Solitons, Peakons, and Periodic Cuspons of a Generalized Degasperis-Procesi Equation

Abstract

We employ the bifurcation theory of planar dynamical systems to investigate the exact travelling wave solutions of a generalized Degasperis-Procesi equation 𝑢_𝑡−𝑢_𝑥𝑥𝑡+4𝑢𝑢_𝑥+𝛾(𝑢−𝑢_𝑥𝑥)_𝑥=3𝑢_𝑥𝑢_𝑥𝑥+𝑢𝑢_𝑥𝑥𝑥. The implicit expression of smooth soliton solutions is given. The explicit expressions of peaked soliton solutions and periodic cuspon solutions are also obtained. Further, we show the relationship among the smooth soliton solutions, the peaked soliton solutions, and the periodic cuspon solutions. The physical relevance of the found solutions and the reason why these solutions can exist in this equation are also given.

1. Introduction

Recently, Degasperis and Procesi [1] derived a nonlinear dispersive equation 𝑢_𝑡−𝑢_𝑥𝑥𝑡+4𝑢𝑢_𝑥=3𝑢_𝑥𝑢_𝑥𝑥+𝑢𝑢_𝑥𝑥𝑥 (1.1) which is called the Degasperis-Procesi equation. Here 𝑢(𝑡,𝑥) represents the fluid velocity at time 𝑡 in the 𝑥 direction in appropriate nondimensional units (or, equivalently the height of the water's free surface above a flat bottom). The nonlinear convection term 𝑢𝑢_𝑥 in (1.1) causes the steepening of wave form, whereas the nonlinear dispersion effect term 3𝑢_𝑥𝑢_𝑥𝑥+𝑢𝑢_𝑥𝑥𝑥=((1/2)𝑢²)_𝑥𝑥𝑥 in (1.1) makes the wave form spread. Equation (1.1) can be regarded as a model for nonlinear shallow water dynamics. Degasperis et al. [2] showed that the (1.1) is integrable by deriving a Lax pair and a bi-Hamiltonian structure for it. Yin proved local well posedness to (1.1) with initial data 𝑢₀∈𝐻^𝑠(ℝ), 𝑠>3 on the line [3] and on the circle [4]. The global existence of strong solutions and weak solutions to (1.1) is investigated in [4–10]. The solution to Cauchy problem of (1.1) can also blow up in finite time when the initial data satisfies certain sign condition[7–10]. Vakhnenko and Parkes [11] obtained periodic and solitary-wave solutions of (1.1). Matsuno [12, 13] obtained multisoliton, cusp and loop soliton solutions of (1.1). Lundmark and Szmigielski [14] investigated multipeakon solutions of (1.1). Lenells [15] classified all weak travelling wave solutions. The shock wave solutions of (1.1) are investigated in [16, 17].

Yu and Tian [18] investigated the following generalized Degasperis-Procesi equation: 𝑢_𝑡−𝑢_𝑥𝑥𝑡+4𝑢𝑢_𝑥=3𝑢_𝑥𝑢_𝑥𝑥+𝑢𝑢_𝑥𝑥𝑥−𝛾𝑢_𝑥𝑥𝑥, (1.2) where 𝛾 is a real constant, and the term 𝑢_𝑥𝑥𝑥 denotes the linear dispersive effect. They obtained peaked soliton solutions and period cuspon solutions of (1.2). Unfortunately, they did not obtain smooth soliton solutions of (1.2).

In this paper, we are interesting in the following generalized Degasperis-Procesi equation: 𝑢_𝑡−𝑢_𝑥𝑥𝑡+4𝑢𝑢_𝑥+𝛾(𝑢−𝑢_𝑥𝑥)_𝑥=3𝑢_𝑥𝑢_𝑥𝑥+𝑢𝑢_𝑥𝑥𝑥, (1.3) where 𝛾 is a real constant, the term 𝑢_𝑥 denotes the dissipative effect and the term 𝑢_𝑥𝑥𝑥 represents the linear dispersive effect. Employing the bifurcation theory of planar dynamical systems, we obtain the analytic expressions of smooth solitons, peaked solitons, and period cuspons of (1.3). Our work covers and supplements the results obtained in [18].

The remainder of the paper is organized as follows. In Section 2, using the travelling wave transformation, we transform (1.3) into the planar dynamical system (2.3) and then discuss bifurcations of phase portraits of system (2.3). In Section 3, we obtain the implicit expression of smooth solitons, the explicit expressions of peaked solitons and periodic cuspon solutions. At the same time, we show that the limits of smooth solitons and periodic cusp waves are peaked solitons. In Section 4, we discuss the physical relevance of the found solutions and give the reason why these solutions can exist in (1.3).

2. Bifurcations of Phase Portraits of System (2.3)

We look for travelling wave solutions of (1.3) in the form of 𝑢(𝑥,𝑡)=𝜑(𝑥−𝑐𝑡)=𝜑(𝜉), where 𝑐 is the wave speed and 𝜉=𝑥−𝑐𝑡. Substituting 𝑢=𝜑(𝜉) into (1.3), we obtain −𝑐𝜑^′+𝑐𝜑^′′′+4𝜑𝜑^′−3𝜑^′𝜑^′′−𝜑𝜑^′′′+𝛾𝜑^′−𝛾𝜑^′′′=0. (2.1)

By integrating (2.1) once we have 𝜑^′′(𝜑−𝑐+𝛾)=𝑔−(𝑐−𝛾)𝜑+2𝜑²−(𝜑^′)², (2.2) where 𝑔 is the integral constant.

Let 𝑦=𝜑^′, then we get the following planar dynamical system: 𝑑𝜑𝑑𝜉=𝑦,𝑑𝑦=𝑑𝜉𝑔−(𝑐−𝛾)𝜑+2𝜑²−𝑦²,𝜑−𝑐+𝛾 (2.3) with a first integral 𝐻(𝜑,𝑦)=(𝜑−𝑐+𝛾)²(𝑦²−𝜑²−𝑔)=ℎ, (2.4) where ℎ is a constant.

Note that (2.3) has a singular line 𝜑=𝑐−𝛾. To avoid the line temporarily we make transformation 𝑑𝜉=(𝜑−𝑐+𝛾)𝑑𝜁. Under this transformation, (2.3) becomes 𝑑𝜑𝑑𝜁=(𝜑−𝑐+𝛾)𝑦,𝑑𝑦𝑑𝜁=𝑔−(𝑐−𝛾)𝜑+2𝜑²−𝑦². (2.5)

System (2.3) and system (2.5) have the same first integral as (2.4). Consequently, system (2.5) has the same topological phase portraits as system (2.3) except for the straight line 𝜑=𝑐−𝛾. Obviously, 𝜑=𝑐−𝛾 is an invariant straight-line solution for system (2.5).

For a fixed ℎ, (2.4) determines a set of invariant curves of system (2.5). As ℎ is varied, (2.4) determines different families of orbits of system (2.5) having different dynamical behaviors. Let 𝑀(𝜑_𝑒,𝑦_𝑒) be the coefficient matrix of the linearized system of (2.5) at the equilibrium point (𝜑_𝑒,𝑦_𝑒), then

𝑀(𝜑_𝑒,𝑦_𝑒𝑦)=^𝑒𝜑_𝑒−𝑐+𝛾4𝜑_𝑒−𝑐+𝛾−2𝑦_𝑒, (2.6) and at this equilibrium point, we have 𝐽(𝜑_𝑒,𝑦_𝑒)=det𝑀(𝜑_𝑒,𝑦_𝑒)=−2𝑦²_𝑒−(𝜑_𝑒−𝑐+𝛾)(4𝜑_𝑒−𝑐+𝛾),𝑝(𝜑_𝑒,𝑦_𝑒)=trace(𝑀(𝜑_𝑒,𝑦_𝑒))=−𝑦_𝑒. (2.7) By the qualitative theory of differential equations (see [19]), for an equilibrium point of a planar dynamical system, if 𝐽<0, then this equilibrium point is a saddle point; it is a center point if 𝐽>0 and 𝑝=0; if 𝐽=0 and the Poincaré index of the equilibrium point is 0, then it is a cusp.

By using the first integral value and properties of equilibrium points, we obtain the bifurcation curves as follows: 𝑔₁(𝑐)=(𝑐−𝛾)²8,𝑔₂(𝑐)=−(𝑐−𝛾)². (2.8) Obviously, the two curves have no intersection point and 𝑔₂(𝑐)<0<𝑔₁(𝑐) for arbitrary constants 𝑐≠𝛾.

Using bifurcation method of vector fields (e.g., [19]), we have the following result which describes the locations and properties of the singular points of system (2.5).

Theorem 2.1. For a given constant wave speed 𝑐≠0, let 𝜑^±₀=𝑐−𝛾±(𝑐−𝛾)²−8𝑔4for𝑔≤𝑔₁(𝑐), (2.9)𝑦^±₀=±(𝑐−𝛾)²+𝑔for𝑔≥𝑔₂(𝑐). (2.10) When 𝑐=𝛾,(1)if 𝑔<0, then system (2.5) has two equilibrium points (−−𝑔/2,0) and (−𝑔/2,0), which are saddle points;(2) if 𝑔=0, then system (2.5) has only one equilibrium point (0,0), which is a cusp;(3) if 𝑔>0, then system (2.5) has two equilibrium points (0,−√𝑔) and (0,√𝑔), which are saddle points.When 𝑐≠𝛾,(1) if 𝑔<𝑔₂(𝑐), then system (2.5) has two equilibrium points (𝜑⁻₀,0) and (𝜑⁺₀,0). They are saddle points:(i)if 𝑐>𝛾, then 𝜑⁻₀<−1/2(𝑐−𝛾)<1/4(𝑐−𝛾)<𝑐−𝛾<𝜑⁺₀,(ii)if 𝑐<𝛾, then 𝜑⁻₀<𝑐−𝛾<1/4(𝑐−𝛾)<−1/2(𝑐−𝛾)<𝜑⁺₀;(2)if 𝑔=𝑔₂(𝑐), then system (2.5) has three equilibrium points (𝜑⁻₀,0), (𝜑⁺₀,0), and (𝑐−𝛾,0). (𝑐−𝛾,0) is a cusp:(i)if 𝑐>𝛾, then 𝜑⁻₀=−(1/2)(𝑐−𝛾)<(1/4)(𝑐−𝛾)<𝑐−𝛾=𝜑⁺₀. (𝜑⁻₀,0) is a saddle point, while (𝜑⁺₀,0) is a degenerate center point,(ii)if 𝑐<𝛾, then 𝜑⁻₀=𝑐−𝛾<1/4(𝑐−𝛾)<−1/2(𝑐−𝛾)=𝜑⁺₀. (𝜑⁻₀,0) is a degenerate center point, while (𝜑⁺₀,0) is a saddle point;(3) if 𝑔₂(𝑐)<𝑔<𝑔₁(𝑐), then system (2.5) has four equilibrium points (𝜑⁻₀,0), (𝜑⁺₀,0), (𝑐−𝛾,𝑦⁻₀), and (𝑐−𝛾,𝑦⁺₀). (𝑐−𝛾,𝑦⁻₀) and (𝑐−𝛾,𝑦⁺₀) are two saddle points:(i)if 𝑐>𝛾, then 𝜑⁻₀<1/4(𝑐−𝛾)<𝜑⁺₀≤𝑐−𝛾. (𝜑⁻₀,0) is a saddle point, while (𝜑⁺₀,0) is a center point,(ii)if 𝑐<𝛾, then 𝑐−𝛾≤𝜑⁻₀<1/4(𝑐−𝛾)<𝜑⁺₀. (𝜑⁻₀,0) is a center point, while (𝜑⁺₀,0) is a saddle point.
Specially, when 𝑔=0,
(i)if 𝑐>𝛾, then the three saddle points (𝜑⁻₀,0), (𝑐−𝛾,𝑦⁻₀), and (𝑐−𝛾,𝑦⁺₀) form a triangular orbit which encloses the center point (𝜑⁺₀,0),(ii)if 𝑐<𝛾, then the three saddle points (𝜑⁺₀,0), (𝑐−𝛾,𝑦⁻₀), and (𝑐−𝛾,𝑦⁺₀) form a triangular orbit which encloses the center point (𝜑⁻₀,0);(4) if 𝑔=𝑔₁(𝑐), then system (2.5) has three equilibrium points ((𝑐−𝛾)/4,0), (𝑐−𝛾,𝑦⁻₀), and (𝑐−𝛾,𝑦⁺₀). ((𝑐−𝛾)/4,0) is a degenerate center point, while (𝑐−𝛾,𝑦⁻₀) and (𝑐−𝛾,𝑦⁺₀) are two saddle points;(5) if 𝑔>𝑔₁(𝑐), then system (2.5) has two equilibrium points (𝑐−𝛾,𝑦⁻₀) and (𝑐−𝛾,𝑦⁺₀), which are saddle points.

Corresponding to the case 𝑐=𝛾 and the case 𝑐≠𝛾, we show the phase portraits of system (2.5) in Figures 1 and 2, respectively.

(a)

(b)

(c)

Figure 1 

The phase portraits of system (2.5)(𝑐=𝛾). (a) 𝑔<0; (b) 𝑔=0; (c) 𝑔>0.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

Figure 2 

The phase portraits of system (2.5) (𝑐≠𝛾). (a) 𝑔<𝑔2(𝑐),𝑐>𝛾; (b) 𝑔<𝑔2(𝑐),𝑐<𝛾; (c) 𝑔=𝑔2(𝑐),𝑐>𝛾; (d)𝑔=𝑔2(𝑐),𝑐<𝛾; (e) 𝑔2(𝑐)<𝑔<0,𝑐>𝛾; (f) 𝑔2(𝑐)<𝑔<0,𝑐<𝛾; (g) 𝑔=0,𝑐>𝛾; (h) 𝑔=0,𝑐<𝛾; (i) 0<𝑔<𝑔1(𝑐),𝑐>𝛾; (j) 0<𝑔<𝑔1(𝑐),𝑐<𝛾; (k) 𝑔=𝑔1(𝑐),𝑐>𝛾; (l) 𝑔=𝑔1(𝑐),𝑐<𝛾; (m)𝑔>𝑔1(𝑐),𝑐>𝛾; (n)𝑔>𝑔1(𝑐),𝑐<𝛾.

3. Solitons, Peakons, and Periodic Cusp Wave Solutions

Theorem 3.1. Given arbitrary constant 𝑐≠𝛾, let 𝜉=𝑥−𝑐𝑡, then(1)when 0<𝑔<𝑔₁(𝑐),(i)if 𝑐>𝛾, then (1.3) has the following smooth hump-like soliton solutions: 𝛽₁(𝜑⁺₁)=𝛽₁(𝜑)𝑒^−|𝜉|for𝜑⁻₀<𝜑<𝜑⁺₁, (3.1)(ii) if 𝑐<𝛾, then (1.3) has the following smooth valley-like soliton solutions: 𝛽₂(𝜑)=𝛽₂(𝜑⁻₁)𝑒^−|𝜉|for𝜑⁻₁<𝜑<𝜑⁺₀, (3.2)(2)when 𝑔=0, (1.3) has the following peaked soliton solutions: 𝜑=(𝑐−𝛾)𝑒^−|𝜉|, (3.3)(3)when 𝑔₂(𝑐)<𝑔<0,(i)if 𝑐>𝛾, then (1.3) has the following periodic cusp wave solutions: 𝑢(𝑥,𝑡)=𝜑₃(𝑥−𝑐𝑡−2𝑛𝑇)for(2𝑛−1)𝑇<𝑥−𝑐𝑡<(2𝑛+1)𝑇, (3.4)(ii)if 𝑐<𝛾, then (1.3) has the following periodic cusp wave solutions: 𝑢(𝑥,𝑡)=𝜑₄(𝑥−𝑐𝑡−2𝑛𝑇)for(2𝑛−1)𝑇<𝑥−𝑐𝑡<(2𝑛+1)𝑇, (3.5) where 𝛽₁(𝜑)=(2𝜑²+𝑙₁𝜑+𝑙₂+2𝜑+𝑙₁)(𝜑−𝜑⁻₀)^𝛼1(2√𝑎₁𝜑²+𝑙₁𝜑+𝑙₂+𝑏₁𝜑+𝑙₃)^𝛼1𝛽₂,(𝜑)=(2𝜑²+𝑚₁𝜑+𝑚₂+2𝜑+𝑚₁)(𝜑−𝜑⁺₀)^𝛼2(2√𝑎₂𝜑²+𝑚₁𝜑+𝑚₂+𝑏₂𝜑+𝑚₃)^𝛼2𝑙₁,=−3(𝑐−𝛾)+(𝑐−𝛾)²−8𝑔2𝑙₂,=3(𝑐−𝛾)²−4𝑔+5(𝑐−𝛾)(𝑐−𝛾)²−8𝑔8𝑙₃,=(𝑐−𝛾)²−4𝑔+3(𝑐−𝛾)(𝑐−𝛾)²−8𝑔2𝑚₁,=−3(𝑐−𝛾)−(𝑐−𝛾)²−8𝑔2𝑚₂,=3(𝑐−𝛾)²−4𝑔−5(𝑐−𝛾)(𝑐−𝛾)²−8𝑔8𝑚₃,=(𝑐−𝛾)²−4𝑔−3(𝑐−𝛾)(𝑐−𝛾)²−8𝑔2𝑎₁,=(𝑐−𝛾)²−8𝑔+3(𝑐−𝛾)(𝑐−𝛾)²−8𝑔4𝑎₂,=(𝑐−𝛾)²−8𝑔−3(𝑐−𝛾)(𝑐−𝛾)²−8𝑔4𝑏₁,=−(𝑐−𝛾)−(𝑐−𝛾)²𝑏−8𝑔,₂=−(𝑐−𝛾)+(𝑐−𝛾)²𝛼−8𝑔,₁=−3(𝑐−𝛾)+(𝑐−𝛾)²−8𝑔2(𝑐−𝛾)²−8𝑔+3(𝑐−𝛾)(𝑐−𝛾)²𝛼−8𝑔₂,=−3(𝑐−𝛾)+(𝑐−𝛾)²−8𝑔2(𝑐−𝛾)²−8𝑔−3(𝑐−𝛾)(𝑐−𝛾)²𝜑−8𝑔₃,(𝜉)=𝑙₊𝑒^−|𝜉|+𝑙₋𝑒^|𝜉|for√−𝑔≤𝜑₃𝜑≤𝑐−𝛾,₄(𝜉)=𝑙₊𝑒^|𝜉|+𝑙₋𝑒^−|𝜉|for𝑐−𝛾≤𝜑₄≤−√𝑙−𝑔,_±=𝑐−𝛾±(𝑐−𝛾)²+𝑔2√,𝑇=|ln(−𝑔+−2𝑔)−ln(2𝑙₋𝜑)|._±1=341(𝑐−𝛾)±4(𝑐−𝛾)²1−8𝑔∓2(𝑐−𝛾)²∓(𝑐−𝛾)(𝑐−𝛾)²−8𝑔, (3.6)(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)𝜑⁺₀ and 𝜑⁻₀ are as in (2.9).

Before proving this theorem, we take a set of data and employ Maple to display the graphs of smooth solion, peaked soliton and periodic cuspon solutions of (1.3), see Figures 3, 4, 5, 6 and 7.

(a)

(b)

(c)

(d)

Figure 3 

Smooth hump-like soliton solutions of (1.3)(𝑐=2, 𝛾=1). (a) 𝑔=0.12; (b) 𝑔=0.1; (c) 𝑔=0.075; (d) g=0.05.

(a)

(b)

(c)

(d)

Figure 4 

Smooth valley-like soliton solutions of (1.3) (𝑐=1, 𝛾=2). (a) 𝑔=0.12; (b) 𝑔=0.1; (c) 𝑔=0.075; (d) 𝑔=0.05.

(a)

(b)

Figure 5 

Peaked soliton solutions of (1.3). (a) 𝑐=2, 𝛾=1; (b) 𝑐=1, 𝛾=2.

(a)

(b)

(c)

(d)

Figure 6 

Period cuspon solutions of (1.3) (𝑐=2, 𝛾=1). (a) 𝑔=−0.4; (b) 𝑔=−0.1; (c) 𝑔=−0.01; (d) 𝑔=−0.0000001.

(a)

(b)

(c)

(d)

Figure 7 

Period cuspon solutions of (1.3) (𝑐=1, 𝛾=2). (a) g =-0.4; (b) 𝑔=−0.1; (c) 𝑔=−0.01; (d) 𝑔=−0.0000001.

Proof. Usually, a solion solution of (1.3) corresponds to a homoclinic orbit of system (2.5), and a periodic travelling wave solution of (1.3) corresponds to a periodic orbit of system (2.5). The graphs of homoclinic orbit, periodic orbit of system (2.5), and their limit curves are shown in Figure 8.
(1) When 0<𝑔<𝑔₁(𝑐),𝑐>𝛾, system (2.5) has a homoclinic orbit (see Figure 8(a)). This homoclinic orbit can be expressed as
𝑦=±(𝜑−𝜑⁻₀)𝜑²+𝑙₁𝜑+𝑙₂𝑐−𝛾−𝜑for𝜑⁻₀<𝜑<𝜑⁺₁. (3.7) Substituting (3.7) into the first equation of system (2.3) and integrating along this homoclinic orbit, we obtain (3.1).
When 0<𝑔<𝑔₁(𝑐),𝑐<𝛾, we can obtain (3.2) in similar way.
(2) When 𝑔=0,𝑐>𝛾, system (2.5) has a homoclinic orbit that consists of the following three line segments (see Figure 8(c)):
𝑦=±𝜑for𝜑⁻₀≤𝜑≤𝜑⁺₁, (3.8)𝜑=𝑐−𝛾for−(𝑐−𝛾)²+𝑔≤𝑦≤(𝑐−𝛾)²+𝑔. (3.9) Substituting (3.8) into the first equation of system (2.3) and integrating along this orbit, we obtain (3.3).
When 𝑔=0,𝑐<𝛾, we can also obtain (3.3).
(3) When 𝑔₂(𝑐)<𝑔<0,𝑐>𝛾, system (2.5) has a periodic orbit (see Figure 8(e)). This periodic orbit can be expressed as
𝑦=±𝜑²+𝑔for√−𝑔≤𝜑≤𝑐−𝛾, (3.10)𝜑=𝑐−𝛾for−(𝑐−𝛾)²+𝑔≤𝑦≤(𝑐−𝛾)²+𝑔. (3.11) Substituting (3.10) into the first equation of system (2.3) and integrating along this periodic orbit, we obtain (3.4).
When 𝑔₂(𝑐)<𝑔<0,𝑐<𝛾, we can obtain (3.5).

(a)

(b)

(c)

(d)

(e)

(f)

Figure 8 

The orbits of system (2.5) connecting with the saddle points. (a) 0<𝑔<𝑔1(𝑐),𝑐>𝛾; (b) 0<𝑔<𝑔1(𝑐),𝑐<𝛾; (c) 𝑔=0,𝑐>𝛾; (d) 𝑔=0,𝑐<𝛾; (e) 𝑔2(𝑐)<𝑔<0,𝑐>𝛾; (f) 𝑔2(𝑐)<𝑔<0,𝑐<𝛾.

Remark 3.2. From the above discussion, we can see that when 𝑔<0,𝑔→0, the period of the periodic cusp wave solution becomes bigger and bigger, and the periodic cuspon solutions (3.4) and (3.5) tend to the peaked soliton solutions (3.3). When 𝑔>0,𝑔→0, the smooth hump-like soliton solutions (3.1) and the smooth valley-like soliton solutions (3.2) lose their smoothness and tend to the peaked soliton solutions (3.3).

4. Discussion

In this paper, we obtain the solitons, peakons, and periodic cuspons of a generalized Degasperis-Procesi equation (1.3). These solitons denote the nonlinear localized waves on the shallow water's free surface that retain their individuality under interaction and eventually travel with their original shapes and speeds. The balance between the nonlinear steepening and dispersion effect under (1.3) gives rise to these solitons.

The peakon travels with speed equal to its peak amplitude. This solution is nonanalytic, having a jump in derivative at its peak. Peakons are true solitons that interact via elastic collisions under (1.3). We claim that the existence of a singular straight line for the planar dynamical system (2.3) is the original reason why the travelling waves lose their smoothness.

Also, the periodic cuspon solution is nonanalytic, having a jump in derivative at its each cusp.

Acknowledgments

The authors are deeply grateful to the anonymous referee for the careful reading of our manuscript and many constructive comments and valuable suggestions, which have helped them to improve it. This research was supported by NSF of China (No. 10771088) and Startup Fund for Advanced Talents of Jiangsu University (No. 09JDG013).