Dynamics of Evolutionary Differential Equations with Several Spatial Variables

Abstract

The article is devoted to a method for constructing exact and approximate solutions of evolutionary partial differential equations with several spatial variables. The method is based on the theory of completely integrable distributions. Examples of applying this method to calculating exact solutions of the generalized Kolmogorov–Petrovskii–Piskunov–Fishev equations with two space variables are given.

1. Introduction

The term “evolutionary equation” is usually used in relation to partial differential equations involving time. This article deals with equations that are solved with respect to time derivatives and whose right-hand sides do not depend on time, i.e., equations of the form

$${\displaystyle \frac{\partial u}{\partial t}}=f\left(x,u,{\displaystyle \frac{\partial u}{\partial x}},\dots ,{\displaystyle \frac{{\partial }^{k}u}{\partial x^k}}\right).$$

(1)

Here x is a vector of independent variables $x_1,\dots ,x_n$ (we call them spatial), t is time, $u=u(t,x)$ is a scalar function, and the symbol ${\partial }^{i}u/\partial x^i$$(i=1,\dots ,k)$ means the set of all partial derivatives of order i by x. We suppose that the function f belongs to the class $C^{\infty }$ within its domain.

If the function f is non-linear, then for such equations, there are no general theorems on the existence of solutions. Moreover, even for simple nonlinearities, solutions destruction effect is often observed, i.e., solutions exist only on finite time intervals.

The method that makes it possible to single out finite dimensional subspaces in the entire infinite set of solutions of Equation (1) for $n=1$ was proposed in [1,2]. This is the so-called method of finite dimensional dynamics.

It was developed in [3,4]. The method makes it possible to select finite dimensional submanifolds of solutions from the infinite set of all solutions of evolutionary equations. These submanifolds are “numbered” by solutions of ordinary differential equations. However, this method works for equations with only one spatial variable and cannot be directly generalized to equations with multiple spatial variables. In such cases, the use of ordinary differential equations is not enough anymore.

In this article, we propose a method that allows us to construct solutions of evolutionary equations with several spatial variables. The importance of constructing exact solutions of nonlinear partial differential equations has been repeatedly emphasized (see [5], for example).

Main ideas were described in the conference paper [6]. This method is based on the theory of overdetermined systems of partial differential equations and on the symmetry theory of completely integrable distributions. Instead of overdetermined systems of equations, general systems of the finite type [7] can be considered.

The paper is organized as follows.

The brief introduction to the symmetries theory of distributions is given in the second section. For details, see [8,9,10].

In the third section, symmetries of systems of the finite type are studied. Such systems are understood as overdetermined systems of partial differential equations, the solutions of which are numbered by points in the jet spaces. The fulfillment of the conditions of the Frobenius theorem guarantees that such systems generate completely integrable distributions. Two main theorems on the structure of shuffling symmetries of systems of the finite type are proved there.

The fourth section is the main one. It gives a definition of finite dimensional dynamics for evolutionary equations with several space variables and indicates how to calculate them.

The fifth section contains two examples of calculating the dynamics and exact solutions of the generalized Kolmogorov–Petrovsky–Piskunov–Fisher equations. Such equations arise in many branches of physics and biology (see [11,12,13,14], for example).

Note that the main definitions and results of the paper can be easily extended to smooth manifolds. We use the arithmetic space only to simplify the formulations.

2. Symmetries of Distributions

Let $\mathcal{P}$ be a p-dimensional completely integrable distribution on the arithmetical space ${\mathbb{R}}^m$:

$$\mathcal{P}:{\mathbb{R}}^m\ni a\mapsto \mathcal{P}\left(a\right)\subset T_a{\mathbb{R}}^m.$$

Here, $T_a{\mathbb{R}}^m$ is the tangent space to ${\mathbb{R}}^m$ at the point a. Assume that the distribution is generated by the vector fields $X_1,\dots ,X_p$ or by the differential 1-forms ${\omega }_1,\dots ,{\omega }_{m-p}$, i.e.,

$$\mathcal{P}\left(a\right)=\mathrm{Span}\left(X_{1,a},\dots ,X_{p,a}\right)$$

and

$$\mathcal{P}\left(a\right)=\bigcap _{i=1}^{m-p}ker{\omega }_{i,a}.$$

A diffeomorphism $\Phi :{\mathbb{R}}^m\to {\mathbb{R}}^m$ is called a symmetry of the distribution $\mathcal{P}$ if it preserves this distribution, i.e., ${\Phi }_{*}\left(\mathcal{P}\right)=\mathcal{P}$. This means that ${\Phi }_{*}\left(\mathcal{P}\left(a\right)\right)=\mathcal{P}(\Phi \left(a\right))$ for any point $a\in {\mathbb{R}}^m$ or, equivalently,

$${\Phi }^{*}\left({\omega }_i\right)\wedge {\omega }_1\wedge \dots \wedge {\omega }_{m-p}=0(i=1,\dots ,m-p).$$

This definition has an infinitesimal counterpart. Namely, a vector field X on ${\mathbb{R}}^m$ is called an infinitesimal symmetry of the distribution $\mathcal{P}$ if translations along X are symmetries of $\mathcal{P}$.

A vector field X is an infinitesimal symmetry of the distribution $\mathcal{P}$ if

$${\mathcal{L}}_X\left({\omega }_i\right)\wedge {\omega }_1\wedge \dots \wedge {\omega }_{m-p}=0$$

(2)

for any $i=1,\dots ,m-p$. Here, ${\mathcal{L}}_X$ is the operator of the Lie derivative.

All infinitesimal symmetries of the distribution $\mathcal{P}$ form the Lie $\mathbb{R}$-algebra $\mathrm{Sym}\left(\mathcal{P}\right)$ with respect to the commutator of vector fields. This means that the following conditions hold:

-

If $X,Y\in \mathrm{Sym}\left(\mathcal{P}\right)$, then $X+Y\in \mathrm{Sym}\left(\mathcal{P}\right)$ and $[X,Y]\in \mathrm{Sym}\left(\mathcal{P}\right)$;

-

If $X\in \mathrm{Sym}\left(\mathcal{P}\right)$ and $\lambda \in \mathbb{R}$, then $\lambda X\in \mathrm{Sym}\left(\mathcal{P}\right)$.

A vector field X belongs to the distribution $\mathcal{P}$ if for any point $a\in {\mathbb{R}}^m$ the tangent vector $X_a\in \mathcal{P}\left(a\right)$. Let $D\left(\mathcal{P}\right)$ be the set of all vector fields that belong to $\mathcal{P}$. This set is a $C^{\infty }\left({\mathbb{R}}^m\right)$-module.

We say that an infinitesimal symmetry is characteristic symmetry if it belongs to the distribution $\mathcal{P}$.

The set of all characteristic symmetries form the $C^{\infty }\left({\mathbb{R}}^m\right)$-module, which we denote by $\mathrm{Char}\left(\mathcal{P}\right)$, i.e.,

$$\mathrm{Char}\left(\mathcal{P}\right)=\mathrm{Sym}\left(\mathcal{P}\right)\cap D\left(\mathcal{P}\right).$$

Any characteristic symmetry is tangent to maximal integral manifolds of the distribution $\mathcal{P}$. Moreover, $\mathrm{Char}\left(\mathcal{P}\right)$ is an ideal in the Lie algebra $\mathrm{Sym}\left(\mathcal{P}\right)$, i.e., $[X,Y]\in \mathrm{Char}\left(\mathcal{P}\right)$ for any $X\in \mathrm{Char}\left(\mathcal{P}\right)$ and any $Y\in \mathrm{Sym}\left(\mathcal{P}\right)$. Therefore, we can define the quotient Lie algebra of shuffling symmetries of the distribution $\mathcal{P}$:

$$\mathrm{Shuf}\left(\mathcal{P}\right)=\mathrm{Sym}\left(\mathcal{P}\right)/\mathrm{Char}\left(\mathcal{P}\right).$$

Elements of this Lie algebra “shuffle” maximal integral manifolds of the distribution $\mathcal{P}$.

3. Symmetries of Finite Type Differential Equations

A system of differential equations is called a system of finite type if the space of its solutions is a finite dimensional. Ordinary differential equations give examples of such systems.

Consider the following overdetermined system of $(q+1)$th order partial differential equations that are resolved with respect to higher derivatives:

$${\displaystyle \frac{{\partial }^{q+1}v}{\partial x^{\sigma +1_i}}}=V_{\sigma +1_i}\left(x,v,{\displaystyle \frac{\partial v}{\partial x}},\dots ,{\displaystyle \frac{{\partial }^{q}v}{\partial x^q}}\right),\left|\sigma \right|=q;i=1,\dots n.$$

(3)

In this system, all partial derivatives of order $q+1$ must be expressed in terms of lower order derivatives. Here, q is a non-negative integer number, v is a scalar function of $x=(x_1,\dots ,x_n)$, the symbol ${\partial }^{i}v/\partial x^i$$(i=1,\dots ,k)$ means the set of all partial derivatives of order i by x, $\sigma =({\sigma }_1,\dots ,{\sigma }_n)$ is a multi-index, ${\sigma }_i\in \{0,1,\dots ,q\}$, $\left|\sigma \right|={\sigma }_1+\dots +{\sigma }_n$,

$$\sigma +1_i=({\sigma }_1,\dots ,{\sigma }_{i-1},{\sigma }_i+1,{\sigma }_{i+1},\dots ,{\sigma }_n),$$

and

$${\displaystyle \frac{{\partial }^{q+1}v}{\partial x^{\sigma +1_i}}}={\displaystyle \frac{{\partial }^{q+1}v}{\partial x_1^{{\sigma }_1}\dots \partial x_i^{{\sigma }_i+1}\dots \partial x_n^{{\sigma }_n}}}.$$

The number of equations in system (3) is

$${\displaystyle \frac{(n+q)!}{(q+1)!(n-1)!}}.$$

Let $J^q$ be the q-jets space of smooth functions on ${\mathbb{R}}^n$ (see [15]). Coordinates $x,v_{\sigma }$ on this space are defined as follows:

$$x\left({\left[h\right]}_a^q\right)=a,v_{\sigma }\left({\left[h\right]}_a^q\right)={\displaystyle \frac{{\partial }^{\left|\sigma \right|}h}{\partial x_1^{{\sigma }_1}\dots \partial x_n^{{\sigma }_n}}}\left(a\right).$$

Here ${\left[h\right]}_a^q$ is the q-jet at the point $a\in {\mathbb{R}}^n$ of the function $h\in C^{\infty }\left({\mathbb{R}}^n\right)$ and $0\le \left|\sigma \right|\le q$. Note that

$$dim{J}^q=m=n+p,\mathrm{where}p=\left(\genfrac{}{}{0pt}{}{n+q}{q}\right).$$

(4)

Define the differential 1-forms

$${\omega }_{\sigma }=\left\{\begin{array}{cc}d{v}_{\sigma }-\sum _{i=1}^n{v}_{\sigma +1_i}d{x}_i,\hfill & \hfill \mathrm{if} 0\le \left|\sigma \right|<q,\\ d{v}_{\sigma }-\sum _{i=1}^n{V}_{\sigma +1_i}(x,v_{\sigma })d{x}_i,\hfill & \hfill \mathrm{if} \left|\sigma \right|=q.\end{array}\right.$$

(5)

These1-forms generate the n-dimensional distribution $\mathcal{P}$ on $J^q$:

$$\mathcal{P}:J^q\ni \theta ⟼\mathcal{P}\left(\theta \right)=\bigcap _{0\le \left|\sigma \right|\le q}ker{\omega }_{\sigma }{|}_{\theta }\subset T_{\theta }{J}^q.$$

The number of the differential 1-forms in (5) is $m-n$. Therefore, $dim\mathcal{P}=n$.

The n-dimensional manifold

$${\Gamma }_h^q=\left\{v_{\sigma }={\displaystyle \frac{{\partial }^{q}h}{\partial x^{\sigma }}},0\le \left|\sigma \right|\le q\right\}\subset J^q$$

is called the q-graph of the function $h\in C^{\infty }\left({\mathbb{R}}^n\right)$.

Suppose that the distribution $\mathcal{P}$ is completely integrable. Then, its maximal integral manifolds are the solution q-graphs of system (3) and vice versa. Indeed, let some function h be a solution of (3). Then, the restriction of the form ${\omega }_{\sigma }$ to ${\Gamma }_h^q$ is zero:

-

for $0\le \left|\sigma \right|<q$ we have

$$\begin{array}{cc}\hfill {\omega }_{\sigma }{|}_{{\Gamma }_h^q}& =d\left({\displaystyle \frac{{\partial }^{\left|\sigma \right|}h\left(x\right)}{\partial x^{\sigma }}}\right)-\sum _{i=1}^n{\displaystyle \frac{\partial }{\partial x_i}}\left({\displaystyle \frac{{\partial }^{\left|\sigma \right|}h\left(x\right)}{\partial x^{\sigma }}}\right)d{x}_i\hfill \\ & =\sum _{i=1}^n\left({\displaystyle \frac{{\partial }^{\left|\sigma \right|+1_i}h\left(x\right)}{\partial x^{\sigma +1_i}}}-{\displaystyle \frac{{\partial }^{\left|\sigma \right|+1_i}h\left(x\right)}{\partial x^{\sigma +1_i}}}\right)d{x}_i=0;\hfill \end{array}$$

-

due to (3) for $\left|\sigma \right|=q$ we have:

$$\begin{array}{cc}\hfill {\omega }_{\sigma }{|}_{{\Gamma }_h^q}& =d\left({\displaystyle \frac{{\partial }^{\left|\sigma \right|}h\left(x\right)}{\partial x^{\sigma }}}\right)-\sum _{i=1}^n{V}_{\sigma +1_i}\left(x,{\displaystyle \frac{{\partial }^{\left|\sigma \right|}h\left(x\right)}{\partial x^{\sigma }}}\right)d{x}_i\hfill \\ & =\sum _{i=1}^n\left({\displaystyle \frac{{\partial }^{q+1}h\left(x\right)}{\partial x^{\sigma +1_i}}}-V_{\sigma +1_i}\left(x,{\displaystyle \frac{{\partial }^{\left|\sigma \right|}h\left(x\right)}{\partial x^{\sigma }}}\right)\right)d{x}_i=0.\hfill \end{array}$$

For this reason, by the symmetries of system (3), we mean the symmetries of the distribution $\mathcal{P}$.

The module $D\left(\mathcal{P}\right)$ is generated by the following vector fields:

$${\mathcal{D}}_i={\displaystyle \frac{\partial }{\partial x_i}}+\sum _{0\le \left|\sigma \right|<q}{v}_{\sigma +1_i}{\displaystyle \frac{\partial }{\partial v_{\sigma }}}+\sum _{\left|\sigma \right|=q}{V}_{\sigma +1_i}(x,v_{\sigma }){\displaystyle \frac{\partial }{\partial v_{\sigma }}}(i=1,\dots n).$$

(6)

Since the distribution $\mathcal{P}$ is completely integrable, pairwise commutators $[{\mathcal{D}}_i,{\mathcal{D}}_j]$ are linear combinations of the vector fields ${\mathcal{D}}_1,\dots ,{\mathcal{D}}_n$. However, there are no vector fields ${\partial }_{x_i}$ and ${\partial }_{x_j}$ in the coordinate representations of commutators. This means that the vector fields commute:

$$[{\mathcal{D}}_i,{\mathcal{D}}_j]=0(i,j=1,\dots ,n).$$

(7)

Lemma 1.

The module of characteristic symmetries $\mathrm{Char}\left(\mathcal{P}\right)$ is generated by the vector fields ${\mathcal{D}}_1,\dots {\mathcal{D}}_n$, i.e.,

$$\mathrm{Char}\left(\mathcal{P}\right)=\left\{h_1{\mathcal{D}}_1+\dots +h_n{\mathcal{D}}_n\right|h_1,\dots ,h_n\in C^{\infty }\left(J^q\right)\}.$$

Proof. 

The Lemma follows from the assumption that the distribution $\mathcal{P}$ is completely integrable and the Frobenius theorem. □

Lemma 2.

The exterior differential of a function $g\in C^{\infty }\left(J^k\right)$ can be represented as

$$dg=\sum _{i=1}^n{\mathcal{D}}_i\left(g\right)d{x}_i+\sum _{0\le \left|\sigma \right|\le q}{\displaystyle \frac{\partial g}{\partial v_{\sigma }}}{\omega }_{\sigma }.$$

Proof. 

From (5), it follows that

$$d{v}_{\sigma }={\omega }_{\sigma }+\sum _{i=1}^n{v}_{\sigma +1_i}d{x}_i\mathrm{for} 0\le \left|\sigma \right|<q,$$

and

$$d{v}_{\sigma }={\omega }_{\sigma }+\sum _{i=1}^n{V}_{\sigma +1_i}d{x}_i\mathrm{for} \left|\sigma \right|=q.$$

Therefore,

$$\begin{array}{cc}\hfill dg=& \sum _{i=1}^n{\displaystyle \frac{\partial g}{\partial x_i}}d{x}_i+\sum _{0\le \left|\sigma \right|<q}{\displaystyle \frac{\partial g}{\partial v_{\sigma }}}d{v}_{\sigma }+\sum _{\left|\sigma \right|=q}{\displaystyle \frac{\partial g}{\partial v_{\sigma }}}d{v}_{\sigma }\hfill \\ \hfill =& \sum _{i=1}^n{\displaystyle \frac{\partial g}{\partial x_i}}d{x}_i+\sum _{0\le \left|\sigma \right|<q}{\displaystyle \frac{\partial g}{\partial v_{\sigma }}}\left({\omega }_{\sigma }+\sum _{i=1}^n{v}_{\sigma +1_i}d{x}_i\right)+\sum _{\left|\sigma \right|=q}{\displaystyle \frac{\partial g}{\partial v_{\sigma }}}\left({\omega }_{\sigma }+\sum _{i=1}^n{V}_{\sigma +1_i}d{x}_i\right)\hfill \\ \hfill =& \sum _{i=1}^n{\mathcal{D}}_i\left(g\right)d{x}_i+\sum _{0\le \left|\sigma \right|\le q}{\displaystyle \frac{\partial g}{\partial v_{\sigma }}}{\omega }_{\sigma }.\hfill \end{array}$$

 □

Introduce the following notation:

$${\mathcal{D}}^{\sigma }={\mathcal{D}}_1^{{\sigma }_1}\circ \dots \circ {\mathcal{D}}_n^{{\sigma }_n},$$

where ${\mathcal{D}}_i^0\left(g\right)=g$ and ${\mathcal{D}}_i^k$ is the kth degree of the operator ${\mathcal{D}}_i$, i.e.,

$${\mathcal{D}}_i^k=\underset{k\mathrm{times}}{\underbrace{{\mathcal{D}}_i\circ \dots \circ {\mathcal{D}}_i}}.$$

Theorem 1.

Any element of the quotient Lie algebra $\mathrm{Shuf}\left(\mathcal{P}\right)$ has a unique representative of the form

$$S_{\phi }=\sum _{0\le \left|\sigma \right|\le q}{\mathcal{D}}^{\sigma }\left(\phi \right){\displaystyle \frac{\partial }{\partial v_{\sigma }}},$$

(8)

where φ is a function on $J^q$.

The vector field $S_{\phi }$ is also called a shuffling symmetry of the distribution.

Proof. 

Due to Lemma 1, any shuffling symmetry of the distribution $\mathcal{P}$ has a representative of the form

$$S=\sum _{0\le \left|\sigma \right|\le q}{a}_{\sigma }{\displaystyle \frac{\partial }{\partial v_{\sigma }}},$$

where $a_{\sigma }$ are some functions on $J^q$. Let us calculate the Lie derivatives of the differential 1-forms ${\omega }_{\sigma }$ for $0\le \left|\sigma \right|<q$. Due to Lemma 2, we have:

$${\mathcal{L}}_S\left({\omega }_{\sigma }\right)=\sum _{i=1}^n({\mathcal{D}}_i\left(a_{\sigma }\right)-a_{\sigma +1_i})d{x}_i+\sum _{0\le \left|\sigma \right|\le q}{\displaystyle \frac{\partial a_{\sigma }}{\partial v_{\sigma }}}{\omega }_{\sigma }.$$

(9)

Define the differential p-form (see (4))

$$\Omega =\underset{0\le \left|\sigma \right|\le q}{\bigwedge }{\omega }_{\sigma }={\omega }_{(0,\dots ,0)}\wedge {\omega }_{(1,\dots ,0)}\wedge \dots \wedge {\omega }_{(0,\dots ,q)}.$$

Using formula (2), we obtain $\Omega \wedge {\mathcal{L}}_S\left({\omega }_{\sigma }\right)=0,$ i.e.,

$$\Omega \wedge \sum _{i=1}^n({\mathcal{D}}_i\left(a_{\sigma }\right)-a_{\sigma +1_i})d{x}_i=0.$$

(10)

Since the $(p+1)$-forms $\Omega \wedge d{x}_1,\dots ,\Omega \wedge d{x}_n$ are linearly independent, we have

$$\sum _{i=1}^n({\mathcal{D}}_i\left(a_{\sigma }\right)-a_{\sigma +1_i})d{x}_i=0.$$

Therefore,

$$a_{\sigma +1_i}={\mathcal{D}}_i\left(a_{\sigma }\right),i=1,\dots ,n.$$

This formula shows that the vector field S is determined by only one function $a_{(0,\dots ,0)}$. We denote this function by $\phi$ and call it the generating function for the vector field S. Therefore, the vector field S we denote by $S_{\phi }$.

Taking into account that vector fields (6) pairwise commute, we obtain:

$$a_{\sigma }={\mathcal{D}}^{\sigma }\left(\phi \right).$$

(11)

Therefore,

$$S_{\phi }=\sum _{0\le \left|\sigma \right|\le q}{\mathcal{D}}^{\sigma }\left(\phi \right){\displaystyle \frac{\partial }{\partial v_{\sigma }}}.$$

(12)

 □

The following Theorem shows how we can find the generating function $\phi$.

Theorem 2.

The generating function φ is a solution of the following system:

$${\mathcal{D}}^{\sigma +1_i}\left(\phi \right)-\sum _{\left|\sigma \right|=q}{\mathcal{D}}^{\sigma }\left(\phi \right){\displaystyle \frac{\partial V_{\sigma +1_i}}{\partial v_{\sigma }}}=0,i=1,\dots ,n;\left|\sigma \right|=q.$$

(13)

Proof. 

Let us calculate the Lie derivative of the differential 1-forms ${\omega }_{\sigma }$ for $\left|\sigma \right|=q$:

$${\mathcal{L}}_{S_{\phi }}\left({\omega }_{\sigma }\right)=d{a}_{\sigma }-\sum _{i=1}^n{S}_{\phi }\left(V_{\sigma +1_i}\right)d{x}_i=\sum _{i=1}^n\left({\mathcal{D}}_i\left(a_{\sigma }\right)-S_{\phi }\left(V_{\sigma +1_i}\right)\right)d{x}_i+\sum _{\left|\sigma \right|=q}{\displaystyle \frac{\partial a_{\sigma }}{\partial v_{\sigma }}}{\omega }_{\sigma }.$$

As above, we obtain

$${\mathcal{D}}_i\left(a_{\sigma }\right)-S_{\phi }\left(V_{\sigma +1_i}\right)=0,\left|\sigma \right|=q;i=1,\dots ,n.$$

Using formulae (11) and (12), we obtain (13). □

System (13) gives us conditions for the generating function $\phi$. Solving it, we find the shuffling symmetry $S_{\phi }$.

Example 1.

Let $n=2$ and $q=1$. Then, system (3) has the form

$$\left\{\begin{array}{ccc}\hfill {\displaystyle \frac{{\partial }^{2}v}{\partial x_1^2}}& =\hfill & \hfill V_{20}\left(x_1,x_2,v,{\displaystyle \frac{\partial v}{\partial x_1}},{\displaystyle \frac{\partial v}{\partial x_2}}\right),\\ \hfill {\displaystyle \frac{{\partial }^{2}v}{\partial x_1\partial x_2}}& =\hfill & \hfill V_{11}\left(x_1,x_2,v,{\displaystyle \frac{\partial v}{\partial x_1}},{\displaystyle \frac{\partial v}{\partial x_2}}\right),\\ \hfill {\displaystyle \frac{{\partial }^{2}v}{\partial x_2^2}}& =\hfill & \hfill V_{02}\left(x_1,x_2,v,{\displaystyle \frac{\partial v}{\partial x_1}},{\displaystyle \frac{\partial v}{\partial x_2}}\right).\end{array}\right.$$

(14)

Differential forms (5) and vector fields (6) are

$$\begin{array}{c}{\omega }_{00}=d{v}_{00}-v_{10}d{x}_1-v_{01}d{x}_2,\hfill \\ {\omega }_{10}=d{v}_{10}-V_{20}d{x}_1-V_{11}d{x}_2,\hfill \\ {\omega }_{01}=d{v}_{01}-V_{11}d{x}_1-V_{02}d{x}_2,\hfill \end{array}$$

and

$$\begin{array}{c}{\mathcal{D}}_1={\displaystyle \frac{\partial }{\partial x_1}}+v_{10}{\displaystyle \frac{\partial }{\partial v_{00}}}+V_{20}{\displaystyle \frac{\partial }{\partial v_{10}}}+V_{11}{\displaystyle \frac{\partial }{\partial v_{01}}},\hfill \\ {\mathcal{D}}_2={\displaystyle \frac{\partial }{\partial x_2}}+v_{01}{\displaystyle \frac{\partial }{\partial v_{00}}}+V_{11}{\displaystyle \frac{\partial }{\partial v_{10}}}+V_{02}{\displaystyle \frac{\partial }{\partial v_{01}}}\hfill \end{array}$$

respectively. By the Frobenius theorem, the distribution $\mathcal{P}$ is completely integrable if and only if the following conditions hold:

$$\left\{\begin{array}{c}-V_{11}{\displaystyle \frac{\partial V_{11}}{\partial v_{01}}}+V_{02}{\displaystyle \frac{\partial V_{20}}{\partial v_{01}}}-V_{20}{\displaystyle \frac{\partial V_{11}}{\partial v_{10}}}+V_{11}{\displaystyle \frac{\partial V_{20}}{\partial v_{10}}}-v_{10}{\displaystyle \frac{\partial V_{11}}{\partial v_{00}}}+v_{01}{\displaystyle \frac{\partial V_{20}}{\partial v_{00}}}-{\displaystyle \frac{\partial V_{11}}{\partial x_1}}+{\displaystyle \frac{\partial V_{20}}{\partial x_2}}=0,\hfill \\ -V_{11}{\displaystyle \frac{\partial V_{02}}{\partial v_{01}}}+V_{02}{\displaystyle \frac{\partial V_{11}}{\partial v_{01}}}-V_{20}{\displaystyle \frac{\partial V_{02}}{\partial v_{10}}}+V_{11}{\displaystyle \frac{\partial V_{11}}{\partial v_{10}}}-v_{10}{\displaystyle \frac{\partial V_{02}}{\partial v_{00}}}+v_{01}{\displaystyle \frac{\partial V_{11}}{\partial v_{00}}}-{\displaystyle \frac{\partial V_{02}}{\partial x_1}}+{\displaystyle \frac{\partial V_{11}}{\partial x_2}}=0.\hfill \end{array}\right.$$

(15)

The vector field $S_{\phi }$ is

$$S_{\phi }=\phi {\displaystyle \frac{\partial }{\partial v_{00}}}+{\mathcal{D}}_1\left(\phi \right){\displaystyle \frac{\partial }{\partial v_{10}}}+{\mathcal{D}}_2\left(\phi \right){\displaystyle \frac{\partial }{\partial v_{01}}}.$$

Equation (13) is

$$\left\{\begin{array}{c}{\mathcal{D}}_1^2\left(\phi \right)-S_{\phi }\left(V_{20}\right)=0,\hfill \\ {\mathcal{D}}_1{\mathcal{D}}_2\left(\phi \right)-S_{\phi }\left(V_{11}\right)=0,\hfill \\ {\mathcal{D}}_2^2\left(\phi \right)-S_{\phi }\left(V_{02}\right)=0.\hfill \end{array}\right.$$

(16)

4. Finite Dimensional Dynamics

Let us go back to evolutionary Equation (1). For our purposes, it is more convenient to write it in the form

$${\displaystyle \frac{\partial u}{\partial t}}=f\left(x,{\displaystyle \frac{{\partial }^{\left|\sigma \right|}u}{\partial x^{\sigma }}}\right),$$

(17)

where the symbol ${\displaystyle \frac{{\partial }^{\left|\sigma \right|}u}{\partial x^{\sigma }}}$ means the set of all partial derivatives of order $0\le \left|\sigma \right|\le k$ by x. Suppose that $k\le q$. The function f generates the function $\psi =f(x,v_{\sigma })$. We simply replace the partial derivatives of the function $u(t,x)$ of $n+1$ variables with the partial derivatives of the function $v\left(x\right)$ of n variables.

Let $\phi$ be the restriction of the function $\psi$ to system (3):

$$\phi =\psi (x,v_{\mu },V_{\sigma +1_i}),\mathrm{where} \left|\mu \right|\le q,\left|\sigma \right|=q,i=1,\dots n,$$

where $\mu =({\mu }_1,\dots ,{\mu }_n)$ is a multi-index. Then, $\phi \in C^{\infty }\left(J^q\right)$.

Definition 1.

Overdetermined system (3) is called (finite dimensional) dynamics of evolutionary Equation (17) if φ is a generating function of some shuffling symmetry of system (3).

Let ${\Phi }_t:J^q\to J^q$ be the translation along trajectories of the vector field $S_{\phi }$ from $t=0$ to t (here, t belongs to some open interval I that includes 0). Let L be a maximal integral manifold of the distribution $\mathcal{P}$. Then, ${\Phi }_t\left(L\right)$ is an integral manifold of this distribution, too.

For any solution $v=h\left(x\right)$ of system (3), its q-graph is a maximal integral manifold of the distribution $\mathcal{P}$. Since ${\Phi }_t$ is a symmetry of system (3), the manifold ${\Phi }_t\left({\Gamma }_h^q\right)$ is a q-graph of another solution of (3) for any $t\in I$, too.

Let $J^q\left({\mathbb{R}}^{n+1}\right)$ be the space of q-jets of functions on ${\mathbb{R}}^{n+1}$ with canonical coordinates $t,x,u_{\delta }$. Here, $\delta =({\delta }_0,{\delta }_1,\dots ,{\delta }_n)$ is a multi-index, $0\le \left|\delta \right|\le q$. Let the function $h\left(x\right)$ be a solution of system (3). Construct the $(n+1)$-dimensional manifold

$$L=\bigcup _{t\in I}{\Phi }_t\left({\Gamma }_h^q\right)\subset J^q\left({\mathbb{R}}^{n+1}\right).$$

The manifold L is a q-graph of some solution $u=U(t,x)$ of the evolutionary equation.

The function $U(t,x)$ can be found by the following way. First, apply the transformation ${\left({\Phi }_t^{-1}\right)}^{*}$ to the system

$$v_{\sigma }-{\displaystyle \frac{{\partial }^{\left|\sigma \right|}h}{\partial x^{\sigma }}}=0,0\le \left|\sigma \right|\le q.$$

We obtain the following system:

$${\Psi }_i(t,x,v_{\sigma })=0,i=1,\dots ,p.$$

(18)

Second, solve this system with respect to $v_{\sigma }$:

$$v_{\sigma }=W_{\sigma }(t,x),0\le \left|\sigma \right|\le q.$$

The function

$$U(t,x)=W_{(0,\dots ,0)}(t,x)$$

is a required solution of evolutionary Equation (17). The remaining functions correspond to its partial derivatives:

$${\displaystyle \frac{{\partial }^{\left|\sigma \right|}U}{\partial x^{\sigma }}}=W_{\sigma }(t,x),0<\left|\sigma \right|\le q.$$

Note that $U(0,x)=h\left(x\right)$.

5. Some Examples

In this section, we consider two examples of calculating the dynamics and constructing exact solutions of equations

$$u_t=\Delta u+g(u,u_x,u_y),$$

were

$$\Delta ={\displaystyle \frac{{\partial }^2}{\partial x^2}}+{\displaystyle \frac{{\partial }^2}{\partial y^2}}$$

is the Laplace operator. This equation is a generalization of the Kolmogorov–Petrovsky–Piskunov–Fisher equation

$$u_t=\Delta u+g\left(u\right),$$

which arises in many branches of physics and biology (see [11,12,14], for example).

5.1. Equation $u_t=\Delta u+u_x+u{u}_y$

Consider the following equation:

$${\displaystyle \frac{\partial u}{\partial t}}={\displaystyle \frac{{\partial }^{2}u}{\partial x^2}}+{\displaystyle \frac{{\partial }^{2}u}{\partial y^2}}+{\displaystyle \frac{\partial u}{\partial x}}+u{\displaystyle \frac{\partial u}{\partial y}}.$$

(19)

Then, $n=2$, $\psi =v_{20}+v_{02}+v_{10}+v_{00}{v}_{01}$, and $\phi =V_{20}+V_{02}+v_{10}+v_{00}{v}_{01}$. Let us find dynamics with $q=1$. It is easy to check that the functions

$$V_{20}=a{v}_{x_1}+b{v}_{x_2},V_{11}=V_{02}=0$$

satisfy Equation (13). Here, $a,b$ are arbitrary real number, $a\ne 0$, $x_1=x,x_2=y$. So, the system

$$v_{x_1{x}_1}=a{v}_{x_1}+b{v}_{x_2},v_{x_1{x}_2}=v_{x_2{x}_2}=0$$

are dynamics. The general solution of this system is

$$v(x_1,x_2)=C_1{e}^{a{x}_1}+C_2(a{x}_2-b{x}_1)+C_3,$$

where $C_1,C_2,C_3$ are arbitrary constants. Then, $\phi =v_{00}{v}_{01}+b{v}_{01}+(a+1)v_{10}$ and the vector field

$$S_{\phi }=\left((b+v_{00})v_{01}+(a+1)v_{10}\right){\displaystyle \frac{\partial }{\partial v_{00}}}+\left(a(a+1)v_{10}+v_{01}(v_{10}+ba+b)\right){\displaystyle \frac{\partial }{\partial v_{10}}}+v_{01}^2{\displaystyle \frac{\partial }{\partial v_{01}}}.$$

The flow of this vector field is

$${\Phi }_t:\left\{\begin{array}{c}{x}_1=x_1,\hfill \\ x_2=x_2,\hfill \\ v_{00}={\displaystyle \frac{a(b{v}_{01}t+v_{10})+b{v}_{01}-a^2{v}_{00}-(a{v}_{10}+b{v}_{01})e^{a(a+1)t}}{a^2(t{v}_{01}-1)}},\hfill \\ v_{10}={\displaystyle \frac{(a{v}_{10}+b{v}_{01}-b{v}_{01}{e}^{-a(a+1)t})e^{a(a+1)t}}{a(1-t{v}_{01})}},\hfill \\ v_{01}={\displaystyle \frac{v_{01}}{1-t{v}_{01}}}.\hfill \end{array}\right.$$

The inverse transformation is

$${\Phi }_t^{-1}:\left\{\begin{array}{c}{x}_1=x_1,\hfill \\ x_2=x_2,\hfill \\ v_{00}={\displaystyle \frac{\left(-b{v}_{01}(e^{a(a+1)t}-1)+((bt{v}_{01}-v_{10}+a{v}_{00})e^{a(a+1)t}+v_{10})a\right)e^{-a(a+1)t}}{a^2(1+t{v}_{01})}},\hfill \\ v_{10}={\displaystyle \frac{(a{v}_{10}+b{v}_{01}-b{v}_{01}{e}^{a(a+1)t})e^{-a(a+1)t}}{a(1+t{v}_{01})}},\hfill \\ v_{01}={\displaystyle \frac{v_{01}}{1+t{v}_{01}}}.\hfill \end{array}\right.$$

Applying the last transformation to the system

$$\left\{\begin{array}{c}{v}_{00}-C_1{e}^{a{x}_1}-C_2(a{x}_2-b{x}_1)-C_3=0,\hfill \\ v_{10}-C_{1}a{e}^{a{x}_1}+C_{2}b=0,\hfill \\ v_{01}-C_{2}a=0,\hfill \end{array}\right.$$

we obtain system (18). This system is cumbersome, and we do not specify it here. Solving this system, we obtain

$$\begin{array}{cc}\hfill v_{00}=& {\displaystyle \frac{C_2(a{x}_2-bt-b{x}_1)+C_1{e}^{a(a+1)t+a{x}_1}+C_3}{1-C_{2}at}},\hfill \\ \hfill v_{10}=& {\displaystyle \frac{e^{a(a+1)t}(C_{1}a{e}^{a{x}_1}-b{C}_2{e}^{-a(a+1)t})}{1-C_{2}at}},\hfill \\ \hfill v_{01}=& {\displaystyle \frac{C_{2}a}{1-C_{2}at}}.\hfill \end{array}$$

So we get the 5-parametric solutions family of Equation (19):

$$u(t,x,y)={\displaystyle \frac{C_2(ay-b(t+x))+C_1{e}^{a(a+1)t+ax}+C_3}{1-C_{2}at}}.$$

5.2. Equation $u_t=\Delta u+u+\beta$

Consider the following linear equation

$${\displaystyle \frac{\partial u}{\partial t}}={\displaystyle \frac{{\partial }^{2}u}{\partial x^2}}+{\displaystyle \frac{{\partial }^{2}u}{\partial y^2}}+u+\beta ,$$

(20)

where $\beta \in \mathbb{R}$. Equation (20) admits dynamics

$$v_{x_1{x}_1}=0,v_{x_1{x}_2}=v_{x_1},v_{x_2{x}_2}=v+a{v}_{x_1}+{\displaystyle \frac{b}{v_{x_1}}}+\beta .$$

Solutions of these dynamics are

$$v(x_1,x_2)=C_1{e}^{x_2}+{\displaystyle \frac{C_2}{e^{x_2}}}+C_3\left({\displaystyle \frac{a(2x_2-1)}{4}}+x_1\right)-{\displaystyle \frac{b(2x_2+1)}{4C_3{e}^{x_2}}}-\beta ,$$

where $C_1,C_2,C_3$ are arbitrary constants. Moreover,

$$\phi =-a{v}_{10}-{\displaystyle \frac{b}{v_{10}}}$$

and

$$S_{\phi }=-{\displaystyle \frac{a{v}_{10}^2+b}{v_{10}}}{\displaystyle \frac{\partial }{\partial v_{00}}}-{\displaystyle \frac{a{v}_{10}^2-b}{v_{10}}}{\displaystyle \frac{\partial }{\partial v_{01}}}.$$

The flow of this vector field is

$${\Phi }_t:\left\{\begin{array}{c}{x}_1=x_1,\hfill \\ x_2=x_2,\hfill \\ v_{00}=v_{00}-{\displaystyle \frac{(a{v}_{10}^2+b)t}{v_{10}}},\hfill \\ v_{10}=v_{10},\hfill \\ v_{01}=v_{01}-{\displaystyle \frac{(a{v}_{10}^2-b)t}{v_{10}}}.\hfill \end{array}\right.$$

The inverse transformation is

$${\Phi }_t^{-1}:\left\{\begin{array}{c}{x}_1=x_1,\hfill \\ x_2=x_2,\hfill \\ v_{00}=v_{00}+{\displaystyle \frac{(a{v}_{10}^2+b)t}{v_{10}}},\hfill \\ v_{10}=v_{10},\hfill \\ v_{01}=v_{01}+{\displaystyle \frac{(a{v}_{10}^2-b)t}{v_{10}}}.\hfill \end{array}\right.$$

Omitting intermediate calculations, we write the final form of the five-parametric family of the solutions of Equation (20):

$$u(t,x,y)=(a{C}_3(2y-4t-1)+4C_{3}x+4C_1)e^y+(4C_2-b(4t+2y-1)C_3^{-1})e^{-y}-4\beta .$$

In the considered examples, we have indicated only particular types of dynamics. Of course, there are other dynamics for Equations (19) and (20), which lead us to other solutions.

6. Conclusions

The described method makes it possible to find exact solutions of evolution equations, even in those cases when they cannot be found using symmetry theory of partial differential equations (see [15], for example). However, sometimes one can find the dynamics, but the flow of the vector field $S_{\phi }$ cannot be calculated explicitly. In such cases, numerical methods can be applied to approximate the flow.