Symbolic computation of exact solutions expressible in rational formal hyperbolic and elliptic functions for nonlinear partial differential equations

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Abstract

With the aid of symbolic computation, some algorithms are presented for the rational expansion methods, which lead to closed-form solutions of nonlinear partial differential equations (PDEs). The new algorithms are given to find exact rational formal polynomial solutions of PDEs in terms of Jacobi elliptic functions, solutions of the Riccati equation and solutions of the generalized Riccati equation. They can be implemented in symbolic computation system Maple. As applications of the methods, we choose some nonlinear PDEs to illustrate the methods. As a result, we not only can successfully obtain the solutions found by most existing Jacobi elliptic function methods and Tanh-methods, but also find other new and more general solutions at the same time.

Introduction

The recent development of the nonlinear wave theory clarified the role of soliton in various system. The appearance of solitary wave solutions in nature is quite common. Bell-shaped sech-solutions and kink-shaped tanh-solutions model wave phenomena in fluids, plasmas, elastic media, electrical circuits, optical fibers, chemical reactions, bio-genetics, etc., [1], [2], [3], [4].

As we known, travelling wave solutions of many nonlinear PDEs from soliton theory (and beyond) can often be expressed as polynomials of the hyperbolic tangent and secant functions. The Tanh-method provides a straightforward algorithm to compute such particular solutions for a large class of nonlinear PDEs [5], [6], [7], [8]. The Tanh-method for, say, a single PDE in u(x, t) works as follows: in a travelling frame of reference, ξ = k(x + λt), one transforms the PDE into an ODE in the new independent variable T = tanh ξ. Since the derivative of tanh is polynomial in tanh, i.e., T′ = 1 − T2, all derivatives of T are polynomials of T. Via a chain rule, the polynomial PDE in u(x, t) is transformed into an ODE in U(T), which has polynomial coefficients in T. One then seeks polynomial solutions of the ODE, thus generating a subset of the set of all solutions. The appeal and success of the Tanh-method lies in the fact that one circumvents integration to get explicit solutions. Recently, the Tanh-methods have been applied to many nonlinear PDEs with various extension [5], [6], [7], [8], [9], [10], [11], [12], [13], [14].

Making use of idea of Tanh-method, we use some variables whose derivatives are also polynomial in themselves to replace the hyperbolic functions and present some more general solution form so that it can be used to bring out more types and general formal solutions which contain not only the results obtained by using the various Tanh-methods [11], [12], [13], [14], but also other types of solutions. We summary our methods and call them rational expansion methods. For illustration, we apply the generalized method to some nonlinear PDEs and successfully construct new and more general solutions including rational form Jacobi elliptic function solutions, solitary wave solutions and triangular periodic solutions.

The paper is organized as follows: in Section 2, we give the main steps of our algorithms for computing exact solutions of nonlinear polynomial PDEs; In Section 3, we consider the Jacobi elliptic function rational expansion method; In Section 4, we show the Riccati equation rational expansion method; In Section 5, we further extend the Riccati equation rational expansion method to a generalized form. Finally, conclusions are presented.

Section snippets

Summary of the rational expansion method

In the following we would like to outline the main steps of our method:

  • Step 1.

    Given a system of nonlinear polynomial PDEs with constant coefficients, with some physical fields ui(x, y, t) in three variables x, y, t,Δ(ui,uit,uix,uiy,uitt,uixt,uiyt,uixx,uiyy,uixy,…)=0use the wave transformationui(x,y,t)=Ui(ξ),ξ=k(x+ly+λt),where k, l and λ are constants to be determined later. Then the nonlinear partial differential system (2.1) is reduced to an ordinary differential systemΘUi,Ui′,Ui″,…=0.

  • Step 2.

    We introduce a

Jacobi elliptic function rational expansion method

In this section we would like to apply our method to obtain rational formal Jacobi elliptic function solutions of nonlinear PDEs, i.e. restricting F and G in Jacobi elliptic functions.

Here snξ, cnξ, dnξ, nsξ, csξ, scξ, ncξ, sdξ and ndξ are the Jacobian elliptic sine function, the Jacobian elliptic cosine function and the Jacobian elliptic function of the third kind and other Jacobian functions which is denoted by Glaisher symbols and are generated by these three kinds of functions, namely [15],

Riccati equation rational expansion method

In this section we would like apply our method to obtain rational form soliton solutions of nonlinear PDEs, i.e., restricting F and G in the solutions of the Riccati equation.

The six main steps of the Riccati equation rational expansion method are illustrated with the (2 + 1)-dimensional dispersive long wave equation (DLWE), i.e.,uyt+vxx+(uux)y=0,vt+ux+(uv)x+uxxy=0.The (2 + 1)-dimensional DLWE (4.1) was first derived by Boiti et al. [26] as a compatibility for a “weak” Lax pair. Recently

Generalized Riccati equation rational expansion method

In this section we would like extend the Riccati equation rational expansion method, i.e., using generalized ansätz to replace (2.4) and solutions of generalized Riccati equation to replace the variables in (2.4).

Consider the (2 + 1)-dimension Burgers system, i.e.,-ut+uuy+αvux+βuyy+αβuxx=0,ux-vy=0.

According to the Step 1 in Section 2, we make the following travelling wave transformationu(x,y,t)=U(ξ),v(x,y,t)=V(ξ),ξ=k(x+ly+λt),where k and λ are constants to be determined later, and thus system

Summary and conclusions

In summary, we have proposed a unified algebraic method: rational expansion method with symbolic computation, which greatly exceeds the applicability of the existing methods in obtaining travelling wave solutions of the nonlinear PDEs.

In Section 3, when μ1 = μ2 = 0, our method just reduces to the Jacobi elliptic function expansion method [33].

In Section 4, when μ1 = μ2 = 0, our method just reduces to the Tanh-method [5], [6], [7], [8], [11], [12]; when μ1 = 0, our method just reduce to the projective

Acknowledgements

The work was supported by China Postdoctoral Science Foundation, Nature Foundation of Zhejiang Province of China (Y604056) and Ningbo Doctoral Foundation of China (2005A610030).

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