Symmetry Reductions and Exact Solutions to the Kudryashov–Sinelshchikov Equation

1 Introduction

In 2010, Kudryashov and Sinelshchikov presented the following equation [1]:

where u is a density and which are the terms which model heat transfer and viscosity, α, β are real parameters. Equation (1) is called Kudryashov–Sinelshchikov equation and is for describing the pressure waves in a mixture liquid and gas bubbles taking into consideration the viscosity of liquid and the heat transfer, it is a generalisation of the KdV and the BKdV equation and similar but not identical to the Camassa–Holm equation. Equation (1) was studied by many researchers in various methods. Ryabov found four families of solitary of (1) when β=−3 or β=−4 by using a modification of the truncated expansion method [2]; Randrüüt obtained a kind of new periodic wave solution which is called meandering solution in a more straightforward manner [3]; Li et al. investigated bifurcations of travelling wave solutions of the Kudryashov–Sinelshchikov equation and proved the existence of the peakon, solitary waves, and smooth and nonsmooth periodic waves [4]. He et al. [5] investigated the periodic loop solutions and their limit forms. Different kinds of other exact solutions are obtained in [6], [7], [8]. In [9], Chen et al. studied travelling wave solutions of (1) in the special case β=2. Here, we will investigate the travelling wave solutions of (1) for the other special cases.

Our aim in this article is to perform the Kudryashov–Sinelshchikov equation with the help of the compatibility method ([10], [11]) and the integral bifurcation method [9]. The remainder of this article is organized as follows: in Section 2, two types of nonclassical symmetries of the Kudryashov–Sinelshchikov equation (1) are presented. In Section 3, different types of symmetry reductions of the Kudryashov–Sinelshchikov equation (1) are obtained. In Section 4, some new exact explicit solutions that include travelling wave solutions and nontravelling wave solutions of the Kudryashov–Sinelshchikov equation (1) are derived from the reduced equations. In Section 5, some new exact travelling wave solutions of the Kudryashov–Sinelshchikov equation (1) are derived by integral bifurcation method. The last section is a short summary and discussion.

2 Symmetries of the Kudryashov–Sinelshchikov Equation

The main purpose of the compatibility method is to seek the nonclassical symmetry of (1) in the form

where a(x, t), b(x, t), and γ(x, t) are functions to be determined later by the compatibility of (1) and (2). Firstly, we can obtain the highest-order derivative term u_xxx of (1) by substituting (2) into (1) as follows:

From the equality of u_t in (1), we can get at the derivatives of (1) and (2) with respect to t. Then in terms of the equality of u_tt of (1) and (2), one can get

Substituting (2) and (3) into the expansion of (4) yields a polynomial of u and its derivatives. Setting all the coefficients of this polynomial to zero yields a set of differential equations with regard to unknown functions a, b, γ as following

bx(β+4)+3axx=0,γxxx+γt+γ2−3axγ=0,(b+γ)(β+1)=0,−γx(β+1)+3axx+3bx=0,(β+1)(axx+2bx)=0,4bxx+axxx+2αax+βbxx=0,−6axx−bx(β+7)+γx(β+1)=0,γxxx−αγx+3axb+αbx−2bxxx−bt+b2=0,−2γxxx+bxxx−γt+αγx+3ax(γ−b)+bt+2bγ=0,axxx+3bxx−γxx(β+1)+at−3aax+γ(a+α)=0,−2axxx−bxx(β+7)+γxx(β+1)−at+ax(3a−2α) +b(a+α)=0.

Solving the above-mentioned system of differential equations, we can get the following results

Case 1 α=0, β=β, a=−C1x+3C3−x3C1t+3C2, b=1C1t+C2, γ=−1C1t+C2,

and the corresponding nonclassical symmetry of (1) expressed by

where β, C₁, C₂ and C₃ are arbitrary constants.

Case 2 α=α,  β=β,  a=−α+C2t+C1,  b=−1t+C1, γ=1t+C1.

and the corresponding nonclassical symmetry of (1) expressed by

where α, β, C₁, and C₂ are arbitrary constants.

3 Symmetry Reductions for the Kudryashov–Sinelshchikov Equation

In order to obtain the invariant transformation, we write the characteristic equation in the form

4 Exact Explicit Solutions to the Kudryashov–Sinelshchikov Equation

For Case 3.1.1, according to simple calculation, obviously we obtain the rational function solution of the reduced equation (10) as

where C is an arbitrary constant. Therefore, when α=0, we have the exact solution of (1) as follows

For Case 3.1.2, we will give the exact analytic solutions to the reduced equation (12) using the power series method ([12], [13]). Now, we seek a solution of (12) in a power series of the following form

Substituting (24) into (12), we have

From (25), comparing coefficients, we obtain (for n=0)

with arbitrary chosen A₀≠0, A₁ and A₂, we have

and (for n=1)

Generally, for n≥1, in view of (25), we can get all the coefficients A_n(n≥0) of the power series (24) as follows

This implies that for (12), there exists a power series solutions (24). In addition, it is easy to prove that the convergence of the power series (24) with the coefficients given by (25–27), we can see [12], [13] and references cited therein. Thus, this power series solution is an exact analytic solution.

Hence, the power series solution of (12) can be written as

f(ξ)=A0+A1ξ+A2ξ2+A3ξ3+∑n = 1∞An + 3ξn + 3=A0+A1ξ+A2ξ2+A0−21+β)C2A1A26C2A0ξ3+ ⋯

Thus, we can obtain that the power series solution of (1) as

here ξ=xe− t3C2, and C₂ is an arbitrary constant.

Remark 1. For Case 3.1.3 and Case 3.1.4, we can obtain the exact analytic solution of the reduced equations (14) and (16) by using the power series method similarly, here we do not list them for simplicity.

For Case 3.2.1, suppose that the solution of the reduced equation (18) is in the form

and ϕ satisfies the equation

where ϕ=ϕ(ξ) and a₀, a₁, a₂, A, B, C are constants to be determined later. Substituting (31) and (32) into (18), and equating the coefficients of like powers of ϕⁱ, (i=0, 1, 2, …) to zero, which give rise to the system of algebraic equations to a₀, a₁, a₂, A, B, C. With the aid of Maple, we obtain the following solutions

From (32) and (33), it is easy to see that ϕ(ξ)=Aξ+ C₀, and C₀ is an arbitrary constant. The solution of the reduced equation (18) is obviously obtained as

and the solution of (1) is expressed as

where ξ=x−αt, and a₀, A, α, C₀ are arbitrary constants.

For Case 3.2.2, suppose that the solution of the reduced equation (20) is in the form of (31) and ϕ satisfies (32) and a₀, a₁, a₂, A, B, C are constants to be determined later. Substituting (31) and (32) into (20). Similar to Case 3.2.1, we obtain the following solutions

From (32) and (36), it is easy to see that ϕ′(ξ)=A+Bϕ, solving this first-order ordinary differential equation, we have ϕ(ξ)=C0eBξ−BA and C₀ is an arbitrary constant. The solution of the reduced equation (20) is obviously obtained as

and the solution of (1) is expressed as

where ξ=x−αt−C₂ ln t, and a₀, A, B, α, and C₀ are arbitrary constants.

5 Exact Travelling Wave Solutions of (1) by Integral Bifurcation Method

In this section, we consider travelling wave solutions of (1) with the special case β=1 by using the integral bifurcation method.

Substituting u(x, t)=1−f(ξ) with ξ=x−ct into (1) and integrating it once, we have

where “′” is the derivative with respect to ξ and g an integral constant.

Letting y=dfdξ, we get the following planar system

Using the transformation dξ=2fdτ, it carries (41) into the Hamiltonian system

For β≠−2, −1, 0, system (40) and (41) have the same first integral

Taking β=1 and h=0, (42) can be rewritten as follow:

From (43) and the first equation of (41), we have

Denote that Δ=13(13α−4g)α, ϵ=±1, with the aid of Table 1 of [14], the following exact solutions of (44) are obtained.

Using (45−57), transformations dξ=2fdτ and u(x, t)=1−f(ξ) with ξ=x−ct, we can obtain the exact travelling wave solutions of (1) as follows:

Remark 2. For the cases β=−4, −3, we can obtain the travelling wave solutions of (1) similarly, we omit them here.

6 Conclusions

We have discussed the symmetry reductions and exact explicit solutions of the Kudryashov–Sinelshchikov equation in this article. It is shown that (1) can be reduced to constant coefficients partial differential equations (10), (18), (20) and variable coefficients partial differential equations (12), (14), and (16) by the compatibility method. Furthermore, some new exact explicit solutions of (1) can be constructed by solving the reduced nonlinear partial differential equations, including travelling wave solutions and nontravelling wave solutions. At last, we also obtain some new exact explicit travelling wave solutions by using the integral bifurcation method.