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

Authors: Jianping Shi; Jibin Li
Journal: Abstract and Applied Analysis
 (2014)

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## 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 parametern 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.


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## Body

## 1. Introduction

A hydrodynamic system of balance equations for mass and momentum is considered by Vladimirov et al. [1]:
(1)ut+px=ℑρ,ρt+ρ2ux=0,τ^pt+p=βν+2ρν+2+σ[ρν+1ρxx+(ν+1)ρν(ρx)2],
where ν=n-2, n=1+ΓV∞, ΓV∞ 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:
(2)R˙=Y,σRν+2Y˙=GR-[D2+βν+2Rν+3+σ(ν+1)Rν+1Y2]-τϵD3RY,G˙=ϵ[f(R)-ξ(C1+1-νD1+νR)],
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.Whenϵ=0, it immediately obtains that G=G1=const, and the system (2) reduces to the following two-dimensional system:
(3)dRdζ=Y,dYdζ=G1R-[D2+(β/(ν+2))Rν+3+σ(ν+1)Rν+1Y2]σRν+2,
where ζ=x-Dt.Assume thatAj(Rj,0), j=1,2, are two equilibrium points of system (3). Vladimirov et al. obtained the following conclusion (see [1, 3]).Theorem A.
Ifν>-2 and D2>βR1ν+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 fixedν, 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 ofν, we use n=ν+2 to rewrite (3) as follows:
(4)dRdζ=Y,dYdζ=G1R-[D2+(β/n)Rn+1+σ(n-1)Rn-1Y2]σRn.This system has the first integral:(5)H(R,Y)=Y2R2n-2+2Rn-1nσ[nD2n-1-G1R+β2nRn+1]=h.
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 Section2, 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):
(6)dRdη=σRnY,dYdη=G1R-[D2+βnRn+1+σ(n-1)Rn-1Y2],
where dζ=σRndη. 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:(7)f(R)=βnRn+1-G1R+D2,f′(R)=(n+1)βnRn-G1.
Obviously, for n=2m, m=1,2,…, when βG1>0, and R=RM±≡±(nG1/(n+1)β)1/n, f′(RM±)=0, while for n=2m+1, m=1,2,…, when R=RM≡(nG1/(n+1)β)1/n, f′(RM)=0; we have f(RM)=f(RM+)=f(RM-)=D2-(nG1/(n+1))RM. When D2=Dcr2≡(nG1/(n+1))(nG1/(n+1)β)1/n, f(RM)=0.Since every real rootRj of the function f(R) gives rise to an equilibrium point Aj(Rj,0) of system (6), by using (7), we can analyse the equilibrium points Aj(Rj,0) of system (6) in different parameter conditions and have the following conclusions.


### 2.1. In the Case ofn=2m.

Consider the following.(1)
Forβ>0 and G1>0, when D2<(nG1/(n+1))RM, andf(RM-)>0, f(RM+)<0. There exist three equilibrium points Aj(Rj,0), j=1,2,3,of system (6), satisfying -∞<R1<RM-<0<R2<RM+<R3<∞. When D2=(nG1/(n+1))RM, f(RM-)>0 and f(RM+)=0. There exist two equilibrium points Aj(Rj,0), j=1,2, of system (6), satisfying -∞<R1<RM-<0<R2=RM+<∞. When D2>(nG1/(n+1))RM, f(RM-)>0, andf(RM+)>0. There exists only one equilibrium point A1(R1,0) of system (6), satisfying -∞<R1<RM-<0.(2)
Forβ<0, and G1<0, when D2<(n|G1|/(n+1))RM, f(RM-)<0, and f(RM+)>0. There exist three equilibrium points Aj(Rj,0), j=1,2,3, of system (6), satisfying -∞<R1<RM-<R2<0<RM+<R3<∞. When D2=(n|G1|/(n+1))RM, f(RM-)=0, and f(RM+)>0. There exist two equilibrium points Aj(Rj,0), j=1,2, of system (6), satisfying -∞<R1=RM-<0<RM+<R2<∞. When D2>(n|G1|/(n+1))RM, f(RM-)>0, and f(RM+)>0. There exists only one equilibrium point A1(R1,0) of system (6), satisfying 0<RM+<R1<∞.(3)
Forβ>0, G1<0, and any givenD, σ, because the number RMis not real, there exists one equilibrium point A1(R1,0) of system (6), satisfying -∞<R1<0.(4)
Forβ<0, G1>0, and any given D, σ, there exists one equilibrium point A1(R1,0) of system (6), satisfying 0<R1<∞.LetM(Rj,0) be the coefficient matrix of the linearized system of system (6) at an equilibrium point Aj(Rj,0). We have
(8)J(Rj,0)=det⁡M(Rj,0)=σRjnf′(Rj).By the theory of planar dynamical systems, for an equilibrium point of a planar integrable system, ifJ<0, then the equilibrium point is a saddle point; if J>0, and (TraceM(Rj,0))2-4J(Rj,0)>0, then it is a node point; if J>0, and TraceM(Rj,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 hj=H(Rj,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.The phase portraits of system (6) for n=2m, β>0, G1>0, and σ>0.
(a)
D
2
>
D
c
r
2
(b)
D
2
=
D
c
r
2
(c)
D
2
<
D
c
r
2The phase portraits of system (6) for n=2m, β>0, G1>0, and σ<0.
(a)
D
2
>
D
c
r
2
(b)
D
2
=
D
c
r
2
(c)
D
m
2
<
D
2
<
D
c
r
2The phase portraits of system (6) for n=2m, β<0, G1<0, and σ>0.
(a)
D
2
>
D
c
r
2
(b)
D
2
=
D
c
r
2
(c)
D
m
2
<
D
2
<
D
c
r
2The phase portraits of system (6) for n=2m, β<0, G1<0, and σ<0.
(a)
D
2
>
D
c
r
2
(b)
D
2
=
D
c
r
2
(c)
D
2
<
D
c
r
2The phase portraits of system (6) for n=2m, and βG1<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<0For the case ofβ<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 ofn=2m+1

Consider the following.(1)
Forβ>0, and G1>0, when D2<(nG1/(n+1))RM, and f(RM)<0, there exist two equilibrium points Aj(Rj,0), j=1,2, of system (6), satisfying 0<R1<RM<R2<∞. When D2=(nG1/(n+1))RM, and f(RM)=0, there exists one equilibrium point A1(R1,0) of system (6), satisfying 0<R1=RM<∞. When D2>(nG1/(n+1))RM, and f(RM)>0, system (6) has no equilibrium point.(2)
Forβ>0, G1<0, when D2<(nG1/(n+1))RM, and f(RM)<0, there exist two equilibrium points Aj(Rj,0), j=1,2, of system (6), satisfying -∞<R1<RM<R2<0. When D2=(nG1/(n+1))RM, and f(RM)=0, there exists one equilibrium point A1(R1,0) of (6), satisfying -∞<R1=RM<0. When D2>(nG1/(n+1))RM, and f(RM)>0, system (6) has no equilibrium point.(3)
Forβ<0, G1>0, and any given D, σ, there exist two equilibrium points Aj(Rj,0), j=1,2, of system (6), satisfying -∞<R1<RM<0<R2.(4)
Forβ<0, G1<0, and any given D, σ, there exist two equilibrium points Aj(Rj,0), j=1,2, of system (6), satisfying -∞<R1<0<RM<R2<∞.Same as Section2.1, we have the following bifurcations of phase portraits of system (6) shown in Figures 6–10.The phase portraits of system (6) for n=2m+1, β>0, G1>0, and σ>0.
(a)
D
2
>
D
c
r
2
(b)
D
2
=
D
c
r
2
(c)
D
2
<
D
c
r
2The phase portraits of system (6) for n=2m+1, β>0, G1>0, and σ<0.
(a)
D
2
>
D
c
r
2
(b)
D
2
=
D
c
r
2
(c)
D
m
2
<
D
2
<
D
c
r
2The phase portraits of system (6) for n=2m+1, β>0, G1<0, and σ>0.
(a)
D
2
>
D
c
r
2
(b)
D
2
=
D
c
r
2
(c)
D
2
<
D
c
r
2The phase portraits of system (6) for n=2m+1, β>0, G1<0, and σ<0.
(a)
D
2
>
D
c
r
2
(b)
D
2
=
D
c
r
2
(c)
D
m
2
<
D
2
<
D
c
r
2The phase portraits of system (6) for n=2m+1, and β<0.
(a)
G
1
>
0
,
σ
>
0
(b)
G
1
>
0
,
σ
<
0
(c)
G
1
<
0
,
σ
>
0
(d)
G
1
<
0
,
σ
<
0


## 2.1. In the Case ofn=2m.

Consider the following.(1)
Forβ>0 and G1>0, when D2<(nG1/(n+1))RM, andf(RM-)>0, f(RM+)<0. There exist three equilibrium points Aj(Rj,0), j=1,2,3,of system (6), satisfying -∞<R1<RM-<0<R2<RM+<R3<∞. When D2=(nG1/(n+1))RM, f(RM-)>0 and f(RM+)=0. There exist two equilibrium points Aj(Rj,0), j=1,2, of system (6), satisfying -∞<R1<RM-<0<R2=RM+<∞. When D2>(nG1/(n+1))RM, f(RM-)>0, andf(RM+)>0. There exists only one equilibrium point A1(R1,0) of system (6), satisfying -∞<R1<RM-<0.(2)
Forβ<0, and G1<0, when D2<(n|G1|/(n+1))RM, f(RM-)<0, and f(RM+)>0. There exist three equilibrium points Aj(Rj,0), j=1,2,3, of system (6), satisfying -∞<R1<RM-<R2<0<RM+<R3<∞. When D2=(n|G1|/(n+1))RM, f(RM-)=0, and f(RM+)>0. There exist two equilibrium points Aj(Rj,0), j=1,2, of system (6), satisfying -∞<R1=RM-<0<RM+<R2<∞. When D2>(n|G1|/(n+1))RM, f(RM-)>0, and f(RM+)>0. There exists only one equilibrium point A1(R1,0) of system (6), satisfying 0<RM+<R1<∞.(3)
Forβ>0, G1<0, and any givenD, σ, because the number RMis not real, there exists one equilibrium point A1(R1,0) of system (6), satisfying -∞<R1<0.(4)
Forβ<0, G1>0, and any given D, σ, there exists one equilibrium point A1(R1,0) of system (6), satisfying 0<R1<∞.LetM(Rj,0) be the coefficient matrix of the linearized system of system (6) at an equilibrium point Aj(Rj,0). We have
(8)J(Rj,0)=det⁡M(Rj,0)=σRjnf′(Rj).By the theory of planar dynamical systems, for an equilibrium point of a planar integrable system, ifJ<0, then the equilibrium point is a saddle point; if J>0, and (TraceM(Rj,0))2-4J(Rj,0)>0, then it is a node point; if J>0, and TraceM(Rj,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 hj=H(Rj,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.The phase portraits of system (6) for n=2m, β>0, G1>0, and σ>0.
(a)
D
2
>
D
c
r
2
(b)
D
2
=
D
c
r
2
(c)
D
2
<
D
c
r
2The phase portraits of system (6) for n=2m, β>0, G1>0, and σ<0.
(a)
D
2
>
D
c
r
2
(b)
D
2
=
D
c
r
2
(c)
D
m
2
<
D
2
<
D
c
r
2The phase portraits of system (6) for n=2m, β<0, G1<0, and σ>0.
(a)
D
2
>
D
c
r
2
(b)
D
2
=
D
c
r
2
(c)
D
m
2
<
D
2
<
D
c
r
2The phase portraits of system (6) for n=2m, β<0, G1<0, and σ<0.
(a)
D
2
>
D
c
r
2
(b)
D
2
=
D
c
r
2
(c)
D
2
<
D
c
r
2The phase portraits of system (6) for n=2m, and βG1<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<0For the case ofβ<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 ofn=2m+1

Consider the following.(1)
Forβ>0, and G1>0, when D2<(nG1/(n+1))RM, and f(RM)<0, there exist two equilibrium points Aj(Rj,0), j=1,2, of system (6), satisfying 0<R1<RM<R2<∞. When D2=(nG1/(n+1))RM, and f(RM)=0, there exists one equilibrium point A1(R1,0) of system (6), satisfying 0<R1=RM<∞. When D2>(nG1/(n+1))RM, and f(RM)>0, system (6) has no equilibrium point.(2)
Forβ>0, G1<0, when D2<(nG1/(n+1))RM, and f(RM)<0, there exist two equilibrium points Aj(Rj,0), j=1,2, of system (6), satisfying -∞<R1<RM<R2<0. When D2=(nG1/(n+1))RM, and f(RM)=0, there exists one equilibrium point A1(R1,0) of (6), satisfying -∞<R1=RM<0. When D2>(nG1/(n+1))RM, and f(RM)>0, system (6) has no equilibrium point.(3)
Forβ<0, G1>0, and any given D, σ, there exist two equilibrium points Aj(Rj,0), j=1,2, of system (6), satisfying -∞<R1<RM<0<R2.(4)
Forβ<0, G1<0, and any given D, σ, there exist two equilibrium points Aj(Rj,0), j=1,2, of system (6), satisfying -∞<R1<0<RM<R2<∞.Same as Section2.1, we have the following bifurcations of phase portraits of system (6) shown in Figures 6–10.The phase portraits of system (6) for n=2m+1, β>0, G1>0, and σ>0.
(a)
D
2
>
D
c
r
2
(b)
D
2
=
D
c
r
2
(c)
D
2
<
D
c
r
2The phase portraits of system (6) for n=2m+1, β>0, G1>0, and σ<0.
(a)
D
2
>
D
c
r
2
(b)
D
2
=
D
c
r
2
(c)
D
m
2
<
D
2
<
D
c
r
2The phase portraits of system (6) for n=2m+1, β>0, G1<0, and σ>0.
(a)
D
2
>
D
c
r
2
(b)
D
2
=
D
c
r
2
(c)
D
2
<
D
c
r
2The phase portraits of system (6) for n=2m+1, β>0, G1<0, and σ<0.
(a)
D
2
>
D
c
r
2
(b)
D
2
=
D
c
r
2
(c)
D
m
2
<
D
2
<
D
c
r
2The phase portraits of system (6) for n=2m+1, and β<0.
(a)
G
1
>
0
,
σ
>
0
(b)
G
1
>
0
,
σ
<
0
(c)
G
1
<
0
,
σ
>
0
(d)
G
1
<
0
,
σ
<
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) Ifβ>0, G1>0, σ>0, and D2<Dcr2≡(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 anyn=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) Ifβ>0, G1>0, σ<0, and Dm2<D2<Dcr2≡(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 anyn=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) Ifβ<0, G1<0, σ<0(or β>0, G1<0, σ>0), andD2<Dcr2≡(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) Ifβ<0, G1<0, σ>0 (or β>0, G1<0, σ<0), and Dm2<D2<Dcr2≡(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∈(hj,0), enclosing the center point Aj(Rj,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∈(hj,0), j=1,2, enclosing the center point Aj(Rj,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 Figure3(c), for every fixed n,β, σ, and G1, when parameter D2 decreases from Dcr2 to 0, there exists a value D2=Dm2 (where Dm2 satisfies H(R1,0)=h1=0) such that when D2<Dm2, 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 Dm2can be numerically determined.The changes of phase portraits of system (6) when D2 are decreased. Parameters: n=2m, β<0, G1<0, and σ>0.
(a)
D
2
<
D
c
r
2
(b)
D
m
2
<
D
2
<
D
c
r
2
(c)
D
2
<
D
m
2We see from Section2 that if n=2m, when β>0, G1>0, σ<0, and Dm<D2<Dcr2, system (4) has the phase portrait Figure 2(c), while when β<0, G1<0, σ>0, and Dm<D2<Dcr2, system (4) has the phase portrait Figure 3(c). Similarly, if n=2m+1, when β>0, G1>0, σ<0, and Dm<D2<Dcr2, system (4) has the phase portrait Figure 7(c), while when β>0, G1<0, σ<0and Dm<D2<Dcr2, 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 Dcr2 to Dm, 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∈(Dm2,Dcr2), 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)).The profiles of pseudo-peakon and periodic cusp wave solutions of (1).
(a)
Peakon
(b)
Smooth wave
(c)
Cusp wave
(d)
Cusp waveHence, we have the following conclusion.Theorem 4.
Under the above mentioned parameter conditions of Figure2(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 Dm, there exist solitary cusp wave solutions (pseudo-peakons) and periodic cusp wave solutions of (1).


## 3.1. The Existence of Periodic Solutions

Theorem 1.
(1) Ifβ>0, G1>0, σ>0, and D2<Dcr2≡(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 anyn=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) Ifβ>0, G1>0, σ<0, and Dm2<D2<Dcr2≡(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 anyn=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) Ifβ<0, G1<0, σ<0(or β>0, G1<0, σ>0), andD2<Dcr2≡(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) Ifβ<0, G1<0, σ>0 (or β>0, G1<0, σ<0), and Dm2<D2<Dcr2≡(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∈(hj,0), enclosing the center point Aj(Rj,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∈(hj,0), j=1,2, enclosing the center point Aj(Rj,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 Figure3(c), for every fixed n,β, σ, and G1, when parameter D2 decreases from Dcr2 to 0, there exists a value D2=Dm2 (where Dm2 satisfies H(R1,0)=h1=0) such that when D2<Dm2, 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 Dm2can be numerically determined.The changes of phase portraits of system (6) when D2 are decreased. Parameters: n=2m, β<0, G1<0, and σ>0.
(a)
D
2
<
D
c
r
2
(b)
D
m
2
<
D
2
<
D
c
r
2
(c)
D
2
<
D
m
2We see from Section2 that if n=2m, when β>0, G1>0, σ<0, and Dm<D2<Dcr2, system (4) has the phase portrait Figure 2(c), while when β<0, G1<0, σ>0, and Dm<D2<Dcr2, system (4) has the phase portrait Figure 3(c). Similarly, if n=2m+1, when β>0, G1>0, σ<0, and Dm<D2<Dcr2, system (4) has the phase portrait Figure 7(c), while when β>0, G1<0, σ<0and Dm<D2<Dcr2, 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 Dcr2 to Dm, 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∈(Dm2,Dcr2), 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)).The profiles of pseudo-peakon and periodic cusp wave solutions of (1).
(a)
Peakon
(b)
Smooth wave
(c)
Cusp wave
(d)
Cusp waveHence, we have the following conclusion.Theorem 4.
Under the above mentioned parameter conditions of Figure2(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 Dm, there exist solitary cusp wave solutions (pseudo-peakons) and periodic cusp wave solutions of (1).


## 4. Analytical Travelling Wave Solutions forn=2

According to (5), for a fixed integral constant h, we have
(9)y=1Rn-1(h-2Rn-1nσ[nD2n-1-G1R+β2nRn+1])1/2.
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
(10)ζ-ζ0=∫R0R‍Rn-1dRF(R)≡∫R0R‍Rn-1dRh-(2Rn-1/nσ)[nD2/(n-1)-G1R+(β/2n)Rn+1].
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 Caseβ>0, G1>0, and σ>0

Suppose thatD2<Dcr2≡(2G1/3)(2G1/3β)1/2 (see Figure 1(c)).(1) Corresponding to the homoclinic orbit to the saddle pointA2(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(Rm-R)(R-R2)2(R-Rl) with Rl<0<R2<R3<Rm. Thus, (10) has the following form:
(11)12βσζ=∫RmR‍RdR(R-R2)(Rm-R)(R-Rl)=∫RmR[1(Rm-R)(R-Rl)‍vvvvvvv+R2(R-R2)(Rm-R)(R-Rl)]dR.The following analytical parametric representation of the bright solitary wave solution of (1) can be obtained:
(12)R(χ)=R2+2(Rm-R2)(R2-Rl)(Rm-Rl)cosh⁡(ωζ)-(Rm-2R2+Rl),ζ(χ)=2σβ[R2χ-π2-arcsin(Rm+Rl-2R(χ)Rm-Rl)],vvvvvvvvvvvvvvvvvvvvvvvvvwvforχ∈(-∞,0],ζ(χ)=2σβ[R2χ+π2+arcsin(Rm+Rl-2R(χ)Rm-Rl)],vvvvvvvvvvvvvvvvvvvvvvvvvvvvforχ∈[0,∞),
where ω=(Rm-R2)(R2-Rl).(2) Corresponding to the periodic orbits defined byH(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<Rm.Hence, (10) gives that
(13)12βσζ=∫r2R‍RdR(r1-R)(R-r2)(R-r3)(R-r4)=∫r2R[r3(r1-R)(R-r2)(R-r3)(R-r4)R-r3(r1-R)(R-r2)(R-r4)‍vvvvvi+r3(r1-R)(R-r2)(R-r3)(R-r4)R-r3(r1-R)(R-r2)(R-r4)]dR.
It implies the following analytical parametric representation of periodic wave solutions of (1):
(14)R(χ)=r3+r2-r31-α2sn2(χ,k),ζ(χ)=2σβ[r3gχ+(r2-r3)Π(arcsin(sn(χ,k)),α2,k)],
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 byH(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<Rm. 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:
(15)12βσζ=∫Rr3‍RdR(r1-R)(r2-R)(r3-R)(R-r4)=∫Rr3‍[r2(r1-R)(r2-R)(r3-R)(R-r4)-r2-R(r1-R)(r3-R)(R-r4)-r2-R(r1-R)(r3-R)(R-r4)vvvvvvv+r2(r1-R)(r2-R)(r3-R)(R-r4)-r2-R(r1-R)(r3-R)(R-r4)]dR.
It implies the following analytical parametric representation of the left family of periodic wave solutions of (1):
(16)R(χ)=r3-α12(r2-r3)sn2(χ,k)1-α12sn2(χ,k),ζ(χ)=2σβ[r2gχ-(r2-r3)Π(arcsin(sn2(χ,k)),α12,k)],
where g=2/(r1-r3)(r2-r4), α12=(r3-r4)/(r2-r4), k2=(r1-r2)(r3-r4)/(r1-r3)(r2-r4).(4) Corresponding to two families of open level curves defined byH(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<Rm. Thus, the left family of open level curves has the same parametric representation as (16) with -χ0<χ<χ0,χ0=sn-1(r3/α12r2,k).The level curves and profiles of breaking wave solutions of (1).
(a)
Level curves
(b)
(c)Figures13(b) and 13(c) show the two families of uncountably infinitely many breaking wave solutions (compactons) of (1).


### 4.2. In the Caseβ>0, G1>0, σ<0

Suppose thatDm<D2<Dcr2≡(2G1/3)(2G1/3β)1/2 (see Figure 2(c)).(1) Corresponding to the homoclinic orbit to the saddle pointA3(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-RM)(R-Rl) with Rl<0<Rm<R2<R3. Thus, formula (10) has the following form:
(17)12β|σ|ζ=∫RmR‍RdR(R3-R)(R-Rm)(R-Rl)=∫RmR[-1(R-Rm)(R-Rl)‍vvvvvvv+R3(R3-R)(R-Rm)(R-Rl)]dR.It implies the following analytical parametric representation of the dark solitary wave solution of (1):
(18)R(χ)=R3-2(R3-Rm)(R3-Rl)(Rm-Rl)cosh⁡(ω1ζ)+(2R3-Rm-Rl),vvvvvvvforχ∈(-∞,0],χ∈[0,∞),respectively,ζ(χ)=2|σ|β[R3χ∓(+R(χ)-Rm+Rl2))ln⁡(Rm-Rl2)vvvvvvvvvvvvvvvvv-ln⁡(Rm-Rl2(R(χ)-Rm)(R(χ)-Rl)vvvvvvvvvvvvvvvvvvvvvvv+R(χ)-Rm+Rl2))].(2) Corresponding to the periodic orbits defined byH(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
(19)12βσζ=∫r3R‍RdR(r1-R)(r2-R)(R-r3)(R-r4)=∫r3R[r4(r1-R)(r2-R)(R-R3)(R-r4)R-r4(r1-R)(r2-R)(R-r3)‍vvvvvv+r4(r1-R)(r2-R)(R-R3)(R-r4)R-r4(r1-R)(r2-R)(R-r3)]dR.
It implies the following analytical parametric representation of periodic wave solutions of (1):
(20)R(χ)=r3+α22(r3-r4)sn2(χ,k)1-α22sn2(χ,k),ζ(χ)=2|σ|β[r4gχ+(r3-r4)Π(arcsin(sn(χ,k)),α22,k)],
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 Dm2 value, (18) and (20) give rise to solitary cusp wave solution (antipeakon) and periodic cusp wave solutions (see Section 3), respectively.


## 4.1. In the Caseβ>0, G1>0, and σ>0

Suppose thatD2<Dcr2≡(2G1/3)(2G1/3β)1/2 (see Figure 1(c)).(1) Corresponding to the homoclinic orbit to the saddle pointA2(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(Rm-R)(R-R2)2(R-Rl) with Rl<0<R2<R3<Rm. Thus, (10) has the following form:
(11)12βσζ=∫RmR‍RdR(R-R2)(Rm-R)(R-Rl)=∫RmR[1(Rm-R)(R-Rl)‍vvvvvvv+R2(R-R2)(Rm-R)(R-Rl)]dR.The following analytical parametric representation of the bright solitary wave solution of (1) can be obtained:
(12)R(χ)=R2+2(Rm-R2)(R2-Rl)(Rm-Rl)cosh⁡(ωζ)-(Rm-2R2+Rl),ζ(χ)=2σβ[R2χ-π2-arcsin(Rm+Rl-2R(χ)Rm-Rl)],vvvvvvvvvvvvvvvvvvvvvvvvvwvforχ∈(-∞,0],ζ(χ)=2σβ[R2χ+π2+arcsin(Rm+Rl-2R(χ)Rm-Rl)],vvvvvvvvvvvvvvvvvvvvvvvvvvvvforχ∈[0,∞),
where ω=(Rm-R2)(R2-Rl).(2) Corresponding to the periodic orbits defined byH(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<Rm.Hence, (10) gives that
(13)12βσζ=∫r2R‍RdR(r1-R)(R-r2)(R-r3)(R-r4)=∫r2R[r3(r1-R)(R-r2)(R-r3)(R-r4)R-r3(r1-R)(R-r2)(R-r4)‍vvvvvi+r3(r1-R)(R-r2)(R-r3)(R-r4)R-r3(r1-R)(R-r2)(R-r4)]dR.
It implies the following analytical parametric representation of periodic wave solutions of (1):
(14)R(χ)=r3+r2-r31-α2sn2(χ,k),ζ(χ)=2σβ[r3gχ+(r2-r3)Π(arcsin(sn(χ,k)),α2,k)],
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 byH(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<Rm. 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:
(15)12βσζ=∫Rr3‍RdR(r1-R)(r2-R)(r3-R)(R-r4)=∫Rr3‍[r2(r1-R)(r2-R)(r3-R)(R-r4)-r2-R(r1-R)(r3-R)(R-r4)-r2-R(r1-R)(r3-R)(R-r4)vvvvvvv+r2(r1-R)(r2-R)(r3-R)(R-r4)-r2-R(r1-R)(r3-R)(R-r4)]dR.
It implies the following analytical parametric representation of the left family of periodic wave solutions of (1):
(16)R(χ)=r3-α12(r2-r3)sn2(χ,k)1-α12sn2(χ,k),ζ(χ)=2σβ[r2gχ-(r2-r3)Π(arcsin(sn2(χ,k)),α12,k)],
where g=2/(r1-r3)(r2-r4), α12=(r3-r4)/(r2-r4), k2=(r1-r2)(r3-r4)/(r1-r3)(r2-r4).(4) Corresponding to two families of open level curves defined byH(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<Rm. Thus, the left family of open level curves has the same parametric representation as (16) with -χ0<χ<χ0,χ0=sn-1(r3/α12r2,k).The level curves and profiles of breaking wave solutions of (1).
(a)
Level curves
(b)
(c)Figures13(b) and 13(c) show the two families of uncountably infinitely many breaking wave solutions (compactons) of (1).


## 4.2. In the Caseβ>0, G1>0, σ<0

Suppose thatDm<D2<Dcr2≡(2G1/3)(2G1/3β)1/2 (see Figure 2(c)).(1) Corresponding to the homoclinic orbit to the saddle pointA3(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-RM)(R-Rl) with Rl<0<Rm<R2<R3. Thus, formula (10) has the following form:
(17)12β|σ|ζ=∫RmR‍RdR(R3-R)(R-Rm)(R-Rl)=∫RmR[-1(R-Rm)(R-Rl)‍vvvvvvv+R3(R3-R)(R-Rm)(R-Rl)]dR.It implies the following analytical parametric representation of the dark solitary wave solution of (1):
(18)R(χ)=R3-2(R3-Rm)(R3-Rl)(Rm-Rl)cosh⁡(ω1ζ)+(2R3-Rm-Rl),vvvvvvvforχ∈(-∞,0],χ∈[0,∞),respectively,ζ(χ)=2|σ|β[R3χ∓(+R(χ)-Rm+Rl2))ln⁡(Rm-Rl2)vvvvvvvvvvvvvvvvv-ln⁡(Rm-Rl2(R(χ)-Rm)(R(χ)-Rl)vvvvvvvvvvvvvvvvvvvvvvv+R(χ)-Rm+Rl2))].(2) Corresponding to the periodic orbits defined byH(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
(19)12βσζ=∫r3R‍RdR(r1-R)(r2-R)(R-r3)(R-r4)=∫r3R[r4(r1-R)(r2-R)(R-R3)(R-r4)R-r4(r1-R)(r2-R)(R-r3)‍vvvvvv+r4(r1-R)(r2-R)(R-R3)(R-r4)R-r4(r1-R)(r2-R)(R-r3)]dR.
It implies the following analytical parametric representation of periodic wave solutions of (1):
(20)R(χ)=r3+α22(r3-r4)sn2(χ,k)1-α22sn2(χ,k),ζ(χ)=2|σ|β[r4gχ+(r3-r4)Π(arcsin(sn(χ,k)),α22,k)],
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 Dm2 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, forn=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.



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