Exact Solutions for a New Nonlinear KdV-Like Wave Equation Using Simplest Equation Method and Its Variants

The simplest equationmethod presents wide applicability to the handling of nonlinear wave equations. In this paper, we focus on the exact solution of a newnonlinearKdV-likewave equation bymeans of the simplest equationmethod, themodified simplest equation method and, the extended simplest equation method. The KdV-like wave equation was derived for solitary waves propagating on an interface (liquid-air) with wave motion induced by a harmonic forcing which is more appropriate for the study of thin filmmass transfer. Thus finding the exact solutions of this equation is of great importance and interest. By these three methods, many new exact solutions of this equation are obtained.

Partially, the simplest equation method is a very powerful mathematical technique to seek more new solutions of NLEEs that can be expressed as polynomial in an elementary function which satisfies a subequation like Riccati equation, auxiliary ordinary equation, elliptic equation, and generalized Riccati equation.It has been developed by Fan [20] and Kudryashov [21] and the method optimizes the use of the auxiliary equation and effectively avoids producing some duplication solutions.There are many applications and generalizations of the method [22][23][24][25].
Recently, Bilige introduced a method called the extended simplest equation method as an extension of the simplest equation method, to look for the exact traveling wave solutions of nonlinear evolution equations (NLEEs) [26,27].In this method, a second order linear ordinary differential equation (ODE) is taken as the auxiliary equation.This method can construct different forms of exact traveling wave solutions which cannot be obtained by using the tanh-function method, -expansion method, and the Expfunction method.
The nonlinear shallow water surface waves satisfy the Korteweg-de Vries (KdV) equation: This equation is only valid for long waves.Solitary waves in film flows were studied by [18,19].These flows also show a transition to turbulence.This process is better understood, if the dynamics of nonlinear waves are traceable.In [28] Rees and Zimmerman derived a new nonlinear KdV-like evolution equation for surface solitary waves propagating on a liquid-air interface where the wave motion is induced by harmonic forcing: The nonlinearity term   in (1) does not appear in (2), and the reconstruction of this term from ( 2  −   ) is not possible by order approximation equivalences; also the nonlinearity term in (2) is more steep.
It was observed by Sohail et al. [29] that wave data obtained from an online measurement technique satisfies this evolution equation, when the wave length is just a few multiples of the fluid depth.In [30], the integrability of this new nonlinear partial differential equation was discussed with a focus on the Painlevé property, the compatibility condition, and the Bäcklund transformation.
The exact traveling wave solutions of (2) are not reported in related references.We have tried many other algebraic methods to seek exact solutions of (2).However, we find that they are either not efficient or invalid.In this work, we apply the simplest equation method, the modified simplest equation method, and the extended simplest equation method for obtaining exact traveling solutions of (2).From the results we see that these methods are more effective.
The organization of the paper is as follows.In Section 2, a brief description of three simplest equation methods for finding traveling wave solutions of nonlinear equations is given.In Section 3, we will study (2) by these methods.Finally conclusions are given in Section 4.
Step 2. Suppose that the solution of ODE (4) can be written as follows: where () are the functions that satisfy some ordinary differential equations.The simplest equation has two properties: first it is of lesser order than (4) and second we know the general solution of the simplest equation. is a positive integer that can be determined by balancing procedure, and   ( = 0, 1, 2, . ..) are parameters to be determined.
In this paper, we will use as simplest equation the equations of Bernoulli and Riccati which are well known nonlinear ordinary differential equations and their solutions can be expressed by elementary functions.
For the Bernoulli equation equation ( 6) admits the following exact solutions: for the case  > 0,  < 0 and for the case  < 0,  > 0, where  0 is a constant.For the Riccati equation Equation ( 9) admits the following exact solutions: when  < 0 and when  > 0, where  0 is a constant.
Step 3. Substituting ( 5) along with (6) (or ( 9)) into ( 4), then the left-hand side of ( 4) is converted into a polynomial in (); equating each coefficient of the polynomial to zero yields a set of algebraic equations for   , , .
Step 4. Solving the algebraic equations obtained in Step 3 and substituting the results into (5), we obtain the exact traveling wave solutions for (3).

The Modified Simplest Equation Method.
In the modified version, one makes an ansatz for the solution () as where   ( = 0, 1, 2, . . ., ) are arbitrary constants to be determined, such that   ̸ = 0 and  = () is an unspecified function to be determined afterward.
Substitute ( 12) into ( 4) and then we account the function .As a result of this substitution, we get a polynomial of   / and its derivatives.In this polynomial, we equate the coefficients of the same power of  − to zero, where  > 0. This procedure yields a system of equations which can be solved to find   ( = 0, 1, 2, . . ., ), , and   .Then the substitution of the results into (12) completes the determination of exact solutions of (3).

Using Simplest Equation Method. Suppose that (17) owns the solutions in the form
For the Bernoulli equation, substituting ( 18) into ( 17) and making use of the Bernoulli equation ( 6) and then setting all the coefficients of   () of the resulting system to zero, we obtain an algebraic system of equations in terms of   ( = 0, 1), , , , and .Solving this system of algebraic equations, we obtain Therefore, using solutions ( 7) and ( 8) of ( 6) and ansatz (18), we obtain the following exact solution of (2): for the case  > 0,  < 0 and for the case  < 0,  > 0, where  =  − ((2 2  2 −  2 )/ 2 2 ).
For the Riccati equation, substituting (18) into (17) and making use of the Riccati equation ( 9) and then setting all the coefficients of   () of the resulting system to zero, we obtain an algebraic system of equations in terms of   ( = 0, 1), , , , and .Solving this system of algebraic equations, we obtain Therefore, using solutions ( 10) and ( 11) of ( 9) and ansatz (18), we obtain the following exact solution of (2): for the case  < 0, and for the case  > 0, where  =  + ((8 2  +  2 )/2 2 ).
By substituting (34) into (17), using the second order linear ODE expressions ( 14) and ( 16), collecting all terms with the same order of 1/  and (1/  )(  /) together, the lefthand side of ( 17) is converted into another polynomial in 1/  and (1/  )(  /).Equating each coefficient of these different power terms to zero yields a set of algebraic equations for  0 ,  1 ,  1 , , , , and ].Solving these equations, we obtain the following results.
function, trigonometric function, and rational functions are obtained.The correctness of all the solutions is verified by substituting them into original equation (2).It is easy to see that the simplest equation methods are direct, concise, effective, and reliable, which can be used for many other NLEEs in mathematical physics.