Symbolic computation of exact solutions for nonlinear evolution equations

Abstract

Based on the invariant subspace method, a symbolic computation scheme and its corresponding MAPLE package are developed to construct exact solutions for nonlinear evolution equations. In the symbolic computation scheme, a crucial step is constructing the linear differential equations as invariant subspaces that systems of evolution equations admin and taking their solutions as subspaces to construct exact solutions. The MAPLE package is proved to provide an easy way for constructing exact solutions of evolution equations automatically by only inputting several necessary parameters. Three different types of examples are given to illustrate the scope and demonstrate the validity of our package, especially for wave equation. The results of the examples reveal that there are polynomial subspaces, trigonometric subspaces, exponential subspaces and other complex subspaces as invariant subspaces that evolutions equations admit. In addition, our MAPLE software package provides a helpful and easy-to-use tool in science and engineering to deal with a wide variety of (1+1) dimensional nonlinear evolution equations.

Appendix: The usage of the software package ISM

The main interface is $Isolve(Eqs, var, dim)$, where the parameter $Eqs$ is a set of nonlinear evolution equations, $var$ represents the independent variable appearing in the invariant subspaces, $dim$ is the desired order of ordinary differential equations.

As an example to show how to use our software package ISM, we consider a Kuramoto–Sivashinsky equation [1]

$$\begin{aligned} u_t +a uu_x +b u_{xx}+ku_{4x}=0, \end{aligned}$$

(75)

To load the package ISM, one can proceed as follows:

• $>restart:$    initializing Maple.

• $>read \,\,``ISM.mpl'':$ reading the program file into memory.

• $>with(ISM);$    loading the package ADMP.

To solve Eq. (75), one can run the main procedure as follows:

• $>eq:=diff(u(t,x),t)+a*u*diff(u(t,x),x)+b*diff(u(t,x),x\$2)+k*diff(u(t,x),x\$4)=0;$

• $>Isolve([eq1], x, [2]);$

Our package automatically outputs the following results:

[1] Obtain the evolution equation(s) is/are

$$\begin{aligned} u_t +a uu_x +b u_{xx}+ku_{4x}=0, \end{aligned}$$

which admits an invariant subspace defined by

$$\begin{aligned} \frac{dy_1(x)}{d x^2} = 0 \end{aligned}$$

The invariant condition is

$$\begin{aligned} \left\{ \_a_0 = 0, \_a_1 = 0, a = a\right\} . \end{aligned}$$

Then, the exact solutions are listed as follows:

$$\begin{aligned} \left[ u(t, x) = \_F_1(t)x+\_F_2(t)\right] , \end{aligned}$$

where the coefficients can be determined by the following

$$\begin{aligned} \left\{ \begin{aligned}&a\_F_1(t)^2+\frac{\partial \_F_1(t)}{\partial t} 0, \\&a\_F_1(t)\_F_2(t)+\frac{\partial \_F_2(t)}{\partial t} = 0. \end{aligned} \right. \end{aligned}$$

Its solution reads

$$\begin{aligned} \left\{ \_F_1(t) = \frac{1}{a t+\_C_2},\quad \_F_2(t) = \frac{\_C_1}{a t+\_C_2}\right\} . \end{aligned}$$