Peaked wave solutions of Camassa–Holm equation☆
Introduction
Camassa and Holm [1] derived a shallow water wave equationwhich is called Camassa–Holm equation. For k=0, they showed that it has peaked solitary wave solutionswhich have discontinuous first derivative at the wave peak in contrast to the smoothness of most previously known species of solitary waves and thus are called peakons.
Boyd [2] studied peaked periodic waves which have discontinuous first derivative at each peak and thus are called coshoidal waves or periodic cusp waves. Here we called them periodic cusp waves. Boyd gave three analytical representations for them. Qian and Tang [3] investigated the two peaked waves of the general Camassa–Holm equation
For k≠0 and a≠0 they showed that (1.3) has:
- (1)
Two peakons of the formandwhen a≠3.
- (2)
One peakon of the formwhen a=3.
They also pointed out that if w0(3w0−4ac)>0, then (1.3) has periodic cusp wave solutionswhere i=1,2,3, and n=0,±1,±2,…, andfor a>0, α0≠0 and ,for a<0 and ,for a<0 and ,Thus the Camassa–Holm equation has been studied successively by many authors (see for instance [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18]).
In this paper we study the peakons and the periodic cusp waves of Eq. (1.3) in more detail. First of all we transform Eq. (1.3) into a Hamiltonian system by several transformations. Then, using bifurcation method and qualitative analysis [19], [20], we obtain bifurcation curves and phase portraits of the systems and the analysis expressions of peaked wave solutions. Our main results are in the following theorem. Theorem 1 Consider Eq. (1.3). For given parameters a, k, constant wave speed c and integral constant g, letwhere a>0 and g1(c)<g<g2(c). Letwhere −T1⩽ξ<T1, g1(c)<g<g2(c) and a, c, k satisfy one of the following conditions: a>1 and ; a=1 and k>0; 0<a<1 and . a>1 and ; a=1 and k<0; 0<a<1 and .
Letwhere −T2⩽ξ<T2, g1(c)<g<g2(c) and a, c, k satisfy one of the following conditions:
Letwhere a=0, c≠2k, g>g1(c) and −T3⩽ξ<T3. Letwhere a<0, g>g1(c) and −T4⩽ξ<T4. Letwhere a<0, g>g1(c) and −T5⩽ξ<T5. Letwhere a<0 and −∞<ξ<∞.
Then we have the following results:
- (1)
If a>0 and , then Eq. (1.3) has peakons
- (2)
If g1(c)<g<g2(c) and a, c, k satisfy one of the conditions (i), (ii) and (iii), then Eq. (1.3) has periodic cusp wave solutionsfor n=0,±1,±2,… and ξ∈[(2n−1)T1,(2n+1)T1).
- (3)
If g1(c)<g<g2(c) and a, c, k satisfy one of the conditions (iv), (v) and (vi), then Eq. (1.3) has periodic cusp wave solutionsfor n=0,±1,±2,… and ξ∈[(2n−1)T2,(2n+1)T2).
- (4)
If a=0, c≠2k and g>g1(c), then Eq. (1.3) has periodic cusp wave solutionsfor n=0,±1,±2,… and ξ∈[(2n−1)T3,(2n+1)T3).
- (5)
If a<0 and g>g1(c), then Eq. (1.3) has periodic cusp wave solutionsfor i=4,5; n=0,±1,±2,… and ξ∈[(2n−1)Ti,(2n+1)Ti).
- (6)
If a>0, g1(c)<g<g2(c) and g tends to g2(c), then Tj (j=1,2) tend to ∞ and the periodic cusp wave solutions ucj(ξ) (j=1,2) become peakons uck(ξ).
- (7)
If a<0, , g>g1(c) and g tends to g1(c), then the periodic cusp wave solutions uc4(ξ) disappear and the uc5(ξ) become the smooth periodic wave solutions .
- (8)
If a<0, , g>g1(c) and g tends to g1(c), then the periodic cusp wave solutions uc5(ξ) disappear and the uc4(ξ) become the smooth periodic wave solution .
Remark 1
It needs many preparations to prove the Theorem 1. So we will give the preparations in 2 Singular points and their properties, 3 Relative locations of bifurcation curves, 4 Bifurcation diagrams of phase portraits, 5 The peakons, 6 The periodic cusp wave solutions and the proof of Theorem 1 in Section 7.
Remark 2
From Theorem 1 the following facts easily are seen:
- (1)
If a=3, then in (3.4) the uck(x,t) becomewhich are peakons of Eq. (1.1) and include uc0(x,t) in (1.2). Furthermore Eq. (1.1) has a special solution
- (2)
If a>0, a≠3 and c=c1=6k/(3−a), then uck(ξ) become .
- (3)
If a>0 and c=c2=2k/(1+a), then uck(ξ) become .
- (4)
If a=3 and c=c3=k/2, then uck(ξ) become .
- (5)
If g=0, then the uc1(ξ) become , uc4(ξ) become and uc5(ξ) become .
From above facts one easily sees that our results are wider than those in [1], [2], [3], [4].
In Section 2 we discuss singular points of travelling wave system. The relative locations of bifurcation curves are given in Section 3, and the bifurcation diagram of phase portraits of travelling wave system are shown in Section 4. The peakons and periodic cusp wave solutions are derived in 5 The peakons, 6 The periodic cusp wave solutions respectively. The proof of Theorem 1 and conclusion including a conjecture are given in Section 7.
Section snippets
Singular points and their properties
Consider the equationwhere a and k are real.
Substituting u=φ(ξ) with ξ=x−ct into (2.1), one gets the following ordinary differential equationIntegrating once it follows thatwhere the g is integral constant. Letting y=φ′ we obtain a planar systemwith first integral
From above derivation we see that the phase portraits of (2.4)
Relative locations of bifurcation curves
For given a and k, consider the following three curves:
In this section we only discuss their relative location. In next sections we will show that the three curves are bifurcation curves of travelling waves. Lemma 2 Consider the gi(c) (i=1,2,3). LetandWe have: If a≠1, then the three curves g1(c), g2(c) and g3(c) have unique intersection point and
Bifurcation diagrams of phase portraits
From Lemma 1, Lemma 2 we have the following bifurcation theorem about phase portraits of system (2.7). Theorem 2 For given a, k and arbitrary c, g, the bifurcation diagrams of the phase portraits of system (2.7) are given in Fig. 3. Proof Consider system (2.7). For a>1, from Lemma 1, Lemma 2 and H(φ,y) we have: (1) When g>g3(c), y−<y+ and φ± are not defined. So there are two singular points (c,y−) and (c,y+) which are saddle points. Because H(c,y−)=H(c,y+), the two saddle points are connected by a heteroclinic
The peakons
In this section we derive the peakons. Firstly we give a lemma to indicate the relationship of solitary waves of (2.1) and homoclinic orbits of (2.4). Lemma 3 Assume that Γ is a homoclinic orbit of (2.4) and it’s parameter expression is φ=φ(ξ) and y=y(ξ). Then u=φ(ξ) with ξ=x−ct is a solitary wave solution of (2.1). Proof From Fig. 3 one sees that the Γ surrounds (φ−,0) and connects with (φ+,0), or surrounds (φ+,0) and connects with (φ−,0). Therefore we haveor On the other hand,
The periodic cusp wave solutions
In this section we derive the analytic expressions of the periodic cusp wave solutions. Similar to Lemma 3 we have: Lemma 4 Assume that l is a periodic orbit of (2.4) and the analytic expression is φ=φ(ξ) and y=y(ξ). Then u=φ(ξ) with ξ=x−ct is a periodic wave solution of (2.1).
If letandthen from Fig. 3 one sees that when h tends to hc from h+ or from h−, the limiting curves of some periodic orbits of (2.4) consist of a arc and a line segment as in Fig. 6.
In Fig. 6 the
The proof of Theorem 1 and conclusion
Based on previous results now we can easily prove the Theorem 1. Proof of Theorem 1 The (1) of Theorem 1 is proved by using the Lemma 3 and the formula (5.5). The (2)–(5) of Theorem 1 are shown based on the Lemma 4 and formulae , , , , , , , , , . For the (6) of Theorem 1, from , , , , , , , , the following facts easily are shown: under one of the conditions (i), (ii) and (iii) in Theorem 1, the α→2(ac−c+2k), w1→0 and T1→∞ when g1(c)<g<g2(c) and g→g2(c); under one of the conditions (iv), (v) and (vi) in Theorem 1
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Research is supported by the Natural Science Foundation Found (10261008) of China and Yunnan Province and “Creative Project” (KZCX2-SW-118) in Chinese Academy of Sciences.