Bifurcation Approach to Analysis of Travelling Waves in Nonlocal Hydrodynamic-Type Models

Abstract

The paper considers the nonlocal hydrodynamic-type systems which are two-dimensional travelling wave systems with a five-parameter group. We apply the method of dynamical systems to investigate the bifurcations of phase portraits depending on the parameters of systems and analyze the dynamical behavior of the travelling wave solutions. The existence of peakons, compactons, and periodic cusp wave solutions is discussed. When the parameter 𝑛 equals 2, namely, let the isochoric Gruneisen coefficient equal 1, some exact analytical solutions such as smooth bright solitary wave solution, smooth and nonsmooth dark solitary wave solution, and periodic wave solutions, as well as uncountably infinitely many breaking wave solutions, are obtained.

1. Introduction

A hydrodynamic system of balance equations for mass and momentum is considered by Vladimirov et al. [1]: 𝑢_𝑡+𝑝_𝑥=ℑ𝜌,𝜌_𝑡+𝜌²𝑢_𝑥=0,𝜏𝑝_𝑡𝛽+𝑝=𝜌𝜈+2^𝜈+2+𝜎𝜌^𝜈+1𝜌_𝑥𝑥+(𝜈+1)𝜌^𝜈𝜌_𝑥², (1) where 𝜈=𝑛−2, 𝑛=1+Γ_𝑉∞, Γ_𝑉∞ is the isochoric Gruneisen coefficient [2], and 𝜏, 𝛽, and 𝜎 are parameters. This system is closed by the dynamic equation of state, taking into account the effects of spatiotemporal nonlocalities. Using group theory reduction, the authors of [1] obtained the following system of the ordinary differential equations: 𝑅̇=𝑌,𝜎𝑅^𝜈+2𝑌̇=𝐺𝑅−𝐷²+𝛽𝑅𝜈+2^𝜈+3+𝜎(𝜈+1)𝑅^𝜈+1𝑌²𝐷−𝜏𝜖³𝑅̇𝑓(𝑅)𝐶𝑌,𝐺=𝜖−𝜉₁+1−𝜈𝐷,1+𝜈𝑅 (2) which describes a set of approximate travelling wave solutions to the source system (1) (see the initial system (1) of PDEs in [1]), where 𝜏, 𝜉 and 𝐶₁, 𝐷 are constant parameters; 𝜖≪0.

When 𝜖=0, it immediately obtains that 𝐺=𝐺₁=const, and the system (2) reduces to the following two-dimensional system: 𝑑𝑅𝑑𝜁=𝑌,𝑑𝑌=𝐺𝑑𝜁¹𝑅−𝐷²+𝛽/(𝜈+2)𝑅^𝜈+3+𝜎(𝜈+1)𝑅^𝜈+1𝑌²𝜎𝑅^𝜈+2, (3) where 𝜁=𝑥−𝐷𝑡.

Assume that𝐴_𝑗(𝑅_𝑗,0), 𝑗=1,2, are two equilibrium points of system (3). Vladimirov et al. obtained the following conclusion (see [1, 3]).

Theorem A. If 𝜈>−2 and 𝐷²>𝛽𝑅₁^𝜈+3, then system (3) possesses a one-parameter family of periodic solutions, localized around the critical point 𝐴₂(𝑅₂,0) in a bounded set 𝑀. The boundary of this set is formed by the homoclinic intersection of separatrices of the saddle point 𝐴₁(𝑅₁,0).

We notice that for a fixed 𝜈, system (3) is a four-parameter system depending on the parameter group (𝛽,𝜎,𝐷,𝐺₁). The bifurcations and dynamical behavior of solutions of system (3) have not be studied by [1, 3]. The conclusion of Theorem A is incomplete (see Theorems 1–3 of Section 3 below). In fact, system (3) is the first class of singular travelling wave systems defined by [4, 5], which has the singular straight line 𝑅=0. Depending on the changes of parameters, there are very interesting bifurcations and dynamical behaviors of the travelling wave solutions in this kind of singular systems, for example, it gives rise to so-called peakons, compactons, loop solutions, and others. As Fokas stated that peakons are peaked solitons [6], that is, solitons with discontinuous first derivative, compactons are solitons with compact support.

In this paper, we will give complete description for the dynamics of solutions of system (3).

Instead of 𝜈, we use 𝑛=𝜈+2 to rewrite (3) as follows: 𝑑𝑅𝑑𝜁=𝑌,𝑑𝑌=𝐺𝑑𝜁¹𝑅−𝐷²+𝛽/𝑛𝑅^𝑛+1+𝜎(𝑛−1)𝑅^𝑛−1𝑌²𝜎𝑅^𝑛. (4)

This system has the first integral: 𝐻(𝑅,𝑌)=𝑌²𝑅^2𝑛−2+2𝑅^𝑛−1𝑛𝜎𝑛𝐷²𝑛−1−𝐺₁𝛽𝑅+𝑅2𝑛^𝑛+1=ℎ. (5) Clearly, in order to make 𝐻(𝑅,𝑌) well defined, we assume that 𝑛>1; that is, 𝜈>−1.

This paper is organized as follows. In Section 2, we analyse the bifurcations of phase portraits of system (4) under different parameter conditions. Section 3 discusses the existence of periodic solutions in different parameter conditions. In particular, it discusses the existence of solitary cusp wave solutions (peakons) and periodic cusp wave solutions. In Section 4, for the case 𝑛=𝜈+2=2, namely, let the isochoric Gruneisen coefficient equal 1, we figure out explicit parametric expressions for the solitary wave solutions, periodic wave solutions, and uncountably infinitely many breaking wave solutions (compactons). Finally, we give a conclusion of this paper.

2. Bifurcations of the Phase Portraits of System (4)

In this section, we study the phase portraits of system (4) in the (𝑅,𝑌)-phase plane. Consider the associated regular system of (4): 𝑑𝑅𝑑𝜂=𝜎𝑅^𝑛𝑌,𝑑𝑌𝑑𝜂=𝐺₁𝑅−𝐷²+𝛽𝑛𝑅^𝑛+1+𝜎(𝑛−1)𝑅^𝑛−1𝑌², (6) where 𝑑𝜁=𝜎𝑅^𝑛𝑑𝜂. Clearly, now the straight line 𝑅=0 is a solution of system (6). On the straight line 𝑅=0, system (6) has no equilibrium point.

Consider the following formulas: 𝑓(𝑅)=𝛽𝑛𝑅^𝑛+1−𝐺₁𝑅+𝐷²,𝑓^′(𝑅)=(𝑛+1)𝛽𝑛𝑅^𝑛−𝐺₁. (7) Obviously, for 𝑛=2𝑚, 𝑚=1,2,…, when 𝛽𝐺₁>0, and 𝑅=𝑅_𝑀±≡±(𝑛𝐺₁/(𝑛+1)𝛽)^1/𝑛, 𝑓^′(𝑅_𝑀±)=0, while for 𝑛=2𝑚+1, 𝑚=1,2,…, when 𝑅=𝑅_𝑀≡(𝑛𝐺₁/(𝑛+1)𝛽)^1/𝑛, 𝑓^′(𝑅_𝑀)=0; we have 𝑓(𝑅_𝑀)=𝑓(𝑅_𝑀+)=𝑓(𝑅_𝑀−)=𝐷²−(𝑛𝐺₁/(𝑛+1))𝑅_𝑀. When 𝐷²=𝐷²_𝑐𝑟≡(𝑛𝐺₁/(𝑛+1))(𝑛𝐺₁/(𝑛+1)𝛽)^1/𝑛, 𝑓(𝑅_𝑀)=0.

Since every real root 𝑅_𝑗 of the function 𝑓(𝑅) gives rise to an equilibrium point 𝐴_𝑗(𝑅_𝑗,0) of system (6), by using (7), we can analyse the equilibrium points 𝐴_𝑗(𝑅_𝑗,0) of system (6) in different parameter conditions and have the following conclusions.

2.1. In the Case of 𝑛=2𝑚.

Consider the following.(1)For 𝛽>0 and 𝐺₁>0, when 𝐷²<(𝑛𝐺₁/(𝑛+1))𝑅_𝑀, and𝑓(𝑅_𝑀−)>0, 𝑓(𝑅_𝑀+)<0. There exist three equilibrium points 𝐴_𝑗(𝑅_𝑗,0), 𝑗=1,2,3,of system (6), satisfying −∞<𝑅₁<𝑅_𝑀−<0<𝑅₂<𝑅_𝑀+<𝑅₃<∞. When 𝐷²=(𝑛𝐺₁/(𝑛+1))𝑅_𝑀, 𝑓(𝑅_𝑀−)>0 and 𝑓(𝑅_𝑀+)=0. There exist two equilibrium points 𝐴_𝑗(𝑅_𝑗,0), 𝑗=1,2, of system (6), satisfying −∞<𝑅₁<𝑅_𝑀−<0<𝑅₂=𝑅_𝑀+<∞. When 𝐷²>(𝑛𝐺₁/(𝑛+1))𝑅_𝑀, 𝑓(𝑅_𝑀−)>0, and𝑓(𝑅_𝑀+)>0. There exists only one equilibrium point 𝐴₁(𝑅₁,0) of system (6), satisfying −∞<𝑅₁<𝑅_𝑀−<0.(2)For 𝛽<0, and 𝐺₁<0, when 𝐷²<(𝑛|𝐺₁|/(𝑛+1))𝑅_𝑀, 𝑓(𝑅_𝑀−)<0, and 𝑓(𝑅_𝑀+)>0. There exist three equilibrium points 𝐴_𝑗(𝑅_𝑗,0), 𝑗=1,2,3, of system (6), satisfying −∞<𝑅₁<𝑅_𝑀−<𝑅₂<0<𝑅_𝑀+<𝑅₃<∞. When 𝐷²=(𝑛|𝐺₁|/(𝑛+1))𝑅_𝑀, 𝑓(𝑅_𝑀−)=0, and 𝑓(𝑅_𝑀+)>0. There exist two equilibrium points 𝐴_𝑗(𝑅_𝑗,0), 𝑗=1,2, of system (6), satisfying −∞<𝑅₁=𝑅_𝑀−<0<𝑅_𝑀+<𝑅₂<∞. When 𝐷²>(𝑛|𝐺₁|/(𝑛+1))𝑅_𝑀, 𝑓(𝑅_𝑀−)>0, and 𝑓(𝑅_𝑀+)>0. There exists only one equilibrium point 𝐴₁(𝑅₁,0) of system (6), satisfying 0<𝑅_𝑀+<𝑅₁<∞.(3)For 𝛽>0, 𝐺₁<0, and any given𝐷, 𝜎, because the number 𝑅_𝑀is not real, there exists one equilibrium point 𝐴₁(𝑅₁,0) of system (6), satisfying −∞<𝑅₁<0.(4)For 𝛽<0, 𝐺₁>0, and any given 𝐷, 𝜎, there exists one equilibrium point 𝐴₁(𝑅₁,0) of system (6), satisfying 0<𝑅₁<∞.

Let 𝑀(𝑅_𝑗,0) be the coefficient matrix of the linearized system of system (6) at an equilibrium point 𝐴_𝑗(𝑅_𝑗,0). We have 𝐽𝑅_𝑗,0=det𝑀𝑅_𝑗,0=𝜎𝑅^𝑛_𝑗𝑓^′𝑅_𝑗. (8)

By the theory of planar dynamical systems, for an equilibrium point of a planar integrable system, if 𝐽<0, then the equilibrium point is a saddle point; if 𝐽>0, and (Trace𝑀(𝑅_𝑗,0))²−4𝐽(𝑅_𝑗,0)>0, then it is a node point; if 𝐽>0, and Trace𝑀(𝑅_𝑗,0)=0, then, it is a center point; if 𝐽=0 and the Poincaré index of the equilibrium point is 0, then this equilibrium point is cusp [5].

According to formula (5), we write that ℎ₀=𝐻(0,0)=0 and ℎ_𝑗=𝐻(𝑅_𝑗,0), 𝑗=1,2,3.

By using the above results, we have the following bifurcations of phase portraits of system (6) shown in Figures 1–5.

(a) 𝐷2>𝐷2𝑐𝑟

(b) 𝐷2=𝐷2𝑐𝑟

(c) 𝐷2<𝐷2𝑐𝑟

Figure 1 

The phase portraits of system (6) for 𝑛=2𝑚, 𝛽>0, 𝐺1>0, and 𝜎>0.

(a) 𝐷2>𝐷2𝑐𝑟

(b) 𝐷2=𝐷2𝑐𝑟

(c) 𝐷2𝑚<𝐷2<𝐷2𝑐𝑟

Figure 2 

The phase portraits of system (6) for 𝑛=2𝑚, 𝛽>0, 𝐺1>0, and 𝜎<0.

(a) 𝐷2>𝐷2𝑐𝑟

(b) 𝐷2=𝐷2𝑐𝑟

(c) 𝐷2𝑚<𝐷2<𝐷2𝑐𝑟

Figure 3 

The phase portraits of system (6) for 𝑛=2𝑚, 𝛽<0, 𝐺1<0, and 𝜎>0.

(a) 𝐷2>𝐷2𝑐𝑟

(b) 𝐷2=𝐷2𝑐𝑟

(c) 𝐷2<𝐷2𝑐𝑟

Figure 4 

The phase portraits of system (6) for 𝑛=2𝑚, 𝛽<0, 𝐺1<0, and 𝜎<0.

(a) 𝛽>0, 𝐺1<0, 𝜎>0

(b) 𝛽>0, 𝐺1<0, 𝜎<0

(c) 𝛽<0, 𝐺1>0, 𝜎>0

(d) 𝛽<0, 𝐺1>0, 𝜎<0

Figure 5 

The phase portraits of system (6) for 𝑛=2𝑚, and 𝛽𝐺1<0.

For the case of 𝛽<0, and 𝐺₁<0, we see from system (6) that when 𝑛=2𝑚, by the transformation 𝑅→−𝑅, 𝑌→−𝑌, and 𝜎→−𝜎, it can become the case of 𝛽>0, 𝐺₁>0. Thus, the phase portraits of system (6) in the case of 𝛽<0, and 𝐺₁<0 just are the reflections of the phase portraits in Figures 1 and 2 with respect to the 𝑌-axis, which are shown in Figures 3 and 4.

2.2. In the Case of 𝑛=2𝑚+1

Consider the following.(1)For 𝛽>0, and 𝐺₁>0, when 𝐷²<(𝑛𝐺₁/(𝑛+1))𝑅_𝑀, and 𝑓(𝑅_𝑀)<0, there exist two equilibrium points 𝐴_𝑗(𝑅_𝑗,0), 𝑗=1,2, of system (6), satisfying 0<𝑅₁<𝑅_𝑀<𝑅₂<∞. When 𝐷²=(𝑛𝐺₁/(𝑛+1))𝑅_𝑀, and 𝑓(𝑅_𝑀)=0, there exists one equilibrium point 𝐴₁(𝑅₁,0) of system (6), satisfying 0<𝑅₁=𝑅_𝑀<∞. When 𝐷²>(𝑛𝐺₁/(𝑛+1))𝑅_𝑀, and 𝑓(𝑅_𝑀)>0, system (6) has no equilibrium point.(2)For 𝛽>0, 𝐺₁<0, when 𝐷²<(𝑛𝐺₁/(𝑛+1))𝑅_𝑀, and 𝑓(𝑅_𝑀)<0, there exist two equilibrium points 𝐴_𝑗(𝑅_𝑗,0), 𝑗=1,2, of system (6), satisfying −∞<𝑅₁<𝑅_𝑀<𝑅₂<0. When 𝐷²=(𝑛𝐺₁/(𝑛+1))𝑅_𝑀, and 𝑓(𝑅_𝑀)=0, there exists one equilibrium point 𝐴₁(𝑅₁,0) of (6), satisfying −∞<𝑅₁=𝑅_𝑀<0. When 𝐷²>(𝑛𝐺₁/(𝑛+1))𝑅_𝑀, and 𝑓(𝑅_𝑀)>0, system (6) has no equilibrium point.(3)For 𝛽<0, 𝐺₁>0, and any given 𝐷, 𝜎, there exist two equilibrium points 𝐴_𝑗(𝑅_𝑗,0), 𝑗=1,2, of system (6), satisfying −∞<𝑅₁<𝑅_𝑀<0<𝑅₂.(4)For 𝛽<0, 𝐺₁<0, and any given 𝐷, 𝜎, there exist two equilibrium points 𝐴_𝑗(𝑅_𝑗,0), 𝑗=1,2, of system (6), satisfying −∞<𝑅₁<0<𝑅_𝑀<𝑅₂<∞.

Same as Section 2.1, we have the following bifurcations of phase portraits of system (6) shown in Figures 6–10.

(a) 𝐷2>𝐷2𝑐𝑟

(b) 𝐷2=𝐷2𝑐𝑟

(c) 𝐷2<𝐷2𝑐𝑟

Figure 6 

The phase portraits of system (6) for 𝑛=2𝑚+1, 𝛽>0, 𝐺1>0, and 𝜎>0.

(a) 𝐷2>𝐷2𝑐𝑟

(b) 𝐷2=𝐷2𝑐𝑟

(c) 𝐷2𝑚<𝐷2<𝐷2𝑐𝑟

Figure 7 

The phase portraits of system (6) for 𝑛=2𝑚+1, 𝛽>0, 𝐺1>0, and 𝜎<0.

(a) 𝐷2>𝐷2𝑐𝑟

(b) 𝐷2=𝐷2𝑐𝑟

(c) 𝐷2<𝐷2𝑐𝑟

Figure 8 

The phase portraits of system (6) for 𝑛=2𝑚+1, 𝛽>0, 𝐺1<0, and 𝜎>0.

(a) 𝐷2>𝐷2𝑐𝑟

(b) 𝐷2=𝐷2𝑐𝑟

(c) 𝐷2𝑚<𝐷2<𝐷2𝑐𝑟

Figure 9 

The phase portraits of system (6) for 𝑛=2𝑚+1, 𝛽>0, 𝐺1<0, and 𝜎<0.

(a) 𝐺1>0,𝜎>0

(b) 𝐺1>0,𝜎<0

(c) 𝐺1<0,𝜎>0

(d) 𝐺1<0,𝜎<0

Figure 10 

The phase portraits of system (6) for 𝑛=2𝑚+1, and 𝛽<0.

3. The Existence of Solutions of System (4)

Because system (6) has the same phase orbits as system (4), therefore, to sum up, by the above discussion, we have the following results.

3.1. The Existence of Periodic Solutions

Theorem 1. (1) If 𝛽>0, 𝐺₁>0, 𝜎>0, and 𝐷²<𝐷²_𝑐𝑟≡(𝑛𝐺₁/(𝑛+1))(𝑛𝐺₁/(𝑛+1)𝛽)^1/𝑛, then, for any 𝑛=𝜈−2=2𝑚>1, system (4) possesses a one-parameter family of periodic solutions defined by 𝐻(𝑅,𝑌)=ℎ, ℎ∈(ℎ₃,ℎ₂), enclosing the center point 𝐴₃(𝑅₃,0). The boundary of this set is formed by the homoclinic orbit defined by 𝐻(𝑅,𝑌)=ℎ₂ to the saddle point 𝐴₂(𝑅₂,0) (see Figure 1(c)).
And for any 𝑛=𝜈−2=2𝑚+1, system (4) possesses a one-parameter family of periodic solutions defined by 𝐻(𝑅,𝑌)=ℎ, ℎ∈(ℎ₂,ℎ₁), enclosing the center point 𝐴₂(𝑅₂,0). The boundary of this set is formed by the homoclinic orbit defined by 𝐻(𝑅,𝑌)=ℎ₁ to the saddle point 𝐴₁(𝑅₁,0) (see Figure 6(c)).
(2) If 𝛽>0, 𝐺₁>0, 𝜎<0, and 𝐷²_𝑚<𝐷²<𝐷²_𝑐𝑟≡(𝑛𝐺₁/(𝑛+1))(𝑛𝐺₁/(𝑛+1)𝛽)^1/𝑛, then, for any 𝑛=𝜈−2=2𝑚>1, system (4) possesses a one-parameter family of periodic solutions defined by 𝐻(𝑅,𝑌)=ℎ, ℎ∈(ℎ₂,ℎ₃), enclosing the center point 𝐴₂(𝑅₂,0). The boundary of this set is formed by the homoclinic orbit defined by 𝐻(𝑅,𝑌)=ℎ₃ to the saddle point 𝐴₃(𝑅₃,0) (see Figure 2(c)).
And for any 𝑛=𝜈−2=2𝑚+1, system (4) possesses a one-parameter family of periodic solutions defined by 𝐻(𝑅,𝑌)=ℎ, ℎ∈(ℎ₁,ℎ₂), enclosing the center point 𝐴₁(𝑅₁,0). The boundary of this set is formed by the homoclinic orbit defined by 𝐻(𝑅,𝑌)=ℎ₂ to the saddle point 𝐴₂(𝑅₂,0) (see Figure 7(c)).

Theorem 2. (1) If 𝛽<0, 𝐺₁<0, 𝜎<0(or 𝛽>0, 𝐺₁<0, 𝜎>0), and𝐷²<𝐷²_𝑐𝑟≡(𝑛𝐺₁/(𝑛+1))(𝑛𝐺₁/(𝑛+1)𝛽)^1/𝑛, then, for any 𝑛=𝜈−2>1, system (4) possesses a one-parameter family of periodic solutions defined by 𝐻(𝑅,𝑌)=ℎ, ℎ∈(ℎ₁,ℎ₂), enclosing the center point 𝐴₁(𝑅₁,0). The boundary of this set is formed by the homoclinic orbit defined by 𝐻(𝑅,𝑌)=ℎ₂ to the saddle point 𝐴₂(𝑅₂,0) (see Figure 4(c) or Figure 8(c)).
(2) If 𝛽<0, 𝐺₁<0, 𝜎>0 (or 𝛽>0, 𝐺₁<0, 𝜎<0), and 𝐷²_𝑚<𝐷²<𝐷²_𝑐𝑟≡(𝑛𝐺₁/(𝑛+1))(𝑛𝐺₁/(𝑛+1)𝛽)^1/𝑛, then, for any𝑛=𝜈−2>1, system (4) possesses a one-parameter family of periodic solutions defined by 𝐻(𝑅,𝑌)=ℎ, ℎ∈(ℎ₂,ℎ₁), enclosing the center point 𝐴₂(𝑅₂,0). The boundary of this set is formed by the homoclinic orbit defined by 𝐻(𝑅,𝑌)=ℎ₁ to the saddle point 𝐴₁(𝑅₁,0) (see Figure 3(c) or Figure 9(c)).

Theorem 3. (1) If one of the following conditions holds:(a)𝛽>0, 𝐺₁>0, 𝜎>0;(b)𝛽<0, 𝐺₁<0, 𝜎<0;(c)𝛽>0, 𝐺₁<0, 𝜎>0;(d)𝛽<0, 𝐺₁>0, 𝜎<0,then, for any 𝑛=2𝑚>1 and 𝐷²>0, system (4) possesses a one-parameter family of periodic solutions defined by 𝐻(𝑅,𝑌)=ℎ, ℎ∈(ℎ_𝑗,0), enclosing the center point 𝐴_𝑗(𝑅_𝑗,0) (see Figures 1, 4, and 5(a), 5(d)). As ℎ→0, the closed orbits of system (4) expand outwards and approach to the singular straight line 𝑅=0.(2)If one of the following conditions holds:(a)𝛽<0, 𝐺₁>0, 𝜎<0;(b)𝛽<0, 𝐺₁<0, 𝜎<0,then, for any 𝑛=2𝑚+1 and 𝐷²>0, system (4) possesses two families of periodic solutions defined by 𝐻(𝑅,𝑌)=ℎ, ℎ∈(ℎ_𝑗,0), 𝑗=1,2, enclosing the center point 𝐴_𝑗(𝑅_𝑗,0), 𝑗=1,2 (see Figures 10(b) and 10(d)). As ℎ→0, the closed orbits of system (4) expand outwards and approach to the singular straight line 𝑅=0.

3.2. The Existence of Uncountably Infinitely Periodic Cusp Wave Solutions and Pseudo-Peakons

We consider Figure 3(c), for every fixed 𝑛,𝛽, 𝜎, and 𝐺₁, when parameter 𝐷² decreases from 𝐷²_𝑐𝑟 to 0, there exists a value 𝐷²=𝐷²_𝑚 (where 𝐷²_𝑚 satisfies 𝐻(𝑅₁,0)=ℎ₁=0) such that when 𝐷²<𝐷²_𝑚, the stable and unstable manifolds to a saddle point cannot connect in 𝑅-axis. Corresponding to Figure 3(c), the changes of phase portraits of system (6) are shown in Figure 11. For a fixed parameter group of 𝑛,𝛽,𝐺₁, and 𝜎, the value of 𝐷²_𝑚can be numerically determined.

(a) 𝐷2<𝐷2𝑐𝑟

(b) 𝐷2𝑚<𝐷2<𝐷2𝑐𝑟

(c) 𝐷2<𝐷2𝑚

Figure 11 

The changes of phase portraits of system (6) when 𝐷2 are decreased. Parameters: 𝑛=2𝑚, 𝛽<0, 𝐺1<0, and 𝜎>0.

We see from Section 2 that if 𝑛=2𝑚, when 𝛽>0, 𝐺₁>0, 𝜎<0, and 𝐷_𝑚<𝐷²<𝐷²_𝑐𝑟, system (4) has the phase portrait Figure 2(c), while when 𝛽<0, 𝐺₁<0, 𝜎>0, and 𝐷_𝑚<𝐷²<𝐷²_𝑐𝑟, system (4) has the phase portrait Figure 3(c). Similarly, if 𝑛=2𝑚+1, when 𝛽>0, 𝐺₁>0, 𝜎<0, and 𝐷_𝑚<𝐷²<𝐷²_𝑐𝑟, system (4) has the phase portrait Figure 7(c), while when 𝛽>0, 𝐺₁<0, 𝜎<0and 𝐷_𝑚<𝐷²<𝐷²_𝑐𝑟, system (4) has the phase portrait Figure 9(c).

According to all the above, for a fixed parameter group (𝑛,𝛽,𝐺₁,𝜎), when parameter 𝐷² decreases from 𝐷²_𝑐𝑟 to 𝐷_𝑚, the homoclinic orbit loop will gradually expand and approache to the straight line 𝑅=0 (see Figure 11). By using the rapid-jump property of 𝑑𝑅/𝑑𝜁 near the straight line given by [4], it implies that depending on 𝐷²∈(𝐷²_𝑚,𝐷²_𝑐𝑟), there exists a family of solitary cusp wave solutions (peakons) of (1). For example, corresponding to the homoclinic orbit in Figure 11(b), we have the peakon shown in Figure 12(a). Corresponding to the periodic orbits defined by 𝐻(𝑅,𝑌)=ℎ, ℎ∈(ℎ₂,ℎ₁)in Figure 11(b), when ℎ is varied from ℎ₂ to ℎ₁, the periodic wave solutions gradually lose their smoothness, from smooth periodic waves become periodic cusp waves (see Figures 12(b)–12(d)).

(a) Peakon

(b) Smooth wave

(c) Cusp wave

(d) Cusp wave

Figure 12 

The profiles of pseudo-peakon and periodic cusp wave solutions of (1).

Hence, we have the following conclusion.

Theorem 4. Under the above mentioned parameter conditions of Figure 2(c), Figure 3(c), Figure 7(c), and Figure 8(c), for a fixed parameter group (𝑛,𝛽,𝐺₁,𝜎), depending on the parameter 𝐷², closing to the value 𝐷_𝑚, there exist solitary cusp wave solutions (pseudo-peakons) and periodic cusp wave solutions of (1).

4. Analytical Travelling Wave Solutions for 𝑛=2

According to (5), for a fixed integral constant ℎ, we have 1𝑦=𝑅^𝑛−1ℎ−2𝑅^𝑛−1𝑛𝜎𝑛𝐷²𝑛−1−𝐺₁𝛽𝑅+𝑅2𝑛^𝑛+11/2. (9) Using the first equation of (4) and taking integration on a branch of the invariant curve 𝐻(𝑅,𝑌)=ℎ with initial value 𝑅(𝜁₀)=𝑅₀, we have 𝜁−𝜁₀=^𝑅_𝑅0𝑅^𝑛−1𝑑𝑅𝐹(𝑅)≡^𝑅_𝑅0𝑅^𝑛−1𝑑𝑅ℎ−2𝑅^𝑛−1/𝑛𝜎𝑛𝐷²/(𝑛−1)−𝐺₁𝑅+𝛽/2𝑛𝑅^𝑛+1. (10) In general, for 𝑛≥3, the right side of (10) cannot be integrated explicitly. However, for 𝑛=2, introducing a new parametric variable 𝜒, we can obtain analytical parametric representations for solitary wave solutions, breaking wave solutions and periodic wave solutions of (4).

4.1. In the Case 𝛽>0, 𝐺₁>0, and 𝜎>0

Suppose that 𝐷²<𝐷²_𝑐𝑟≡(2𝐺₁/3)(2𝐺₁/3𝛽)^1/2 (see Figure 1(c)).

(1) Corresponding to the homoclinic orbit to the saddle point 𝐴₂(𝑅₂,0), as a closed branch of the level set 𝐻(𝑅,𝑌)=ℎ₂, enclosing the center 𝐴₃(𝑅₃,0) in Figure 1(c), it gives rise to a bright solitary wave solution of (1). Now, we have 𝐹(𝑅)=(1/2)𝛽/𝜎(𝑅_𝑚−𝑅)(𝑅−𝑅₂)²(𝑅−𝑅_𝑙) with 𝑅_𝑙<0<𝑅₂<𝑅₃<𝑅_𝑚. Thus, (10) has the following form: 12𝛽𝜎𝜁=^𝑅_𝑅𝑚𝑅𝑑𝑅𝑅−𝑅₂𝑅_𝑚−𝑅𝑅−𝑅_𝑙=^𝑅_𝑅𝑚1𝑅_𝑚−𝑅𝑅−𝑅_𝑙+𝑅₂𝑅−𝑅₂𝑅_𝑚−𝑅𝑅−𝑅_𝑙𝑑𝑅. (11)

The following analytical parametric representation of the bright solitary wave solution of (1) can be obtained: 𝑅𝜒=𝑅₂+2𝑅_𝑚−𝑅₂𝑅₂−𝑅_𝑙𝑅_𝑚−𝑅_𝑙cosh(𝜔𝜁)−𝑅_𝑚−2𝑅₂+𝑅_𝑙,𝜁𝜒=2𝜎𝛽𝑅₂𝜋𝜒−2𝑅−arcsin^𝑚+𝑅_𝑙−2𝑅𝜒𝑅_𝑚−𝑅_𝑙,(],𝜁𝜒for𝜒∈−∞,0=2𝜎𝛽𝑅₂𝜋𝜒+2𝑅+arcsin^𝑚+𝑅_𝑙−2𝑅𝜒𝑅_𝑚−𝑅_𝑙,for𝜒∈[0,∞), (12) where 𝜔=(𝑅_𝑚−𝑅₂)(𝑅₂−𝑅_𝑙).

(2) Corresponding to the periodic orbits defined by 𝐻(𝑅,𝑌)=ℎ, ℎ∈(ℎ₃,ℎ₂), enclosing the center 𝐴₃(𝑅₃,0) in Figure 1(c), we have a family of periodic wave solutions of (1). Now, 𝐹(𝑅)=(1/2)𝛽/𝜎(𝑟₁−𝑅)(𝑅−𝑟₂)(𝑅−𝑟₃)(𝑅−𝑟₄) with 𝑟₄<0<𝑟₃<𝑅₂<𝑟₂<𝑅₃<𝑟₁<𝑅_𝑚.

Hence, (10) gives that 12𝛽𝜎𝜁=^𝑅_𝑟2𝑅𝑑𝑅𝑟₁−𝑅𝑅−𝑟₂𝑅−𝑟₃𝑅−𝑟₄=^𝑅_𝑟2𝑅−𝑟₃𝑟₁−𝑅𝑅−𝑟₂𝑅−𝑟₄+𝑟₃𝑟₁−𝑅𝑅−𝑟₂𝑅−𝑟₃𝑅−𝑟₄𝑑𝑅. (13) It implies the following analytical parametric representation of periodic wave solutions of (1): 𝑅𝜒=𝑟₃+𝑟₂−𝑟₃1−𝛼²sn²,𝜁𝜒𝜒,𝑘=2𝜎𝛽𝑟₃𝑔𝜒+𝑟₂−𝑟₃Πarcsinsn𝜒,𝑘,𝛼²,𝑘, (14) where 𝑔=2/(𝑟₁−𝑟₃)(𝑟₂−𝑟₄), 𝛼²=(𝑟₁−𝑟₂)/(𝑟₁−𝑟₃), 𝑘²=(𝑟₁−𝑟₂)(𝑟₃−𝑟₄)/(𝑟₁−𝑟₃)(𝑟₂−𝑟₄) and Π(⋅,𝛼²,𝑘) is the elliptic integral of the third kind, sn(𝑢,𝑘) is the Jacobian elliptic function (see [7]).

(3) Corresponding to two families of the periodic orbits defined by 𝐻(𝑅,𝑌)=ℎ, ℎ∈(ℎ₁,ℎ₃), ℎ₁<ℎ₃<0<ℎ₂ enclosing the center 𝐴₁(𝑅₁,0) and 𝐴₃(𝑅₃,0), respectively, in Figure 1(c), we have 𝐹(𝑅)=(1/2)𝛽/𝜎(𝑟₁−𝑅)(𝑅−𝑟₂)(𝑅−𝑟₃)(𝑅−𝑟₄) and 𝐹(𝑅)=(1/2)𝛽/𝜎(𝑟₁−𝑅)(𝑟₂−𝑅)(𝑟₃−𝑅)(𝑅−𝑟₄), respectively, with 𝑟₄<𝑅₁<𝑟₃<0<𝑅₂<𝑟₂<𝑅₃<𝑟₁<𝑅_𝑚. In this case, the right family of the periodic orbits has the same parametric representation as (14). For the left family of periodic orbits, formula (10) has the following form: 12𝛽𝜎𝜁=^𝑟3_𝑅𝑅𝑑𝑅𝑟₁−𝑅𝑟₂−𝑅𝑟₃−𝑅𝑅−𝑟₄=^𝑟3_𝑅−𝑟₂−𝑅𝑟₁−𝑅𝑟₃−𝑅𝑅−𝑟₄+𝑟₂𝑟₁−𝑅𝑟₂−𝑅𝑟₃−𝑅𝑅−𝑟₄𝑑𝑅. (15) It implies the following analytical parametric representation of the left family of periodic wave solutions of (1): 𝑅𝜒=𝑟₃−𝛼²₁𝑟₂−𝑟₃sn²𝜒,𝑘1−𝛼²₁sn²,𝜁𝜒𝜒,𝑘=2𝜎𝛽𝑟₂𝑔𝜒−𝑟₂−𝑟₃Πarcsinsn²𝜒,𝑘,𝛼²₁,𝑘, (16) where 𝑔=2/(𝑟₁−𝑟₃)(𝑟₂−𝑟₄), 𝛼²₁=(𝑟₃−𝑟₄)/(𝑟₂−𝑟₄), 𝑘²=(𝑟₁−𝑟₂)(𝑟₃−𝑟₄)/(𝑟₁−𝑟₃)(𝑟₂−𝑟₄).

(4) Corresponding to two families of open level curves defined by 𝐻(𝑅,𝑌)=ℎ, ℎ∈(0,ℎ₂), ℎ₂>0in the two sides of the straight line𝑅=0in Figure 1(c) (see also Figure 13(a)), we have 𝐹(𝑅)=(1/2)𝛽/𝜎(𝑟₁−𝑅)(𝑟₂−𝑅)(𝑟₃−𝑅)(𝑅−𝑟₄) with 𝑟₄<𝑅₁<𝑟₃<0<𝑅₂<𝑟₂<𝑅₃<𝑟₁<𝑅_𝑚. Thus, the left family of open level curves has the same parametric representation as (16) with −𝜒₀<𝜒<𝜒₀,𝜒₀=sn⁻¹(𝑟₃/𝛼²₁𝑟₂,𝑘).

(a) Level curves

(b)

(c)

Figure 13 

The level curves and profiles of breaking wave solutions of (1).

Figures 13(b) and 13(c) show the two families of uncountably infinitely many breaking wave solutions (compactons) of (1).

4.2. In the Case 𝛽>0, 𝐺₁>0, 𝜎<0

Suppose that 𝐷_𝑚<𝐷²<𝐷²_𝑐𝑟≡(2𝐺₁/3)(2𝐺₁/3𝛽)^1/2 (see Figure 2(c)).

(1) Corresponding to the homoclinic orbit to the saddle point 𝐴₃(𝑅₃,0), as a closed branch of the level set 𝐻(𝑅,𝑌)=ℎ₂, enclosing the center 𝐴₂(𝑅₂,0) in Figure 2(c), it gives rise to a dark solitary wave solution of (1). In this case, we have 𝐹(𝑅)=(1/2)𝛽/|𝜎|(𝑅₃−𝑅)²(𝑅−𝑅_𝑀)(𝑅−𝑅_𝑙) with 𝑅_𝑙<0<𝑅_𝑚<𝑅₂<𝑅₃. Thus, formula (10) has the following form: 12𝛽|𝜎|𝜁=^𝑅_𝑅𝑚𝑅𝑑𝑅𝑅₃−𝑅𝑅−𝑅_𝑚𝑅−𝑅_𝑙=^𝑅_𝑅𝑚−1𝑅−𝑅_𝑚𝑅−𝑅_𝑙+𝑅₃𝑅₃−𝑅𝑅−𝑅_𝑚𝑅−𝑅_𝑙𝑑𝑅. (17)

It implies the following analytical parametric representation of the dark solitary wave solution of (1): 𝑅𝜒=𝑅₃−2𝑅₃−𝑅_𝑚𝑅₃−𝑅_𝑙𝑅_𝑚−𝑅_𝑙cosh𝜔₁𝜁+2𝑅₃−𝑅_𝑚−𝑅_𝑙,(][)𝜁𝜒for𝜒∈−∞,0,𝜒∈0,∞,respectively,=2|𝜎|𝛽𝑅₃𝑅𝜒∓ln^𝑚−𝑅_𝑙2−ln𝑅𝜒−𝑅_𝑚𝑅𝜒−𝑅_𝑙𝜒−𝑅+𝑅^𝑚+𝑅_𝑙2. (18)

(2) Corresponding to the periodic orbits defined by 𝐻(𝑅,𝑌)=ℎ, ℎ∈(ℎ₂,ℎ₃), enclosing the center 𝐴₂(𝑅₂,0) in Figure 2(c), we have a family of periodic wave solutions of (1). Now, 𝐹(𝑅)=(1/2)𝛽/|𝜎|(𝑟₁−𝑅)(𝑟₂−𝑅)(𝑅−𝑟₃)(𝑅−𝑟₄)with 𝑟₄<0<𝑟₃<𝑅₂<𝑟₂<𝑅₃<𝑟₁.

Hence, (10) gives that 12𝛽𝜎𝜁=^𝑅_𝑟3𝑅𝑑𝑅𝑟₁−𝑅𝑟₂−𝑅𝑅−𝑟₃𝑅−𝑟₄=^𝑅_𝑟3𝑅−𝑟₄𝑟₁−𝑅𝑟₂−𝑅𝑅−𝑟₃+𝑟₄𝑟₁−𝑅𝑟₂−𝑅𝑅−𝑅₃𝑅−𝑟₄𝑑𝑅. (19) It implies the following analytical parametric representation of periodic wave solutions of (1): 𝑅𝜒=𝑟₃+𝛼²₂𝑟₃−𝑟₄sn²𝜒,𝑘1−𝛼²₂sn²,𝜁𝜒𝜒,𝑘=2|𝜎|𝛽𝑟₄𝑔𝜒+𝑟₃−𝑟₄Πarcsinsn𝜒,𝑘,𝛼²₂,𝑘, (20) where 𝑔=2/(𝑟₁−𝑟₃)(𝑟₂−𝑟₄), 𝛼²₂=(𝑟₂−𝑟₃)/(𝑟₂−𝑟₄), 𝑘²=(𝑟₂−𝑟₃)(𝑟₁−𝑟₄)/(𝑟₁−𝑟₃)(𝑟₂−𝑟₄).

Notice that for a given parameter group of (𝛽,𝐺₁,𝜎), when 𝐷² is close to 𝐷²_𝑚 value, (18) and (20) give rise to solitary cusp wave solution (antipeakon) and periodic cusp wave solutions (see Section 3), respectively.

5. Conclusion

In this paper, we use the method of dynamical systems to investigate the bifurcations of phase portraits of nonlocal hydrodynamic-type models. Depending on the different parameters conditions of the systems, the existence of various travelling wave solutions is discussed. Specially, for 𝑛=2, namely, let the isochoric Gruneisen coefficient equal 1, some analytical parametric representations of solutions such as smooth bright solitary, smooth and nonsmooth dark solitary wave, and periodic wave solutions, as well as uncountably infinitely many breaking wave solutions are obtained. These results develop and complete the corresponding studies for the nonlocal hydrodynamic-type models.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (10831003). The authors thank the anonymous reviewers for their valuable comments.