theta }{b}^{\left(1\right)}\right)\end{array}\right)+\right.\\ +\left.\left(\begin{array}{cc}0& 1\\ -a^{\left(0\right)}& b^{\left(1\right)}\end{array}\right)\left(\begin{array}{cc}0& 1\\ -a^{\left(0\right)}& -b^{\left(1\right)}\end{array}\right)\right]\left(\begin{array}{c}{v}_1\\ v_1\end{array}\right)+\\ +\left(\begin{array}{cc}0& 1\\ -a^{\left(0\right)}& b^{\left(1\right)}\end{array}\right)\left(\begin{array}{c}0\\ -g^{\left(1\right)}\end{array}\right)+\left(\begin{array}{c}0\\ -\left(D_{\theta }{g}^{\left(1\right)}\right)\end{array}\right)\end{array}$$

(156)

can be equivalently rewritten as a non-uniform vector equation

$$\partial v/\partial x=\tilde{\mathcal{L}}(x,\theta )v+\tilde{g}(x,\theta )$$

(157)

on the real axis $\mathbb{R}$ for the vector $v:={(v_0,v_1)}^{⊺}\in C^1({\mathbb{R}}^{1|1};{\Lambda }_1^{\left(1\right)}\times {\Lambda }_0^{\left(1\right)}),$ where the matrix

$$\tilde{\mathcal{L}}:=\left(\begin{array}{cc}0& 1\\ -\left(D_{\theta }{a}^{\left(0\right)}\right)& -\left(D_{\theta }{b}^{\left(1\right)}\right)\end{array}\right)+\left(\begin{array}{cc}0& 1\\ -a^{\left(0\right)}& b^{\left(1\right)}\end{array}\right)\left(\begin{array}{cc}0& 1\\ -a^{\left(0\right)}& -b^{\left(1\right)}\end{array}\right)$$

(158)

and the vector

$$\tilde{g}:=\left(\begin{array}{cc}0& 1\\ -a^{\left(0\right)}& b^{\left(1\right)}\end{array}\right)\left(\begin{array}{c}0\\ -g^{\left(1\right)}\end{array}\right)+\left(\begin{array}{c}0\\ -\left(D_{\theta }{g}^{\left(1\right)}\right)\end{array}\right).$$

(159)

To solve now the ordinary differential vector Equation (157), we need to determine the corresponding fundamental matrix $F(x,x_0)\in \mathrm{End}\left({\Lambda }_1^{\left(1\right|1)}\times {\Lambda }_0^{\left(1\right|1)}\right),$ parametrically depending on the supervariable $\theta \in {\Lambda }_1^{\left(1\right)}$ and satisfying the following conditions:

$$\partial F(x,x_0)/\partial x=\tilde{\mathcal{L}}(x,\theta )F(x,x_0)$$

(160)

and

$$F(x,x_0){|}_{x=x_0}=I$$

(161)

for arbitrary $x,x_0\in \mathbb{R},$ where $I\in \mathrm{End}\left({\Lambda }_1^{\left(1\right|1)}\times {\Lambda }_0^{\left(1\right|1)}\right)$ denotes the unit matrix. Then arbitrary solution to the vector Equation (157) looks as follows:

$$v(x,\theta )=F(x,x_0)c_0+{\int }_{x_0}^{x}F(x,s)\tilde{g}(s,\theta )ds,$$

(162)

where $c_0{:=v(x,\theta )|}_{x=x_0}\in {\left({\Lambda }_1^{\left(1\right|1)}\times {\Lambda }_0^{\left(1\right|1)}\right)}^{⊺}$ is an arbitrary constant.

Concerning a general linear vector superdifferential equation

$$D_{\theta }v=\mathcal{L}(x,\theta )v$$

(163)

for $v\in C^1({\mathbb{R}}^{1|1};{\left({\Lambda }^{\left(1\right|1)}\right)}^n),n\in \mathbb{N},$ with a matrix $\mathcal{L}(x,\theta )$∈$\mathrm{End}\left({\Lambda }_1^{\left(1\right|1)}\times {\Lambda }_0^{\left(1\right|1)}\right)$ at any point $(x,\theta )\in {\mathbb{R}}^{1|1}$, we can apply the same scheme as above and obtain, similarly to [41], the following ordinary linear differential equation:

$$\begin{array}{ccc}\hfill \partial v/\partial x& =& \left(D_{\theta }\mathcal{L}\right)v+\widehat{\mathcal{L}}{D}_{\theta }v=\left(D_{\theta }\mathcal{L}\right)v+\widehat{\mathcal{L}}\mathcal{L}v=\hfill \\ & =& \left(D_{\theta }\mathcal{L}+\widehat{\mathcal{L}}\mathcal{L}\right)v:=\tilde{\mathcal{L}}(x,\theta )v,\hfill \end{array}$$

(164)

where the matrix $\widehat{\mathcal{L}}(x,\theta ):=\mathrm{alt}\mathcal{L}(x,\theta )\in \mathrm{End}{\left({\Lambda }^{\left(1\right)}\right)}^n$ is the odd-elements sign-altered matrix $\mathcal{L}(x,\theta ).$ Then solutions to the obtained above ordinary differential equation

$$\partial v/\partial x=\tilde{\mathcal{L}}(x,\theta )v$$

(165)

on the real axis $\mathbb{R}$ for the vector $v\in C^1({\mathbb{R}}^{1|1};{\left({\Lambda }^{\left(1\right|1)}\right)}^n)$ are completely described by means of the respectively constructed fundamental matrix $F(x,x_0)\in \mathrm{End}{\left({\Lambda }^{\left(1\right|1)}\right)}^n,$ parametrically depending on the supervariable $\theta \in {\Lambda }_1^{\left(1\right)}$ and satisfying the matrix equation

$$\partial F(x,x_0)/\partial x=\tilde{\mathcal{L}}(x,\theta )F(x,x_0)$$

(166)

under the Cauchy condition

$$F(x,x_0){|}_{x=x_0}=I$$

(167)

for arbitrary $x,x_0\in \mathbb{R},$ where $I\in \mathrm{End}{\left({\Lambda }_0^{\left(1\right|1)}\right)}^n$ denotes the unit matrix.

As for a general nonlinear non-uniform n–th order evolution partial super-differential equation

$$D_{\theta }^{n}u=K(x,\theta ;u,D_{\theta }u,\dots ,D_{\theta }^{n-1}u)=0,$$

(168)

where $u\in C^n({\mathbb{R}}^{1|1};{\Lambda }_1^{\left(1\right)}),n\in \mathbb{N},$ and the mapping $K:J^{n-1}({\mathbb{R}}^{1|1};{\Lambda }^{\left(1\right|1)})\to {\Lambda }^{\left(1\right|1)}$ above is analytic [15,17] on the functional jet-supermanifold $J^{n-1}({\mathbb{R}}^{1|1};{\Lambda }^{\left(1\right|1)}),$ parameterized by points $(x,\theta )\in {\mathbb{R}}^{1|1},$ it can be represented, albeit ambiguously, as a super vector field

$$D_{\theta }v=\mathcal{K}(x,\theta ;v)$$

(169)

on the suitably constructed supermanifold ${\mathfrak{M}}^n$≃$J^{n-1}({\mathbb{R}}^{1|1};{\Lambda }^{\left(1\right|1)}),$ parameterized by a super-vector variable $v\in {\mathfrak{M}}^n.$ We now introduce the following useful definition.

Definition 2.

We will call the super-differential equations like (168) quasi-solvable, if its super-vector field form (169) possesses analytical functional quasi-invariants, that is super-functions $\gamma \in \mathcal{D}\left({\mathfrak{M}}^n\right),$ invariant with respect to the super-derivation $D_{\theta }$:

$$\left(\partial /\partial \theta +\theta \partial /\partial x\right)\gamma +\langle \mathit{grad}\gamma |\mathcal{K}\rangle =0$$

(170)

on the whole supermanifold ${\mathfrak{M}}^n,$ where $\phi :=\mathit{grad}\gamma \in T^{*}\left({\mathfrak{M}}^n\right)$ denotes the usual super-gradient of the functional $\gamma \in \mathcal{D}\left({\mathfrak{M}}^n\right)$ subject to the vector super-variable $v\in {\mathfrak{M}}^n.$

Corollary 1.

It is easy to observe that any quasi-invariant $\gamma \in \mathcal{D}\left({\mathfrak{M}}^n\right)$ is invariant with respect to the vector field $\partial /\partial x:\mathbb{R}\to {\mathfrak{M}}^n$ on the real axis $\mathbb{R}:$ if $D_{\theta }\gamma (x,\theta )=0,$$(x,\theta )\in {\mathbb{R}}^{1|1},$ then

$$D_{\theta }^2\gamma (x,\theta )=d\gamma (x,\theta )/dx=0$$

(171)

for all $x\in \mathbb{R}.$

Moreover, based on the reasonings of Section 3, the following generalization of the classical Noether–Lax Proposition, which makes it possible to describe quasi-invariants of the vector vector superfield (169), holds.

Proposition 3.

Let $\gamma \in \mathcal{D}\left({\mathfrak{M}}^n\right)$ be an invariant of the vector super-field (169) on the supermanifold ${\mathfrak{M}}^n$ and $\phi =\mathsf{grad}\gamma \in T^{*}\left({\mathfrak{M}}^n\right)$ be its gradient. Then the following functional super-differential evolution equation

$$D_{\theta }\phi +{\mathcal{K}}^{\prime ,*}\phi =0$$

(172)

holds on the whole super-space ${\mathfrak{M}}^n$ for all $(x,\theta )\in {\mathbb{R}}^{1|1},$ where ${\mathcal{K}}^{\prime ,*}$: $T^{*}\left({\mathfrak{M}}^n\right)\to T^{*}\left(\mathfrak{M}\right)$ is the adjoint operator to the Frechet derivative ${\mathcal{K}}^{\prime }:T\left(\mathfrak{M}\right)\to T\left(\mathfrak{M}\right)$ of the vector super-field (169) with respect to the natural bilinear super-form $\langle ·|·\rangle :$$T^{*}\left({\mathfrak{M}}^n\right)\times T\left({\mathfrak{M}}^n\right)\to {\Lambda }^{1|1}$ on the Euclidean product $T^{*}\left({\mathfrak{M}}^n\right)\times T\left({\mathfrak{M}}^n\right).$

It is well known that very interesting super-differential quasi-solvable nonlinear partial super-differential equations (168) can be constructed [25,42,43,44,45] by means of the super-variational analysis [46,47] on supermanifolds. In particular, let a smooth nondegenerate super-functional $\mu \in \mathfrak{D}\left(J^m\right)\simeq \mathcal{D}\left(J^{m/2}\right)/\mathsf{mod}{D}_{\theta }$ and consider its critical super-submanifold

$$J_{\mu }^{2m-1}:=\{u\in J^{2m-1}:\mathsf{grad}\mu \left[u\right]=0\},$$

(173)

which can be represented as the super vector field (169) for $n=2m-1$ by means of the following new super-variables:

$$v:=(u,{\mathsf{grad}}_{D_{\theta }u}\mu \left[u\right],{\mathsf{grad}}_{D_{\theta }^{2}u}\mu \left[u\right],\dots ,{\mathsf{grad}}_{D_{\theta }^{m}u}\mu \left[u\right])\in {\mathfrak{M}}^{2m-1},$$

(174)

on the supermanifold ${\mathfrak{M}}^{2m-1}.$

Example 9.

As an interesting example, we consider the super-Korteweg–de Vries type [19,48] invariant functional

$$\mu =\int [D_{\theta }^{2}u{D}_{\theta }^{3}u+u{\left(D_{\theta }u\right)}^2/2]dxd\theta ,$$

(175)

whose super-submanifold

$${\mathfrak{M}}^5\simeq \{u\in J^5:\mathit{grad}\mu \left[u\right]=-2D_{\theta }^{5}u+3/2{\left(D_{\theta }u\right)}^2-u{D}_{\theta }^{2}u=0\}$$

(176)

can be equivalently represented by means of the set of super-orbits, generated by the following vector superfield

$$D_{\theta }\left(\begin{array}{c}{v}_0\\ v_1\\ v_2\\ v_3\\ v_4\end{array}\right)=\left(\begin{array}{c}{v}_1,\\ v_2\\ 1/2v_4\\ v_1^2/2\\ v_0{v}_1+v_3\end{array}\right):=\mathcal{K}(u,p),$$

(177)

on the super-submanifold ${\mathfrak{M}}^5,$ parameterized by means of the new supervariables $(v_0=u, v_1=D_{\theta }u,v_3=D_{\theta }^{2}u,v_4=u{D}_{\theta }u+2D_{\theta }^{4}u,v_5=2D_{\theta }^{3}u)\in {\mathfrak{M}}^5.$

The generating Nooether–Lax equation (172) looks as

$$D_{\theta }\left(\begin{array}{c}{\phi }_1\\ {\phi }_2\\ {\phi }_3\\ {\phi }_4\\ {\phi }_5\end{array}\right)+\left(\begin{array}{ccccc}0& 0& 0& 0& 1/2\\ 1& 0& 0& v_1& v_0\\ 0& 1& 0& 0& 0\\ 0& 0& 0& 0& 1\\ 0& 0& 1/2& 0& 0\end{array}\right)\left(\begin{array}{c}{\phi }_1\\ {\phi }_2\\ {\phi }_3\\ {\phi }_4\\ {\phi }_5\end{array}\right),$$

(178)

and allows a simple polynomial super-vector solution $\phi ={({\phi }_1,{\phi }_2,{\phi }_3,{\phi }_4,{\phi }_5)}^{⊺}\in T\left({\mathfrak{M}}^5\right)$ as

$$\phi ={\left(\begin{array}{ccccc}{v}_1^2/2,& v_3+v_0{v}_1,& -v_4,& v_1,& -v_2\end{array}\right)}^{⊺},$$

(179)

giving rise via the Volterra homotopy formula

$$H={\int }_0^{1}d\lambda \langle \phi (\lambda u,\lambda p)|{(u,p)}^{⊺}\rangle$$

(180)

to the first integral super-invariant

$$H_1=u_1(v_0-u_0{u}_1/2)-v_1{u}_2,$$

(181)

satisfying a priori the super-invariance condition $D_{\theta }{H}_1=0$ on the whole super-submanifold ${\mathfrak{M}}^5.$ Solving further the determining Equation (178), one can obtain additional super-invariants to the super-vector field (177) and state its integrability. Moreover, it is easy to check that the super-vector field (177) is Hamiltonian [10,11,25,32,47] with respect to the quasi-canonical super-symplectic structure

$${\omega }^{\left(2\right)}=d{v}_0\wedge d{u}_0+d{v}_1\wedge d{u}_1-d{u}_2\wedge d{u}_2$$

(182)

on the super-submanifold ${\mathfrak{M}}^5.$ On theses and geometrically related aspects of the ordinary super equations (168) we plan to stop in more detail in another work in preparation.

7. Supplement

Consider the Bernoulli equation from (105):

$${\omega }^{\prime }+\omega b\left(x\right)-2c_0\left(x\right){\omega }^3=0$$

(183)

and make the substitution $\omega =z_1^{-{\displaystyle \frac{1}{2}}}:$

$$z_1^{\prime }-2b{z}_1+4c_0=0,$$

(184)

whose solution equals

$$z_1=H_{1}exp\left(2{\partial }^{-1}b\right)-4exp\left(2{\partial }^{-1}b\right){\partial }^{-1}{c}_{0}exp(-2{\partial }^{-1}b),$$

(185)

where $H_1\in \mathbb{R}\left(\left(x\right)\right)$ denotes the first invariant to the Equation (99). Let now substitute the expression

$$\omega \left(x\right)={\left(H_{1}exp\left(2{\partial }^{-1}b\right)-4exp\left(2{\partial }^{-1}b\right){\partial }^{-1}{c}_{0}exp(-2{\partial }^{-1}b)\right)}^{-{\displaystyle \frac{1}{2}}}$$

(186)

into the second Bernoulli Equation (106) and similarly obtain the next analytic expression:

$$\begin{array}{c}u\left(x\right)=[H_{2}exp(-{\partial }^{-1}(2b-6{\omega }^2{c}_0))-\\ -2exp(-{\partial }^{-1}(2b-6{\omega }^2{c}_0)){\partial }^{-1}{c}_0\times exp(-{\partial }^{-1}(2b-6{\omega }^2{c}_0)){]}^{1/2}.\end{array}$$

(187)

Recalling the above substitutions (100), (103) and (105), one finds finally the equality (108), describing two invariants $H_1,H_2\in \mathbb{R}$ to the second order differential Equation (99). It is worth to mention here that our general functional relationship (108) contains as particular cases the results previously presented in [2,8]. Moreover, taking into account the obtained general relationship (68), one can successfully construct all time-dependent invariants to the second order differential Equation (99), which we are going to present in another work under preparation.

8. Conclusions

We presented a new effective geometric-analytic approach to studying invariants to nonlinear second order ordinary differential equations. They were considered interesting from the application point of view examples, and moreover, they constructed exact analytic expression for a special case second order differential equation, generalizing those previously considered in the literature. Especially, we analyzed in detail geometric-analytic properties of invariants and their determining Noether-Lax evolution equation, including its asymptotic properties. We also studied in detail, interesting from a practical point, examples of the second ordinary differential equations, including the classical van der Pol and Painlevé equations. As a natural and interesting for modern mathematical physics generalization of the devised approach is studying integrability properties of ordinary super-differential equations on the super-axis [14,15,17], initiated in [18,19], and having applications for studying quantum mechanical fermionic models of quantum physics, known as instantons and quantum solitoms.