Exact Solutions of the Kudryashov–sinelshchikov Equation and Nonlinear Telegraph Equation via the First Integral Method

In this article we find the exact traveling wave solutions of the Kudryashov–Sinelshchikov equation and nonlinear telegraph equation by using the first integral method. This method is based on the theory of commutative algebra. This method can be applied to nonintegrable equations as well as to integrable ones.


Introduction
Nonlinear evolution equations are widely used to describe complex phenomena in various sciences such as fluid physics, condensed matter, biophysics, plasma physics, nonlinear optics, quantum field theory and particle physics, etc.In recent years, various powerful methods have been presented for finding exact solutions of the nonlinear evolution equations in mathematical physics, such as, tanh method [1][2][3], multiple exp-function method [4], transformed rational function method [5], Hirotas direct method [6,7], extended tanh-function method [8] and so on.
The first integral method, which is based on the ring theory of commutative algebra, was first proposed by Feng [9].This method was further developed by the same author in [10][11][12][13].
The aim of this work is to find new exact solutions of Kudryashov-Sinelshchikov equation by using the first integral method.
The rest of this paper is organized as follows.In Section 2, we give the description of the first integral method.In Sections 3 and 4, we apply this method to nonlinear telegraph equation and Kudryashov-Sinelshchikov equation.
Using traveling wave u(x, t) = U (ξ), ξ = x − ct, from Eq. ( 1), we obtain the ordinary differential equation (ODE): where prime denotes the derivative with respect to the same variable ξ.
Suppose that the solution of ODE (2) can be written as follows: Next, we introduce a new independent variable which leads a system of nonlinear ordinary differential equations By the qualitative theory of ordinary differential equations [14], if we can find the integrals to Eq. ( 4) under the same conditions, then the general solutions to Eq. ( 4) can be solved directly.However, in general, it is really difficult for us to realize this even for one first integral, because for a given plane autonomous system, there is no systematic theory that can tell us how to find its first integrals, nor is there a logical way for telling us what these first integrals are.We will apply the division theorem to obtain one first integral to Eq. ( 4), which reduces Eq. ( 2) to a first order integrable ordinary differential equation.An exact solution to Eq. ( 1) is then obtained by solving this equation.Now, let us recall the division theorem: Division theorem.Suppose that P (w, z) and Q(w, z) are polynomials in C[w, z] and

Nonlinear telegraph equation
Let us consider the nonlinear telegraph equation [15] Using traveling wave u(x, t) = U (ξ), ξ = x − ct carries (5) into an ODE as follows: where prime denotes the derivative with respect to the same variable ξ.Using ( 3) and ( 4), we get Now, we apply the above division theorem to look for the first integral of system (7).
Similarly, in the case of (24), from ( 8), we obtain and then the exact solution of nonlinear telegraph equation can be written as where ξ 0 is an arbitrary constant.
Comparing our results with Wang's results [15] then it can be seen that the results are same.
Eq. ( 31) describes the pressure waves in the liquid with gas bubbles taking into account the heat transfer and viscosity.More details are presented [18,19].Now, applying the transformation u(x, t) = u(ξ), ξ = x − ct to Eq. ( 31) and integrating the resultant equation once, we get where integration constant is taken to zero and the primes denote derivative with respect to ξ.Using ( 3) and ( 4), we get Now, we make the transformation dξ = (1 − εX) dη in Eq. to avoid the singular line εX = 1 temporarily.Thus, system (33) becomes In this example, we assume that m = 1 in Eq. ( 8).From now on, we shall omit some details because the procedure is the same.Then, by equating the coefficients of Y i (i = 2, 1, 0) on both sides of Eq. ( 9), we have As a 1 (X) and h(X) are polynomials, from Eq. (35), we deduce that h(X) = k/2 and a 1 (X) must be a constant.For simplicity, we can take a 1 (X) = 1.Then Eq. (36) indicates that deg(g(X)) ≤ deg(a 0 (X)).Thus, from Eq. (37), we conclude that deg Substituting g(X) and a 0 (X) into Eq.(37) and setting the coefficients of X i (i = 2, 1, 0) to be zero, we derive a system of nonlinear algebraic equations B 0 , B 1 and c.Solving the resultant system simultaneously, we get the solution set (38) Using the condition (38) in (8), we obtain where ξ 0 is an arbitrary constant.
Comparing our results with Kudryashov's results [16] and Ryabov's results [17] then it can be seen that the results are new.