Abstract
For the generalized mCH equation, we construct a 2-peakon solution on both the line and the circle, and we can control the size of the initial data. The two peaks at different speeds move in the same direction and eventually collide. This phenomenon is that the solution at the collision time is consistent with another solitary peakon solution. By reversing the time, we get two new solutions with the same initial value and different values at the rest of the time, which means the nonuniqueness for the equation in Sobolev spaces is proved for .
1. Introduction
The Camassa-Holm (CH) equation [1–3] is an integrable system with a bi-Hamiltonian structure, which is derived by Camassa and Holm using the asymptotic expansion in the Hamiltonian for Euler’s equation. A special kind of weak solution for this equation describes the solitary wave at the peak, called peakons [4, 5], whose wave slope is discontinuous at the peak. The interactions between any number of peakons were described by the multipeakon solutions [6, 7], in the form of a linear superposition of peakons whose amplitude and velocity change with time.
In recent years, people’s great interest in the research of Camassa-Holm (CH) equation has inspired people to explore the CH-type equation, especially the equations that admit peakons and mutipeakons. The CH, Degasperis-Procesi (DP) [8–13], modified CH (mCH) [14–19], and Novikov (NE) [20–22] equations are all integrable systems that admit peakons and mutipeakons. Of course, there are also some nonintegrable systems that admit peakons and mutipeakons, such as the b-family of equations [23], the modified b-family of equations [24], and the cubic ab-family of equations [25]. It is worth noting that the b-family of equations includes the CH equation and the DP equation, the modified b-family of equations includes the NE equation, and the cubic ab-family of equations includes the mCH equation and the NE equation.
With the development of research, great interest has been aroused in the uniqueness or posedness of solutions, setting initial value . The study of Li and Olver [26] shows that the CH equation is locally well posed in for , and Byers [27] proved the ill-posedness for the CH equation in when . Himonas, Grayshan, and Holliman [28] studied the ill-posedness for the DP equation. Himonas and Holliman [29] proved that the NE equation is well posed in for . Himonas, Kenig, and Holliman [30] demonstrated the nonuniqueness for the NE equation in when by studying the collision of the peakons. Guo et al. [31] studied the ill-posedness for the CH, DP, and NE equations in critical spaces. Himonas and Mantzavinos [32] proved that the FORQ equation (also called mCH) is well posed in for . The nonuniqueness results of Himonas and Holliman [33] show that solutions to the Cauchy problem for the FORQ equation are not unique in when . At present, there is no theory to show the uniqueness for the FORQ equation in when . Holmes and Puri [34] discussed the nonuniqueness for the ab-family of equations. Himonas, Grayshan, and Holliman [35] considered the ill-posedness for the b-family of equations in for when . On this basis, Novruzov [36] studied the ill-posedness for the b-family of equations when .
In this paper, we consider the Cauchy problem for a generalized mCH (gm- CH) equation which has the following form
This equation is obtained by Anco and Recio [37], by extending a Hamiltonian structure of the CH equation. Substituting into the first equation of (1), it infers the following partial differential equation
The results of Anco and Recio [37] show that the gmCH equation admits peakon traveling wave solutions and multipeakon solutions. They studied the existence of the single peakon travelling solutions with and classification of 2-peakon solutions. Recio and Anco [38] considered the conservation laws (energy, momentum, -norm, etc.) of the gmCH equation, by modifying the general multiplier method combined with some tools from variational calculus. They also discussed the Hamiltonian structure and solitary traveling waves of the gmCH equation, by using the conservation laws. One remark is that the Hamiltonian structure for the family (1) corresponds to an energy conservation law that has a local density but a nonlocal flux.
Based on the conservation laws in [38], the Cauchy problem and nonuniqueness of the peakon solutions in this paper are studied. Under this premise, we obtain our main result, and its proof is closely related to the conservation of norms. And based on the existence of peakons in [37], we conduct the research on the peakon solutions. The difference is that we obtain the peakon traveling wave solutions by verifying the weak solution. The peakon traveling wave solutions on the line are given by
On the circle, they are given by where is defined by
On the other hand, the classification of 2-peakon solutions in [37] helps us construct 2-peakon solutions. In contrast to this, we construct a special 2-peakon solution based on the characteristics of the ODE system and study the collision of peakons. The result is summarized in the following theorem.
Theorem 1. Solutions to the Cauchy problem for the gmCH equation (1) are not unique in Sobolev spaces when .
The rest is organized as follows. In Section 2, we study the ODE systems that the 2-peakon solutions of the gmCH Equation (1) need to satisfy. In Section 3, we give the proof of Theorem 1 on the line by constructing a 2-peakon solution. In Section 4, we prove Theorem 1 on the circle.
2. 2-Peakon on the Line and the Circle
In [37], Recio and Anco studied the multipeakon solutions on the line, and they proved the following result.
Theorem 2 (see [37]). The nonperiodic 2-peakon is a solution to Equation (1) if its positions and momenta satisfy Now, we consider the 2-peakon system on the circle, based on the methods in [25].
Theorem 3. The periodic 2-peakon is a solution to Equation (1) if its positions and momenta satisfy where is defined as in (6).
Proof. We can rewrite the equation (1) as the following equivalent form Let be any smooth periodic test function on , and which causes the periodic 2-peakon solution (9) to be rewritten as . So, we have Firstly, we calculate . Note that when , we have and , which leads to . Let . Obviously, . Since for and for , it obtains by integrating by parts Since and , On the other hand, when , we find for . It follows that along with (17) leads to , for all . Analogously, since is odd, , which means that . Moreover, we find Now, we conclude . We use to get Since for , where , we find and , which combined with integration by parts give that Without loss of generality, we assume . So, we have Similar to (22), we obtain Although the above calculation is made by assuming , the result is also valid to any and . We next compute It follows from (20)-(28) that Similar to (29), it infers with Substituting (29)-(32) into the equation (13), also noting and , which is caused by the fact that for , we obtain the system (10).
3. Nonuniqueness on the Line
In this section, we use the ODE system (8) to prove Theorem 1 on the line. To do this, we take a 2-peakon solution of the form (7). From the first two items of the system (8), and are obvious. At the same time, we have if we take the symmetric initial data, . The two peaks move at the same speed which means there is no collision. Therefore, we choose the following initial data with and , where . The selection of these initial data is summarized in Figure 1.
According to (33), and are obtained. We introduce the symbol to represent the difference between the positions of the two peakons, in other words, . It follows from the ODE system (8) that where . Integrating (34), we calculate
Since and , we obtain a collision and a positive collision time when . Using the symbol for the collision time, from (35), we find
Applying expressions and to define the collision location and the collision function , we get the following proposition.
Proposition 4. The limit of as approaches is , or
Proof. We compute the Fourier transform of and , which is denoted by Combining (38) and (39), we have Notice that the equation inside the absolute value can be scaled up to Let . There is no doubt that is integrable when , and dominates the original integrand, which means, we can apply the dominated convergence theorem and put the limit inside the integral. So, we get Proposition 4 is proven.
Proof of Theorem 1. (On the line). In view of the 2-peakon solution we constructed, we need to construct a traveling wave solution that satisfies Reviewing the system (8), we take the following data The system is simplified and easy to be solved as We introduce the symbol to represent corresponding to and to represent corresponding to . It follows from (43) that . So, we find ,which makes the construction complete. In order to prove the nonuniqueness of the solution, we define two new solutions . Since and are two solutions to (1), and are also solutions to (1), which can be obtained by reversing time. These mean and solve (1). Moreover, through Proposition 4, we have . Finally, we note that the initial data for these nonunique solutions can be made arbitrarily small. Since the new initial data is the collision function , we have Therefore, for any , we can find a to make .
4. Nonuniqueness on the Circle
In this section, we use the ODE system (10) to prove the Theorem 1 on the circle. To do this, we take a 2-peakon solution of the form (9). From the first two items of the system (10), and are obvious. At the same time, we have if we take the symmetric initial data, . The two peaks move at the same speed which means there is no collision. Therefore, we choose the same initial data as the line case with and , where . The selection of these initial data is summarized in Figure 2.
According to (47), and are obtained. We introduce the symbol to represent the difference between the positions of the two peakons, in other words, . It follows from the ODE system (10) that where
These are easy to get from (49) and (50) that and . So, we have , which leads to . Integrating (48), we calculate where
Differentiating (52), we have
Since , we find . It follows that increases in . Combined with , we obtain a collision and a positive collision time when . Using the symbol for the collision time, from (51), we find
Applying and to define the collision location and the collision function , we get the following proposition.
Proposition 5. The limit of as approaches is , or
Proof. We compute the Fourier transform of and , which is denoted by Combining (56) and (57), we have Notice that the equation inside the absolute value can be scaled up to Let . is addable when and dominate the original integrand, which means that we can apply the dominated convergence theorem and put the limit inside the integral. So, we get
Proof of Theorem 1 (On the circle). In view of the 2-peakon solution we constructed, we need to construct a traveling wave solution that satisfies Reviewing the system (10), we take the following data The system is simplified and easy to be solved as We introduce the symbol to represent corresponding to and to represent corresponding to . It follows from (61) that . So, we get which makes the construction complete. In order to prove the nonuniqueness of the solution, we define two new solutions . Since and are two solutions to (1), and are also solutions to (1), which can be obtained by reversing time. These mean and solve (1). In addition, similar to the line case, through Proposition 5, we have . Finally, we note that the initial data for these nonunique solutions can be made arbitrarily small. Since the new initial data is the collision function , we have Therefore, for any , we can find a to make .
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare no conflict of interest.
Authors’ Contributions
All authors read and approved the final manuscript.
Acknowledgments
The authors would like to thank Jifeng Chu for illuminating discussions. This work is supported by the National Natural Science Foundation of China (Grant Nos. 11971163 and 12061016) and the Natural Science Foundation of Hunan Province (No. 2021JJ30166).