Exact Solution and Conservation Laws for Fifth-Order Korteweg-de Vries Equation

With the aid of Mathematica, new exact travelling wave solutions for fifth-order KdV equation are obtained by using the solitary wave ansatz method and the Wu elimination method. The derivation of conservation laws for a fifth-order KdV equation is considered.


Introduction
It is well-known that nonlinear complex physical phenomena are related to nonlinear partial differential equations (NLPDEs) which are involved in many fields from physics to biology, chemistry, mechanics, etc.As mathematical models of the phenomena, the investigation of exact solutions to the NLPDEs reveals to be very important for the understanding of these physical problems.Many mathematicians and physicists have well understood this importance when the importance of this so they decided to pay special attention to the development of sophisticated methods for constructing exact solutions to the NLPDEs.Thus, a number of powerful methods have been presented.
The notion of conservation laws is important in the study of nonlinear evolution equations (NLEEs) appearing in mathematical physics [19].The mathematical origin of conservation laws results from the formulation of familiar physical laws such as for mass, energy and momentum [20].As is known, the investigation of conservation laws of the Korteweg-de Vries (KdV) equation led to the discovery of a number of techniques to solve NLEEs [21], e.g., Miura transformation, Lax pair, inverse scattering technique and bi-Hamiltonian structures.
On the other hand, it is useful in the numerical integration of NLEEs [22] (e.g., to control numerical errors); particularly with regard to integrability and linearization, constants of motion, analysis of solutions, and numerical solution methods [23].Consider a dynamical system,

 
, , , , , , (1) where while the components V and of the conserved vector x t and derivatives of .The equality (2) is assumed to be satisfied for any solution of the corresponding system of equations, is called conserved density and G is called conserved flow.With the assumption that the function and its derivatives with respect to x go to zero sufficiently fast as is obtained to be a constant of motion.It has already been proved that a large number of NLEEs possess an infinite number of conservation laws such as the fifth-order KdV equation

Exact Solution for Fifth-Order KdV Equation
With In order to obtain the soliton solution of ( 4), the solitary wave ansatz is assumed as where A is the soliton amplitude, is the width of the soliton, is the soliton velocity and B is constant to be determined later, the unknown index n will be determined during the course of derivation of the solution of Equation ( 4).From Equation ( 5), I obtain Equation (6).
Open Access JAMP E. M. AL-ALI 51 (8) 0 , (9) 0. (10) Solving the above system by the aid of Wu elimination method [25], I obtain the two solutions 4 (11) Then the soliton solutions of the fifth order KdV equais given by tion isfy the structure equations of a pss, i.e. sat , se Gaussian curvature is constant, equal to −1.Moreover, the above definition is equivalent to saying that u where denotes exterior differentiation, d  is a col- um r and the 2 × 2 matrix n vecto from Equations ( 18) and ( 19), we obtain where S and T are two 2 × 2 null-trace matrices .
Here  is a pa eter, independent of ram x and t , while q and r are functions of x and t .Now which requires the vanishing of the two form where Chern and Tenenblat [27] obtain rectly from the structure equations (17).By suitably choosing and in (24), we shall obtain various fifth V uation wh Konno and Wadati introduced the function [28] ed Equation ( 24) di-, , r A B order Kd this function first appeared used and explained in the uations in [11,13], and see also the classical pap 27].Then Equati : geometric context of pseudo spherical eq ers by Sasaki [29] on ( 20) is reduced and Chern-Tenenblat [ to the Riccati equations Equations ( 28) and (29) imply that Open Access JAMP to both sides and using the expression   A q r   from (24), Equation (30) take rm s the fo let us show how an infinite number of conservatio result from these results.The Riccati equations for in the n laws Г x variable can be rearranged to take the form A similar pair of equations can be obtained for the t derivatives.Expand Г r into a power series in the verse of in- so that  are unknown at this point, however a recursion relation can be obtained for the n  by using (32), substituting (33) into the Г equation in (32), I find that Now equate powers of  on both sides of this expression to produce the set of recursions, Substituting (33) into (31), the followin conservation laws appears


This procedure generates an infinite number of conservation laws for the equation under examination.To obtain conservation laws using (37) in a particular example using this procedure, let us consider the fifth-order KdV Equation (4), for Equation ( 4) x nto (24), I obtain the fifth-order tting (38) into (37), it is found that ear wave that possesses remarkable stability properties.Typically, problems that admit soliton solutions are in the form o lution equations that describe how some variable or set of variables evolve in time from a given state.The equatio tions, partial difference equations, and integro-differential equations, as well as coupled ODEs of finite order.
In this paper, we considered the construction of exact so hysics and applied mathematics.Solitons are found in various areas of physics from hydrodynamics and plasma physics, nonlinear optics and solid state physics, to field theory and gravitation.NLEEs which describe soliton ph have a universal character.
A travelling wave of permanent form has already been met; this is the solitary wave solution of the NLEE itself.Such a wave is a special solution of the governing equatio The Soliton equations play a central role in the field of integrable systems and also play a fundamental role in several other areas of mathematics and physics.

Conclusions
A soliton is a localized pulse-like nonlin f evons may take a variety of forms, for example, PDEs, differential difference equa lutions to fifth-order KdV equation.We obtain travelling wave solutions for the above equation by using the solitary wave ansatz method with the aid of Mathematica.
The soliton phenomena and conservation laws of NLEEs represent an important and well established field of modern physics, mathematical p enomena n which does not change its shape and which propagates at constant speed.

Systematic Construction Method nfinitely Many Conservation L ifth-Order KdV Equation
n 3.