Abstract
This paper investigates the continuation of solutions to the modified coupled two-component Camassa-Holm system after wave breaking. The underlying problem is rather challenging due to the mutual coupling effect between two components in the system. By introducing a novel transformation that makes use of a skillfully defined characteristic and a set of newly defined variables, the original system is converted into a Lagrangian equivalent system, from which the global conservative solution is obtained, which further allows for the establishment of the multipeakon conservative solution of the system. The results obtained herein are deemed useful for understanding the inevitable phenomenon near wave breaking.
1. Introduction
We consider here the following modified coupled two-component Camassa-Holm system with peakons [1]: System (1) is a modified version of the new coupled two-component Camassa-Holm system in the following equation; namely, which, as an extension of the Camassa-Holm (CH) equation, has been established by Fu and Qu to allow for peakon solitons in the form of a superposition of multipeakons. By parameterizing for system (2), it then takes the form of (1), which can be rewritten as a Hamiltonian system with the Hamiltonian , where , , and .Particularly, when (or ), the degenerated (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:
Note that, when , system (1) is reduced to the scalar Camassa-Holm equation as follows: The CH equation, which models the unidirectional propagation of shallow water waves over a flat bottom, has a bi-Hamiltonian structure [3] and is completely integrable [4–6]. The CH equation has attracted considerable attention because it has peaked solitons [4, 7] and experiences wave breaking [4, 8]. 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 can be continued uniquely as either global conservative [10–13] or global dissipative solutions [14].
As one of the integrable multicomponent generalizations of the CH equation, system (1) has been shown to be locally well posed with global strong solutions which blow up in finite time [1, 2]. Moreover, the existence issue for a class of local weak solutions for the modified coupled CH2 system was also addressed in [1]. It has been known that the continuation of solutions for the system beyond wave breaking has been a challenging problem. In our recent work [15], the problem of continuation beyond wave breaking for the modified coupled CH2 system was studied by applying an approach that reformulates the system (1) into a semilinear system of O.D.E. taking values in a Banach space. Such treatment makes it possible to investigate the continuity of the solutions beyond collision time, leading to the uniquely global solutions of this system. Also the global dissipative and multipeakon dissipative solutions of this system have been established in [16, 17], while, as far as the authors’ concern, there is no effort made in the literature on the study of the global conservative as well as multipeakon conservative solutions of such system, another important feature associated with the system. Motivated by our recent work [15–17], in this paper we develop a new approach to establish a global and stable solution for the modified coupled CH2 system, which is conservative and further allows for the construction of the multipeakon conservative solution. The approach utilized in this paper makes use of a novel system transformation, which is different from [15] and is based on a skillfully defined characteristic and a set of newly introduced variables, where the associated energy is introduced as an additional variable so as to obtain a well-posed initial-value problem, facilitating the study on the behavior of wave breaking. It should be stressed that both global stable solution and multipeakon solution are important aspects related to the solutions near wave breaking, while there is no effort made in the literature on the study of multipeakon property of system (1), which is another motivation of this work. Our inspiration of investing the underlying issue mainly also stems from the early work [10, 11] in the study of the global conservative solution of the CH equation and [13] where the multipeakon solution is obtained for the CH equation. In this work a coupled system is dealt with where the mutual effect between two components makes the analysis more complicated than a single one as considered in [10, 11, 13]. By utilizing the novel transformation method, the inherent difficulty is circumvented and then the global conservative solutions of (1) are obtained, which then allows for the establishment of the multipeakon conservative solution of system (1).
The remainder of this paper is organized as follows. Section 2 presents the basic equations. In Section 3, by introducing a set of Lagrangian variables, we transform the original system into an equivalent semilinear system and derive the global solutions of the equivalent system. We obtain a global continuous semigroup of weak conservative solutions for the original system in Section 4 and the multipeakon conservative solution in Section 5.
2. The Original System
We first introduce an operator , which can be expressed by its associated Green’s function such as , for all , where denotes the spatial convolution. Thus we can rewrite (1) as a form of a quasilinear evolution equation: Let us define , , , and as Then (1) can be rewritten as For regular solutions, we get that the total energy is constant in time. Thus (8) possesses the -norm conservation law defined as where denotes the solution of system (8). Note that , and so Young’s inequality ensures that .
3. Global Solutions of the Lagrangian Equivalent System
We reformulate system (8) as follows. For a given initial data , we define the corresponding characteristic as the solution of and we define the Lagrangian cumulative energy distribution as It is not hard to check that Then it follows from (11) and (13) that
Throughout the following, we use the notation
In the following, we drop the variable for simplification. Here, we take as an increasing function for any fixed time for granted (later on we will prove this). Then after the change of variables and , we obtain the following expressions for and (); namely, Since , then , , , and can be rewritten as From the definition of the characteristic, it is not hard to check that We introduce another variable with . It will turn out that . With these new variables, we now derive an equivalent system of (8) as follows: where and are given in (18), while , , , and are given in (19). We regard system (21) as a system of ordinary differential equations in the Banach space endowed with the norm for any . Here is a Banach space with the norm given by . Note that .
Differentiating (21) with respect to the variable yields which are semilinear with respect to the variables , , , and .
To obtain the uniqueness of solutions, one proceeds as follows. By proving that all functions on the right-hand side of (21) are locally Lipschitz continuous, the local existence of solutions will follow from the standard theory of ordinary differential equations in Banach spaces. In a second step, we will then prove that this local solution can be extended globally in time. Note that global solutions of (21) 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 (i) (ii) (iii) where and .
Lemma 2. Let and let , or let be two locally Lipschitz maps. Then, the product is also a locally Lipschitz map from to or from to .
Theorem 3. Given initial data , there exists a time depending only on such that the system (21) admits a unique solution in .
Proof. To obtain the local existence of solutions, it suffices to show that , given by
with , is a Lipchitz function on any bounded set of which is a Banach space.
Our main task is to prove the Lipschitz continuity of and () given by (18) and (19) from to . We first prove that given in (19) is locally Lipschitz continuous from to and the others follow in the same way. Let us write
where denotes the indicator function of a given set . Let
We rewrite as
where is the operator from to given as
Since the operator (given in Section 2) is linear and continuous from to and is continuously embedded in , we have . It is not hard to check that is locally Lipschitz continuous from to and therefore from to . Thus is locally Lipschitz continuous from to . Since the mapping is locally Lipschitz continuous from to , by Lemma 2, we deduce that is locally Lipschitz continuous from to . Similarly, is also locally Lipschitz continuous and therefore is locally Lipschitz continuous. One proceeds in the same way and proves that , , , and defined by (18) and , , and defined by (19) are locally Lipschitz continuous from to . We rewrite the solutions of (21) as
Thus the theorem follows from the standard contraction argument of ordinary differential equations.
It remains to prove the existence of global solutions of (21). Theorem 3 gives us the existence of local solutions of (21) for initial data in . In the following, we will only consider initial data that belongs to given by . To obtain that the solution of (21) belongs to , we have to specify the initial condition for (24). Let We have . For , is taken as , and is given as , for .
The global existence of the solution for initial data in relies essentially on the fact that the set is preserved by the flow as the next lemma shows.
Lemma 4. Given initial data , one considers the local solution of (21) with initial data for some . One then gets that for all . Moreover, for a.e. , , for a.e. , and exists and is independent of time for all .
Proof. We first show that for all . For any given initial data , we get that the local solution of (21) belongs to , which satisfies (25) for all . We now show that (27) holds for any and therefore a,e.. Consider a fixed and drop it in the notation if there is no ambiguity. On the one hand, it follows from (24) that
and, on the other hand,
Hence, . Notice that ; then for all and (27) has been proved. We now prove the inequalities in (26). Set . Assume that . Since is continuous with respect to , we have . It follows from (27) that . Furthermore, (24) implies that and . If , then which implies that for all by the uniqueness of the solution of system (24). This contradicts the fact that for all . If , then . Since , there exists a neighborhood of such that for all . This contradicts the definition of . Hence, . We now have , which conversely implies that for all , which contradicts the fact that . Thus we have proved for all . We now prove that for all . This follows from (27) when . If , then from (27). As we have seen, would imply that for some in a punctured neighborhood of , which is impossible. Hence, for all . Now we get that for all . If for some , it then follows that which implies that for all , which contradicts the fact that for all . Hence, . This completes the proof that for all .
We now prove that for almost all . Define the set . It follows from Fubini’s theorem that
where and . From the above proof, we know that, for all , consists of isolated points that are countable. This means that . Since , it thus follows from (37) that for almost every . This implies that for almost all and therefore is strictly increasing and invertible with respect to .
For any given , since and , we know that exist. We have the following:
Let . Since , , , are bounded in and as for all , (38) implies that for all . Since , it follows that for all .
Theorem 5. For any initial data , there exists a unique global solution for the system (21). Moreover, for all , we have , which constructs a continuous semigroup.
Proof. To ensure that the local solution of system (21) can be extended to a global solution, it suffices to show that Since is an increasing function with respect to for all and , we have . We now consider a fixed and drop it for simplification. Since when and , for a.e. , it follows from (27) that which implies that and therefore Similarly, We can obtain from the governing equation (21) that and then . We can also get from the governing equation (21) that From the identity , we can deduce that which implies that Therefore, . It is not hard to know that . Similarly, one can obtain that the bounds hold for , , , , , , and . Let Using the integrated version of (21) and (24), after taking the -norms on both sides, we obtain It follows from Gronwall’s inequality that . Hence, we infer that the map defined as generates a continuous semigroup from the standard theory of ordinary differential equations.
4. Global Conservative Solutions of the Original System
We show that the global solution of the equivalent system (21) yields a global conservative solution of the original system (8), which constructs a continuous semigroup in this section.
To obtain the global conservative solution of the original system, we have to establish the correspondence between the Lagrangian equivalent system and the original system.
Let us first introduce the subsets and of given by where is defined as And, for any , the subsets of are given by with a useful characterization. If (), then a.e. Conversely, if is absolutely continuous, and there exists such that a.e., and then for some depending only on and . With this useful characterization of , it is not hard to prove that the space is preserved by the governing equation (21). Notice that the map defines a group action of on ; we consider the quotient space of with respect to the group action. The equivalence relation on is defined as follows: for any , if there exists such that , we claim that and are equivalent.
We denote the projection by . For any , we introduce the mapping given by . It is not hard to prove that when and for any and . Hence, we can define the map as , for any representative of . For any , we have . Hence, and any topology defined on is naturally transported into by this isomorphism. That is, if we equip with the metric induced by the -norm; that is, , for all , which is complete, then the topology on is defined by a complete metric given by for any . Let us denote by the continuous semigroup which to any initial data associates the solution of (21). The system (8) is invariant with respect to relabeling. That is, for any , , for an and . Thus the map given by is well-defined, which generates a continuous semigroup.
To obtain a semigroup of solution for (8), we have to consider the space , which characterizes the solutions in the original system: where and is a positive finite Radon measure with as its absolute continuous part.
We now establish a bijection between and to transport the continuous semigroup obtained in the Lagrangian equivalent system (functions in ) into the original system (functions in ).
We first introduce the mapping , which transforms the original system into the Lagrangian equivalent system defined as follows.
Definition 6. For any , let with . We define as the equivalence class of .
Remark 7. From the definition of , , , , , in (55)–(57), we can check that , which also satisfies (25). Moreover, we have from (57), which implies that . Furthermore, if is absolutely continuous, then and
for all .
We are led to the mapping , which corresponds to the transformation from the Lagrangian equivalent system into the original system. In the other direction, we obtain the energy density in the original system, by pushing forward by the energy density in the Lagrangian equivalent system, where the push-forward of a measure by a measurable function is defined as
for all the Borel set . Give any element , and let be defined as
where and . We get that , which does not depend on the representative of that we choose. We denote by the map to any and given by (60)-(61), which conversely transforms the Lagrangian equivalent system into the original system.
We claim that the transformation from the original system into the Lagrangian equivalent system is a bijection.
Theorem 8. The maps and are well-defined and . That is,
Proof. Let in be given. We consider as a representative of and given by (60)-(61) for this particular . From the definition of , we have . Let be the representative of in given by Definition 6. We have to prove that and therefore . Let
Using the fact that is increasing and continuous, it follows that
and . From (61) and since , for any , we get the following:
Since and , we have
From the definition of , it follows that
For any given , using the fact that is increasing and (64), it follows that . If , there then exists such that and (67) implies that . Conversely, since is increasing, then implies that , which gives us a contradiction. Hence, we have . Since , it follows directly from the definitions that , , , , and . We thus proved that .
Given in , we denote by the representative of in given by Definition 6. Let and be defined as before by (63). The same computation that leads to (66) now gives
Given , we consider an increasing sequence converging to which is guaranteed by (55) such that . Let tend to infinity. Since is lower semi-continuous, we have . Take and then we get
By the definition of , there exists an increasing sequence converging to such that . It follows from the definition of in (55) that . Passing to the limit, we obtain which, together with (69), yields
We obtain that by comparing (70) and (68). It is clear from the definitions that . Hence, and .
The topology defined in can be transported into , which is guaranteed by the fact that we have established a bijection between the two equivalent systems. We define the metric on as which makes the bijection between and into an isometry. Since equipped with is a complete metric space, equipped with the metric is also a complete metric space. For each , we define the mapping as Then a continuous semigroup of conservative weak solutions for the original system is obtained as the following theorem shows.
Theorem 9. Let be given. If one denotes by the corresponding trajectory, then is a weak solution of the modified coupled two-component Camassa-Holm equation (8), which constructs a continuous semigroup. Moreover, is a weak solution of the following transport equation for the energy density: Furthermore, for all , it holds that and, for almost all , Thus the unique solution described here is a conservative weak solution of the system (8).
Proof. To prove that is a weak solution of the original system (8), it suffices to show that, for all with compact support,
where , , , and are given by (8). Let the solution of (21) be a representative of . On the one hand, since is Lipschitz continuous and invertible with respect to , for almost all , we then can use the change of variables and obtain
By using the identities and , it then follows from (21) that
On the other hand, using the change of variables and and since is increasing with respect to , we have the following:
We obtain from (27) that
By comparing (78) and (80), we know that
Hence, the first identity in (76) holds. The second identity in (76) follows in the same way. One can easily check that is solution of (73). From the definition in (61), we get
which is constant in time from Lemma 2. Thus, we have proved (74).
Since a.e., for almost every , it then follows from (27) that
for any Borel set . Since is one-to-one and , a.e. and then (83) implies that
Hence, (75) is proved (and the solution is conservative), which completes the proof.
5. Multipeakon Conservative Solutions of the Original System
In this section, we will derive a new system of ordinary differential equations for the multipeakon solutions which is well posed even when collisions occur, and the variables will be 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 following form: 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 (86) allows for a global smooth solution. It is not hard to see that is a global weak solution of system (8) by inserting that solution into (85). In the case where and have the same sign for all , (86) admits a unique global solution, where the remain distinct and the peakons are traveling in the same direction. However, when two peakons have opposite signs, collisions may occur, and, if so, the system (86) blows up.
Let us consider initial data given by 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 9 with initial data (87) explicitly. Since the variables and blow up at collisions, they are not appropriate to define a multipeakon in the form of (85). 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 and . The variables denote the position of the peaks, and the variablesdenote the values of at the peaks. In the following we will show that this property persists for conservative solutions.
Let us set . The next lemma gives us the functions , , and which belong to (they even belong to ).
Lemma 10. Let such that is given, and then the solution of (21) with initial data belongs to .
Proof. To prove this Lemma, one proceeds as in Theorem 3 by using the contraction argument and replacing by
endowed with the norm
Our main task is to prove the Lipschitz continuity of and () from to . We first show that is Lipschitz continuous from to and the others follow in the same way. Given a bounded set where is a positive constant, from Theorem 3, we get that
for a constant depending only on . We can compute the derivative of given as
From Lemma 2, is Lipschitz continuous from to and therefore is Lipschitz continuous from to . Similarly, we obtain the same results for , , , and and , , . We can also compute the derivative of on as follows:
Since is locally Lipschitz maps from to , we then get that is locally Lipschitz continuous from to . The same results can be obtained for the other and () by the same way. From the standard contraction argument, the local existence of solutions of (21) can be proved in . As far as global existence is concerned, does not blow up for initial data in . For the second derivative, for any , we get that
The system (94) is affine with respect to , , , . Thus, we get that
where is a constant depending only on which is bounded on any time interval . It follows from Gronwall's lemma that does not blow up and therefore the solution is globally defined in .
We now prove that is a representative of in the Lagrangian system; that is, , where is given byWe first check that . Since is a multipeakon, we get that from (87). Hence, , , and all belong to while is identically zero. Due to the exponential decay of and and , we get that . The properties (25)–(27) are straightforward to check. It is not hard to check that and, therefore, since , we get that .
Theorem 11. Let initial data be given in (87). The solution given by Theorem 9 satisfies between the peaks.
Proof. Let us first prove that . Assuming that , we get that and therefore We set For a given , differentiating (99) with respect to and after using (21), (24), and (94), we obtain Differentiating (27) with respect to , we get After inserting the value of given by (101) into (100) and multiplying the equation by , we obtain that It follows from (27), and since , that We claim that, for any time such that , We have to prove that is in time. Since for some polynomial and , we get that , , and are in time. Since remains in , for all , from (26), we have and therefore is in time, which implies that is in time. Hence, it holds that for some constant which is independent of time, which leads to For the multipeakons at time , we have and for all . Hence, for all time and all . Thus, . Similarly, .
For solutions with multipeakon initial data, we have the following result. If vanishes at some point in the interval , then vanishes everywhere in . Moreover, for given initial multipeakon solution , let be the solution of system (21) with initial data given by (96a), (96b) and (96c), and then, between adjacent peaks, if , the solution is twice differentiable with respect to the space variable and we have , for .
We now start the derivation of a system of ordinary differential equations for multipeakons.
From (21), we get that, for each , where , , and , , respectively. Since the function is invertible, for almost every , we can use the change of variables such that and can be rewritten as
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 where and . At this point, we can get some more understanding of what is happening at a time of collision. Let be a time when the two peaks located at and collide, that is, such that . Since the solution remains in for all time, the function remains continuous so that we have , and, when tends to , , , , and may have a finite limit. However, we know that the first derivative blows up, which implies that and . Thus, and tend to zero, respectively, but are slower than . Indeed, let tend to in (114), and then, to first order in , we obtain that which implies that and tend to zero at the same rate as . We now turn to the computation of () given by (110). Let us write as We have sets , , , , , , , , and , , , . We have We set By inserting (117) into (110), we get It follows from (112) and (114) that Thus, from (120)–(126), we can obtain that It thus follows from (127)–(134) that We can also write and as where The terms , , and () can be computed in the same way and we have
The result can be summarized in the following theorem.
Theorem 12. Assume that , and for with a multipeakon initial data as given by (87). Then, there exists a global solution of (109), (136), and (138) with initial data . On each interval , one defines as the solution of the Dirichlet problem with boundary conditions and for each time . Thus is a conservative solution of the modified coupled two-component Camassa-Holm system, which is the multipeakon conservative solution.
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 Key Laboratory of Dependable Service Computing in Cyber Physical Society (Chongqing University), Ministry of Education.