Abstract

New approach for tanh and extended-tanh methods with applications on Hirota-Satsuma equations

Hassan A. Zedan

Mathematics Department, Faculty of Education, Kafr El-Sheikh, University Kafr El-Sheikh, Egypt, E-mail: hassanzedan2003@yahoo.com

ABSTRACT

In this work, we establish new exact solutions for the Hirota-Satsuma equations. New approach for the tanh is used and extended tanh methods to construct traveling wave solutions in terms of a hyperbolic tangent functions. New families of solitary wave solutions and periodic solutions are also obtained for Hirota-Satsuma equations. Our approach is reduce the size of the computational adopted in other techniques without any conditions to apply on any system of partial differential equations.

Mathematical subject classification: 35K99, 35P05, 35P99.

Key words: Tanh method, extended tanh method, Hirota-Satsuma equations, traveling wave solutions, solitary wave solutions.

1 Introduction

The world around us is inherently nonlinear. The Hirota-Satsuma equations are widely used as models to describe complex physical phenomena in various fields of science, especially in fluid mechanics, solid stat physics, plasma physics. Various methods have been used to explore different kinds of solutions of physical models described by nonlinear PDEs. One of the basic physical problems for those models is obtaining their traveling wave solutions. Concepts like solitons, peakons, kinks, breathers, cusps and compacton are now being thoroughly investigated in the scientific literature [1-3]. During the past decades, quite a few methods for obtaining explicit traveling and solitary wave solutions of the Hirota-Satsuma equations have led to a variety of powerfull methods, such as inverse scattering method [4], bilinear transformation [5], the tanh-sech method [6], Backlund and Darboux transform [7], Hirota [8], tanh-function method [9-10]. Moreover these methods, extended tanh-function method [11-12], modified extended tanh-function method [13-14] and homogeneous balance method [15].

The extended tanh-function method and the modified extended tanh-function method belong to a class of methods called sub-equation method for which these appears few basic relationships among the complicated NLPDEs in the study and some simple and solvable nonlinear ordinary equations.

In this work, we introduce a matrix spectral problem with three potentials and propose a corresponding hierarchy of nonlinear evaluation equations. An interesting equation in the hierarchy is a generalization of the Hirota-Satsuma coupled kdv equations:

2 Tanh method

In this section, we will try to search for a new analytical solutions for system (1)-(3), by using tanh method [12, 16]. This technique has been proven to be very powerful in finding travelling-wave solutions. We represent the tanh method for the problems (1)-(3) which is a workable and universal solution method that can be used to find exact as well as approximate solutions. This technique is based on the fact that in many cases traveling-wave solutions can be written in terms of a hyperbolic tangents.

Let the boundary conditions be

To find possible travelling-wave solution, first we introduce the independent variable ξ = c(x - vt), then, substituting ξ = c(x - vt) into Eqs. (1)-(3),we get:

where the constants (c and v) represent (positive) wave number and (positive) velocity of the travelling-wave, u(x, t) = U(ξ), v(x, t) = V(ξ) and w(x, t) = W(ξ).

Secondly, substituting Y = tanh ξ into Eqs. (6)-(8) gives:

where U(ξ) → U(Y), V(ξ) → V(Y) and W(ξ) → W(Y).

Consequently, the boundary conditions reduce to the conditions with respect to the variable with respect to the variable Y:

To find an exact solution,we consider a finite expansion of Y in the following form:

Substitution of Eq. (14) into Eqs. (9)-(11) and balancing the highest order derivative terms with the nonlinear terms, gives

Taking account of Eq. (15) with Eq. (14), we obtain

By using Eqs. (12) and (13) into Eqs. (16)-(18), we get:

where A₁, B₁, A₂, B₂, A₃ and B₃ are constants and can be determined.

After substitution of Eqs. (19)-(21) into Eqs. (6)-(8), and eliminating terms having equal power in Y, we get the following:

In order to obtain A₁, B₁, A₂, B₂, A₃, B₃, we can solve Eqs. (22)-(33). Consequently we get

Substituting the above solution into Eqs. (19)-(21), we have the following analytical solutions:

where A₃ = 2c².

To find the wave number (c), we use the boundary condition

It is well known that (wave-number) c = 109737.3

V_∞ = 4.816910004516×10¹⁰,

v = 4.816910004516×10¹⁰

Then, Eqs.(1)-(3) have the following analytical solutions:

with c, v, V_∞, are given.

3 Extended tanh method

In this section, we solve the Hirota-Satsuma coupled KdV equations by extended tanh method. This method is used to derive explicit travelling-wave solutions. New families of solitary wave solutions and periodic solutions are formally obtained for these equations. The method is used to reduce the computational size compared to the other techniques [16-20] without any conditions to apply on any system of partial differential equations.

Using the wave variable ξ = x - ct, Eqs. (1)-(3) reduce to ODEs in the following form:

where u(x, t) → U(ξ), v(x, t) → V(ξ) and w(x, t) → W(ξ).

Integrating Eq. (41) with respect to (ξ), we get:

where the constant of integration is equal to zero.

The solution of the reduced equations can be expressed as a finite power series in Y in the form:

where M, N and H can be determined. By balancing between the highest derivative with the nonlinear terms, in Eqs. (42)-(44) gives

Substituting Eq. (46) into Eqs. (45), we find that:

Substituting Y = tanh (µξ) in Eqs. (42)-(44), with the aid of Eqs. (47)-(49), and equating the coefficients of each power of Y to zero, we obtain a system of algebraic equations of the parameters a₀, a₁, a₂, b₀, b₁, b₂, c₀, c₁, c₂, namely:

Using symbolic software Mathematica to solve the algebraic equations (50)-(62), we obtain the following set of distinct solutions of parameters. The set is given the following cases:

Case 1:

where

and a₀, c₂ are arbitrary constants. In this case, the soliton solutions take the form:

where a₀, c₂ are arbitrary constants.

Figs. (4)-(6) represent the soliton solutions for Eq. (63), by taking different values for constants, for example a₀ = -1, µ = 0.5, c₂ = -1.

Case 2:

where

and a₀, c₁ are arbitrary constants. Thus, the soliton solutions take the form:

where a₀, c₁ are arbitrary constants.

Figs. (7)-(9) represent the soliton solutions for Eq. (64) as taking different values for constants, for example a₀ = -2.5, µ = 0.5, c₁ = -1.

Acknowledgement. The author would like to thank the referee for his suggestions and comments on this article.

Received: 28/I/08. Accepted: 11/VI/08.

#750/08.

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