# On the Multipeakon Dissipative Behavior of the Modified Coupled Camassa-Holm Model for Shallow Water System

Authors: Zhixi Shen; Yujuan Wang; Hamid Reza Karimi; Yongduan Song
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

This paper investigates the multipeakon dissipative behavior of the modified coupled two-component Camassa-Holm system arisen from shallow water waves moving. To tackle this problem, we convert the original partial differential equations into a set of new differential equations by using skillfully defined characteristic and variables. Such treatment allows for the construction of the multipeakon solutions for the system. The peakon-antipeakon collisions as well as the dissipative behavior (energy loss) after wave breaking are closely examined. The results obtained herein are deemed valuable for understanding the inherent dynamic behavior of shallow water wave breaking.


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## Body

## 1. Introduction

The study of the dynamic behavior of shallow water wave represents an important research topic in view of its potential application in surface and underwater vehicle systems design, control, deployment, and monitoring. There are several classical models describing the motion of waves at the free surface of shallow water under the influence of gravity, the best known of which are the Korteweg-de Vries (KdV) equation [1, 2] and the Camassa-Holm (CH) equation [3–5]. The KdV equation admits solitary wave solutions but does not model the phenomenon of breaking for water waves. The CH equation, modeling the unidirectional propagation of shallow water waves over a flat bottom [4–6] as well as water waves moving over an underlying shear flow [7], has many remarkable properties like solitary waves with singularities called peakons [4, 6] and breaking waves [4, 8] which set it apart from KdV. The peaked solitary waves mean that they are smooth except at the crests, where they are continuous but have a jump discontinuity in the first derivative, while the presence of breaking waves means that the solution remains bounded while its slope becomes unbounded in finite time [8, 9]. After wave breaking the solutions of the CH equation, as shown by several works [10–16], become uniquely as either global conservative or global dissipative solutions.Recently, the CH equation has been extended to many multicomponent generalizations, which can better reflect the feature of the shallow water moving. In this paper, we consider the following modified coupled two-component Camassa-Holm system [17]:
(1)mt+2mux+mxu+(mv)x+nvx=0,t>0,x∈R,nt+2nvx+nxv+(nu)x+mux=0,t>0,x∈R,m=u-uxx,t>0,x∈R,n=v-vxx,t>0,x∈R,
which is a modified version of the coupled two-component Camassa-Holm system as established by Fu and Qu in [18], allowing for peakon solitons in the form of a superposition of multipeakons. System (1) can be rewritten as a Hamiltonian system,
(2)∂∂t(mn)=-(∂m+m∂∂m+n∂∂n+m∂∂n+n∂)(δHδm=uδHδn=v)
with the Hamiltonian H=(1/2)∫(mG*m+nG*n)dx, where G*m=u, G*n=v, and G=(1/2)e-|x|. Particularly, when u=0 (or v=0), the degenerated equation (1) has the same peakon solitons as the CH equation. We are interested in such system because it exhibits the following conserved quantities, as can be easily verified:
(3)E1(u)=∫Rudx,E2(v)=∫Rvdx,E3(u)=∫Rmdx,E4(u)=∫Rndx,E5(u,v)=∫R(u2+ux2+v2+vx2)dx.It has been shown that system (1) is locally well-posed and also has global strong solutions which blow up in finite time [17, 18]. Moreover, the existence issue for a class of local weak solutions for such system was also addressed in [17]. It is interesting to know that whether the two remarkable properties associated with the original CH equation persist in this modified coupled two-component Camassa-Holm system. In our recent work [19, 20], we studied the problem of solution continuation beyond wave breaking of system (1), where it was established that the system admits either global conservative solutions or global dissipative solutions.Just as with the CH equation, the multipeakon dissipative solution represents an important aspect related to the solutions near wave breaking, it is interesting to know whether or not system (1) also exhibits the similar feature. Thus far very little effort has been made on studying the multipeakon dissipative solution associated with the modified coupled two-component Camassa-Holm system of the form as expressed in (1) in the literature. Based on our recent work  [20] where a global continuous semigroup of dissipative solutions of system (1) is established, in this paper we show how to construct globally defined multipeakon solutions in the dissipative case for the modified coupled two-component Camassa-Holm system.It should be stressed that the system considered in this work is a heavily coupled one; it is the mutual effect between the two components that makes the analysis and computation much more involved than the system with single component as studied in [16]. The key to circumvent the difficulty is to utilize a skillfully defined characteristic and several new variables to obtain a new set of ordinary differential equations, from which the dissipative multipeakons are globally determined. Such feature discovered is deemed useful in further understanding the dynamic behavior of the wave breaking associated with the system. Examples are presented to illustrate the feature of the multipeakons with peakon-antipeakon collisions.The rest of this paper is organized as follows. Section2 represents the construction of the global dissipative solutions of the modified coupled Camassa-Holm system. Section 3 is devoted to the establishment of the dissipative multipeakon solutions of system (1). The method is illustrated by explicit calculations in the case n=1 and by numerical computations when n=2 with peakon-antipeak on collisions in Section 4. The paper is concluded in Section 5.


## 2. Global Dissipative Solutions of the Modified Coupled Camassa-Holm System

We represent the construction of the global dissipative solutions of system (1) obtained in [20] in this section. System (1) can be rewritten as(4)ut+(u+v)ux+P1+P2,x=0,t>0,x∈R,vt+(u+v)vx+P3+P4,x=0,t>0,x∈R,
where P1, P2, P3, P4 are given by
(5)P1(t,x)=G*(uvx)=12·∫Re-|x-x′|(uvx)(t,x′)dx′,P2(t,x)=G*(u2+ux22+uxvx+v22-vx22)=12·∫Re-|x-x′|(u2+ux22+uxvx+v22-vx22)×(t,x′)dx′,P3(t,x)=G*(vux)=12·∫Re-|x-x′|(vux)(t,x′)dx′,P4(t,x)=G*(v2+vx22+uxvx+u22-ux22)=12·∫Re-|x-x′|(v2+vx22+uxvx+u22-ux22)×(t,x′)dx′,
with G=e-|x|/2 the Green’s function such as G*f(x)=1/2·∫Re-|x-x′|f(x′)dx′ for all f∈L2(R) and * the spatial convolution.By using a skillfully defined characteristicyt(t,ξ)=(u+v)(t,y(t,ξ)), which can be decomposed as y(t,ξ)=ς(t,ξ)+ξ, and a new set of Lagrangian variables; namely,
(6)h(t,ξ)=(u2+ux2+v2+vx2)∘yyξ,U(t,ξ)=u(t,y(t,ξ)),V(t,ξ)=v(t,y(t,ξ)),M(t,ξ)=ux(t,y(t,ξ)),N(t,ξ)=vx(t,y(t,ξ)),
where h corresponds to the Lagrangian energy density and U, V the Lagrangian velocity, we derive an equivalent system of the modified coupled Camassa-Holm system,
(7)ςt=U+V,Ut=-P1-P2,x,Vt=-P3-P4,x,Mt=(-M22-N22+U2+V22-P1,x-P2),Nt=(-N22-M22+V2+U22-P3,x-P4),ht=(3U2-2P2)Uξ-2UP2,xyξ+(3V2-2P4)Vξ-2VP4,xyξ.It is shown in [20] that the existence, uniqueness, and stability of solutions of system (7) are obtained in a Banach space, which is transformed into the conservative solution of the original system (4), while dissipative solutions differ from conservative solutions when particles collide, that is, when yξ(t,ξ)=0 for ξ in an interval of positive length. To obtain the dissipative solution, we impose that when particles collide, they lose their energy; that is, if yξ(τ,ξ)=0 for some τ, then we set h(τ,ξ)=0. Thus we define τ(ξ) as the first time when yξ(t,ξ) vanishes; namely,(8)τ(ξ)=sup{t∈R+∣yξ(t′,ξ)>0∀0≤t′<t},
and the expressions for Pi and Pi,x(i=1,2,3,4) become
(9)P1(t,ξ)=12·∫t<τ(ξ)e-|y(ξ)-y(ξ′)|×[(UN)yξ](ξ′)dξ′,P1,x(t,ξ)=-12·∫t<τ(ξ)sgn(ξ-ξ′)e-|y(ξ)-y(ξ′)|×[(UN)yξ](ξ′)dξ′,P2(t,ξ)=14·∫t<τ(ξ)e-|y(ξ)-y(ξ′)|×[h+(U2+2MN-N2)yξ]×(ξ′)dξ′,P2,x(t,ξ)=-14·∫t<τ(ξ)sgn(ξ-ξ′)e-|y(ξ)-y(ξ′)|×[h+(U2+2MN-N2)yξ]×(ξ′)dξ′,P3(t,ξ)=12·∫t<τ(ξ)e-|y(ξ)-y(ξ′)|×[(VM)yξ](ξ′)dξ′,P3,x(t,ξ)=-12·∫t<τ(ξ)sgn(ξ-ξ′)e-|y(ξ)-y(ξ′)|×[(VM)yξ](ξ′)dξ′,P4(t,ξ)=14·∫t<τ(ξ)e-|y(ξ)-y(ξ′)|×[h+(V2+2MN-M2)yξ]×(ξ′)dξ′,P4,x(t,ξ)=-14·∫t<τ(ξ)sgn(ξ-ξ′)e-|y(ξ)-y(ξ′)|×[h+(V2+2MN-M2)yξ]×(ξ′)dξ′,
and the modified system to be solved here reads
(10)ςt=U+V,Ut=-P1-P2,x,Vt=-P3-P4,x,Mt=(-M22-N22+U2+V22-P1,x-P2),Nt=(-N22-M22+V2+U22-P3,x-P4).(11)ςξt={Uξ+Vξift<τ(ξ),0otherwise,Uξt={h2+(U22+MN-N2-P2-P1,xU22)yξift<τ(ξ),0otherwise,Vξt={h2+(V22+MN-M2-P4-P3,xV22)yξift<τ(ξ),0otherwise,ht={(3U2-2P2)Uξ-2UP2,xyξ+(3V2-2P4)Vξ-2VP4,xyξift<τ(ξ),0otherwise.
Note that, in this definition, we do not reset the energy density h to zero for t≥τ(ξ) but keep the value it reached just before the collision, which has the advantage of rendering the right hand of (11) continuous across the value t=τ(ξ) and the behavior of the system remains unchanged.The local existence of solutions is proved in the Banach spaceE where
(12)E=L∞∩W∩W∩W∩W∩W∩W∩W∩L1,W=L2∩L∞.
The global solutions of (10) may not exist for all initial data in E; however, they exist when the initial data X0=(ς0,U0,V0,M0,N0,h0)∈Γ, where Γ is defined as follows.Definition 1.
The setΓ is composed of all (ς,U,V,M,N,h) such that
(13)X=(ς,U,V,M,N,ςξ,Uξ,Vξ,h)∈E,(14)yξ≥0,h≥0almosteverywhere,(15)yξh=yξ2U2+Uξ2+yξ2V2+Vξ2,almosteverywhere,(16)1(yξ+h)∈L∞(R),(17)g(y,U,V,yξ,Uξ,Vξ,h)-1∈W,
where g(x) is given by
(18)g(x)={|x5|+|x6|+2(1+x22+x32)x4,ifx∈Ω,x4+x7,otherwise,
for x=(x1,x2,x3,x4,x5,x6,x7)∈R7, where Ω is the following subset of R7(19)Ω={x∈R7∣|x5|+|x6|+2(1+x22+x32)×x4≤x4+x7,x5,x6≤0{x22}}.The main result in [20] is stated in the following theorem.Theorem 2.
Letz0=(u0,v0)∈H1×H1 be given. If one denotes t→z(t)=Tt(z0) the corresponding trajectory, then z=(u,v) is a weak dissipative solution of the modified coupled two-component Camassa-Holm system, which constructs a continuous semigroup with respect of the metric dH1 on bounded sets of H1; that is, for any M>0 and any sequence zn∈H1 such that ∥zn∥H1≤M, one has that limn→∞dH1(zn,z)=0 implies limn→∞dH1(Tt(zn),Tt(z))=0.


## 3. Multipeakon Dissipative Solutions of the Original System

In this section, we derive a new system of ordinary differential equations for the multipeakon solutions which is well-posed even when collisions occur, and the variables(y,U,V,M,N,H) are used to characterize multipeakons in a way that avoids the problems related to blowing up.Solutions of the modified coupled two-component Camassa-Holm system may experience wave breaking in the sense that the solution develops singularities in finite time, while keeping theH1 norm finite. Continuation of the solution beyond wave breaking imposes significant challenge as can be illustrated in the case of multipeakons, which are special solutions of the modified coupled two-component Camassa-Holm system of the form
(20)(u,v)(t,x)=(∑i=1npi(t)e-|x-qi(t)|,∑i=1nri(t)e-|x-qi(t)|),
where (pi(t),ri(t),qi(t)) satisfy the explicit system of ordinary differential equations
(21)p˙i=∑j=1,i≠jn(pipj+rirj)sgn(qj-qi)e-|qi-qj|,r˙i=∑j=1,i≠jn(pipj+rirj)sgn(qj-qi)e-|qi-qj|,q˙i=-∑j=1n(pj+rj)e-|qi-qj|.
Let us consider initial data z0=(u0,v0) given by
(22)(u0,v0)(x)=(∑i=1npie-|x-ξi|,∑i=1nrie-|x-ξi|).
Peakons interact in a way similar to that of solitons of the CH equation, and wave breaking may appear when at least two of the qi coincide. In the case that pi(0) and ri(0) have the same sign for all i=1,2,…n and qi(t) remain distinct, (21) allows for a unique global solution, where the peakons are traveling in the same direction. By inserting that solution into (20), it is not hard to know that z=(u,v) is a global weak solution of system (4). However, when two peakons have opposite signs, collisions may occur, and if so, the system (21) blows up. Without loss of generality, we assume that the pi and ri are all nonzero, and that the ξi are all distinct. The aim is to characterize the unique and global weak solution from Theorem 2 with initial data (22) explicitly. Since the variables pi and ri blow up at collisions, they are not appropriate to define a multipeakon in the form of (20). We consider the following characterization of multipeakons given as continuous solutions z=(u,v), which are defined on intervals [yi,yi+1] as the solutions of the Dirichlet problem
(23)z-zxx=0,
with boundary conditions z(t,yi(t))=zi(t), z(t,yi+1(t))=zi+1(t). The variables yi denote the position of the peaks, and the variables zi denote the values of z at the peaks. In the following part we will show that this property persists for dissipative solutions.We introduceX-=(y-,U-,V-,M-,N-,h-) as a representative of z=(u,v) in Lagrangian equivalent system; that is, X-=L(z-), which is given by
(24)y-(ξ)=ξ,(25)U-(ξ)=u-(ξ),V-(ξ)=v-(ξ),M-(ξ)=u-x(ξ),N-(ξ)=v-x(ξ),(26)h-(ξ)=u-2+u-x2+v-2+v-x2.LetI=⋃i=0nIi, where Ii denote the open interval (ξi,ξi+1) with the conventions that ξ0=-∞ and ξn+1=∞. For each interval Ii, we define τi=inf{τ(ξ)∣ξ∈(ξi,ξi+1)} such that τi>0 for all i=1,…,n. By the linearity of the governing equations (10), and the bounds which hold on the solution X and Pi,Pi,x(i=1,2,3,4), it is not hard to check that y,U,V∈C([0,τi],C2(Ii)), while h∈C([0,τi],C1(Ii)).Thus the existence of multipeakon solutions is given by the next theorem.Theorem 3.
For any given multipeakon initial dataz-(x)=(u-,v-)(x)=(∑i=1npie-|x-ξi|,∑i=1nrie-|x-ξi|), let (y,U,V,M,N,h) be the solution of system (10), (11) with initial data (y-,U-,V-,M-,N-,H-) given by (24), (25), and (26). Between adjacent peaks, if xi=y(t,ξi)≠xi+1=y(t,ξi+1), the solution z(t,x)=(u,v)(t,x) is twice differentiable with respect to the space variable, and one has
(27)(z-zxx)(t,x)=0forx∈(xi,xi+1).Proof.
For a given timet, we consider two adjacent peaks xi=y(t,ξi) and xi+1=y(t,ξi+1). If xi=xi+1, then the two peaks have collided and, since yξ is positive, we must have yξ(t,ξ)=0 for all ξ∈Ii. Hence, t≥τi, which conversely implies that t<τi when xi(t)<xi+1(t). There exists ξ∈Ii such that x=y(t,ξ) for any x∈(xi(t),xi+1(t)). Since ξ∈Ii and t<τi, we have yξ(t,ξ)≠0. It follows from the implicit function theorem that y(t,·) is invertible in a neighborhood of ξ and its inverse is C2, and therefore (u,v)(t,x′)=(U,V)(t,y-1(t,x′)) are C2 with respect to the spatial variable and the quantity (z-zxx)(t,x) is defined in the classical sense.
We now prove that(z-zxx)(t,x)=0 for x∈(xi,xi+1). Let us first prove that u-uxx=0. Assuming that yξ(t,ξ)≠0, we have that
(28)ux∘y=M,uxx∘y=Mξyξ=(Uξξyξ-yξξUξ)yξ3,
and therefore
(29)(u-uxx)∘y=(Uyξ3-Uξξyξ+yξξUξ)yξ3.
We set
(30)R=Uyξ3-Uξξyξ+yξξUξ.
For a given ξ∈I and t<τi, differentiating (30) with respect to t, it then follows from (10) and (11) that
(31)dRdt=3Uyξ2yξt+Utyξ3-Uξξtyξ-Uξξyξt+yξξtUξ+yξξUξt=2U(Uξ+Vξ)yξ2-2N(Mξ-Nξ)yξ2-hξyξ2+hyξξ2.
Differentiating (15) with respect to ξ, we get
(32)yξξh+yξhξ=2yξyξξU2+2yξ2UUξ+2UξUξξ+2yξyξξV2+2yξ2VVξ+2VξVξξ.
We have, after inserting the value of yξhξ given by (32) into (31) and multiplying the equation by yξ, that
(33)yξdRdt=UUξyξ3-UξUξξyξ+(hyξ-yξ2U2-yξ2V2-Vξ2)yξξ+UVξyξ3-UξξVξyξ+UξVξyξξ.
Since yξt=(Uξ+Vξ), it follows from (15) that
(34)yξdRdt=yξt·R.
For any ξ∈I, as u- is a multipeakon initial data, we have R(0,ξ)=(u--u-xx)·yξ3=0. It thus follows from Gronwall’s lemma that R(t,ξ)=0 and therefore (u-uxx)(t,ξ)=0 for t∈[0,τi]. Similarly, we can obtain that (v-vxx)(t,ξ)=0.Thus, the system of ordinary differential equations that the dissipative multipeakon solutions satisfy can be derived based on the fact that the multipeakon structure is preserved by the semigroup of dissipative solutions.Let us define(35)Hi=∫ξiξi+1h(ξ)dξ.
For each i=1,2…,n, by using (11), we obtain the following system of O.D.E.; namely,
(36)dyidt=ui+vi,duidt=-P1,i-P2,xi,dvidt=-P3,i-P4,xi,dHidt=(ui+13-2ui+1P2,i+1+vi+13-2vi+1P4,i+1)-(ui3-2uiP2,i+vi3-2viP4,i),
where (yi,ui,vi)=(y,U,V)(t,ξi), Pk,i=Pk(t,ξi), Pk,xi=Pk,x(t,ξi), (k=1,2,3,4), respectively. We have
(37)P2,i=14·∫{t<τ(ξ′)}e-|yi-y(ξ′)|×[h+(U2+2MN-N2)yξ](ξ′)dξ′.
We denote B={ξ′∈R∣yξ(t,ξ′)>0}. Thus, we get
(38)P2,i=12·∫Be-|yi-y(ξ′)|(u2+ux22+uxvx+v22-vx22)∘y(ξ′)yξ(ξ′)dξ′=12·∫y(B)e-|yi-x|(u2+ux22+uxvx+v22-vx22)×(x)dx,
where we have used the fact that h=(u2+ux2+v2+vx2)∘yyξ on B and t<τ(ξ′) if and only if yξ(t,ξ′)>0. Since yξ(ξ)=0 on Bc, that means (y(Bc))=∫y(Bc)dx=∫Bcyξ(ξ)dξ=0, which implies that the domain of integration in (37) can be extended to the whole axis,
(39)P2,i=12·∫Re-|yi-x|(u2+ux22+uxvx+v22-vx22)dx.
Similarly, we can get that
(40)P1,i=12·∫Re-|yi-x|(uvx)dx,P1,xi=-12·∫Rsgn(yi-x)e-|yi-x|(uvx)dx,P2,xi=-12·∫Rsgn(yi-x)e-|yi-x|×(u2+ux22+uxvx+v22-vx22)dx,P3,i=12·∫Re-|yi-x|(vux)dx,P3,xi=-12·∫Rsgn(yi-x)e-|yi-x|(vux)dx,P4,i=12·∫Re-|yi-x|×(v2+vx22+uxvx+u22-ux22)dx,P4,xi=-12·∫Rsgn(yi-x)e-|yi-x|×(v2+vx22+uxvx+u22-ux22)dx.Between two adjacent peaks located atyi and yi+1, we know that z=(u,v) satisfies (z-zxx)=0 and therefore z=(u,v) can be written as
(41)z(x)=(u,v)(x)=(Aiex+Bie-x,Ciex+Die-x)
for x∈[yi,yi+1], i=1,2,…,n-1, where the constants Ai, Bi, Ci, and Di depend on ui, ui+1, vi, vi+1,  yi, and yi+1 and read
(42)Ai=e-y-i2[u-icosh(δyi)+δuisinh(δyi)],Bi=ey-i2[u-icosh(δyi)-δuisinh(δyi)],Ci=e-y-i2[v-icosh(δyi)+δvisinh(δyi)],Di=ey-i2[v-icosh(δyi)-δvisinh(δyi)],
where
(43)y-i=12·(yi+yi+1),δyi=12·(yi-yi+1),u-i=12·(ui+ui+1),δui=12·(ui-ui+1),v-i=12·(vi+vi+1),δvi=12·(vi-vi+1).
Thus, the constants Ai, Bi, Ci, and Di uniquely determine z=(u,v) on the interval [yi,yi+1], and we compute
(44)δHi=Hi+1-Hi=∫yiyi+1(u2+ux2+v2+vx2)dx=2u-i2tanh(δyi)+2δui2coth(δyi)+2v-i2tanh(δyi)+2δvi2coth(δyi)=δH1i+δH2i,
with δH1i=2u-i2tanh(δyi)+2δui2cosh(δyi) and δH2i=2v-i2tanh(δyi)+2δvi2cosh(δyi). We now turn to the computation of Pk,i(k=1,2,3,4) given by (39) and (40). Let us write z=(u,v) as
(45)z(t,x)=(u,v)(t,x)=(∑j=0n(Ajex+Bje-x)χ(yj,yj+1)(x),∑j=0n(Cjex+Dje-x)χ(yj,yj+1)(x)).
Set y0=-∞, yn+1=∞, u0=un+1=0, v0=vn+1=0, A0=u1e-y1, B0=0, An=0, Bn=uneyn, C0=v1e-y1, D0=0, Cn=0, and Dn=vneyn. We have
(46)uvx=∑j=0n(AjCje2x-AjDj+BjCj-BjDje-2x)χ(yj,yj+1),u2+ux22+uxvx+v22-vx22=∑j=0n((32·Aj2+AjCj)e2x+(AjBj-AjDj-BjCj+2CjDj)+(32·Bj2+BjDj)e-2x)χ(yj,yj+1),vux=∑j=0n(AjCje2x-CjBj+DjAj-BjDje-2x)χ(yj,yj+1),v2+vx22+uxvx+u22-ux22=∑j=0n((32·Cj2+AjCj)e2x+(CjDj-CjBj-DjAj+2AjBj)+(32·Dj2+DjBj)e-2x)χ(yj,yj+1).
By inserting (46) into (39) and (40), we get
(47)P1,i=12·∑j=0n∫yjyj+1e-kij(yi-x)×(AjCje2x-AjDj+BjCj-BjDje-2x)dx,P2,i=12∑j=0n∫yjyj+1e-kij(yi-x)×((32Aj2+AjCj)e2x+(AjBj-AjDj-BjCj+2CjDj)+(32Bj2+BjDj)e-2x)dx,P3,i=12·∑j=0n∫yjyj+1e-kij(yi-x)×(AjCje2x-CjBj+DjAj-BjDje-2x)dx,P4,i=12∑j=0n∫yjyj+1e-kij(yi-x)×((32Cj2+AjCj)e2x+(CjDj-CjBj-DjAj+2AjBj)+(32Dj2+DjBj)e-2x)dx,
where kij=-1 if  i≤j, kij=1 if i>j. It then follows from (42) and (44) that
(48)Aj2=e-2y-jsinh2(2δyj)[u-j2sinh2(δyj)+2u-jδujsinh(δyj)×cosh(δyj)+δuj2cosh2(δyj)]=e-2y-j4sinh(2δyj)·[δH1j+4u-jδuj],AjBj=14sinh(2δyj)·[4u-j2tanh(δyj)-δH1j],AjCj=e-2y-j2sinh(2δyj)[u-jv-jtanh(δyj)+δujv-j+δvju-j+δujδvjcoth(δyj)],AjDj=12sinh(2δyj)[u-jv-jtanh(δyj)+δujv-j-δvju-j-δujδvjcoth(δyj)],BjCj=12sinh(2δyj)[u-jv-jtanh(δyj)+δvju-j-δujv-j-δujδvjcoth(δyj)],BjDj=e2y-j2sinh(2δyj)[u-jv-jtanh(δyj)-δujv-j-δvju-j+δujδvjcoth(δyj)],Bj2=e2y-j4sinh(2δyj)·[δH1j-4u-jδuj].
Thus, from (48), we can obtain that
(49)∫yjyj+1e-kij(yi-x)Aj2e2xdx=e-kijyi·ekijy-j2(2+kij)sinh(2δyj)×sinh((2+kij)δyj)[δH1j+4u-jδuj],∫yjyj+1e-kij(yi-x)AjCje2xdx=e-kijyi·ekijy-j(2+kij)sinh(2δyj)×sinh((2+kij)δyj)·[u-jv-jtanh(δyj)+δujv-j+δvju-j+δujδvjcoth(δyj)],∫yjyj+1e-kij(yi-x)AjBjdx=e-kijyi·ekijy-j2sinh(2δyj)sinh(δyj)×[4u-j2tanh(δyj)-δH1j],∫yjyj+1e-kij(yi-x)AjDjdx=e-kijyi·ekijy-jsinh(2δyj)sinh(δyj)·[u-jv-jtanh(δyj)+δujv-j-δvju-j-δujδvjcoth(δyj)],∫yjyj+1e-kij(yi-x)BjCjdx=e-kijyi·ekijy-jsinh(2δyj)sinh(δyj)·[u-jv-jtanh(δyj)+δvju-j-δujv-j-δujδvjcoth(δyj)],∫yjyj+1e-kij(yi-x)CjDjdx=e-kijyi·ekijy-j2sinh(2δyj)sinh(δyj)×[4v-j2tanh(δyj)-δH2j],∫yjyj+1e-kij(yi-x)Bj2e-2xdx=e-kijyi·ekijy-j2(kij-2)sinh(2δyj)×sinh((kij-2)δyj)[δH1j-4u-jδuj],∫yjyj+1e-kij(yi-x)BjDje-2xdx=e-kijyi·ekijy-j(kij-2)sinh(2δyj)×sinh((kij-2)δyj)·[u-jv-jtanh(δyj)-δujv-j-δvju-j+δujδvjcoth(δyj)].
It thus follows from (49) that
(50)P1,i=∑j=0nP1,ij,P2,i=∑j=0nP2,ij,
where
(51)P1,ij={16·u1v1ey1-yi,δujδujforj=0,e-kijyi·ekijy-j6cosh(δyj)×[2kiju-jv-jsinh2(δyj)tanh(δyj)+2kijδujδvjsinh(δyj)cosh(δyj)+2δujv-jcosh2(δyj)+2δvju-jcosh2(δyj)+δujv-j+δvju-j-3u-jv-jtanh(δyj)+3δujδvjcoth(δyj)],δujδujforj=1,…,n-1,-16·unvne-yn+yiforj=n,P2,ij={14·u12ey1-yi+16·u1v1ey1-yi,forj=0,e-kijyi·ekijy-j4cosh(δyj)×[δH1jcosh2(δyj)+4kiju-jδujsinh2(δyj)+2u-j2tanh(δyj)+4v-j2tanh(δyj)-δH2j-43u-jv-jtanh(δyj)+83δujδvj×coth(δyj)+43u-jv-jcosh(δyj)sinh(δyj)+43δujδvj·coth(δyj)cosh2(δyj)+43kijδujv-jsinh2(δyj)+43×kijδvju-jsinh2(δyj)],forj=1,…,n-1,14·un2e-yn+yi+16·unvne-yn+yi,forj=n.
The terms P3,i, P4,i, and Pk,ix(k=1,2,3,4) can be computed in the same way and we have
(52)Pk,ix=-∑j=0nkijPk,ij·(k=1,2,3,4).The result can be summarized in the following theorem.Theorem 4.
Assumey-i=ξi, z-i=(u-i,v-i)=(u-(ξi),v-(ξi)) and H-i=∫ξiξi+1(u-2+u-x2+v-2+v-x2)dx for i=1,…,n with a multipeakon initial data z-=(u-,v-) as given by (22). Then, there exists a global solution (yi,ui,vi,Hi) of (36), (50), and (52) with initial data (y-i,u-i,v-i,H-i). On each interval [yi(t),yi+1(t)], one defines z(t,x)=(u,v)(t,x) as the solution of the Dirichlet problem z-zxx=0 with boundary conditions z(t,yi(t))=zi(t), z(t,yi+1(t))=zi+1(t) for each time t. Thus z=(u,v) is a dissipative solution of the modified coupled two-component Camassa-Holm system, which is the dissipative multipeakon solution.


## 4. Examples

In this section, we give the examples with the casen=1 by explicit calculations and the case n=2 by numerical computations with peakon-antipeakon collisions.(i)
Letn=1. From (50) and (52), we can compute that P1,1=∑j=0,1P1,1j=u1v1/6-u1v1/6=0 and P2,x1=-k1j∑j=0,1P2,1j=-u12/4-u1v1/6+u12/4+u1v1/6=0, which imply that u1t=-P1,1-P2,x1=0 and therefore u1=c1. Similarly, we can get that v1=c2. Thus from (36), we can obtain that y1t=u1+v1=c1+c2=c, which yields y1=ct+a with c1,c2,c,a some constants. There is no collision and we find the familiar one peakon (u,v)(t,x)=(c1e-|x-c1t-a|,c2e-|x-c2t-a|).(ii)
Letn=2. We first consider the case of an antisymmetric pair of peakons where the two peakons collide. We take the initial conditions as
(53)y2(0)=-y1(0)=y-,u2(0)=-u1(0)=u-,v2(0)=-v1(0)=v-,δH1(0)=E2for some strictly positive constants y-,u-,v-, and E the initial total energy of the system, that is, the H1 norm of the solution; we denote τ=τ1, the time of collision. For t<τ, the solution is identical to the conservative case. After collision, for t≥τ, the solution remains antisymmetric. Let us assume this for the moment and write
(54)y=y2=-y1,u=u2=-u1,v=v2=-v1,h=δH1,P2=P2,1=P2,2,P4=P4,1=P4,2,P1=P1,1=-P1,2,P3=P3,1=-P3,2,P2,x=P2,x1=-P2,x2,P4,x=P4,x1=-P4,x2.
By using (50) and (52) and after some calculations, we can compute Pk and Pk,x(k=1,2,3,4) and obtain that
(55)P1=∑j=0,2P1,1j=16·uv(1-e-2y),P2=∑j=0,2P2,1j=(14·u2+16·uv)(1+e-2y),P3=∑j=0,2P3,1j=16·uv(1-e-2y),P4=∑j=0,2P4,1j=(14·v2+16·uv)(1+e-2y),P2,x=-(-∑j=0,2k1,jP2,1j)=-(14·u2+16·uv)(1-e-2y),P4,x=-(-∑j=0,2k1,jP4,1j)=-(14·v2+16·uv)(1-e-2y).
Thus we are led to the following system of ordinary differential equations:
(56)yt=u+v,ut=14·u2(1-e-2y),vt=14·v2(1-e-2y),ht=(u3+v3)(1-e-2y)-23·uv(u+v)(1+e-2y).
Note that this system holds before collision. With the initial condition y(τ)=u(τ)=v(τ)=0, the solution of (56) is y(t)=u(t)=v(t)=0 and h(t)=h(τ). It means that the multipeakon solution remains identically equal to zero after the collision.If we consider a more general case with two colliding peakons by using the Hamiltonian system before collision, then from (50) and (52), the system (36) can be rewritten as
(57)dy1dt=u1+v1,dy2dt=u2+v2,du1dt=14·u12-14·u22ey1-y2,du2dt=14·u12ey1-y2-14·u22,dv1dt=14·v12-14·v22ey1-y2,dv2dt=14·v12ey1-y2-14·v22,dH1dt=(u23-2u2P2,2+v23-2v2P4,2)-(u13-2u1P2,1+v13-2v1P4,1).
Thus we have
(58)y1=ln(c1-c2c1e-c1(t-τ)-c2e-c2(t-τ)),y2=ln(c1ec1(t-τ)-c2ec2(t-τ)c1-c2),u1=(c11)2-(c21)2e(c11-c21)(t-τ)c11-c21e(c11-c21)(t-τ),u2=(c21)2-(c11)2e(c11-c21)(t-τ)c21-c11e(c11-c21)(t-τ),v1=(c12)2-(c22)2e(c12-c22)(t-τ)c12-c22e(c12-c22)(t-τ),v2=(c22)2-(c12)2e(c12-c22)(t-τ)c22-c12e(c12-c22)(t-τ),
where c1=c11+c12 and c2=c21+c22 denote the speed of the peaks y1 and y2, respectively. At collision time, we have
(59)y1(τ)=y2(τ)=0,u1(τ)=u2(τ)=c11+c21,v1(τ)=v2(τ)=c12+c22.
For the initial data given by (59), the solution of (57) is
(60)u1(t)=u2(t)=c11+c21,v1(t)=v2(t)=c12+c22,y1=y2=(c1+c2)t,H1(t)=H1(τ),
with c1=c11+c12, c2=c21+c22, and after the collision we obtain a single peakon traveling at speed c1+c2.We know that ifpi(0) and ri(0) have the same sign for all i=1,2,…,n and qi(t) remain distinct, (57) admits a unique global solution, where the peakons are traveling in the same direction. However, when two peakons have opposite signs, collisions may occur, and if so, the system (57) blows up. In Figures 1 and 2, the solution is plotted with c1=15, c2=-5, and τ=1.Figure 1
Position of the peakons.(a) Initial configuration. (b) The two peakons collide.
(a)
(b)


## 5. Conclusion

Considered in this paper is the dissipative property of the modified coupled two-component Camassa-Holm system after wave breaking. Based on the obtained global dissipative solutions of the modified coupled two-component Camassa-Holm system, we construct the dissipative multipeakon solutions, a useful result for understanding the inevitable multipeakon phenomenon near wave breaking. Some interesting issues worthy of further investigation include the computational complexity of the algorithms and the impact of the multipeakon dissipative behavior on the system performance when blended into control system design.



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*Source: 107450-2013-08-20.xml*