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 .
(a) ,
(b) ,
(c) ,
(d) ,
(e) ,
(f) ,
(g) ,
(h) ,
Let be the coefficient matrix of the linearized system of (13) at equilibrium point , , we have By the bifurcation theory of dynamical system, we know that is a saddle point if , a center point if , and a cusp if and the Poincaré index of is zero.
By using the properties of equilibrium points and the bifurcation theory of dynamical system, we can show that bifurcation sets and phase portraits of (13) are as drawn in Figure 2.
(a) ,
(b) ,
(c) ,
(d) ,
(e) ,
(f) ,
3. Existences of Traveling Wave Solutions of (9)
Liu and Guo [15] investigated the periodic blow-up solutions and their limit forms of a generalized Camassa-Holm equation. We consider existences of periodic blow-up wave solutions and other traveling wave solutions of (9) in this section. Denote that , , and from Figure 2, we have the following results.
Theorem 1. (i) When , , for , there exists a family of uncountably infinite many smooth periodic wave solutions of (9). Moreover, the smooth periodic wave solutions converge to a solitary wave solution of peak type as approaches .
(ii) When , , for , there exists a family of uncountably infinite many smooth periodic wave solutions of (9). Moreover, the smooth periodic wave solutions converge to a solitary wave solution of valley type as approaches .
Theorem 2. (i) When , , for , there exists a family of uncountably infinite many periodic blow-up wave solutions of (9). Moreover, the periodic blow-up wave solutions converge to a blow-up wave solution as approaches .
(ii) When , , for , there exists a family of uncountably infinite many periodic blow-up wave solutions of (9). Moreover, the periodic blow-up wave solutions converge to a blow-up wave solution as approaches .
Theorem 3. When , , for , there exists a family of uncountably infinite many periodic blow-up wave solutions of (9). Moreover, the periodic blow-up wave solutions converge to a blow-up wave solution as approaches .
4. Explicit Exact Traveling Solutions of (9)
4.1. Solitary Wave Solutions
From Figure 2(e), we see that there is a homoclinic orbit connecting with the saddle point and passing point when , .
When , expression of the homoclinic orbit is where , .
Substituting (15) into the and integrating it along the homoclinic orbit yields Completing above integral, we can get a solitary wave solution of peak type of (9) as follows: where .
When , expression of the homoclinic orbit is where , . For example, letting , , we get that , , , and .
Substituting (18) into the and integrating it along the homoclinic orbit yields Completing above integral, we can get the implicit representation of the solitary wave solution of peak type of (9) for as follows: where , are the elliptic integrals of the first and third kind, respectively, with the modulus [16] and , , , , , and /.
From Figure 2(f), we see that there is a homoclinic orbit connecting with the saddle point and passing point when , .
When , expression of the homoclinic orbit is where , .
Substituting (21) into the and integrating it along the homoclinic orbit yields Completing above integral, we can get a solitary wave solution of valley type of (9) as follows: where .
When , expression of the homoclinic orbit is where , . For example, letting , , we get that , , , and .
Substituting (24) into the and integrating it along the homoclinic orbit yields Completing above integral, we can get the implicit representation of the solitary wave solution of valley type of (9) for as follows: where , , , , , and /.
4.2. Blow-Up Wave Solutions
From Figure 2(c), we see that there is an open orbit passing the cusp when , .
When , expression of the open orbit is where .
Substituting (27) into the and integrating it along the open orbit yields Completing above integral, we can get a blow-up wave solution of (9) as follows: where .
When , expression of the open orbit is where , . For example, letting , , we get that , , and .
Substituting (30) into the and integrating it along the open orbit yields Completing above integral, we can get the implicit representation of the blow-up wave solution of (9) for as follows:where is the elliptic integral of the second kind with the modulus [16] and , , .
From Figure 2(d), we see that there is an open orbit passing the cusp when , .
When , expression of the open orbit is where .
Substituting (33) into the and integrating it along the open orbit yields Completing above integral, we can get a blow-up wave solution of (9) as follows: where .
When , expression of the open orbit is where , . For example, letting , , we get that , , and .
Substituting (36) into the and integrating it along the open orbit yields Completing above integral, we can get the implicit representation of the blow-up wave solution of (9) for as follows: where , , and .
From Figure 2(e), we see that there is an open orbit passing the saddle point when , .
When , expression of the open orbit is where , are given in (15).
Substituting (39) into the and integrating it along the open orbit yields Completing above integral, we can get a blow-up wave solution of (9) as follows: where is given in (17).
When , expression of the open orbit is where , , , and are given in (18).
Substituting (42) into the and integrating it along the open orbit yields Completing above integral, we can get the implicit representation of the blow-up wave solution of (9) for as follows: where and , , , , and are given in (20).
From Figure 2(f), we see that there is an open orbit passing the saddle point when , .
When , expression of the open orbit is where , are given in (21).
Substituting (45) into the and integrating it along the open orbit yields Completing above integral, we can get a blow-up wave solution of (9) as follows: where is given in (23).
When , expression of the open orbit is where , , , and are given in (24).
Substituting (48) into the and integrating it along the open orbit yields Completing above integral, we can get the implicit representation of the blow-up wave solution of (9) for as follows: where and , , , , and are given in (26).
4.3. Smooth Periodic Wave Solutions
From Figure 2(e), we see that there is a periodic orbit passing points and when , , .
When , expression of the periodic orbit is where , , and are three real roots of . For example, taking , , we have , . Letting , we get that , , and .
Substituting (51) into the and integrating it along the periodic orbit yields Completing above integral, we can get a smooth periodic wave solution of (9) as follows: where is the Jacobian elliptic function with the modulus [16], and , .
From Figure 2(f), we see that there is a periodic orbit passing points and when , , .
When , expression of the periodic orbit is where , , and are three real roots of . For example, taking , , we have , . Letting , we get that , , and .
Substituting (54) into the and integrating it along the periodic orbit yields Completing above integral, we can get a smooth periodic wave solution of (9) as follows: where , .
4.4. Periodic Blow-Up Wave Solutions
From Figure 2(e), we see that there is an open orbit passing point when , , .
When , expression of the open orbit is where , , and are given in (51).
Substituting (57) into the and integrating it along the open orbit yields Completing above integral, we can get a periodic blow-up wave solution of (9) as follows: where is the Jacobian elliptic function with the modulus [16], and , are given in (53).
From Figure 2(f), we see that there is an open orbit passing point when , , .
When , expression of the periodic orbit is where , , and are given in (54).
Substituting (60) into the and integrating it along the open orbit yields Completing above integral, we can get a periodic blow-up wave solution of (9) as follows: where , are given in (56).
5. Conclusion
In this paper, the modified Novikov equation is studied by using the bifurcation theory of dynamical system and the method of phase portraits analysis. For a special case, the existences of different kinds of nonlinear waves and limit forms of two periodic wave solutions of the equation are stated in Theorems 1–3. All possible exact parametric representations of the nonlinear waves also are presented in (17), (20), (23), and (29) and so forth. The previous results of the equation are enriched.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work is supported by the National Natural Science Foundation of China under Grant no. 11461022 and the Natural Science Foundations of Yunnan Province, China, under Grant nos. 2014FA037, 2013FZ117.