Symmetry Transformations of a Nonlinear Model of Optical Wave Transmission

Abstract

The full symmetry group is found for a system of nonlinear schrödinger equations describing the propagation of optical pulses in an isotropic media. It is shown, in particular, that the six-dimensional symmetry group found is composed of a scaling transformation and a rotation of the four-dimensional space, thereby proving that the symmetry group preserves the shape of solutions. A symmetry classification of one-dimensional subalgebras of the Lie algebra is performed and yields, in particular, the symmetry reduction to the most general system of equations satisfied by the solitary waves of the equation. Explicit soliton solutions of the equation are found by largely autonomous technics. The found solitons are used to recursively generate two new ones by means of two iterations using the symmetry group. Other properties of the system are also highlighted, as well as the possible connections between the theories of symmetry groups and Darboux transformations inspired by this study.

1. Introduction

Optical waves have found a wide range of applications in various fields of science and engineering and, in fact, in our daily lives. Such applications range from optical instruments, medical imaging, optical sensors, quantum computing, through to fibre optic cables and many others [1,2,3,4,5,6]. Over the last decades a growing interest from the mathematical and engineering fields has been devoted to the latter type of applications, namely optical fibres, for the efficient transmission of information in the form of short optical pulses, yielding high-speed internet and data network and more secure data transmission.

The transmission of short optical pulses for telecommunication purposes can be achieved in anomalous dispersion fibres where optical pulses are realized as solitons by balancing the anomalous group velocity parameter with the self-phase modulation parameter describing the nonlinearity of the fibre [7,8]. Due to the stability of solitons with regard to their velocity and shape, even after collision, this form of data transmission certainly provides the most suitable response to the telecommunication demands of modern times given that soliton pulses can effectively maintain their shapes and waveforms while travelling over long distances.

With regard to mathematical modelling, optical wave transmission in anomalous dispersion fibres can be described as a system of coupled nonlinear Schrödinger equations (NLSEs) due to the anisotropic nature of such a medium of transmission [9,10,11].

In this paper, we undertake the study of symmetry transformations of a system of coupled NLSEs describing the propagation of short optical pulses in anomalous dispersion fibres and involving four-wave mixing terms. Such a system takes the form

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill i{u}_t+\beta u_{xx}+\left({\mu }_1{\left|u\right|}^2+{\mu }_2{\left|v\right|}^2\right)u+{\mu }_3{v}^2{u}^{*}& =0\hfill \\ \hfill i{v}_t+\beta v_{xx}+\left({\mu }_1{\left|v\right|}^2+{\mu }_2{\left|u\right|}^2\right)v+{\mu }_3{u}^2{v}^{*}& =0,\hfill \end{array}\end{array}$$

(1)

where partial derivatives with respect to (w.r.t) a variable are denoted by subscripts in the variable, and u and v are the components of the vector pulse while a star as superscript represents complex conjugation. Moreover, ${\mu }_1,{\mu }_2$ and ${\mu }_3$ represent the strength of the self-phase modulation, the cross-phase modulation, and the four-wave mixing effects, respectively. In physical applications, the equality ${\mu }_3={\mu }_1-{\mu }_2$ is generally required. The parameter $\beta$ is the group velocity parameter and will generally be assumed to be a fixed constant. The above model of cubic coupled NLSEs is a version of the equation considered in [12] from an optical physics perspective for the theoretical description of the behaviour of modulated pulses in nonlinear birefringent optical waveguides. Equation (1) differs from the original version from [12] in that the non-instantaneous response of the waveguide has been neglected.

The early Lie group analysis of NLSEs was carried out by Winternitz and Gagnon [13,14] who found the exact solutions of the generalized cubic–quintic NLSE and derived its symmetry group and subgroups. More recently, the Lie group analysis was applied to models of NLSEs describing the propagation of optical pulses in [15,16]. Other studies of the NLSE from the Lie group approach were carried out in [17,18,19,20,21,22,23] for the study of their symmetry generators, reductions, and often for their more general exact solutions or their conservation laws.

The Lie group analysis of (1) was considered in [24] as a first study of this kind involving four-wave mixing terms. In the latter study, the parameters ${\mu }_j(j=1,2,3)$ were all considered to be arbitrary, limiting the symmetry algebra to the smallest in dimension, namely the principal algebra. The symmetry generators found were also applicable only to the real form of the solution. Moreover, the equation was not solved for explicit solutions, but the generation of solitons performed in the study used as seed solution a soliton solution adapted from one found by Park and Shin [25] for a different model describing the propagation of optical waves in isotropic media.

In this paper, we consider a symmetry class of (1) obtained in [26] from a group classification of (1) according the values of the parameters ${\mu }_j,(j=1,2,3)$. The explicit construction of the related six-dimensional symmetry algebra L is presented, first in the real form and then in the original complex form. This is followed by the determination of the full symmetry group and the classification of one-dimensional subalgebras of $L,$ which in turn gives rise in particular to the most general symmetry reduction in (1), yielding solitary wave solutions. It is also shown that the symmetry group consists uniquely of a composition of scalings and a rotation of a four-dimensional space, and thereby preserves the shape of any solution. Soliton solutions are found in terms of the arbitrary values of all parameters of the equation by explicitly solving the NLSE. Such solitons are naturally preserved by the full symmetry group and examples of generation of new solitons from known ones are given through two consecutive iterations of the symmetry group transformations. Other properties of the system of NLSEs are highlighted. Some insights inspired by this study on the possible connections between symmetry and Darboux transformations are also discussed.

2. Basic Concepts

Let G be a local group of transformations acting on the manifold $M.$ Then to each (local) group action of $G,$ there corresponds an infinitesimal action generated by a vector field on $M.$ The set of all infinitesimal generators of G forms a Lie algebra called the Lie algebra of the local Lie group $G.$

Suppose that

$$\Delta \equiv {\Delta }_{\kappa }(x,u_{\left(n\right)})=0,(\kappa =1,\dots ,q)$$

(2)

is a system of q differential equations in the independent variable x and dependent variable u, with $(x,u)\in M$, where M is an open subset of $X\times U,$ with $X={\mathbb{R}}^p$ and $U={\mathbb{R}}^q,$ while $u_{\left(n\right)}$ denotes u and all its derivatives up to the order $n.$ The symmetry group of (2) is the largest local group of transformations G acting on M which leaves the solution set $\mathcal{S}$ of (2) invariant. The symmetry algebra of (2) is just the Lie algebra L of infinitesimal generators of $G.$

Given a symmetry group G of a system of equations of the form (2), there exists a prolongation of G into a local group action on the extended jet space $M^{\left(n\right)}$ of $M.$ Correspondingly, each infinitesimal generator $\mathbf{v}$ of G can be prolonged into a vector field ${\mathbf{v}}^{\left[n\right]}$ also often still denoted again by $\mathbf{v},$ on $M^{\left(n\right)}.$

The infinitesimal criterion of the invariance of a given system of equations of the form (2) under the action of a transformation group G is given by the following result of Olver [27].

Theorem  1.

Suppose

$${\Delta }_{\kappa }(x,u_{\left(n\right)})=0,(\kappa =1,\dots ,q)$$

is a system of differential equations of maximal rank defined over $M\subset X\times U.$ If G is a local group of transformations acting on M, and

$${\mathbf{v}}^{\left(n\right)}\left[{\Delta }_{\kappa }(x,u_{\left(n\right)})\right]=0,\kappa =1,\dots ,qwhenever{\Delta }_i(x,u_{\left(n\right)})=0,$$

for every infinitesimal generator $\mathbf{v}$ of G, then G is a symmetry group of the system.

3. Infinitesimal Symmetry Generators

We undertake in this section the determination of the symmetry algebra $L_c$ of the particular case of (1) in which we are interested, namely the case where ${\mu }_3=-{\mu }_1:=\mu$ and ${\mu }_2=0.$ The resulting form of (1) then takes the form

$$\begin{array}{cc}\hfill i{u}_t+\beta u_{xx}-\mu {\left|u\right|}^{2}u+\mu v^2{u}^{*}& =0,\hfill \\ \hfill i{v}_t+\beta v_{xx}-\mu {\left|v\right|}^{2}v+\mu u^2{v}^{*}& =0.\hfill \end{array}$$

(3)

It turns out to be more practical for the determination of the symmetry algebra and related studies to work with the real form of (3). We thus write the complex variables u and v into their real–imaginary form by setting

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill u& =A+iB,\mathrm{and}v=R+iS,\hfill \end{array}\end{array}$$

(4)

where $A=A(t,x),B=B(t,x),R=R(t,x),\mathrm{and}S=S(t,x)$ are real-valued functions. Substituting (4) into (3) and re-arranging into real and imaginary parts shows that (3) has a real form given by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill 2\mu S(-BR+AS)-B_t+\beta A_{xx}& =0,\hfill \\ \hfill 2\mu R(BR-AS)+A_t+\beta B_{xx}& =0,\hfill \\ \hfill 2\mu B(BR-AS)-S_t+\beta R_{xx}& =0,\hfill \\ \hfill 2\mu A(-BR+AS)+R_t+\beta S_{xx}& =0.\hfill \end{array}\end{array}$$

(5)

Let $\mathbf{v}$ be the generic symmetry generator of (5) which may be represented as

$$\mathbf{v}=\xi {\partial }_t+\eta {\partial }_x+\varphi {\partial }_A+\rho {\partial }_B+\theta {\partial }_R+\Omega {\partial }_S,$$

where the six functions $\xi ,\eta ,\varphi ,\rho ,\theta ,$ and $\Omega$ called infinitesimals are each a function of $t,x,A,B,R$ and $S.$

As Equation (5) is a system of second order, we need to prolong $\mathbf{v}$ to the jet space of the second order so as to find the symmetry generators. Let $\Delta =({\Delta }_1,{\Delta }_2,{\Delta }_3,{\Delta }_4)$ be the left-hand side of (5) so that (5) is given by the system of equations $0={\Delta }_1={\Delta }_2={\Delta }_3={\Delta }_4.$ Let $E=(E_1,\dots ,E_4)$ be the expression resulting from the application of the second prolongation of $\mathbf{v}$ to (5) on the solution surface. That is, $E=(E_1,E_2,E_3,E_4)$ is the expression

$${\mathbf{v}}^{\left[2\right]}\left({\Delta }_j\right){|}_{\Delta =0}=0\mathrm{for}j=1,\dots ,4.$$

(6)

The determining equations of (5) will then consist in viewing E as a polynomial in the derivatives of $A,B,R,$ and S, which must vanish. Some of the coefficients of such a polynomial written in matrix form by the juxtaposition of coefficients and monomials are as follows in the case of some of the derivatives of the variable $\xi .$

$$\left[\begin{array}{cc}-2\beta {\xi }_x& A_{tx}\\ {\xi }_S& B_t{S}_t\\ {\xi }_R& B_t{R}_t\end{array}\right],\left[\begin{array}{cc}-2\beta {\xi }_A& A_x{A}_{tx}\\ {\xi }_B& B_t^2\end{array}\right].$$

(7)

These two $\mathrm{matrices}$ correspond to a total of five $\mathrm{equations}$ and show that

$$0={\xi }_x={\xi }_A={\xi }_B={\xi }_R={\xi }_S$$

(8)

must hold. That is, $\xi =\xi \left(t\right)$ depends on t alone. Proceeding in the same manner for the coefficients in (6) involving $\eta$ and its derivatives shows that $\eta =\eta (t,x)$ must hold. Substituting this expression for $\eta$ back into the latest updated version of the determining equations shows that the remaining infinitesimals $\varphi ,\rho ,\theta ,\mathrm{and}\Omega$ are all linear functions of $A,B,R,\mathrm{and}S.$ Indeed, let us set $x_1=A,x_2=B,x_3=R,\mathrm{and}{x}_4=S.$ Then the determining equations show that

$$0={\varphi }_{x_i{x}_j}={\rho }_{x_i{x}_j}={\theta }_{x_i{x}_j}={\Omega }_{x_i{x}_j},\forall i,j\in \{1,2,3,4\},$$

(9)

where for a given function $q=q(t,x,A,B,R,S)$ one sets

$$q_{x_i}={\displaystyle \frac{\partial q}{{\partial }_{x_i}}},\mathrm{and}{q}_{x_i{x}_j}={\displaystyle \frac{{\partial }^{2}q}{{\partial }_{x_i}{\partial }_{x_j}}}.$$

It follows therefore from (9) that the functions $\varphi ,\rho ,\theta ,\mathrm{and}\Omega$ are linear as specified. Consequently, one has

$$q=\sum _{j=1}^4{q}_j{x}_j,\mathrm{for}\mathrm{all}q\in \{\varphi ,\rho ,\theta ,\Omega \}$$

(10)

where the $q_j=q_j(t,x)$ are unknown functions to be found.

The substitution of the expression for q from (10) into the latest version of the determining equations shows that

$$\begin{array}{cc}{\Omega }_4={\theta }_3,\hfill & {\rho }_4={\varphi }_3,\hfill \end{array}$$

(11)

$$\begin{array}{cc}{\Omega }_3=-{\theta }_4,\hfill & {\rho }_3=-{\varphi }_4,\hfill \end{array}$$

(12)

$$\begin{array}{cccc}\hfill {\Omega }_2& ={\theta }_1,\hfill & \hfill {\rho }_2& ={\varphi }_1,\hfill \end{array}$$

(13)

$$\begin{array}{cccc}\hfill {\Omega }_1& =-{\theta }_2,\hfill & \hfill {\rho }_1& =-{\varphi }_2.\hfill \end{array}$$

(14)

Moreover, one has

$$0={\left({\theta }_1\right)}_x={\left({\theta }_2\right)}_x={\left({\rho }_3\right)}_x={\left({\rho }_4\right)}_x={\left({\varphi }_3\right)}_x={\left({\varphi }_4\right)}_x={\left({\Omega }_1\right)}_x={\left({\Omega }_2\right)}_x$$

(15)

which implies from (11) the following conditions

$$\begin{array}{cc}\hfill f\left(t\right)+{\displaystyle \frac{1}{2}}x{\xi }^{\prime }\left(t\right)=& \eta ,\hfill \\ \hfill {\left({\theta }_3\right)}_x={\left({\varphi }_1\right)}_x=& 0,\hfill \\ \hfill {\eta }_t+2\beta {\theta }_{4x}=& 0,\hfill \\ \hfill {\eta }_t+2\beta {\varphi }_{2x}=& 0,\hfill \end{array}$$

(16)

for a certain function $f=f\left(t\right).$ The latter condition implies that

$$\begin{array}{cc}\hfill {\theta }_4(t,x)& =g\left(t\right)-{\displaystyle \frac{2xf\left(t\right)+\frac{1}{2}{x}^2{\xi }_t^{″}}{4\beta }}\hfill \\ \hfill {\varphi }_4(t,x)& =h\left(t\right)-{\displaystyle \frac{2x{f}^{\prime }\left(t\right)+\frac{1}{2}{x}^2{\xi }_t^{″}}{2\beta }}\hfill \end{array}$$

(17)

for some functions $g=g\left(t\right)$ and $h=h\left(t\right),$ where the prime on a function denotes derivative with respect to the single variable of the function. After the application of (17), the only determining equations left are those depending on the parameter $\mu ,$ i.e., the so-called classifying equations. Given that none of the functions which are still to be found at this stage depend on the functions $A,B,R,\mathrm{and}S,$ the classifying equations which also appear as polynomials in these four variables will vanish only if the resulting coefficients vanish. These considerations give rise to 64 elementary equations, among which many are redundant as usual. It should be recalled that the classifying equations are to be solved under the initial assumption that the parameter $\mu$ satisfies $\mu \ne 0.$ In this way, we first obtain the following set of conditions:

$$\begin{array}{cc}\hfill {\left({\varphi }_1\right)}_t& ={\varphi }_4={\theta }_2,\hfill \\ \hfill \left({\varphi }_3\right)& =-{\theta }_1,\hfill \\ \hfill g& =h.\hfill \end{array}$$

(18)

Substituting (18) back into the classifying equations and solving the resulting new equations shows that

$$\begin{array}{cc}\hfill {\theta }_1& =k_1,\hfill \\ \hfill \xi & =a_0{t}^2+k_{2}t+k_3,\hfill \\ \hfill f& =k_{4}t+k_5,\hfill \\ \hfill h& =k_6,\hfill \end{array}$$

(19)

where $k_j,\mathrm{for}j=1,\dots ,6$, and $a_0$ are arbitrary constants. Substituting (19) back into the latest version of the determining equations shows that

$$\begin{array}{cc}\hfill {\theta }_3& ={\displaystyle \frac{1}{2}}(-k_2-2a_{0}t),\hfill \\ \hfill {\varphi }_1& ={\displaystyle \frac{1}{2}}(-k_2-2a_{0}t)\mathrm{with}{a}_0=0.\hfill \end{array}$$

(20)

The result of solving the determining equations, including the application of all the relations (7)–(20), gives rise to the following expressions for the infinitesimals:

$$\begin{array}{cc}\hfill \xi & =k_3+k_{2}t,\hfill \\ \hfill \eta & =k_5+k_{4}t+{\displaystyle \frac{k_2}{2}}x,\hfill \\ \hfill \varphi & =-{\displaystyle \frac{1}{2}}{k}_{2}A+\left(k_6-{\displaystyle \frac{k_4}{2\beta }}x\right)B-k_{1}R,\hfill \\ \hfill \rho & =\left(-k_6+{\displaystyle \frac{k_4}{2\beta }}x\right)A-{\displaystyle \frac{1}{2}}{k}_{2}B-k_{1}S,\hfill \\ \hfill \theta & =k_{1}A-{\displaystyle \frac{1}{2}}{k}_{2}R+\left(k_6-{\displaystyle \frac{k_4}{2\beta }}\right)S,\hfill \\ \hfill \Omega & =k_{1}B+{\displaystyle \frac{1}{2}}\left[\left(-2k_6+{\displaystyle \frac{k_4}{\beta }}x\right)R-k_{2}S\right].\hfill \end{array}$$

(21)

Normalizing the expression in (21) by replacing $k_2$ with $2k_2$ and $k_1$ with $-k_1$ shows that the symmetry algebra L of (5) is clearly of the sixth dimension and generated by the vector fields

$$\begin{array}{cc}\hfill {\mathbf{v}}_1& =R{\partial }_A+S{\partial }_B-A{\partial }_R-B{\partial }_S,\hfill \\ \hfill {\mathbf{v}}_2& =2t{\partial }_t+x{\partial }_x-A{\partial }_A-B{\partial }_B-R{\partial }_R-S{\partial }_S,\hfill \\ \hfill {\mathbf{v}}_3& ={\partial }_t,\hfill \\ \hfill {\mathbf{v}}_4& =t{\partial }_x-{\displaystyle \frac{Bx}{2\beta }}{\partial }_A+{\displaystyle \frac{Ax}{2\beta }}{\partial }_B-{\displaystyle \frac{Sx}{2\beta }}{\partial }_S+{\displaystyle \frac{Rx}{2\beta }}{\partial }_S,\hfill \\ \hfill {\mathbf{v}}_5& ={\partial }_x,\hfill \\ \hfill {\mathbf{v}}_6& =B{\partial }_A-A{\partial }_B+S{\partial }_R-R{\partial }_S.\hfill \end{array}$$

(22)

Expressing the above symmetries (22) of the real form (5) of (3) in terms of the original complex variables $u,v,u^{*},$ and $v^{*}$ gives the symmetry generators ${\mathbf{v}}_j^0(j=1,\dots ,6)$ of the symmetry algebra $L_c$ of (3), which have the following expressions:

$$\begin{array}{cc}\hfill {\mathbf{v}}_1^0& =v{\partial }_u-u{\partial }_v+v^{*}{\partial }_{u^{*}}-u^{*}{\partial }_{v^{*}},\hfill \\ \hfill {\mathbf{v}}_2^0& =2t{\partial }_t+x{\partial }_x-u{\partial }_u-v{\partial }_v-u^{*}{\partial }_{u^{*}}-v^{*}{\partial }_{v^{*}},\hfill \\ \hfill {\mathbf{v}}_3^0& ={\partial }_t,\hfill \\ \hfill {\mathbf{v}}_4^0& =t{\partial }_x+{\displaystyle \frac{iux}{2\beta }}{\partial }_u+{\displaystyle \frac{ivx}{2\beta }}{\partial }_v-{\displaystyle \frac{i{u}^{*}x}{2\beta }}{\partial }_{u^{*}}-{\displaystyle \frac{i{v}^{*}x}{2\beta }}{\partial }_{v^{*}},\hfill \\ \hfill {\mathbf{v}}_5^0& ={\partial }_x,\hfill \\ \hfill {\mathbf{v}}_6^0& =-iu{\partial }_u-iv{\partial }_v+i{u}^{*}{\partial }_{u^{*}}+i{v}^{*}{\partial }_{v^{*}}.\hfill \end{array}$$

(23)

Let us now set

$$\begin{array}{c}\hfill \begin{array}{cccccc}\hfill {\mathbf{w}}_1& ={\mathbf{v}}_3,\hfill & \hfill {\mathbf{w}}_2& ={\mathbf{v}}_5,\hfill & \hfill {\mathbf{w}}_3& ={\mathbf{v}}_1\hfill \\ \hfill {\mathbf{w}}_4& ={\mathbf{v}}_6,\hfill & \hfill {\mathbf{v}}_5& =-{\mathbf{v}}_2,\hfill & \hfill {\mathbf{w}}_6& ={\mathbf{v}}_4.\hfill \end{array}\end{array}$$

(24)

In the sequel, we shall be working with the two different bases of L according to the study at hand. Namely, the basis ${\mathcal{B}}_{\mathbf{v}}=\left\{{\mathbf{v}}_1,\dots ,{\mathbf{v}}_6\right\}$ given by (22) and the basis ${\mathcal{B}}_{\mathbf{w}}=\left\{{\mathbf{w}}_1,\dots ,{\mathbf{w}}_6\right\}$ given by (24) in terms of (22).

4. Lie Point Symmetry Group

Using the exponentiation of the symmetry generators from the preceding section, we derive in this section the most general symmetry transformation that leaves (5) invariant, in the sense that it transforms a solution of the equation into another solution. Applications of these results will be explored in the next section.

4.1. Flow of a Vector Field

Suppose that $\mathbf{v}$ is a vector field acting on the open subset $M\subset X\times Y,$ with the usual notation. Let $\epsilon$ be a small parameter in the neighbourhood of $0\in ℝ$ and $p\in M$ be a point. The flow of $\mathbf{v}$ can be viewed as a local group action

$$\Psi :G_0\times M\to M:(\epsilon ,p)\mapsto \Psi (\epsilon ,p)=q\in M,$$

(25)

where $G_0$ is a neighbourhood of $0\in ℝ$ and is given by

$$\begin{array}{cc}\hfill {\displaystyle \frac{d\Psi }{d\epsilon }}(\epsilon ,p)& ={\mathbf{v}}_{|\Psi (\epsilon ,p)}\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill \Psi (0,p)& =p.\hfill \end{array}$$

(27)

That is, we have

$$\dot{q}=\mathbf{v}\left(q\right),\forall q=\Psi (\epsilon ,p),\mathrm{with}p\in M.$$

(28)

Note that for a fixed value of $p,$ we may write $q=q\left(\epsilon \right)\equiv {\Psi }_p\left(\epsilon \right).$ Suppose in local coordinates $x=(x_1,\dots ,x_m),$ where $m=dimM$ one has $q=(q_1,\dots ,q_m),$$p=(p_1,\dots ,p_m)$ and

$$\mathbf{v}={\xi }^1{\partial }_{x_1}+\cdots +{\xi }^m{\partial }_{x_m}.$$

Then (28) becomes

$${\dot{q}}_j={\xi }^j\left(q\right),\forall j=1,\dots ,m.$$

(29)

In other words, the flow function satisfies $q\left(0\right)=p$ or equivalently

$$q_j\left(0\right)=p_j,\forall p=(p_1,\dots ,p_m)\in M.$$

(30)

Therefore, finding the flow of $\mathbf{v}$ amounts to solving (29) with initial conditions (30).

Let us consider, for instance, the generator ${\mathbf{v}}_4$ of the symmetry algebra L of (4). That is

$${\mathbf{v}}_4=t{\partial }_x-{\displaystyle \frac{Bx}{2\beta }}{\partial }_A+{\displaystyle \frac{Ax}{2\beta }}{\partial }_B-{\displaystyle \frac{Sx}{2\beta }}{\partial }_S+{\displaystyle \frac{Rx}{2\beta }}{\partial }_S={\mathbf{v}}_4\left(p\right),$$

(31)

where $p\equiv (t,x,A,B,R,S).$ If, as usual, we write $q=(y,z,u,v,w,c),$ the flow $\Psi (\epsilon ,p)=q$ is given by

$$\begin{array}{cccc}\hfill \dot{y}& =0,\hfill & \hfill \dot{v}& ={\displaystyle \frac{zu}{2\beta }},\hfill \end{array}$$

(32)

$$\begin{array}{cccc}\hfill \dot{z}& =y,\hfill & \hfill \dot{w}& =-{\displaystyle \frac{zc}{2\beta }},\hfill \end{array}$$

(33)

$$\begin{array}{cccc}\hfill \dot{u}& =-{\displaystyle \frac{zv}{2\beta }},\hfill & \hfill \dot{c}& ={\displaystyle \frac{zw}{2\beta }}\hfill \end{array}$$

(34)

together with $y\left(0\right)=t,z\left(0\right)=x,u\left(0\right)=A,v\left(0\right)=B,w\left(0\right)=R,c\left(0\right)=S.$ Solving (32) and setting $a={\displaystyle \frac{\epsilon }{4\beta }}(2x+t\epsilon )$ shows that $\forall \epsilon \in G_0,$

$$\begin{array}{cccc}\hfill y\left(\epsilon \right)& =t,\hfill & \hfill v\left(\epsilon \right)& =Bcosa+Asina,\hfill \end{array}$$

(35)

$$\begin{array}{cccc}\hfill z\left(\epsilon \right)& =x+\epsilon t,\hfill & \hfill w\left(\epsilon \right)& =Rcosa-Ssina,\hfill \end{array}$$

(36)

$$\begin{array}{cccc}\hfill u\left(\epsilon \right)& =Acosa-Bsina,\hfill & \hfill c\left(\epsilon \right)& =Scosa+Rsina.\hfill \end{array}$$

(37)

4.2. Symmetry Group

If $\mathbf{v}$ is a generator of the symmetry algebra L of a differential equation, every flow $\Psi (\epsilon ,p)$ of $\mathbf{v}$ defines a one-parameter symmetry group of the differential equation. Suppose that a given system of diffential equations $(\Delta =0)$ has m basis symmetry generators ${\mathbf{v}}_1,\dots ,{\mathbf{v}}_m.$ By abuse of notation, let ${\Psi }^i(r_i,p)\equiv \Psi (r_i,p)$ denote the flow of ${\mathbf{v}}_i$ through $p\in M,$ where $r_i\in ℝ$ is a small parameter. Let ${\Psi }_{r_i}$ denote the partial map given by

$${\Psi }_{r_i}:M\to M:p\mapsto {\Psi }_{r_i}\left(p\right)=\Psi (r_i,p).$$

Then for different values of $r_i$, the maps ${\Psi }_{r_i}$ can be composed. Let us write ${\Psi }_{r_1}\left(p\right)=p^1,$${\Psi }_{r_2}\left(p^1\right)=p^{12},$ and more generally ${\Psi }_{r_j}\left(p^{12\dots (j-1)}\right)=p^{12\dots (j-1)j}.$ Then we have

$${\Psi }_{r_m}\circ {\Psi }_{r_{m-1}}\circ \cdots \circ {\Psi }_{r_1}\left(p\right)=p^{1\cdots (m-1)m}.$$

(38)

Naturally, if we replace $r_i$ by $j_i$ in (38) and let ${\Psi }_{r_{ji}}={\Psi }_{j_i},$ where $j_i\in \{1,\dots ,m\}$ for all $i=1,\dots ,m,$ then (38) becomes

$${\Psi }_{j_m}\circ \cdots \circ {\Psi }_{j_2}\circ {\Psi }_{j_1}\left(p\right)=p^{j_1{j}_2\cdots j_m}.$$

(39)

Equation (39) represents the most general expression of the symmetry group of $(\Delta =0).$ Note that the right-hand side of (39) just provides a procedure for the calculation while the left-hand side is the natural expression of the symmetry transformation.

In the case of (5) under consideration, we note, for instance, that the two-parameter symmetry group generated by ${\mathbf{v}}_2$ and ${\mathbf{v}}_3$ can be obtained by the flow composition ${\Psi }_2\circ {\Psi }_3=q=(y,z,u,v,w,c),$ with

$$\begin{array}{cccc}\hfill y& =e^{2r_2}(r_3+t),\hfill & \hfill v& =e^{-r_2}B,\hfill \\ \hfill z& =e^{r_2}x,\hfill & \hfill v& =e^{-r_2}B,\hfill \\ \hfill u& =e^{-r_2}A,\hfill & \hfill c& =e^{-r_2}S,\hfill \end{array}$$

where $r_2$ and $r_3$ are the group parameters. Similarly, the two-parameter symmetry group generated by ${\mathbf{v}}_5$ and ${\mathbf{v}}_3$ is obtained from the flow composition ${\Psi }_5\circ {\Psi }_3\left(p\right)=p^{35}=q\equiv (y,z,u,v,w,c).$ In this case, again letting $r_5$ and $r_3$ be the group parameters, one has

$$\begin{array}{cccc}\hfill y& =t+r_3,\hfill & \hfill v& =B,\hfill \\ \hfill z& =x+r_5,\hfill & \hfill w& =R,\hfill \\ \hfill u& =A,\hfill & \hfill c& =S.\hfill \end{array}$$

In this way, the full six-parameter symmetry group of (5) can be obtained by the flow composition

$${\Psi }_4\circ {\Psi }_1\circ {\Psi }_6\circ {\Psi }_3\circ {\Psi }_5\circ {\Psi }_2=q=(y,z,u,v,w,c).$$

(40)

To give concise and explicit expressions for the components of q in (40), let us set

$$\vartheta (t,x)={\displaystyle \frac{1}{4\beta }}\left[4\beta r_6-r_4(r_3{r}_4+2r_5)-e^{2r_2}{r}_4^{2}t-e^{r_2}{r}_{4}x\right]\equiv \vartheta .$$

(41)

Then it turns out that the most general symmetry transformation of (5) as defined by (40) is given by

$$\begin{array}{cc}\hfill y& =r_3+e^{2r_2}t,\hfill \end{array}$$

(42)

$$\begin{array}{cc}\hfill z& =r_3{r}_4+r_5+e_2^r(e_2^r{r}_{4}t+x),\hfill \end{array}$$

(43)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill u& =e^{-r_2}[Acos\left(r_1\right)cos\left(\vartheta \right)+Rsin\left(r_1\right)cos\left(\vartheta \right)\hfill \\ \hfill & +Bcos\left(r_1\right)sin\left(\vartheta \right)+Ssin\left(r_1\right)sin\left(\vartheta \right)],\hfill \end{array}\end{array}$$

(44)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill v& =e^{-r_2}[Bcos\left(r_1\right)cos\left(\vartheta \right)+Ssin\left(r_1\right)cos\left(\vartheta \right)\hfill \\ \hfill & -Acos\left(r_1\right)sin\left(\vartheta \right)-Rsin\left(r_1\right)sin\left(\vartheta \right)],\hfill \end{array}\end{array}$$

(45)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill w& =e^{-r_2}[Rcos\left(r_1\right)cos\left(\vartheta \right)-Asin\left(r_1\right)cos\left(\vartheta \right)\hfill \\ \hfill & +Scos\left(r_1\right)sin\left(\vartheta \right)-Bsin\left(r_1\right)sin\left(\vartheta \right)],\hfill \end{array}\end{array}$$

(46)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill c& =e^{-r_2}[Scos\left(r_1\right)cos\left(\vartheta \right)-Bsin\left(r_1\right)cos\left(\vartheta \right)\hfill \\ \hfill & -Rcos\left(r_1\right)sin\left(\vartheta \right)+Asin\left(r_1\right)sin\left(\vartheta \right)].\hfill \end{array}\end{array}$$

(47)

4.3. Symmetry Transformation of Solutions

As a composite of the flow of individual basis symmetry generators, the symmetry group of a system of differential equations $(\Delta =0)$ defined on M can be viewed as a local group of transformations $\Psi$ acting on $M.$ Thus for a pair $(g,p)$ in the domain of $\Psi ,$ let us set $q=\Psi (g,p),$ where $p=(t,x,A,B,R,S)\in M$ and $q=(y,z,u,v,w,c)\in M.$ Assume now that $\Psi$ is the symmetry group of (5) and that

$$A=F(t,x),B=G(t,x),R=H(t,x),S=J(t,x)$$

(48)

represents a solution set of (5). Then the four functions $u=u(y,z),v=v(y,z),w=w(y,z),\mathrm{and}c=c(y,z)$ also represent a solution of (5).

In the expression of $u,v,w$ and c given by (42)–(47), one can make use of the first two equations in (42)–(47) to express t and x in terms of y and $z,$ and then proceed to a renaming of variables for uniformization by calling y and z with the usual names t and $x,$ respectively. In this way, one first obtains $t=-e^{-2r_2}(r_3-y)$ and $x=-e^{-r_2}(r_5+r_{4}y-z),$ and these are inserted in the expressions for $u,v,w$ and c given in (42)–(47). The renaming of y and z to the original names t and x gives rise to the following intermediary variables that henceforth arise in (42)–(47):

$$\begin{array}{cc}\hfill a(t,x)& =-e^{-2r_2}(r_3-t),b(t,x)=-e^{-r_2}(r_5+r_{4}t-x)\hfill \end{array}$$

(49)

$$\begin{array}{cc}\hfill h(t,x)& ={\displaystyle \frac{1}{4\beta }}(r_4^{2}t-2r_{4}x+4r_6\beta ).\hfill \end{array}$$

(50)

It then follows that if the expressions in (48) represent a solution of (5), then so are their transformed versions which, according to the above, take the form

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill u& =e^{-r_2}[Fcos\left(r_1\right)cos\left(h\right)+Hsin\left(r_1\right)cos\left(h\right)\hfill \\ \hfill & +Gcos\left(r_1\right)sin\left(h\right)+Jsin\left(r_1\right)sin\left(h\right)],\hfill \end{array}\end{array}$$

(51)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill v& =e^{-r_2}[Gcos\left(r_1\right)cos\left(h\right)+Jsin\left(r_1\right)cos\left(h\right)\hfill \\ \hfill & -Fcos\left(r_1\right)sin\left(h\right)-Hsin\left(r_1\right)sin\left(h\right)],\hfill \end{array}\end{array}$$

(52)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill w& =e^{-r_2}[Hcos\left(r_1\right)cos\left(h\right)-Fsin\left(r_1\right)cos\left(h\right)\hfill \\ \hfill & +Jcos\left(r_1\right)sin\left(h\right)-Gsin\left(r_1\right)sin\left(h\right)],\hfill \end{array}\end{array}$$

(53)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill c& =e^{-r_2}[Jcos\left(r_1\right)cos\left(h\right)-Gsin\left(r_1\right)cos\left(h\right)\hfill \\ \hfill & -Hcos\left(r_1\right)sin\left(h\right)+Fsin\left(r_1\right)sin\left(h\right)].\hfill \end{array}\end{array}$$

(54)

It should be noted that in (51)–(54), one has

$$F=F(a,b),G=G(a,b),H=H(a,b),J=J(a,b),$$

(55)

where $a=a(t,x)$ and $b=b(t,x)$ and which together with $h=h(t,x)$ are all given by (49).

To gain a better insight into the structure of the transformations in (51)–(54), let us set

$$\begin{array}{cc}\hfill \mathcal{U}& ={\left(u,v,w,c\right)}^T,\mathcal{F}={\left(F,G,H,J\right)}^T\hfill \end{array}$$

(56)

$$\begin{array}{cc}\hfill \mathcal{W}& =\left[\begin{array}{cccc}cos\left(r_1\right)cos\left(h\right)& cos\left(r_1\right)sin\left(h\right)& sin\left(r_1\right)cos\left(h\right)& sin\left(r_1\right)sin\left(h\right)\\ -cos\left(r_1\right)sin\left(h\right)& cos\left(r_1\right)cos\left(h\right)& -sin\left(r_1\right)sin\left(h\right)& sin\left(r_1\right)cos\left(h\right)\\ -sin\left(r_1\right)cos\left(h\right)& -sin\left(r_1\right)sin\left(h\right)& cos\left(r_1\right)cos\left(h\right)& cos\left(r_1\right)sin\left(h\right)\\ sin\left(r_1\right)sin\left(h\right)& -sin\left(r_1\right)cos\left(h\right)& -cos\left(r_1\right)sin\left(h\right)& cos\left(r_1\right)cos\left(h\right)\end{array}\right],\hfill \end{array}$$

(57)

where the symbol T as exponent denotes matrix transpose. It then follows that

$$\mathcal{U}=e^{-r_2}\mathcal{W}·\mathcal{F},\mathrm{and}{\mathcal{W}}^T={\mathcal{W}}^{-1},det\left(\mathcal{W}\right)=1.$$

(58)

In other words, $\mathcal{W}\in \mathfrak{so}\left(4\right)$ is a rotation matrix and the most general symmetry transformation is simply a composite of a scaling and a rotation in the four-dimensional space. Consequently, such a transformation preserves the shape of solutions.

To conclude this section, let us mention that Lie point symmetries can be found for every analytic system that satisfies the local solvability and the Cauchy–Kovalevskaya Theorems [27] (pp. 165, 166). These conditions relate to the form of the system and roughly speaking to the satisfaction by the system of conditions similar to the Implicit Function Theorem. All known integrable systems, in particular those of the above type, usually satisfy such conditions and Lie point symmetries can be found for such systems.

5. Classification of One-Dimensional Subalgebras

In this section the basis of the symmetry algebra L of (5) that we shall consider is ${\mathcal{B}}_{\mathbf{w}}=\left\{{\mathbf{w}}_1,{\mathbf{w}}_2,\dots ,{\mathbf{w}}_6\right\}$ as given by (24) and (22).

Knowledge of the commutation relations of L, especially in the basis ${\mathcal{B}}_{\mathbf{w}}$, will be particularly helpful for the classification problem at hand. The commutator table of L is given in Table 1.

Table 1. Commutator table of L in the basis ${\mathcal{B}}_{\mathbf{w}}$.

$[{\mathbf{w}}_{\mathit{i}},{\mathbf{w}}_{\mathit{j}}]$	${\mathbf{w}}_1$	${\mathbf{w}}_2$	${\mathbf{w}}_3$	${\mathbf{w}}_4$	${\mathbf{w}}_5$	${\mathbf{w}}_6$
${\mathbf{w}}_1$	0	0	0	0	$-2{\mathbf{w}}_1$	${\mathbf{w}}_2$
${\mathbf{w}}_2$	0	0	0	0	$-{\mathbf{w}}_2$	$-{\displaystyle \frac{1}{2\beta }}{\mathbf{w}}_4$
${\mathbf{w}}_3$	0	0	0	0	0	0
${\mathbf{w}}_4$	0	0	$-0$	0	0	0
${\mathbf{w}}_5$	$2{\mathbf{w}}_1$	${\mathbf{w}}_2$	0	0	0	$-{\mathbf{w}}_6$
${\mathbf{w}}_6$	${\mathbf{w}}_2$	${\displaystyle \frac{1}{2\beta }}{\mathbf{w}}_4$	0	0	${\mathbf{w}}_6$	0

5.1. Connections Between Adjoint Representation and Group-Invariant Solutions

As is well-known, the classification of all group-invariant solutions of (5) obtained by a reduction by a one-dimensional subalgebra of L is equivalent to the classification of all one-dimensional subalgebras of L under the group of adjoint representation of L given by

$${Ad}_g:L\to L:\mathbf{v}\mapsto {Ad}_g\left(\mathbf{v}\right),$$

where $g\in G$ for all $g,$ and where G stands for the symmetry group of (5), i.e., the connected Lie group with Lie algebra $L.$

Equivalently, to describe all group-invariant solutions obtained from a reduction in the original equation by an s-dimensional subalgebra, one only needs to find an optimal system of s-dimensional subalgebras of the given system of equations. More precisely, an optimal system of s-dimensional subalgebras of a given Lie algebra $\mathcal{M}$ is a collection of s-dimensional subalgebras of $\mathcal{M}$ such that any other s-dimensional subalgebra $\tilde{\mathfrak{h}}$ of $\mathcal{M}$ is equivalent to precisely one member $\mathfrak{h}$ of the collection, that is, $\tilde{\mathfrak{h}}={Ad}_g\left(\mathfrak{h}\right)$ where g is a member of the Lie group generated by $\mathcal{M}.$

Classifying one-dimensional subalgebras of L amounts therefore to finding an optimal system of one-dimensional subalgebras of $L.$ In practice, to find the adjoint representation of a Lie algebra $\mathcal{M}$ one reconstructs it from the one-parameter subgroups generated by its basis elements. More precisely, if $\mathbf{v},\mathbf{w}\in \mathcal{M},$ then we have

$${Ad}_g\mathbf{w}=\sum _{n=0}^{\infty }{\displaystyle \frac{{\epsilon }^n}{n!}}{(ad\mathbf{v})}^n\left(\mathbf{w}\right),$$

(59)

where $g=exp\left(\epsilon \mathbf{v}\right)$ is an element of the one-parameter subgroup generated by $\mathbf{v}$ and ${ad}_{\mathbf{v}}$ is the vector field on $\mathcal{M}$, generating the one-parameter subgroup of adjoint transformations. That is

$${ad}_{\mathbf{v}}\left({\mathbf{w}}_0\right)={\displaystyle \frac{d}{d\epsilon }}{|}_{\epsilon =0}Ad[exp\left(\epsilon \mathbf{v}\right)]·{\mathbf{w}}_0,{\mathbf{w}}_0\in \mathcal{M}.$$

(60)

We now focus on the determination of an optimal system of one-dimensional subalgebras of the symmetry algebra L of (5). One way of achieving this is to use the method described in [27]. Hence we start by using (59) to construct the table of the adjoint representations of $L.$ This is given in Table 2, in which the table entry in row i and column j represents $Ad(exp\epsilon {\mathbf{w}}_i)·{\mathbf{w}}_j.$

Table 2. Adjoint representation of the symmetry algebra of (5).

${Ad}_{\mathit{g}}$	${\mathbf{w}}_1$	${\mathbf{w}}_2$	${\mathbf{w}}_3$	${\mathbf{w}}_4$	${\mathbf{w}}_5$	${\mathbf{w}}_6$
${\mathbf{w}}_1$	${\mathbf{w}}_1$	${\mathbf{w}}_2$	${\mathbf{w}}_3$	${\mathbf{w}}_4$	${\mathbf{w}}_5+2\epsilon {\mathbf{w}}_1$	${\mathbf{w}}_6-\epsilon {\mathbf{w}}_2$
${\mathbf{w}}_2$	${\mathbf{w}}_1$	${\mathbf{w}}_2$	${\mathbf{w}}_3$	${\mathbf{w}}_4$	${\mathbf{w}}_5+\epsilon {\mathbf{w}}_2$	${\mathbf{w}}_6+{\displaystyle \frac{1}{2\beta }}\epsilon {\mathbf{w}}_4$
${\mathbf{w}}_3$	${\mathbf{w}}_1$	${\mathbf{w}}_2$	${\mathbf{w}}_3$	${\mathbf{w}}_4$	${\mathbf{w}}_5$	${\mathbf{w}}_6$
${\mathbf{w}}_4$	${\mathbf{w}}_1$	${\mathbf{w}}_2$	${\mathbf{w}}_3$	${\mathbf{w}}_4$	${\mathbf{w}}_5$	${\mathbf{w}}_6$
${\mathbf{w}}_5$	$e^{-2\epsilon }{\mathbf{w}}_1$	$e^{-\epsilon }{\mathbf{w}}_2$	${\mathbf{w}}_3$	${\mathbf{w}}_4$	${\mathbf{w}}_5$	$e^{\epsilon }{\mathbf{w}}_6$
${\mathbf{w}}_6$	${\mathbf{w}}_1+\epsilon {\mathbf{w}}_2-{\displaystyle \frac{{\epsilon }^2}{4\beta }}{\mathbf{w}}_4$	${\mathbf{w}}_2-{\displaystyle \frac{\epsilon }{2\beta }}{\mathbf{w}}_4$	${\mathbf{w}}_3$	${\mathbf{w}}_4$	${\mathbf{w}}_5-\epsilon {\mathbf{w}}_6$	${\mathbf{w}}_6$

Next, we let $\mathbf{v}={\sum }_{k=1}^6{a}_k{\mathbf{w}}_k$ be a general vector in $L,$ and we act on $\mathbf{v}$ by elementary adjoint representations $Ad(exp\epsilon {\mathbf{w}}_i),$$i\in \{1,\dots ,6\}$, so as to reduce $\mathbf{v},$ as much as possible, to a simpler expression. We proceed in this discussion by case analysis.

• Case 1: $a_6\ne 0$

In this case, by replacing $\mathbf{v}$ with $(1/a_6)\mathbf{v}$ if necessary, we may assume that $a_6=1.$ Applying $Ad\left({\displaystyle \frac{a_2}{2}}{\mathbf{w}}_1\right)$ to $\mathbf{v}$ cancels the coefficient of ${\mathbf{w}}_2$ in $\mathbf{v}$, and keeping the same name for the coefficients of $\mathbf{v}$ in its transformed expression as for those in its original expression as we shall do here and in the sequel, $\mathbf{v}$ is now reduced to $\mathbf{v}=a_1{\mathbf{w}}_1+{\sum }_{k=0}^5{a}_k{\mathbf{w}}_k+{\mathbf{w}}_6.$

Next, applying $Ad(-2\beta a_4{\mathbf{w}}_2)$ to the latter expression for $\mathbf{v}$ cancels the term in ${\mathbf{w}}_4$ and reduces $\mathbf{v}$ to an expression of the form

$$\mathbf{v}=a_1{\mathbf{w}}_1+a_3{\mathbf{w}}_3+a_5{\mathbf{w}}_5+{\mathbf{w}}_6.$$

Assuming that $a_5\ne 0$ and applying $Ad\left({\displaystyle \frac{a_1}{2a_5}}{\mathbf{w}}_1\right)$ to $\mathbf{v}$ cancels the term in ${\mathbf{w}}_1$ and reduces $\mathbf{v}$ to

$$\mathbf{v}=a_3{\mathbf{w}}_3+a_5{\mathbf{w}}_5+{\mathbf{w}}_6.$$

Else, if $a_5=0,$ then

$$\mathbf{v}=a_1{\mathbf{w}}_1+a_3{\mathbf{w}}_3+{\mathbf{w}}_6.$$

This leads us to the consideration of two cases.

• Case 1.1: $a_6\ne 0,a_5\ne 0$

One has $\mathbf{v}=a_3{\mathbf{w}}_3+a_5{\mathbf{w}}_5+{\mathbf{w}}_6.$ Then $Ad\left({\displaystyle \frac{1}{a_5}}{\mathbf{w}}_6\right)$ reduces this expression to $\mathbf{v}=a_3{\mathbf{w}}_3+a_5{\mathbf{w}}_5.$ So $\mathbf{v}={\mathbf{w}}_5$ or $\mathbf{v}={\mathbf{w}}_3+a_5{\mathbf{w}}_5$ and no further simplification is possible.

• Case 1.2: $a_6\ne 0,a_5=0$

One has $\mathbf{v}=a_1{\mathbf{w}}_1+a_3{\mathbf{w}}_3+{\mathbf{w}}_6$. Applying $Ad\left(\epsilon {\mathbf{w}}_5\right)$ transforms $\mathbf{v}$ into $\mathbf{v}=e^{-2\epsilon }{a}_1{\mathbf{w}}_1+a_3{\mathbf{v}}_3+e^{\epsilon }{\mathbf{w}}_6.$ Hence $\mathbf{v}=c{\mathbf{w}}_1+a_3{\mathbf{w}}_3+b{\mathbf{w}}_6,$ where $c\in \{0,1,-1\}$ and $b>0.$ This completes the case where $a_6\ne 0.$

• Case 2: $a_6=0$

• Case 2.1: $a_6=0,$ $a_5\ne 0$

We may write $\mathbf{v}={\sum }_{k=1}^4{\mathbf{w}}_k+{\mathbf{w}}_5$. Then $Ad(-{\displaystyle \frac{a_1}{2}}{\mathbf{w}}_1)$ cancels the term in ${\mathbf{w}}_1$ in $\mathbf{v}$ and $Ad(-a_2{\mathbf{w}}_2)$ cancels also the term in ${\mathbf{w}}_2$ in the resulting expression for $\mathbf{v}.$ Hence

$$\mathbf{v}=a_3{\mathbf{w}}_3+a_4{\mathbf{w}}_4+{\mathbf{w}}_5.$$

• Case 2.2: $a_6=0,a_5=0$ and $a_4\ne 0$

We may let $\mathbf{v}=a_1{\mathbf{w}}_1+a_2{\mathbf{w}}_2+a_3{\mathbf{w}}_3+{\mathbf{w}}_4.$

• Case 2.2.1: $0=a_6=a_5,$ $0\ne a_4$ and $0\ne a_1$

Applying $Ad\left(\epsilon {\mathbf{w}}_6\right)$ to $\mathbf{v}$ with $\epsilon =-{\displaystyle \frac{a_2}{a_1}}$ cancels the term in ${\mathbf{w}}_2.$ We are left with

$$\mathbf{v}=a_1{\mathbf{w}}_1+a_3{\mathbf{w}}_3+{\mathbf{w}}_4.$$

Then the application of $Ad\left(\pm \sqrt{{\displaystyle \frac{4\beta }{a_1}}}{\mathbf{w}}_6\right)$ to $\mathbf{v}$ cancels the term in ${\mathbf{w}}_4$ and reduces $\mathbf{v}$ to

$$\mathbf{v}=a_1{\mathbf{w}}_1+a_3{\mathbf{w}}_3.$$

Then applying $Ad\left(\epsilon {\mathbf{w}}_5\right)$ to $\mathbf{v}$ reduces it to $\mathbf{v}=e^{-2\epsilon }{a}_1{\mathbf{w}}_1+a_3{\mathbf{w}}_3.$ This means that either $\mathbf{v}=-{\mathbf{w}}_1+a_3{\mathbf{w}}_3$ or $\mathbf{v}={\mathbf{w}}_1+a_3{\mathbf{w}}_3.$

• Case 2.2.2: $0=a_6=a_5=a_1,$ $0\ne a_4$

We have in this case

$$\mathbf{v}=a_2{\mathbf{w}}_2+a_3{\mathbf{w}}_3+{\mathbf{w}}_4.$$

Then assuming $a_2\ne 0,$ $\mathbf{v}$ reduces to

$$\mathbf{v}=a_2{\mathbf{w}}_2+a_3{\mathbf{w}}_3$$

under the action of $Ad\left({\displaystyle \frac{2\beta }{a_2}}{\mathbf{w}}_6\right)$ and then to $e^{-\epsilon }{a}_2{\mathbf{w}}_2+a_3{\mathbf{w}}_3$ under the action of $Ad\left(\epsilon {\mathbf{w}}_5\right).$ Thus $\mathbf{v}={\mathbf{w}}_2+a_3{\mathbf{w}}_3$ or $\mathbf{v}=-{\mathbf{w}}_2+a_3{\mathbf{w}}_3$ in this case. Assuming now that $a_2=0,$ one has $\mathbf{v}=a_3{\mathbf{w}}_3+{\mathbf{w}}_4.$

• Case 2.3: $0=a_6=a_5=a_4$

• Case 2.3.1: $a_3\ne 0$

In this case we may write

$$\mathbf{v}=a_1{\mathbf{w}}_1+a_2{\mathbf{w}}_2+{\mathbf{w}}_3.$$

An application of $Ad\left(\epsilon {\mathbf{w}}_5\right)$ to $\mathbf{v}$ reduces it to

$$\mathbf{v}=e^{-2\epsilon }{a}_1{\mathbf{w}}_1+a_2{\mathbf{w}}_2+{\mathbf{w}}_3.$$

Thus

$$\mathbf{v}=c{\mathbf{w}}_1+a_2{\mathbf{w}}_2+{\mathbf{w}}_3,c\in \{0,1,-1\}.$$

• Case 2.3.2: $a_3=0$

We have

$$\mathbf{v}=a_1{\mathbf{w}}_1+a_2{\mathbf{w}}_2.$$

We may assume that $a_1\ne 0$ (else $a_1=a_2=0)$ which means $\mathbf{v}=0,$ a trivial and meaningless case. Therefore applying $Ad\left(\epsilon {\mathbf{w}}_5\right)$ to $\mathbf{v}$ reduces it to $\mathbf{v}=a_1{e}^{-2\epsilon }{\mathbf{w}}_1+a_2{e}^{\epsilon }{\mathbf{w}}_2.$ Then $\mathbf{v}=a_1{\mathbf{w}}_1+c{\mathbf{w}}_2,c\in \{0,1,-1\}.$

5.2. Optimal System of One-Dimensional Subalgebras

The above discussion and the resulting classification shows that an optimal system of one-dimensional subalgebras of the symmetry algebra L of (5) should be taken from the following preliminary list given in terms of the generator $\mathbf{v}$ of each member of the list:

(a₁)

$\mathbf{v}={\mathbf{w}}_5.$

(a₂)

$\mathbf{v}={\mathbf{w}}_3+a{\mathbf{w}}_5.$

(b)

 $\mathbf{v}=a{\mathbf{w}}_1+b{\mathbf{w}}_3+{\mathbf{w}}_6.$

(c)

 $\mathbf{v}=a{\mathbf{w}}_3+b{\mathbf{w}}_4+{\mathbf{w}}_5.$

(d)

 $\mathbf{v}=c{\mathbf{w}}_1+a{\mathbf{w}}_3.$

(e)

 $\mathbf{v}=c{\mathbf{w}}_2+a{\mathbf{w}}_3.$

(f)

 $\mathbf{v}=a{\mathbf{w}}_3+{\mathbf{w}}_4.$

(g)

 $\mathbf{v}=c{\mathbf{w}}_1+a{\mathbf{w}}_2+{\mathbf{w}}_3.$

(h)

 $\mathbf{v}=c{\mathbf{w}}_1+a{\mathbf{w}}_2.$

In the above list, a and b are arbitrary scalars while $c\in \left\{0,-1,1\right\}.$ Thus each generator $\mathbf{v}$ in the list containing a parameter represents itself a finite or infinite family of one-dimensional subalgebras. It turns out that subalgebras given by (${\mathrm{a}}_1$), (${\mathrm{a}}_2$), and (b)–(h) are pairwise non-equivalent and represent therefore an optimal system of one-dimensional subalgebras of (5).

5.3. Application: Solitary Wave Solutions of (5)

In ($\mathrm{h}$) above, one has $\mathbf{v}=c{\mathbf{w}}_1+a{\mathbf{w}}_2=c{\partial }_t+a{\partial }_x.$ As is well-known, a reduction in an equation by this symmetry leads to the obtention of solitary wave solutions of the original equation. Indeed, the invariant functions of $\mathbf{v}$ are

$$y=t,z=cx-at,\mathrm{and}f=A,g=B,h=R,k=S.$$

In terms of the new variables $y,z,f,g,h,$ and $k,$ (5) reduces to the system of ordinary differential equations (odes) in the independent variable $z=cx-at$ expressing a solitary wave solution and is given as follows:

$$\begin{array}{cc}\hfill 2\mu S(-BR+AS)+a{B}_z+c^2\beta A_{zz}& =0,\hfill \end{array}$$

(61)

$$\begin{array}{cc}\hfill 2\mu R(BR-AS)-a{A}_z+c^2\beta B_{zz}& =0,\hfill \end{array}$$

(62)

$$\begin{array}{cc}\hfill 2\mu B(BR-AS)+a{S}_z+c^2\beta R_{zz}& =0,\hfill \end{array}$$

(63)

$$\begin{array}{cc}\hfill 2\mu A(-BR+AS)-a{R}_z+c^2\beta S_{zz}& =0,\hfill \end{array}$$

(64)

where the new dependent variables $f,g,h$ and k have been renamed to the old ones $A,B,R$ and $S,$ respectively.

6. Solutions of the NLSE

We derive in this section some solutions of (5) and hence for (3). Given that the system of four cubic equations is strongly nonlinear and coupled, we shall restrict to a specific type of solutions in an attempt to reduce it to a single scalar equation. For this purpose, we therefore seek a solution of (5) of the form

$$\begin{array}{cccc}\hfill A& =v\left(t\right)u\left(x\right),\hfill & \hfill B& =-w\left(t\right)u\left(x\right),\hfill \end{array}$$

(65)

$$\begin{array}{cccc}\hfill R& =w\left(t\right)u\left(x\right),\hfill & \hfill S& =v\left(t\right)u\left(x\right),\hfill \end{array}$$

(66)

where the functions $v=v\left(t\right),w=w\left(t\right),\mathrm{and}u=u\left(x\right)$ are unknown functions. Given that we seek to obtain soliton solutions, we let

$$v=sin\left(t\right),w=cos\left(t\right).$$

(67)

Then substituting those fixed values of $v\mathrm{and}w$ from (65) into Equation (5) yields the system

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill sin\left(t\right)\left(\beta u_{xx}-u+2\mu u^3\right)& =0,\hfill \\ \hfill cos\left(t\right)\left(\beta u_{xx}-u+2\mu u^3\right)& =0,\hfill \\ \hfill -cos\left(t\right)\left(\beta u_{xx}-u+2\mu u^3\right)& =0,\hfill \\ \hfill sin\left(t\right)\left(\beta u_{xx}-u+2\mu u^3\right)& =0.\hfill \end{array}\end{array}$$

(68)

In other words, the substitution of (65) and (67) into (5) reduces it to the mere scalar ode

$$-u+\beta u_{xx}+2\mu u^3=0.$$

(69)

A solution of (69) can be obtained using a symbolic system such as Mathematica, in terms of the Jacobi elliptic function Sn. However, since such a function does not yield a soliton-type solution in which we are interested, we resort to one of the common methods for finding such solutions, namely the so-called Homogeneous Balance Method [28,29,30]. The method consists in considering an auxiliary ode, generally a first order Riccati, which we shall choose here to be of the form

$$y^{\prime }=p\left(x\right)+q\left(x\right)y+s\left(x\right)y^2$$

(70)

where the coefficients $p\left(x\right),q\left(x\right),$ and $s\left(x\right)$ need to be appropriately specified for the chosen function $g\left(x\right)$ to be effectively a solution. A solution of the original equation to be solved, namely (69), is then sought in the form

$$u=\sum _{k=0}^m{a}_k{y}^k,$$

(71)

where y is assumed to solve (70) and the coefficients $a_1,\dots ,a_m,$ with $a_m\ne 0,$ as well as the natural number m are unknown parameters to be found. The parameter m is found by substituting $u=y^m$ in (69) with the assumption that y satisfies (70). In this way, (69) can be expressed completely as polynomial in y not involving any of its derivatives. In such a polynomial expression, to determine $m,$ one equates (balances) the resulting degree of the term of highest derivative in the transformed version of (69) with that of the term of highest degree of nonlinearity.

Once the integer m has been found, the resulting expression for (71), with y now replaced by a chosen explicit solution $g\left(x\right)$ of (70), is substituted into (69) to determine the coefficients $a_0,\dots ,a_m$.

We now apply the above described homogeneous balance method to find a solution of (69) of the form (71). We consider the case in which $g\left(x\right)$ is chosen to be of the form

$$g\left(x\right)=sech\left(bx\right),$$

(72)

where b is itself a parameter to be found. The function $g\left(x\right)$ yields a solution of (70), if one sets $p\left(x\right)=q\left(x\right)=0$ and $s\left(x\right)=-2bsinh\left(bx\right).$ In turn, the substitution into (69) of (71) with y replaced with g gives the expression

$$\begin{array}{cc}\hfill & a_1(-1-3b^2\beta +6a_0^2\mu +a{1}^2\mu ){sech}^3\left(bx\right)\hfill \\ \hfill & +a_1(-1+b^2\beta +6a_0^2\mu )cosh\left(2bx\right){sech}^3\left(bx\right)\hfill \\ \hfill & (-3a_0+6a_0(a_0^2+a_1^2)\mu ){sech}^2\left(bx\right)+a_0(-1+2a_0^2\mu )cosh\left(3bx\right)sech{\left(bx\right)}^3=0.\hfill \end{array}$$

(73)

Given that $a_1\ne 0,$ it is implied that $a_0=0,$ and consequently $a_1=\pm 2/\sqrt{\mu }$ and $b=\pm 1/\sqrt{\beta }.$ The corresponding solutions $\mathbf{u}$ of (5) and $(u,v)$ of (3) are given by

$$\begin{array}{cc}\hfill \mathbf{u}=(A,B,R,S)& =\pm {\displaystyle \frac{1}{\sqrt{\mu }}}sech\left(\pm {\displaystyle \frac{x}{\sqrt{\beta }}}\right)(sin\left(t\right),-cos\left(t\right),cos\left(t\right),sin\left(t\right))\hfill \\ \hfill u& =\pm {\displaystyle \frac{1}{\sqrt{\mu }}}sech\left(\pm {\displaystyle \frac{x}{\sqrt{\beta }}}\right)e^{-it}\hfill \\ \hfill v& =\pm {\displaystyle \frac{1}{\sqrt{\mu }}}sech\left(\pm {\displaystyle \frac{x}{\sqrt{\beta }}}\right)e^{-i(t+{\displaystyle \frac{\pi }{2}})}\hfill \end{array}$$

(74)

It should be noted that our version of the Homogeneous Balance Method is innovative in two ways. First, we’ve replaced the usual numeric or constant coefficients Riccati equation with one having arbitrary variable coefficients $p\left(x\right),q\left(x\right)$ and $s\left(x\right).$ This makes it possible for virtually every desired function $g\left(x\right)$ to be made a solution of the Riccati equation by an appropriate choice of the coefficients $p\left(x\right),q\left(x\right)$ and $s\left(x\right).$ Moreover, our choice of the solution $g\left(x\right)$ depends also on an arbitrary parameter, $b,$ to be found together with the coefficients $a_1,\dots ,a_m.$ All this adds more versatility to our approach.

Symmetry Transformation of Solutions

We shall make use in this subsection of the general symmetry transformation of the solution formula given by (51)–(54) to find the transformation of the solutions of (5) and (3) obtained in this section. For more conciseness, let us denote by $G\left[1\right]\left(\mathbf{x}\right)$ the first transformation of a solution $\mathbf{x}$ of (5) by (51)–(54) and similarly by $G\left[N\right]\left(\mathbf{x}\right)=G\left[1\right]\left(G\right[N-1\left]\right(\mathbf{x}\left)\right)$ the N-th recursive transformation of $\mathbf{x},$ where the natural number $N\ge 1$ and $G\left[0\right]$ is the identity map.

It turns out that for the soliton solution $\mathbf{u}$ of (5) found in (74), one has $G\left[1\right]\left(\mathbf{u}\right)=\left(u_{11},u_{12},u_{13},u_{14}\right)$ with

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill u_{11}& ={\displaystyle \frac{e^{-r2}}{\sqrt{\mu }}}Psin\left[b(t,x)\right]sech\left[Pa(t,x)\right]\hfill \\ \hfill u_{12}& =-{\displaystyle \frac{e^{-r2}}{\sqrt{\mu }}}Pcos\left[b(t,x)\right]sech\left[Pa(t,x)\right],\hfill \\ \hfill u_{13}& ={\displaystyle \frac{e^{-r2}}{\sqrt{\mu }}}Pcos\left[b(t,x)\right]sech\left[Pa(t,x)\right],\hfill \\ \hfill u_{14}& ={\displaystyle \frac{e^{-r2}}{\sqrt{\mu }}}Psin\left[b(t,x)\right]sech\left[Pa(t,x)\right],\hfill \end{array}\end{array}$$

(75)

where

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill a(t,x)& ={\displaystyle \frac{e^{r_2}(r_5+r_{4}t-x)}{\sqrt{\beta }}},P=\pm 1\hfill \\ \hfill b(t,x)& =r_1-r_6+e^{-2r_2}(t-r_3)-{\displaystyle \frac{r_4(r_{4}t-2x)}{4\beta }}.\hfill \end{array}\end{array}$$

(76)

In a similar way, the transformation $G\left[2\right]\left(\mathbf{u}\right)=\left(u_{21},u_{22},u_{23},u_{24}\right)$ of the solution $\mathbf{u}$ of (5) given by (74) takes the form

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill u_{21}& ={\displaystyle \frac{e^{-2r_2}}{\sqrt{\mu }}}Psin\left[z(t,x)\right]sech\left[Pd(t,x)\right]\hfill \\ \hfill u_{22}& =-{\displaystyle \frac{e^{-2r_2}}{\sqrt{\mu }}}Pcos\left[z(t,x)\right]sech\left[Pd(t,x)\right],\hfill \\ \hfill u_{23}& ={\displaystyle \frac{e^{-2r_2}}{\sqrt{\mu }}}Pcos\left[z(t,x)\right]sech\left[Pd(t,x)\right],\hfill \\ \hfill u_{24}& ={\displaystyle \frac{e^{-2r_2}}{\sqrt{\mu }}}Psin\left[z(t,x)\right]sech\left[Pd(t,x)\right],\hfill \end{array}\end{array}$$

(77)

where

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill d(t,x)& =P{\displaystyle \frac{e^{-3r_2}}{\sqrt{\beta }}}\left(-r_3{r}_4+e^{2r_2}{r}_5+r_{4}t+e^{r_2}(r_5+r_{4}t-x)\right),\hfill \end{array}\end{array}$$

(78)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill z(t,x)& ={\displaystyle \frac{e^{-4r_2}}{4\beta }}[-2e^{3r_2}{r}_4(r_5+r_{4}t-x)+e^{2r_2}\left(-r_4^{2}t+r_3(r_4^2-4\beta )\right)\hfill \\ \hfill & +4(-r_3+t)\beta +e^{4r_2}\left(-r_4^{2}t+2r_{4}x-8(r_1-r_6)\beta \right)].\hfill \end{array}\end{array}$$

(79)

On the other hand, for a given kth iteration of a solution $\mathbf{u}$ of (5), the resulting kth iteration $(u_k,v_k)$ of the solution $(u,v)\equiv (u_0,v_0)$ of (3) is obtained from $G\left[k\right]\left(\mathbf{u}\right)=(u_{k,1},\dots ,u_{k,4})$ simply by setting $u_k=u_{k,1}+i{u}_{k,2}$ and $v_k=u_{k,3}+i{u}_{k,4}.$ This clearly follows from the formula in (4) establishing the relationship between solutions of (3) and those of (5).

We can generate in this way a possibly infinite sequence ${\left(G\left[k\right]\left(\mathbf{u}\right)\right)}_{k\in \mathbb{N}}$ of solutions $G\left[k\right]\left(\mathbf{u}\right)$ of (5), each of which depends on the six parameters of the symmetry group of (5). In other words, each generated solution $G\left[k\right]\left(\mathbf{u}\right)$ represents a six-parameter class of solutions of (5).

7. Concluding Remarks

One of the critical findings in our Lie group analysis of the coupled nonlinear system of Schrödinger Equation (3) was that its entire six-parameter symmetry group acts on solutions by mere rotations and scalings. Although these scalings and rotations are in general dependent on time and space, it remains true that they do not alter the shape of any given solution. In particular, the symmetry transformations of (3) will map solitons to solitons, especially as far as the shape of the solution function is concerned. The latter property is known to hold for Darboux transformations applied to integrable systems defined by a Lax pair. It would therefore be meaningful to find out if the Schrödinger equation studied in this work is integrable, and more exactly symmetry-integrable, and if it is defined by a Lax pair.

By an autonomous technique, we were able to obtain a reduction in the cubic nonlinear real system of four equations into a mere nonlinear ode, and to find a solution of the latter through the Homogeneous Balance method. This yielded a soliton for (5). The symmetry group tools constructed earlier then allowed a generation of new solitons from old ones. The scheme thus designed showed that one can build a possibly infinite series of solutions ${\left(G\left[k\right]\left(\mathbf{u}\right)\right)}_{k\in \mathbb{N}}$ of (5) starting from a known one $\mathbf{u}.$

Some of the natural questions from this study is whether the sequence of solutions iterated from symmetry transformation consists effectively of an infinite number of distinct and even functionally independent terms. If that is not the general case, it would be naturally desirable to know for which types of equation the sequence does have an infinity of terms, and if there is a general way of finding a closed form expression for the general term. In the negative case, it would still be desirable to know how many distinct iterated solutions can be obtained and if there is a way of predicting the number of such solutions.

It is worthwhile to mention at this point another interesting property of (3) inherited, in fact, from (1). Indeed, if we write the left-hand side of (3) as $\Delta (u,v):=({\Delta }_1(u,v),{\Delta }_2(u,v)),$ then due to the symmetry with respect to u and v in the expression of $\Delta$ one clearly has $\Delta (v,u):=({\Delta }_2(u,v),{\Delta }_1(u,v)),$ showing that for any given pair $(u,v)$ of functions, $(u,v)$ is a solution to $\Delta =0$ if and only if its permutation $(v,u)$ is.

Lastly, the type of Lie symmetries considered in the manuscript and referred to as Lie point symmetries consists of point transformations of the space dependent and independent variables preserving the solution set. However, there are several other types of Lie symmetries, which include contact, potential and other types. The most prominent type with regard to integrability is certainly the generalized symmetries, which have given rise in the literature to the concept of symmetry-integrable systems (see [31] and the references therein). These are systems which possess an infinite sequence of generalized symmetries of increasing orders. Symmetry-integrable systems also possess an infinite number of conservation laws, and in the case of variational symmetries of Lagrangian equations, this is a direct consequence of Noether Theorem [27]. Known integrable systems and generalized models of the type treated in [32,33] usually possess a recursion operator which is a criteria for symmetry-integrability. In addition, as already mentioned, all known integrable systems, including those generated from the AKNS scheme, possess Lie point symmetries. But Lie point symmetry alone seem insufficient to characterize integrable systems, while generalized symmetries can usually be used to identify such systems.