Abstract and Applied Analysis

Volume 2014 (2014), Article ID 348695, 16 pages

http://dx.doi.org/10.1155/2014/348695

Research Article

On the Global Dissipative and Multipeakon Dissipative Behavior of the Two-Component Camassa-Holm System

Yujuan Wang,¹ Yongduan Song,¹ and Hamid Reza Karimi²

¹School of Automation, Chongqing University, Chongqing 400044, China

²Department of Engineering, Faculty of Engineering and Science, University of Agder, 4898 Grimstad, Norway

Received 14 February 2014; Revised 20 March 2014; Accepted 20 March 2014; Published 4 May 2014

Academic Editor: Bo Shen

Copyright © 2014 Yujuan Wang 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.

Abstract.

The global dissipative and multipeakon dissipative behavior of the two-component Camassa-Holm shallow water system after wave breaking was studied in this paper. The underlying approach is based on a skillfully defined characteristic and a set of newly introduced variables which transform the original system into a Lagrangian semilinear system. It is the transformation, together with the associated properties, that allows for the continuity of the solution beyond collision time to be established, leading to a uniquely global dissipative solution, which constructs a semigroup, and the multipeakon dissipative solution.

1. Introduction

In view of the wide applications in fluid dynamics, nonlinear optics, biochemistry, microbiology, physics, and many other fields, the study of the dynamic behavior of shallow water wave represents an important subject of research [1–5]. The Camassa-Holm (CH) equation [1] has been widely used to model the unidirectional propagation of shallow water waves over a flat bottom. The nonstandard properties of the CH equation set it apart from the classical soliton equations such as KdV, the first two remarkable of which are that it has peaked solitons [1, 6] and is able to model wave breaking [1, 2]. The presence of breaking waves means that the solution remains bounded while its slope becomes unbounded in finite time [2, 5]. After wave breaking, the solutions of the CH equation become uniquely as either global conservative [7–9] or global dissipative solutions [10–12].

In this paper, we focus on the two-component Camassa–Holm shallow water system [13–15]:

with , , ( or in the “short wave” limit, ). Here represents the horizontal velocity of the fluid and is in connection with the free-surface elevation from equilibrium (or scalar density). This system appears originally in [16] and then derived by Constantin and Ivanov [13] in the context of shallow water theory. System (1) is an extension of the Camassa-Holm (CH) equation by combing its integrability property with compressibility or free-surface elevation dynamics in its shallow water interpretation [6, 17]; analogously it is formally integrable [13–15] in the sense that it can be written as a compatibility condition of two linear systems (Lax pair) with a spectral parameter :

The Cauchy problem for the two-component Camassa-Holm (CH2) system has been studied extensively [18–22]. It has been shown that the CH2 system is locally well-posed with initial data , [18]. The system also has global strong solutions which blow up in finite time [19, 21, 22] and a global weak solution [23]. However, the problem about continuation of the solutions beyond wave breaking, although interesting and important, has not been explicitly addressed yet to the best of our knowledge. In our recent work [20], we studied the continuation beyond wave breaking by applying an approach based on a novel transformation that transforms system (1) into an equivalent semilinear system of original differential equation. Such treatment makes it possible to investigate the continuity of the solution beyond collision time, leading to the multipeakon conservative solution and a global conservative solution where the energy is conserved for almost all times.

It should be noted that both conservation and dissipation are important features associated with the system. The global conservation property of the CH2 system has been obtained in [20], and the dissipative behavior of the modified coupled two-component Camassa-Holm system, different from the CH2 system, has also been established in [24, 25]; however, there is no effort made in the literature on the study of the global dissipative and the multipeakon dissipative solutions of such system studied in this paper to the best of the authors’ knowledge. In this work, we explore a new approach to establish a global and stable dissipative solution of the CH2 system, which allows for the construction of the global dissipative multipeakon solution of this system. Our study is also motivated by the early work [10, 11] in the study of the global dissipative solution of the CH equation. The main difference is that here we deal with a coupled system and consider explicitly the mutual effect between two components, which literally makes the analysis more challenging as compared to the single one considered in [10, 11]. The key to circumvent the difficulty is to use the skillfully defined characteristic and a new set of variables, which allows for the establishment of the global dissipative solutions of system (1). Furthermore, it is useful to understand whether or not system (1) has the multipeakon solution as with the CH equation [12], an important aspect related to the solutions near wave breaking. And this work develops such multipeakon solution and confirms that the semigroup of the global dissipative solution preserves the multipeakon structure.

The remainder of this paper is organized as follows. Section 2 presents the transformation from the original system to a Lagrangian semilinear system. The global solutions of the equivalent semilinear system are obtained in Section 3, which are transformed into the global dissipative solutions of the original system in Section 4. We establish the multipeakon dissipative solutions for the original system in Section 5. To demonstrate the feature of the solution, two numerical examples are considered in Section 6. Finally the paper is closed in Section 7.

2. The Original System and the Equivalent Lagrangian System

For simplicity, we consider here the associated evolution for positive times (of course, one would get similar results for negative times just by changing the initial condition into ). Let us introduce an operator , which can be expressed by its associated Green’s function such as for all . If we define as

where , then (1) can be rewritten as

Moreover, for regular solutions, we have the total energy

is constant in time. Thus (4) possesses the -norm conservation law defined as

where . Since with , Young’s inequality ensures .

We reformulate system (4) into a Lagrangian equivalent semilinear system as follows. Let denote the solution of system (4). For given initial data , we define the corresponding characteristic as the solution of

where the variable is identified with a “particle.” The change in Lagrangian energy distribution along the particle path is given by

It is not hard to check that

Then it follows from (7) and (9) that

Throughout the following, we use the notation

In the following, we drop the variable for simplicity. After the change of variables and , we obtain the following expressions for and ; namely,

where we have taken that is an increasing function for any fixed time for granted (the validity will be proved later) and have used the fact that . From the definition of the characteristic, it follows that

Let us introduce another variable , such as (it will turn out that ). With these new variables, we now derive an equivalent system of (4):

where and are given by (12). Differentiating the first two equations in (14) with respect to yields

which is semilinear with respect to the variables , , and . Dissipative solutions differ from conservative solutions when particles collide, that is, where for in an interval of positive length. If we solve (14) and (15), we obtain the conservative solution. However, to obtain the dissipative solution, we impose that when particles collide, they lose their energy; that is, if for some , then we set . One can show that implies so that the system (15) implies that for . Thus, we can define to be the first time when vanishes; namely,

However, from this definition in (16), it is not clear whether is measurable or not. We now replace this definition by the following one. Let

where is a dense countable subset of and the sets are measurable for all , with for . Let us set with being the indicator function of the set . It is not hard to know that is by construction measurable and increasing with respect to , which is also bounded by . Thus, we define

and is a measurable function. One can check that this definition coincided with (16). Hence, the expressions for and in (12) become

and the modified system to be solved here reads

where is the indicator function of the set , which can be regarded as an O.D.E. in the Banach space .

3. Global Solutions of the Equivalent System

To obtain the global existence of solutions, we start from a contraction argument that offers the local existence of solutions, which is proved in the Banach space . Note that global solutions of (20) may not exist for all initial data in . However, they exist when the initial data belongs to the set which is defined as follows.

Definition 1. The set is composed of all such that

with , and is given by

for , where is the following subset of :

We write system (21) in a compact form that

where . For , we set . Note that, in this definition, we do not reset the energy density to zero after collision but keep the value it reached just before the collision. Let us define (the subset of ) as follows:

Theorem 2. Given initial data , there exists a time such that the system (20) and (21) admit a unique solution in .

Proof. To obtain the local existence of solutions, it suffices to show that given by (25) is a contraction, and, therefore, there exists a unique fixed point which is solution to (20) and (21).

Our main task is to prove the estimates for and defined by (19). Let us write as with

where

Young’s inequalities imply that

where we have denoted by a generic constant. The estimate for can be obtained in exactly the same way. We thus obtain that given in (25) is a contraction, and the local existence of solutions follows from the standard contraction argument of ordinary differential equations.

When proving existence and stability, a priori control is essential. Let us introduce the set

and therefore there exists an such that

for , and in addition

and the set

and therefore, by taking large enough depending only on and , is a decreasing function and are increasing functions of time for , and in addition

We have because , and which implies .

Lemma 3. Given for some constant , one denotes the solution of (20), (21) by with initial data . Let . Then the following hold.

(i)Consider

for all and almost all . Thus implies .(ii)Consider

for all and a constant .

Proof. (i) We know that (35)–(37) hold for almost every at because . We consider a fixed that we suppress in the notation. On the one hand, it follows from (21) that

and on the other hand

Hence, , which together with yields for all , and (36) has been proved. It follows directly from the definition of and that for . For , we have as . The continuity (with respect to ) of implies and, therefore, for all because the variable does not change for .

(ii) We consider a fixed and drop it in the notation. Let us denote the Euclidean norm of by . Since , we have

We have for a constant depending on the norm chosen for the matrix . We infer from (25) that

for a constant depending only on and . We denote generically by such constant. Thus, . Gronwall’s lemma implies that , and therefore

Since and , it follows from (36) that , and therefore , which yields

We show that the local solution can be extended into the global solution.

Theorem 4. Given initial data , system (20) and (21) admit a unique global solution .

Proof. The local solution described in Theorem 2 does not provide any lower bound on the time of existence of the solution. Let us introduce the maximum time of existence defined as

Let us assume that . To prove global existence of the solution, the basic ingredient is a global bound on the solution .

We begin by showing that, for given , there exists a constant such that, for any , if for some set , then is bounded by a constant ,. It follows from (21) that

for all , where we have used that as . We denote by a general constant and drop for simplification. We have

where we have used the fact that when , . Thus,

We can obtain from the governing equation (20) that

and then . We can also get from the governing equation (20) that

From the identity , we can deduce that

which implies that

Therefore, . To summarize, we have established

We consider a set such that is finite. Thus it is not hard to obtain that

Similarly, one can obtain that the same bound holds for . Let

By taking the -norm on both sides of (20) and (21), it follows that

which, by using Gronwall’s lemma, yields

for some constant depending only on and .

It is not hard to check that there exists a constant such that for all . We know that there exists a constant such that (34) holds. From (20), we get

for a constant depending only on . We denote generically by such constants. From the above proof, we have . It follows from (25) that

We obtain from (22c) that

and, therefore, . For , we have from (34) that for . Thus we have

which implies that exists as is a Banach space and we denote it by . We claim that . Since for all , we have and therefore and the condition (22d) holds. Since or convergence implies almost everywhere convergence up to a subsequence, the conditions (22b), (22c), and (22e) are fulfilled. It remains to check (22f). Note that the mapping defined in (23) is lower-semicontinuous. We consider a sequence such that for almost every . We can check from the definition of that is positive as and are positive. Thus, we get by the lower semicontinuity of that

and because . The composition of an increasing lower semicontinuous function with a lower semicontinuous function is also lower semicontinuous. Hence, since is increasing for and is positive, we get from Fatou’s Lemma that

which implies

as

Hence, fulfills (22f) and . We get from Theorem 2 the existence of a local solution with initial data which, combined with on , gives a solution on for some . The assumption regarding is contradicted, and we have proved the global existence of solutions.

4. Global Dissipative Solutions for the Original System

We show that the global solution of the equivalent system (20) and (21) yields a global dissipative solution of the original system (4), which needs to establish the correspondence between the Lagrangian equivalent system and the original system.

Let us start by introducing the set as . For any , we define

so that , and

for any such that . We consider the pushforward of by and denote it by ; that is, . By the Radon-Nikodym theorem, there exists a unique function in such that

where is the singular part of the decomposition of and the absolutely continuous part. From the definitions (65)–(67), it is not hard to check that . Thus we define the mapping as , which is a continuous mapping with respect to the distance on bounded set of ; that is, for any sequence and in , we have

The system is invariant with respect to relabeling. Let us explain what we mean by relabeling.

Definition 5. If there exists a which satisfies

and such that

one says that is a relabeling of .

Note that and imply that for almost all such that ,

Thus, for any , if is a relabeling of , is a relabeling of . In the dissipative case, we cannot define an equivalence relation between elements that are equal up to a relabeling. If is a relabeling of , then is not necessarily a relabeling of , basically because is either not well defined or not sufficiently regular. However, we have the result that if is a relabeling of , then . Thus we can define an equivalence relation in as follows: and are equivalent if ; that is, if is a relabeling of , then and are equivalent. The set of equivalent classes is in bijection with on which we can define a semigroup as with the semigroup property

We now establish a bijection between Lagrangian equivalent system and original system by introducing two mappings between the original variable and the Lagrangian variable .

The mapping is defined from original system to Lagrangian equivalent system as follows. Given , we denote and define as

and set

Then, . We denote by the mapping from to , which sends bounded set of into bounded sets of ; that is, for any and , implies for some constant depending only on .

Let us introduce the mapping from Lagrangian equivalent system to original system. Given , if , the function

is well defined and belongs to . We denote by the mapping from to with the property that from which we know that implies .

We claim that is in bijection with as the following theorem shows.

Theorem 6. Consider the following

Proof. Given , we denote and . We have . Since , is invertible and therefore . Hence, and .

Given , we denote and . Let . It is not hard to check that . As we have known, and since , we have and . Let . We get

as , which can be rewritten as

or

Since and almost everywhere, after a change of variables in (78), we get

Hence, by definition, and satisfy (72) and therefore they coincide; that is, . We have

almost everywhere.

The bijective mapping allows us to transport the metric on and the semigroup from to . The metric on and the semigroup on are given by

Then a continuous semigroup of dissipative weak solutions for the two-component Camassa-Holm system is obtained as the following theorem shows.

Theorem 7. Let be given. If one denotes to be the corresponding trajectory, then is a weak solution of the two-component Camassa-Holm system (4), which constructs a continuous semigroup with respect to the metric on bounded sets of ; that is, for any and any sequence such that , one has that implies .

Proof. We denote and . To prove is a weak solution of the original system (4), it suffices to show that, for all with compact support,

where is given by (4). On the one hand, since is Lipschitz and invertible with respect to for almost all , we then can use the change of variables and obtain

We have and then as for . By using the identity , and since for , it then follows from (20) that

On the other hand, using the change of variables and , and since is increasing with respect to , we have

We restrict the integration domain to again because for . Then it follows from (22c) that

By comparing (84) and (86), we know that

Hence, the first identity in (82) holds. The second identity in (82) follows in the same way. Given a converging sequence such that , we have, by the definition of , that in and for some , and in and therefore in from the definition of .

5. Multipeakon Solutions of the Original System

We derive a new system of ordinary differential equations for the multipeakon solutions which is well-posed even when collisions occur in this section, and the variables are used to characterize multipeakons in a way that avoids the problems related to blowing up.

Solutions of the 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. Extending the solution beyond wave breaking imposes significant challenge as can be illustrated in the case of multipeakons given by

where satisfy the explicit system of ordinary differential equations

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. Clearly, if the remain distinct, the system (89) allows for a global smooth solution. In the case where has the same sign for all , then remain distinct, and (89) admits a unique global solution. In this case, the peakons are traveling in the same direction. However, when two peakons have opposite signs, collisions may occur, and if so, the system (89) blows up.

We consider initial data given by

Without loss of generality, we assume that the and are all nonzero and that the are all distinct. From Theorem 7 we know that there exists a unique and global weak solution with initial data (90), and the aim is to characterize this solution explicitly. We consider the following characterization of multipeakons. The multipeakons are given as continuous solutions defined on intervals as the solutions of the Dirichlet problem

where the variables denote the position of the peaks and the variables denote the values of at the peaks. In the following we will show that this property persists for dissipative solutions.

Let us define as

which is a representative of in Lagrangian equivalent system; that is, .

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 (20) 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 following theorem.

Theorem 8. For any given multipeakon initial data , let be the solution of system (20), (21) with initial data given by (92a), (92b), and (92c). 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 . Assuming that , we have

and therefore

We set

For a given and , differentiating (96) with respect to , it then follows from (20) and (21) that

Differentiating (22c) with respect to , we get

We have, after inserting the value of given by (98) into (97) and multiplying the equation by , that

Since , it follows from (22c) that

For any , as is multipeakon initial data, we have . It thus follows from Gronwall’s lemma that and therefore for .

Thus, the system of ordinary differential equations that the dissipative multipeakon solutions satisfy can be derived based on the fact that we have known that the multipeakon structure is preserved by the semigroup of dissipative solutions.

Let us define

For each , by using (20), we obtain the following system of O.D.E.; namely,

where , , and . We have

Let us denote and then we get

where we have used the fact that on and if and only if and the domain of integration in (103) has been extended to the whole axis as on which implies that . Similarly, we can get that

For , , we write as

The constants , and depend on , and and read

where

The constants , and uniquely determine on the interval . We now turn to the computation of given by (106). Let us write as

We have set , , , , , , and . We have

Let

Inserting (110) into (106), we obtain

From (108) and (110), we get

It then follows from (113) that

Therefore, the above formulas in (114) imply that

which can also be written in the following form:

with

We compute in the same way and obtain

Now we can summarize the result as follows.

Theorem 9. Let , , and for with multipeakon initial data given by (90). Then, with initial data , there exists a global solution of (105), (116), and (118). For each time , is defined as the solution of the Dirichlet problem with boundary conditions , on each interval . Thus is a dissipative solution of the two-component Camassa-Holm system, which is the dissipative multipeakon solution.

In the following, we give the examples with cases with collision and without collision.

6. Numerical Examples

Two examples are considered here to illustrate the property of the system solution. The first is when = 1, and the second is when .

(i)Let . From (116) and (118), we can compute that and . Thus from (102), we obtain that and with some constants. There is no collision and we find the one peakon .(ii)Let . We consider the case of an antisymmetric pair of peakons when the two peakons collide. We take the initial conditions as

for some strictly positive constants and (the initial total energy of the system). We denote by 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

Using (116) and (118) we can compute and and obtain after some calculations that

Thus we are led to the following system of ordinary differential equations:

Note that this system holds before collision. The solution of (122) with the initial condition is and . It means that the multipeakon solution remains identically equal to zero after the collision.

We can get from (116) that

Thus, for , we have

and therefore the system (102) rewrites

If we consider a more general case with two colliding peakons by using the Hamiltonian system before collision, we have

where , denote the speed of the peaks and . At collision time, we have

Thus, for the initial data given by (127), the solution of (125) is

and after the collision we obtain a single peakon traveling at speed . Figures 1 and 2 represent the solution for , , and , respectively.

(a)

(b)

Figure 1: (a) Initial configuration. (b) The two peakons collide.

Figure 2: Position of the peakons.

7. Conclusion

Considered in this paper is the dissipative property of the two-component Camassa-Holm system after wave breaking. By using a new approach, we obtain the global dissipative solutions of the two-component Camassa-Holm system and then the multipeakon dissipative solutions, a useful result for understanding the inevitable multipeakon phenomenon near wave breaking.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported in part by the Major State Basic Research Development Program 973 (no. 2012CB215202), the National Natural Science Foundation of China (no. 61134001), and the Key Laboratory of Dependable Service Computing in Cyber Physical Society (Chongqing University), Ministry of Education.

References

R. Camassa and D. Holm, “An integrable shallow water equation with peaked solitons,” Physical Review Letters, vol. 71, no. 11, pp. 1661–1664, 1993.

A. Constantin and J. Escher, “Wave breaking for nonlinear nonlocal shallow water equations,” Acta Mathematica, vol. 181, no. 2, pp. 229–243, 1998.

A. Constantin, “The Hamiltonian structure of the Camassa-Holm equation,” Expositiones Mathematicae, vol. 15, no. 1, pp. 53–85, 1997.

A. Constantin, “On the scattering problem for the Camassa-Holm equation,” Proceedings of the the Royal Society of London Series A: Mathematical, Physical and Engineering Sciences, vol. 457, no. 2008, pp. 953–970, 2001.

A. Constantin, “Global existence of solutions and breaking waves for a shallow water equation: a geometric approach,” Annales de l'Institut Fourier, vol. 50, no. 2, pp. 321–362, 2000.

A. Constantin and D. Lannes, “The hydrodynamical relevance of the Camassa–Holm and Degasperis–Procesi equations,” Archive for Rational Mechanics and Analysis, vol. 192, no. 1, pp. 165–186, 2009.

A. Bressan and A. Constantin, “Global conservative solutions of the Camassa–Holm equation,” Archive for Rational Mechanics and Analysis, vol. 183, no. 2, pp. 215–239, 2007.

H. Holden and X. Raynaud, “Global conservative solutions of the Camassa-Holm equation—a lagrangian point of view,” Communications in Partial Differential Equations, vol. 32, no. 10, pp. 1511–1549, 2007.

H. Holden and X. Raynaud, “Global conservative multipeakon solutions of the Camassa-Holm equation,” Journal of Hyperbolic Differential Equations, vol. 4, no. 1, pp. 39–64, 2007.

A. Bressan and A. Constantin, “Global dissipative solutions of the Camassa-Holm equation,” Analysis and Applications, vol. 5, no. 1, pp. 1–27, 2007.

H. Holden and X. Raynaud, “Dissipative solutions for the Camassa-Holm equation,” Discrete and Continuous Dynamical Systems, vol. 24, no. 4, pp. 1047–1112, 2009.

H. Holden and X. Raynaud, “Global dissipative multipeakon solutions of the Camassa-Holm equation,” Communications in Partial Differential Equations, vol. 33, no. 11, pp. 2040–2063, 2008.

A. Constantin and R. Ivanov, “On an integrable two-component Camassa–Holm shallow water system,” Physics Letters A, vol. 372, no. 48, pp. 7129–7132, 2008.

M. Chen, S.-Q. Liu, and Y. Zhang, “A two-component generalization of the Camassa-Holm equation and its solutions,” Letters in Mathematical Physics, vol. 75, no. 1, pp. 1–15, 2006.

G. Falqui, “On a Camassa–Holm type equation with two dependent variables,” Journal of Physics A: Mathematical and General, vol. 39, no. 2, pp. 327–342, 2006.

P. Olver and P. Rosenau, “Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support,” Physical Review E, vol. 53, Article ID 1900, 1996.

R. S. Johnson, “Camassa–Holm, Korteweg–de Vries and related models for water waves,” Journal of Fluid Mechanics, vol. 455, pp. 63–82, 2002.

G. Gui and Y. Liu, “On the Cauchy problem for the two-component Camassa–Holm system,” Mathematische Zeitschrift, vol. 268, no. 1-2, pp. 45–66, 2011.

J. Escher, O. Lechtenfeld, and Z. Y. Yin, “Well-posedness and blow-up phenomena for the 2-component Camassa-Holm equation,” Discrete and Continuous Dynamical Systems, vol. 19, no. 3, pp. 493–513, 2007.

Y. Wang and Y. Song, “Global conservative and multipeakon conservative solutions for the two-component Camassa-Holm system,” Boundary Value Problems, vol. 2013, article 165, 2013.

C. Guan and Z. Y. Yin, “Global existence and blow-up phenomena for an integrable two-component Camassa–Holm shallow water system,” Journal of Differential Equations, vol. 248, no. 8, pp. 2003–2014, 2010.

G. Gui and Y. Liu, “On the global existence and wave-breaking criteria for the two-component Camassa–Holm system,” Journal of Functional Analysis, vol. 258, no. 12, pp. 4251–4278, 2010.

C. Guan and Z. Yin, “Global weak solutions for a two-component Camassa–Holm shallow water system,” Journal of Functional Analysis, vol. 260, no. 4, pp. 1132–1154, 2011.

Y. Wang and Y. Song, “On the global existence of dissipative solutions for the modified coupled Camassa–Holm system,” Soft Computing, vol. 17, no. 1, pp. 2007–2019, 2013.

Z. Shen, Y. Wang, H. Karimi, and Y. Song, “On the multipeakon dissipative behavior of the modiied coupled Camassa-Holm model for shallow water system,” Mathematical Problems in Engineering, vol. 2013, Article ID 107450, 11 pages, 2013.