Travelling Wave Solutions to the Benney-Luke and the Higher-Order Improved Boussinesq Equations of Sobolev Type Ömer

and Applied Analysis 3 Based on this one may use the following changes:


Introduction
The term "Sobolev equation" is used in the Russian literature to refer to any equation with spatial derivatives on the highest order time derivative 1 .In other words, they are characterized by having mixed time and space derivatives appearing in the highest-order terms of the equation and were studied by Sobolev 2 .Equations of Sobolev type describe many physical phenomena 3-7 .In recent years considerable attention has been paid to the study of equations of Sobolev type.For more details we refer the reader to 8 and references therein.
The Benney-Luke equation is as follows: where a and b are positive numbers, such that a − b σ − 1/3 is a Sobolev type equation and studied for a very long time.The dimensionless parameter σ is named the Bond number, which captures the effects of surface tension and gravity force and is a formally valid approximation for describing two-way water wave propagation in the presence of surface tension 9 .In 10 Pego and Quintero studied the propagation of long water waves with small amplitude.They showed that in the presence of a surface tension, the propagation of such waves is governed by 1.Nevertheless, several types of the improved Boussinesq equation were investigated by many researchers and found exact solutions by using exp-function method 20 , modified extended tanh-function method 21 , sine-cosine method 22 , improved G'/G-expansion method 22 , the standard tanh and the extended tanh method 23 , and so forth.
The tanh-coth is a powerful and reliable technique for finding exact travelling wave solutions for nonlinear equations.This method has been used extensively, and it was subjected by some modifications using the Riccati equation.The main features of the tanhcoth method will be outlined in the subsequent section, and this method will be applied to the the Benney-Luke and the Higher-order improved Boussinesq equations.The main purpose of this work is to obtain travelling wave solutions of the above-mentioned equations and to show that the tanh-coth method can be easily applied to Sobolev type equations.Throughout the work, Maple is used to deal with the tedious algebraic operations.

Outline of the Tanh-Coth Method
Wazwaz has summarized the tanh method in the following manner.
i First consider a general form of nonlinear equation ii To find the traveling wave solution of 2.1 , the wave variable ξ x − V t is introduced, so that u x, t U μξ .

2.2
Based on this one may use the following changes: and so on for other derivatives.Using 2.3 changes the PDE 2.1 to an ODE as follows: iii If all terms of the resulting ODE contain derivatives in ξ, then by integrating this equation and by considering the constant of integration to be zero, one obtains a simplified ODE.
iv A new independent variable is introduced that leads to the change of derivatives: where other derivatives can be derived in a similar manner.
v The ansatz of the form is introduced where M is a positive integer, in most cases, that will be determined.
If M is not an integer, then a transformation formula is used to overcome this difficulty.Substituting 2.6 and 2.7 into the ODE, 2.4 yields an equation in powers of Y .
vi To determine the parameter M, the linear terms of highest order in the resulting equation with the highest order nonlinear terms are balanced.With M determined, one collects the all coefficients of powers of Y in the resulting equation where these coefficients have to vanish.This will give a system of algebraic equations involving the a k and b k , k 0, . . ., M , V , and μ.Having determined these parameters, knowing that M is a positive integer in most cases, and using 2.7 one obtains an analytic solution in a closed form.

The Benney-Luke Equation
The Benney-Luke equation can be written as where a and b are positive numbers such that a−b σ −1/3 σ is named the Bond number .In order to solve 3.1 by the tanh-coth method, we use the wave transformation u x, t U μξ with wave variable ξ x − V t; 3.1 takes on the form of an ordinary differential equation as follows: Balancing the order of U with the order of U U in 3.2 we find M 1.Using the assumptions of the tanh-coth method 2.5 -2.7 gives the solution in the form Substituting 3.3 into 3.2 , we obtain a system of algebraic equations for a 0 , a 1 , b 1 , and V in the following form:

3.4
From the output of the Maple packages we find three sets of solutions: where μ is left as a free parameter.The travelling wave solutions are as follows: 3.6

The Higher-Order Improved Boussinesq Equation
We consider the Higher-order improved Boussinesq equation as follows: where α and β are arbitrary non zero real constants.
Using the wave transformation u x, t U μξ with wave variable ξ x − V t then by integrating this equation and considering the constant of integration to be zero, we obtain the ODE as follows: Balancing the first term with the last term in 4.2 we find M 4. Using the assumptions of the tanh-coth method 2.5 -2.7 gives the solution in the form  Using Maple gives six sets of solutions: The travelling wave solutions are as follows:

Y 1 : 2b 3 b 4 − 4 −
360V 2 b 3 αμ 4 0, Y 0 : b 2 840V 2 b 4 αμ 4 0. 4.4 1 , originally derived by Benney and Luke 11 .There are many studies concerning with this equation.Amongst them the stability analysis 9, 12 , Cauchy problem 13-15 , existence and analyticity of solutions 16 , and travelling wave solutions 17 can be mentioned.In 18 , Schneider and Wayne showed that in the longwave limit the water wave problem without surface tension can be described approximately by two decoupled KdV equations.They considered a class of Boussinesq equation which models the water wave problem with surface tension as follows: −u xxxxtt u xxtt − u tt u xx μu xxxx u 2 where x, t, μ ∈ R and u x, t ∈ R. Duruk et al. investigated the well posedness of the Cauchy problem −βu xxxxtt u xxtt − u tt u xx g u xx 0, x ∈ R, t > 0 1.3 and showed that under certain conditions the Cauchy problem is globally well posed 19 .