Mathematical Problems in Engineering

Volume 2015, Article ID 627269, 9 pages

http://dx.doi.org/10.1155/2015/627269

## Periodic Wave Solutions and Their Limit Forms of the Modified Novikov Equation

^{1}Department of Physics, Honghe University, Mengzi, Yunnan 661100, China^{2}College of Mathematics, Honghe University, Mengzi, Yunnan 661100, China

Received 16 December 2014; Accepted 1 April 2015

Academic Editor: K. M. Liew

Copyright © 2015 Qing Meng and Bin He. 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

The modified Novikov equation is studied by using the bifurcation theory of dynamical system and the method of phase portraits analysis. The existences, dynamic properties, and limit forms of periodic wave solutions for being a negative even are investigated. All possible exact parametric representations of the different kinds of nonlinear waves also are presented.

#### 1. Introduction

The Novikov equationwas discovered by Novikov in a symmetry classification of nonlocal PDEs with quadratic or cubic nonlinearity [1]. The perturbative symmetry approach yields necessary conditions for a PDE to admit infinitely many symmetries. Using this approach, Novikov was able to isolate (1) and find its first few symmetries, and he subsequently found a scalar Lax pair for it (also see [2]) and then proved that the equation is integrable. Hone and Wang [3] have shown that (1) admits peakon solutions like the CH and the DP equations. Jiang and Ni [4] have shown that (1) possesses the blow-up phenomenon. The existence and uniqueness of global weak solutions for (1) were studied in [5]. Bozhkov et al. [6] found the Lie point symmetries of (1) and demonstrate that it is strictly self-adjoint. Li [7] obtained exact cuspon wave solution and compactons and found that the corresponding traveling system of (1) has no one-peakon solution. The Cauchy problem of (1) was investigated in [8, 9].

The modified Novikov equation reads as where is a real parameter. Clearly, letting , (2) becomes the Novikov equation (1). Lai and Wu [10] considered the local strong and weak solutions of (2). The global solution and blow-up phenomena of (2) were investigated in [11]. The Cauchy problem of (2) was studied in [12, 13].

In this paper, we consider the existences, dynamic properties, and limit forms of periodic wave solutions of (2) for being a negative even using the bifurcation theory of dynamical system and the method of phase portraits analysis [7, 14], we also will present some new explicit exact solutions of (2).

Using transformation where is the wave speed, (2) can be rewritten as where “′” is the derivative with respect to .

Integrating (4) once, it follows that

Let , and then (5) becomes Differentiating both sides of (6) with respect to , we have It implies that where is the integral constant.

For simplicity, we only consider the special case and in this paper. For this special case, (2) becomes and (8) can be rewritten as We see from (10) that where is an integral constant. Thus, the function is a first integral of (9). The dynamics of (9) is equivalent to the system where ; otherwise system (13) becomes a linear system.

For a fixed , the level curve defined by (12) determines a set of invariant curves of system (13) which contains different branches of curves. As is varied, it defines different families of orbits of (13) with different dynamical behaviors.

The remainder of this paper is organized as follows: In Section 2, we consider bifurcation sets and phase portraits of (13). Existences and limit forms of periodic wave solutions of (9) are stated in Section 3. Some explicit exact traveling wave solutions of (9) are presented in Section 4. A short conclusion will be given in Section 5.

#### 2. Bifurcation Analysis of (13)

Obviously, the equilibrium point of system (13) is just the intersection point of the straight line and the curve defined by (see Figures 1(a)–1(h)). Clearly, system (13) does not have any equilibrium point when . There exists only one equilibrium point of (13) satisfying when , and when , when . System (13) has two equilibrium points , satisfying , , when , and when , when .