Peaked wave solutions of Camassa–Holm equation

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Abstract

The analytic expressions of peaked solitary wave solutions and peaked periodic wave solutions of Camassa–Holm equation are obtained by using bifurcation method of planar dynamical systems. The convergence of the peaked periodic wave solutions is proved. Numerical simulation results show the consistence with the theoretical analysis. The results in this paper are wider than those already known.

Introduction

Camassa and Holm [1] derived a shallow water wave equationut+2kux−uxxt+3uux=2uxuxx+uuxxx,which is called Camassa–Holm equation. For k=0, they showed that it has peaked solitary wave solutionsuc0(x,t)=cexp(−|x−ct|),which 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 equationut+2kux−uxxt+auux=2uxuxx+uuxxx.

For k≠0 and a≠0 they showed that (1.3) has:

  • (1)

    Two peakons of the formūc1k(x,t)=6k3−aexpa3x−6kt3−aandūc2k(x,t)=2k1+a3expa3x−2kt1+a−2,when a≠3.

  • (2)

    One peakon of the formūc3k(x,t)=3k2expx−kt2−k,when a=3.


They also pointed out that if w0(3w0−4ac)>0, then (1.3) has periodic cusp wave solutionsūci(ξ)=v̄ci(ξ−2nTi)forξ∈(2n−1)Ti,(2n+1)Ti,where i=1,2,3, and n=0,±1,±2,…, andv̄c1(ξ)=w0(3w0−4ac)4aα0expa3|ξ|+04aexpa3|ξ|3w02afor a>0, α0≠0 and T1⩽ξ<T1,v̄c2(ξ)=12a−3w0+3w0(3w0−4ac)cos|a|3ξfor a<0 and T2⩽ξ<T2,v̄c3(ξ)=−12a3w0+3w0(3w0−4ac)cos|a|3ξfor a<0 and T3⩽ξ<T3,α0=ac+2k−c+2ac(ac+4k−2c)3forac(ac+4k−2c)⩾0,w0=2k−c+ac3,T1=3aln0|w0(w0−(4ac/3))fora>0,T2=3|a|π2sin−12ac+3w03w0(3w0−4ac)fora<0,T3=3|a|π2+sin−12ac+3w03w0(3w0−4ac)fora<0.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, letξ=x−ct;c=2k/(1−a);ω0=2k−c+ac/3;ω1=w0(3w0−4ac)−8ag;g1(c)=(2c−ac−4k)c2;g2(c)=(ac+c−2k)(3c−ac−6k)8afora≠0;α=ac+2k−c+2[ac(ac+4k−2c)+2ag]/3;where a>0 and g1(c)<g<g2(c). LetT1=3aln3αw1fora>0andg1(c)<g<g2(c);T2=3alnw13αfora>0andg1(c)<g<g2(c);T3=22c(2k−c)+2g|2k−c|fora=0,g>g1(c)andc≠2k;T4=3|a|π2sin−12ac+3w03w1fora<0andg>g1(c);T5=3|a|π2+sin−12ac+3w03w1fora<0andg>g1(c);vc1(ξ)=w14aαexpa3|ξ|+4aexpa3|ξ|3w02a;whereT1ξ<T1, g1(c)<g<g2(c) and a, c, k satisfy one of the following conditions:

  • (i)

    a>1 and c>c;

  • (ii)

    a=1 and k>0;

  • (iii)

    0<a<1 and c<c.


Letvc2(ξ)=4aexpa3|ξ|+w14aαexpa3|ξ|3w02a,whereT2ξ<T2, g1(c)<g<g2(c) and a, c, k satisfy one of the following conditions:
  • (iv)

    a>1 and c<c;

  • (v)

    a=1 and k<0;

  • (vi)

    0<a<1 and c>c.


Letvc3(ξ)=2c(2k−c)+2g−|(2k−c)ξ|/22+c(c−2k)−2g2k−c,where a=0, c≠2k, g>g1(c) andT3ξ<T3. Letvc4(ξ)=12a−3w0+3w1cos|a|3|ξ|,where a<0, g>g1(c) andT4ξ<T4. Letvc5(ξ)=−12a3w0+3w1cos|a|3|ξ|,where a<0, g>g1(c) andT5ξ<T5. Letuc(ξ)=−32aw0+2ac3+w0cos|a|3ξ,where a<0 and −∞<ξ<∞.

Then we have the following results:

  • (1)

    If a>0 and c≠c, then Eq. (1.3) has peakonsuck(ξ)=3(ac−c+2k)2aexpa3|ξ|+3c−ac−6k2a.

  • (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 solutionsuc1(ξ)=vc1(ξ−2nT1)for 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 solutionsuc2(ξ)=vc2(ξ−2nT2)for 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 solutionsuc3(ξ)=vc3(ξ−2nT3)for 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 solutionsuci(ξ)=vci(ξ−2nTi)for 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 toand the periodic cusp wave solutions ucj(ξ) (j=1,2) become peakons uck(ξ).

  • (7)

    If a<0, c<c, 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 uc(ξ).

  • (8)

    If a<0, c>c, 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 uc(ξ).

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) becomeūck(x,t)=(k+c)exp(−|x−ct|)−k,which are peakons of Eq. (1.1) and include uc0(x,t) in (1.2). Furthermore Eq. (1.1) has a special solutionū0k(x,t)=kexp(−|x|)−kfork≠0.

  • (2)

    If a>0, a≠3 and c=c1=6k/(3−a), then uck(ξ) become ūc1k(x,t).

  • (3)

    If a>0 and c=c2=2k/(1+a), then uck(ξ) become ūc2k(x,t).

  • (4)

    If a=3 and c=c3=k/2, then uck(ξ) become ūc3k(x,t).

  • (5)

    If g=0, then the uc1(ξ) become ūc1(x,t), uc4(ξ) become ūc2(x,t) and uc5(ξ) become ūc3(x,t).

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 equationut+2kux−uxxt+auux=2uxuxx+uuxxx,where a and k are real.

Substituting u=φ(ξ) with ξ=xct into (2.1), one gets the following ordinary differential equation−cφ+2kφ+cφ′′′+aφφ=2φφ′′+φφ′′′.Integrating once it follows that(φ−c)φ′′=a2φ2+(2k−c)φ−12)2+g,where the g is integral constant. Letting y=φ we obtain a planar systemdφdξ=y,dydξ=a2φ2+(2k−c)φ−12y2+gφ−c,with first integralH(φ,y)=(φ−c)y2a3φ3+(c−2k)φ2−2gφ=h.

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:g1(c)=(2c−ac−4k)c2,g2(c)=(ac+c−2k)(3c−ac−6k)8afora≠0.g3(c)=(c−2k)22afora≠0.

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). Letc=2k1−a,andg=2ak2(a−1)2.We have:

  • (i)

    If a≠1, then the three curves g1(c), g2(c) and g3(c) have unique intersection point (c,g) andg1(c)<g2(c)<g3(c)fora>0andc≠c,g1(c)>g2(c)>g3(c)fora<0

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=xct 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 havelim|ξ|→∞φ(ξ)=φ+orlim|ξ|→∞φ(ξ)=φ.

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=xct is a periodic wave solution of (2.1).

If leth+=H(φ+,0),h=H(φ,0),andhc=H(c,y±),then 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(acc+2k), w1→0 and T1→∞ when g1(c)<g<g2(c) and gg2(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.

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