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Mathematical Problems in Engineering
Volume 2013 (2013), Article ID 107450, 11 pages
On the Multipeakon Dissipative Behavior of the Modified Coupled Camassa-Holm Model for Shallow Water System
1School of Automation, Chongqing University, Chongqing 400044, China
2Department of Engineering, Faculty of Technology and Science, University of Agder, N-4898 Grimstad, Norway
Received 9 May 2013; Accepted 26 June 2013
Academic Editor: Hongli Dong
Copyright © 2013 Zhixi Shen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.
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 , 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 : which is a modified version of the coupled two-component Camassa-Holm system as established by Fu and Qu in , allowing for peakon solitons in the form of a superposition of multipeakons. System (1) can be rewritten as a Hamiltonian system, with the Hamiltonian , where , , and . Particularly, when (or ), 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:
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 . 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  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 . 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. Section 2 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 and by numerical computations when 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  in this section. System (1) can be rewritten as where , , , are given by with the Green’s function such as for all and the spatial convolution.
By using a skillfully defined characteristic , which can be decomposed as , and a new set of Lagrangian variables; namely, where corresponds to the Lagrangian energy density and , the Lagrangian velocity, we derive an equivalent system of the modified coupled Camassa-Holm system,
It is shown in  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 for in an interval of positive length. To obtain the dissipative solution, we impose that when particles collide, they lose their energy; that is, if for some , then we set . Thus we define as the first time when vanishes; namely, and the expressions for and become and the modified system to be solved here reads Note that, in this definition, we do not reset the energy density to zero for 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 and the behavior of the system remains unchanged.
The local existence of solutions is proved in the Banach space where The global solutions of (10) may not exist for all initial data in ; however, they exist when the initial data , where is defined as follows.
Definition 1. The set is composed of all such that where is given by for , where is the following subset of
The main result in  is stated in the following theorem.
Theorem 2. Let be given. If one denotes the corresponding trajectory, then is a weak dissipative solution of the modified coupled two-component Camassa-Holm system, which constructs a continuous semigroup with respect of the metric on bounded sets of ; that is, for any and any sequence such that , one has that implies .
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 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 the 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 where satisfy the explicit system of ordinary differential equations Let us consider initial data given by 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 coincide. In the case that and have the same sign for all and 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 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 and are all nonzero, and that the 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 and 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 , which are defined on intervals as the solutions of the Dirichlet problem with boundary conditions , . The variables denote the position of the peaks, and the variables denote the values of at the peaks. In the following part we will show that this property persists for dissipative solutions.
We introduce as a representative of in Lagrangian equivalent system; that is, , which is given by
Let , where denote the open interval with the conventions that and . For each interval , we define such that for all . By the linearity of the governing equations (10), and the bounds which hold on the solution and , it is not hard to check that , while .
Thus the existence of multipeakon solutions is given by the next theorem.
Theorem 3. For any given multipeakon initial data , let be the solution of system (10), (11) with initial data given by (24), (25), and (26). Between adjacent peaks, if , the solution is twice differentiable with respect to the space variable, and one has
Proof. For a given time , we consider two adjacent peaks and . If , then the two peaks have collided and, since is positive, we must have for all . Hence, , which conversely implies that when . There exists such that for any . Since and , we have . It follows from the implicit function theorem that is invertible in a neighborhood of and its inverse is , and therefore are with respect to the spatial variable and the quantity is defined in the classical sense.
We now prove that for . Let us first prove that . Assuming that , we have that and therefore We set For a given and , differentiating (30) with respect to , it then follows from (10) and (11) that Differentiating (15) with respect to , we get We have, after inserting the value of given by (32) into (31) and multiplying the equation by , that Since , it follows from (15) that For any , as is a multipeakon initial data, we have . It thus follows from Gronwall’s lemma that and therefore for . Similarly, we can obtain that .
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 For each , by using (11), we obtain the following system of O.D.E.; namely, where , , , , respectively. We have We denote . Thus, we get where we have used the fact that on and if and only if . Since on , that means , which implies that the domain of integration in (37) can be extended to the whole axis, Similarly, we can get that
Between two adjacent peaks located at and , we know that satisfies and therefore can be written as for , , where the constants , , , and depend on , , , , , and and read where Thus, the constants , , , and uniquely determine on the interval , and we compute with and . We now turn to the computation of given by (39) and (40). Let us write as Set , , , , , , , , , , , and . We have By inserting (46) into (39) and (40), we get where if , if . It then follows from (42) and (44) that Thus, from (48), we can obtain that It thus follows from (49) that where The terms , , and can be computed in the same way and we have
The result can be summarized in the following theorem.
Theorem 4. Assume , and for with a multipeakon initial data as given by (22). Then, there exists a global solution of (36), (50), and (52) with initial data . On each interval , one defines as the solution of the Dirichlet problem with boundary conditions , for each time . Thus is a dissipative solution of the modified coupled two-component Camassa-Holm system, which is the dissipative multipeakon solution.
In this section, we give the examples with the case by explicit calculations and the case by numerical computations with peakon-antipeakon collisions.(i) Let . From (50) and (52), we can compute that and , which imply that and therefore . Similarly, we can get that . Thus from (36), we can obtain that , which yields with some constants. There is no collision and we find the familiar one peakon .(ii) Let . We first consider the case of an antisymmetric pair of peakons where the two peakons collide. We take the initial conditions as for some strictly positive constants , and the initial total energy of the system, that is, the norm of the solution; we denote , the time of collision. For , the solution is identical to the conservative case. After collision, for , the solution remains antisymmetric. Let us assume this for the moment and write By using (50) and (52) and after some calculations, we can compute and and obtain that Thus we are led to the following system of ordinary differential equations: Note that this system holds before collision. With the initial condition , the solution of (56) is and . 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 Thus we have where and denote the speed of the peaks and , respectively. At collision time, we have For the initial data given by (59), the solution of (57) is with , , and after the collision we obtain a single peakon traveling at speed .
We know that if and have the same sign for all and 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 , , and .