The travelling wave solutions of nonlinear evolution equations with both a dissipative term and a positive integer power term

1.   Introduction

It is well known that nonlinear waves are governed by nonlinear evolution equations and some problems related to nonlinear equations are very difficult to handle in a general way. So far, there is no unified method that can deal with all types of nonlinear evolution equations. However, some special problems have been solved from time to time by methods suited to the special problems. In the present paper, we shall concentrate on a class of nonlinear evolution equations with both a dissipative term and a positive integer power term of dependent variable. These equations are all generalized form of important model equations in mathematical physics. Several examples of this class of nonlinear equations are listed below:

The generalized KdV-Burgers equation (GKdV-Burgers)

The generalized KP-Burgers equation (GKP-Burgers)

The generalized Zakharov-Kuznetsov-Burgers equation (GZK-Burgers)

The generalized Boussinesq equation (GBoussinesq)

General (m + 1)-dimension Klein-Gordon equation (GKlein-Gordon)

where m is positive integer, and so on.

In particular, when $n = 1$ the Eqs (1)–(5) above become the famous model equations in mathematical physics, to be more precise, Eqs (1)–(5) become KdV-Burgers equation ^[1,2], KP-Burgers equation ^[3], ZKB equation ^[4], Boussinesq equation ^[5] and dissipative Klein-Gordon equation ^[6], respectively. Each of these model equations mentioned has been investigated extensively by many authors and methods during the past four decades or so. Especially, for GKdV-Burgers equation, GZK-Burgers equation, GBoussinesq equation, and GKlein-Gordon equation, some literatures ^[7,8,9,10] have obtained different types of soliton solutions by using HPM and tanh method etc. Nevertheless, as for the Eqs (1)–(5), which contains a dissipative term and a positive integer power term, to our knowledge, there are less systemic methods to solve them. Since Eqs (1)–(5) are all the generalized form of the important model equations, therefore the investigation of them is of great significance both mathematically and physically

In recent years, the development of mathematical physics furtherly provides us with more detailed methods to seek exact solutions of NLDEs, for example, Hirota bilinear method ^[11], $(\frac{{G'}}{G})$-expansion method ^[12,13,14], exp-function method ^[15], complex method ^{[16,17,18,19]}, exp(−ψ(ξ))-expansion method ^[20,21], the sub-ODE method ^[22] and so on. As one of the efficient computing techniques, there is the spectral collocation method for solving equations with a fractional integro-differential ^[23]. In addition, Ahmad El-Ajou also has suggested a new method that relies on a new fractional expansion in the Laplace transform space and residual power series method to construct exact solitary solutions to the nonlinear time-fractional dispersive PDEs in the literature ^[24]. These developments have helped to open up various new mathematical branches and have led to immense enrichment therein.

The main goal of this paper is to obtain travelling wave solutions of Eqs (1)–(5). Obviously the crucial problem is how to solve the travelling wave reduction ODEs of Eqs (1)–(5), since reduction ODEs contain an undetermined coefficient and a positive integer power term of dependent variable. In Section 2, the formula of solution as well as the undetermined coefficient of a nonlinear ODE with an undetermined coefficient and a positive integer power term of dependent variable has been obtained by using transformation of dependent variable and $(\frac{{G'}}{G})$-expansion method, we use a theorem to describe such a result briefly. In subsequent sections, from Section 3 to Section 7, we shall apply the theorem obtained in Section 2 to KdV-Burgers, GKP-Burgers, GZK-Burgers, GBoussinesq and GKlein-Gordon, respectively, to obtain the travelling wave solutions of these equations. In Section 8, the technique used in the paper is summarized briefly.

2.   A nonlinear ODE with an undetermined coefficient and a positive integer power term of dependent variable

In this section, we considered the following ODE

where $n \geqslant 1$, a positive integer; $a, b$ and $V$ are constants, $b > 0, \; V$ to be determined later. Our aim is to find out $V$such that Eq (6) has a positive solution. We shall demonstrate the following:

Theorem. The ODE (6) admits a positive solution

Provided that

Proof. Making transformation of dependent variable

Equation (6) is converted into the nonlinear ODE for $u = u(\xi)$,

We shall use the $(\frac{{G'}}{G})$-expansion method ^[12] to solve ODE (10).

First of all, considering the homogeneous balance between $uu'$ (or ${u'^2}$) and ${u^3}$$(2m + 2 = 3m \Rightarrow m = 2)$, we can suppose that the solution of Eq (10) is of the form

where $G = G(\xi)$ satisfies

$A$ and $\lambda$ are constants to be determined.

Second, from (11) and using (12), it is derived that

Substituting (13) into the left hand side of Eq (6), collecting all terms with ${\left({\frac{{G'}}{G}} \right)^i}(i = 4, 5, 6)$ together and setting the coefficient of ${\left({\frac{{G'}}{G}} \right)^i}(i = 4, 5, 6)$ to zero, yields a set of algebraic equations for $A, \; \lambda$ and $V$ as follows

Solving (14), yields

Third, when $\lambda = \frac{{an}}{{n + 4}}$, Eq (12) admits a solution

Therefore

substituting $A = \frac{{2(n + 2)}}{{b{n^2}}}$ and $\left({\frac{{G'}}{G}} \right)$ in (15) into (9), we obtain the solution of nonlinear ODE (10)

provided

Last of all, substituting (16) into (9), we have the solution of Eq (6), which is expressed by (7), provided $V = \frac{{2(n + 2)}}{{{{(n + 4)}^2}}}{a^2}.$

The discussion above completes the proof of the theorem.

3.   Generalized KdV-Burgers equation

Introducing travelling wave variables

Equation (1) is converted into an ODE for $u = u(\xi)$

Integrating it with respect to $\xi$ once, and taking constant of integration to zero, yields

Comparing Eq (18) with Eq (6) in Section 2, it is found that Eq (18) belongs to the same type ODE as Eq (6). According to the theorem in Section 2, the travelling wave solution of Eq (1) is given by

provided $V = \frac{{2(n + 2)}}{{{{(n + 4)}^2}}}{a^2}$, in view of (17)

In particular, when $n = 1$, (19) becomes

which is the solution of KdV-Burgers equation

When $n = 2$, (19) becomes

which is the solution of mKdV-Burgers equation

4.   Generalized KP-Burgers equation

Introducing travelling wave variables

Equation (2) is converted into an ODE for $u = u(\xi),$

Integrating it with respect to$\xi$ twice, and taking the constants of integration to zero, yields

Comparing Eq (25) with Eq (6) in Section 2, it is found that Eq (25) belongs to the same type ODE as Eq (6), according to the theorem in Section 2,

The travelling solution of general KP-Burgers equation is given by

In particular, when $n = 1$, (27) becomes

which is the solution of KP-Burgers equation

When $n = 2$, (27) becomes

which is the solution of mKP-Burgers equation

5.   Generalized ZK-Burgers equation

Introducing travelling wave variables

where ${l^2} + {m^2} + {p^2} = 1$. Substituting (32) into (3), integrating with respect to $\xi$, and taking the constants of integration to zero, yields

Suppose that $\alpha < 0, \; (\beta - \gamma){l^2} + \gamma > 0$, Eq (33) can be rewritten as

Equation (34) belongs to the same type ODE as Eq (6). According to the theorem in Section 2,

which implies

and the solution of Eq (3) is given by

where $\xi = lx + my + pz + \frac{{2(n + 2){a^2}{l^3}}}{{{{(n + 4)}^2}[(\beta - \gamma){l^2} + \gamma]}}t, \; {\kern 1pt} {l^2} + {m^2} + {p^2} = 1, \, \; {\xi _0} = const.$

In particular, when $n = 1$, (35) becomes

where $\xi = lx + my + pz + \frac{6}{{25}} \cdot \frac{{{a^2}{l^3}}}{{(\beta - \gamma){l^2} + \gamma }}t, \; \; {l^2} + {m^2} + {p^2} = 1, \; \, {\xi _0} = const.$ Then Eq (36) is a bounded solution of ZK-Burgers equation

6.   Generalized Boussinesq equation

Introducing travelling wave variables

thus the Eq (4) can be reduced to ODE for $u = u(\xi),$

Integrating it with respect to $\xi$ twice, and taking the constants of integration to zero, yields

where Eqs (39) and (6) belong to the same type. According to the theorem in Section 2, ${V^2} - {c^2} = \frac{{2(n + 2){a^2}{V^2}}}{{{{(n + 4)}^2}}},$

which implies $V = \pm \frac{c}{{\sqrt {1 - \frac{{2(n + 2){a^2}}}{{{{(n + 4)}^2}}}} }},$and the travelling wave solution of the general Boussinesq equation (4) is given by

where variables $A = \frac{{2(n + 2){a^2}{c^2}}}{{[{{(n + 4)}^2}-2(n + 2){a^2}]b}}, \, \, B = \mp \frac{{acn}}{{\sqrt {{{(n + 4)}^2} - 2(n + 2){a^2}} }}, \,$$\xi = x \mp \frac{c}{{\sqrt {1 - \frac{{2(n + 2){a^2}}}{{{{(n + 4)}^2}}}} }}t$, and ${\xi _0} = const.$

7.   Generalized (m+1)-dimension Klein-Gordon equation

Introducing travelling wave variables

the Eq (5) can be reduced to the ODE for $u = u(\xi),$

Let ${V^2} < {c^2}, \, b > 0, \, d < 0$, the ODE (42) can be rewritten as

where Eqs (43) and (6) belong to the same type. According to the theorem in Section 2,

which implies

and the travelling wave solution of the general Klein-Gordon equation (5) is given by

In particular, when $n = 1$, (44) becomes

Then (45) is a travelling wave solution of Klein-Gordon equation

For ${V^2} > {c^2}, \, b < 0, \, d > 0$, there are some similar conclusions, the same travelling wave solution of Eq (5) is given by

8.   Conclusions

In this paper, the nonlinear ODE equation (6) with an undetermined coefficient and a positive integer power term of dependent variable were introduced and solved. The formula of solution (7) with (8) to the nonlinear ODE (6) have played an important role for finding travelling wave solutions of a class of complicated nonlinear evolution equations with a dissipative term and a positive integer power term of dependent variable. We have successfully illustrated our technique in detail with five important model equations. The results show that the present method performs extremely well in terms of directness efficiency, simplicity and reliability to deal with various differential equations in the applied sciences, especially containing a dissipative term and a positive integer power term. It is worthy of note that the formula of solution (7) with (8) to the nonlinear ODE (6) can be also applied to other similar nonlinear evolution equations in complexity and nonlinear science to obtain their travelling wave solutions. It is expected that other extended and improved $(\frac{{G'}}{G})$-expansion methods will be more applied to solve nonlinear differential equations involving a dissipative term and a positive integer power term of dependent variable.

Acknowledgments

This work is supported in part the National Natural Science Foundation of China (Grant no. 12101193) and the General Project of Education Science Planning in Henan Province (Grant no. 2020YB0075)

Conflict of interest

The authors declare no conflict of interest.