Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 641308, 14 pages

http://dx.doi.org/10.1155/2015/641308

## Frobenius’ Idea Together with Integral Bifurcation Method for Investigating Exact Solutions to a Water Wave Model of the Generalized mKdV Equation

College of Mathematics, Chongqing Normal University, Chongqing 401331, China

Received 19 May 2014; Accepted 30 July 2014

Academic Editor: Salvatore Alfonzetti

Copyright © 2015 Weiguo Rui. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

By using Frobenius’ idea together with integral bifurcation method, we study a third order nonlinear equation of generalization form of the modified KdV equation, which is an important water wave model. Some exact traveling wave solutions such as smooth solitary wave solutions, nonsmooth peakon solutions, kink and antikink wave solutions, periodic wave solutions of Jacobian elliptic function type, and rational function solution are obtained. And we show their profiles and discuss their dynamic properties aim at some typical solutions. Though the types of these solutions obtained in this work are not new and they are familiar types, they did not appear in any existing literatures because the equation + + = is very complex. Particularly, compared with the cited references, all results obtained in this paper are new.

#### 1. Introduction

It has come to light that many problems in nonlinear science associated with mechanical, structural, aeronautical, oceanic, electrical, and control systems can be summarized as solving nonlinear partial differential equations (PDEs) which arise from important models with mathematical and physical significances. Finding their exact solutions has extensive applications in many scientific fields such as hydrodynamics, condensed matter physics, solid-state physics, nonlinear optics, neurodynamics, crystal dislocation, model of meteorology, water wave model of oceanography, and fibre-optic communication. The research methods for solving nonlinear PDEs deal with the inverse scattering transformation [1, 2], the Darboux transformation [3–5], the Bäcklund transformation [5–8], the bilinear method and multilinear method [9, 10], the classical and nonclassical Lie group approaches [11, 12], the Clarkson-Kruskal direct method [13–15], the mixing exponential method [16], the geometrical method [17, 18], the truncated Painlev expansion [19, 20], the function expansion method (including tanh expansion method [21, 22], sine-cosine expansion method [23, 24], exp-function method [25], and multiple exp-function method [26]), the bifurcation theory of planar dynamical system [27, 28], the F-expansion type method [29, 30], method [31, 32], and the integral bifurcation method [33–36]. Among these available methods for solving nonlinear PDEs, some of them employed Frobenius’ idea directly or indirectly. Frobenius’ idea is aso called Frobenius’ integrable decompositions [37]; it can reduce a partial differential equation (PDE) to an ordinary differential equation (ODE) under investigation for solution. Indeed, the F-expansion type methods indirectly employed Frobenius’ idea; crucial points of this method are to choose integrable ODE to start investigation for solution. In fact, the tanh function method and method are special cases of such an idea or general Frobenius’ idea. Direct Frobenius’ idea was also used to establish the transformed rational function method [38] and to solve the KPP equation [39].

In this paper, we will employ Frobenius’ idea together with integral bifurcation method to investigate exact traveling wave solutions of the following integrable generalization of the modified KdV equation: where , and are constants and . The model (1) comes from the physical and asymptotic considerations via the methodology introduced by Fokas [40] in 1995; it can be regarded as a water wave model to describe the motion of water wave. It is worth to point out that the special case of (1), is also an important physical model. The above two equations were studied by many authors. Equation (2) was introduced by Fuchssteiner and Fokas in their previous works [41, 42] in 1981. The Lax pairs of (2) were given by Fokas in [40]. New Lax pairs and Darboux transformation of (2) were introduced by Yang and Rui in [43] recently. In [44], by using the bifurcation theory of dynamical system, the existence conditions of different kinds of traveling wave solutions of (2) were presented by Bi. In [45], by using the same method, Li and Zhang studied (1), the existence of solitary wave, kink and antikink wave solutions, uncountably infinite many smooth, and nonsmooth periodic wave solutions were discussed. However, exact travelling wave solutions of (1) were not obtained in [45] though the authors obtained some results of numerical simulation for smooth and nonsmooth periodic wave solutions in this work. Moreover, the investigations on exact solutions of (1) are few in the existing literatures because (1) is more complex than (2). Therefore, in this paper, employing Frobenius’ idea together with integral bifurcation method, we will investigate different kinds of exact traveling wave solutions of (1).

The rest of this paper is organized as follows. In Section 2, by using Frobenius’ idea, we will derive ordinary differential equation (ODE) which is equivalent to (1). In Section 3, by using the integral bifurcation method combined with factoring technique, we will investigate different kinds of exact traveling wave solutions of (1) and discuss their dynamic properties when the integral constants satisfy different conditions. In Section 4, we will discuss different kinds of exact traveling wave solutions of (1) under the special case of the parameter .

#### 2. Direct Application of Frobenius’ Idea on Reducing the PDE (1) to an Integrable ODE

Frobenius’ idea is about changing a partial differential equation (PDE) into an ordinary differential equation (ODE) and then using integrable decomposition method to investigate its exact solutions. Thus, in this section, we first employ the direct Frobenius’ idea to change (1) into an integrable ordinary differential equation; see the following discussions.

Making a traveling wave transformation with , (1) can be reduced to the following ordinary differential equation (ODE): where is wave velocity which moves along the direction of -axis and . Equation (3) can be rewritten as Integrating (4) once, we obtain where is an integral constant. Employing direct Frobenius’ idea, we need not change (5) into a 2-dimensional planar system as the method in [33–36]. we can directly integrate (5) again; see the following calculus.

Multiplying to the both sides of (5) yields Integrating (6) once, we obtain where is another arbitrary integral constant. When , (7) can be rewritten as

#### 3. Different Kinds of Exact Traveling Wave Solutions of (1)

In this section, by using the integral bifurcation method combined with factoring technique as in [36], we will investigate different kinds of exact traveling wave solutions of (1) and discuss their dynamic properties via (7) and (8).

##### 3.1. Hyperbolic Function Solutions and Periodic Wave Solutions of (1) as the Two Integral Constants and

When and + + + , , (8) can be decomposed in the following form: where the coefficients , and are defined by the following expressions: Thus (9) can be reduced to the following two ordinary differential equations: or where the coefficients , and are given by (10).

Solving (11) under the conditions and , we obtain two hyperbolic function solutions of (1) as follows:

Solving (11) under the conditions and , we obtain two periodic wave solutions of (1) as follows:

Similarly, solving (12) under the conditions and , we obtain two hyperbolic function solutions of (1) as follows:

Solving (12) under the conditions and , we obtain two periodic wave solutions of (1) as follows:

##### 3.2. Hyperbolic Function Solutions and Periodic Wave Solutions of (1) as the Two Integral Constants and

When and , , (8) can be decomposed in the following form: Equation (17) can be reduced to the following two ordinary differential equations: or Solving (18), we obtain two hyperbolic function solutions and two periodic wave solutions of (1) as follows: where the , and are defined by

Similarly solving (19), we also obtain two hyperbolic function solutions and two periodic wave solutions of (1) as follows: where the , and are defined by

##### 3.3. Hyperbolic Function Solutions and Periodic Wave Solutions of (1) as the Two Integral Constants and

When and , , , (8) can be decomposed in the following form: Similarly, solving (26) we obtain four hyperbolic function solutions and four periodic wave solutions of (1) as follows: where the , and are defined bywhere .

##### 3.4. Hyperbolic Function Solutions, Periodic Wave Solutions, and Rational Function Solution of (1) as the Two Integral Constants

When , and , , (8) can be decomposed in the following form: Solving (32) we obtain four hyperbolic function solutions and four periodic wave solutions of (1) as follows: where .

When , and , , (8) can be decomposed in the following form: where , .

Solving (37) we obtain two hyperbolic function solutions and two periodic wave solutions of (1) as follows: where and have been given above.

When and , , (8) can be decomposed in the following form: Solving (42), we obtain a hyperbolic function solution, a periodic wave solution, and a rational function solution as follows: where and , have been given above.

All the above exact solutions which were obtained by us are smooth travelling wave solutions including smooth periodic wave solutions and smooth hyperbolic function solutions. In order to show the dynamical profiles of periodic wave solutions, as examples, we plot the graphs of solutions (14) and (38) for , which are shown in Figures 1(a) and 1(b).