Exact solutions and conservation laws for Ibragimov-Shabat equation which describe pseudo-spherical surface

Exact solutions and conservation laws for Ibragimov-Shabat equation which describe pseudo-spherical surface

S.M. Sayed^{I, II}; A.M. Elkholy^{I, II}; G.M. Gharib^II

^IMathematics Department, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt

^IIMathematics Department, P.O. Box 1144, Tabouk Teacher College, Deputy for Teacher College Ministry of Higher Education, Tabouk, Kingdom of Saudi Arabia, E-mail: [e-mail]

ABSTRACT

Travelling wave solution for Ibragimov-Shabat equation, is obtained by using an improved sine-cosine method and the Wu's elimination method. An infinite number of conserved quantities for the above equation are also obtained by solving a set of coupled Riccati equations.

Mathematical subject classification: 35, 53C, 58J, 58Z05.

Key words: nonlinear evolution equations, conservation laws, pseudo-spherical surfaces.

1 Introduction

Sine-cosine method and the Wu's elimination method have been useful in the calculation of soliton solutions of certain nonlinear evolution equations (NLEEs) of physical significance [1-6] restricted to one space variable x and a time coordinate t.

Khater et al. [7, 8] used the notion of a differential equation (DE) for a function u(x,t) that describes a pseudo-spherical surface (pss), and they derived some Bäcklund transformations and conservation laws for NLEEs which are integrability condition of sl(2, R)-valued linear problems [9-18].

It is well-known [19-23] that a DE for a real-valued function u(x, t) or a differential system for a two-vector valued function u(x, t), is said to describe pss if it is the necessary and sufficient condition for the existence of smooth real functions f_ij, 1 < i < 3, 1 < j < 2, depending only on u and a finite number of its derivatives, such that the one-forms

satisfying the structure equations of a surface of constant Gaussian curvature K = -1,

The inverse scattering method (ISM) was introduced first for the Korteweg-de Vries equation (KdVE) [24]. Later it was extended by Zakharov and Shabat [25] to a 2 × 2 scattering problem for the nonlinear Schrödinger equation (NLSE) and that was subsequently generalized by Ablowitz, Kaup, Newell and Segur (AKNS) [26] to include a variety of NLEEs. Khater et al. [7] generalized the results of Konno and Wadati [27] by considering ν as a three component vector and Ω as a traceless 3 × 3 matrix one-form. The above definition of a DE is equivalent to saying that the DE for u is the integrability condition for the problem

where ν is a vector and the 3 × 3 matrix Ω (Ω_ij, i, j = 1,2,3) is traceless

and consists of a one-paramter (η), family of one-forms in the independent variables (x, t), the dependent variable u and its derivatives.

Khater et al. [7] introduced the inverse scattering problem (ISP):

The integrability condition for Eq. (3) is given by

or in component form

By various choices of the coefficients f_ij, it can be shown that the conditions (8)-(10) are equivalent to a large class of NLEEs. The procedure is clarified in the following example:

(a) The Ibragimov-Shabat equation (ISE)

where

The essence of the first step of the ISM is summarized as follows [7]. Find nine one-forms , i = 1,2,3, j = 1,2,3 consisting of independent and dependent variables and their derivatives, such that the NLEE is given by

It should be noted that the solution of these equations are of very special kind. In general, Eq. (14) gives three different equations, which cannot be satisfied simultaneously by one dependent variable u. It has been pointed out [7, 28] that Ω can be interpreted as a connection one-form for the principle SL(3, R) bundle on R³ and Θ as its curvature two form. The geometrical explanation of the SL(3, R) structure is given in section 2.

The main aim of this paper is to extend the fundamental equations of pseudo-spherical surfaces in reference [28] by considering ν as a three component vector and Ω as a traceless 3 × 3 matrix one-form. In the present paper we use sine-cosine method and the Wu's elimination method derived in [1-6] in the construction of exact soliton solution for ISE describing pss. We also obtain an infinite number of conserved charges by solving a set of coupled Riccati equations and apply the geometrical method to ISE which describe pss.

The paper is organized as follows. In section 2 the geometry of pss is described and the correspondence between the soliton equations and their families of pss are established. In section 3 the sine-cosine method and the Wu's elimination method is used to obtain travelling wave solutions for the ISE. Section 4 contains the derivation of an infinite number of conserved charges from the Riccati equations. Finally, we give some conclusions in section 5.

2 On equations describing pss

In this section we shall show that the fundamental equations of pss, can be written in the form of Eq. (14). Let us start with the general description of a three-dimensional Riemannian manifold S following reference [28]. An orthonormal basis is

with respect to the Riemannian metric introduced on the tangent plane T_p at each point p Є S. Then the structure equations for S read

where ω_i are one forms dual to {e_i} and , (i, j = 1,2,3) are called the connection one form. The integrability conditions are obtained by differentiating Eq. (16) and using (17). These conditions are

To sum up, a set of one forms (ω_i, ) satisfying Eqs. (18) and (19) describes a pss locally through the structure Eqs. (16) and (17). It is easy to show that Eqs. (18) and (19) can be written in the form of Eq. (14) by choosing

from Eqs. (14) and (20) we obtain this relations

Let M² be a two - dimensional differentiable manifold parametrized by coordinates x, t. We consider a metric on M² defined by ω₁, ω₂. The first two equations in (2) are the structure equations which determine the connection form ω₃, and the last equation in (2), the Gauss equation, determines that the Gaussian curvature of M² is – 1, i.e. M² is a pss. Moreover, an evolution equation must be satisfied for the existence of forms (1) satisfying (2). This justifies the definition of a DE which describes a pss that we considered in the introduction.

It has been known, for a long time, that the sine-Gordon equation describes a pss. KdVE and mKdVE, were also shown to describe such surfaces [26]. Here we show that ISE equation also describe pss as well. The latter equation proved to be of great importance in many physical applications [7-14, 29, 30]. The procedure is clarified in the following example: Let M² be a differentiable surface, parametrized by coordinates x, t.

(a) The ISE

Consider

Then M² is a pss iff u satisfies ISE (11).

3 Exact solution for ISE

Now we shall find a travelling wave solution u(x, t) for ISE

Wang [1] has found some exact solutions for compound KdV-Burgers equations by using the homogenous balance method. In this section we obtain travelling wave solution class for IES by using an improved sine-cosine method [4, 5] and Wu's elimination method [2]. The main idea of the algorithm is as follows. Given a partial differential equation (PDE) of the form

where f is a polynomial. By assuming travelling wave solutions of the form

where k, η are constant parameters to be determined, and c is an arbitrary constant, from the two Eqs. (24) and (25) we obtain an ordinary differential equation (ODE)

where Ø' = . According to the sine-cosine method [1-6], we suppose that Eq. (26) has the following formal travelling wave solution

and

where A₀,...,A_n and B₁,...,B_n are constants to be determined. Then we proceed as follows:

(i) Equating the highest order nonlinear term and highest order linear partial derivative in (26), yield the value of n.

(ii) Substituting Eqs. (27), (28) into (26), we obtain a polynomial equation involving cos ψ sin^j ψ, sin^jψ for j = 0, 1, 2,..., n, (with n being positive integer).

(iii) Setting the constant term and coefficients of sin ψ, cos ψ, sin ψ cos ψ, sin² ψ,..., in the equation obtained in (ii) to zero, we obtain a system of algebraic equations about the unknown numbers k, λ, A₀, A_i, B_i for i = 1, 2,..., n.

(iv) Using the Mathematica and the Wu's elimination methods, the algebraic equations in (iii) can be solved.

These yield the solitary wave solutions for the system (26). We remark that the above method yield solutions that includes terms sechρ or tanh ρ, as well as their combinations. There are different forms of those obtained by other methods, such as the homogenous balance method [1-6]. We assume formal solutions of the form

where k, η are constant parameters to be determined later, and c is an arbitrary constant. Substituting from (29) and (23), we obtain an ODE

(i) ,We suppose that equation (30) has the following formal solutions

and

(ii) From two Eqs. (31) and (32), we get

(iii) Setting the coefficients of sin^jψ cosⁱψ for i = 0, 1 and j = 1 to 6, we have the following set of over determined equations in the unknowns A₀, B₁, A₁, λ and k:

(iv) We now solve the above set of equations by using Mathematica and the Wu's elimination method, and obtain the following solution:

by integrating (32) and taking the integration constant equal zero, we obtain

Substituting (35) and (36) into (31), we obtain

where

4 Infinite number of conserved charges for ISE

It will be found that the Riccati form of ISM is sometimes useful. Introducing the variables [31-33]

which are related to the conserved charges α₃₃ (x, t, η) in the following way, from Eq. (5)

Eqs. (5) and (6) can be rewritten as

and

The set of coupled NLPDEs for Γ₁ and Γ₂ in (40)-(43) are known as Riccati equations. It is obvious from (39) that the solutions of Riccati equations eventually determine the conserved quantities. Now in order to solve (40)-(43), we assume a series solutions for Γ₁ and Γ₂ as

Substituting (44) into (40) and (41), the following recursion relations areobtained:

The infinite number of Hamiltonians (conserved quantities) may explicitly bedetermined in terms of smooth real functions f_ij and their derivatives by expanding α₃₃ (x, t, η) in the form

and thus comparing (48) with (39) and (44), H_l becomes

The explicit expressions of the first few order Hamiltonians are

where

such that the functions r, q, A, B and C satisfy the equations [7]

This section ends with the following example:

(a) The ISE

For any solution u of the ISE (11), we consider the functions

Eq. (49) implies that the first few order Hamiltonians are determined by the relation

Whenever u(x, t) is a solution (37) of the ISE. This Hamiltonians relations yields a sequence of conserved charges given by the coefficients of the seriesin η

5 Conclusions

In this paper, the fundamental equations of pseudo-spherical surfaces in reference [28] may be rewritten by considering ν as a three component vector and Ω as a traceless 3 × 3 matrix one-form [7]. The latter yields directly the curvature condition (Gaussian curvature equal to -1, corresponding to pseudo-spherical surfaces). This geometrical method is considered for the ISE.

We obtain travelling wave solution for ISE by using an improved sine-cosine method and Wu's elimination method. This geometrical method allows some further generalization of the work on conserved charges given by Wadati, Sanuki and Konno [34]. An infinite number of conserved charges for ISE mentioned above are derived in this way.

Received: 25/V/07. Accepted: 02/IV/08.

#727/07.

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Publication Dates

• Publication in this collection
  30 Oct 2008
• Date of issue
  2008

History

• Accepted
  02 Apr 2008
• Received
  25 May 2007