Construction of 2-Peakon Solutions and Nonuniqueness for a Generalized mCH Equation

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 Hs is proved for s < 3/2.


Introduction
The Camassa-Holm (CH) equation [1][2][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][9][10][11][12][13], modified CH (mCH) [14][15][16][17][18][19], and Novikov (NE) [20][21][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 abfamily of equations [25]. It is worth noting that the bfamily 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 u 0 ðxÞ ∈ H s ðℝÞ. The study of Li and Olver [26] shows that the CH equation is locally well posed in H s for s > 3/2, and Byers [27] proved the ill-posedness for the CH equation in H s when s < 3/2. 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 H s for s > 3/2. Himonas, Kenig, and Holliman [30] demonstrated the nonuniqueness for the NE equation in H s when s < 3/2 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 H s for s > 5/2. The nonuniqueness results of Himonas and Holliman [33] show that solutions to the Cauchy problem for the FORQ equation are not unique in H s when s < 3/2. At present, there is no theory to show the uniqueness for the FORQ equation in H s when 3/2 ≤ s ≤ 5/2. Holmes and Puri [34] discussed the nonuniqueness for the ab-family of equations. Himonas, Grayshan, and Holliman [35] considered the ill-posedness for the bfamily of equations in H s for s < 3/2 when b > 1. On this basis, Novruzov [36] studied the ill-posedness for the bfamily of equations when b < 1.
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 m = u − u xx 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 c ≠ 0 and classification of 2-peakon solutions. Recio and Anco [38] considered the conservation laws (energy, momentum, H 1 -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 ½: p 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. 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-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 q 1 , q 2 and momenta p 1 , p 2 satisfy Now, we consider the 2-peakon system on the circle, based on the methods in [25].
is a solution to Equation (1) if its positions q 1 , q 2 : and momenta p 1 , p 2 satisfy 2 Advances in Mathematical Physics where ½· p is defined as in (6).

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), p 1 ðtÞ = p 1 ð0Þ and p 2 ðtÞ = p 2 ð0Þ are obvious. At the same time, we have q 1 ′ = q 2 ′ if we take the symmetric initial data, j p 1 ð0Þj = jp 2 ð0Þj. The two peaks move at the same speed which means there is no collision. Therefore, we choose the following initial data with a > 0 and b 2 + ðb + δÞ 2 > 5bðb + δÞ, where b + δ > b > 0. The selection of these initial data is summarized in Figure 1. According to (33), p 1 ðtÞ = −ðb + δÞ and p 2 ðtÞ = b are obtained. We introduce the symbol q to represent the difference between the positions of the two peakons, in 5 Advances in Mathematical Physics other words, q≐q 2 − q 1 . It follows from the ODE system (8) that where −1 < α = 5p 1 p 2 /p 2 1 + p 2 2 = −5bðb + δÞ/b 2 + ðb + δÞ 2 < 0. Integrating (34), we calculate Since q ′ < 0 and q 0 = q 2 ð0Þ − q 1 ð0Þ = a > 0, we obtain a collision and a positive collision time when qðtÞ = 0. Using the symbol τ for the collision time, from (35), we find Applying expressions q τ ≐ lim t⟶τ − q 1 ðtÞ = lim t⟶τ − q 2 ðtÞ and v ðxÞ≐−δe −jx−q τ j to define the collision location q τ and the collision function v, we get the following proposition.
Proof. We compute the Fourier transform of u and v, which is denoted bŷ Combining (38) and (39), we have Notice that the equation inside the absolute value can be scaled up to Let f ðξÞ = 4ðb + δÞ 2 ð1 + ξ 2 Þ s−2 . There is no doubt that f is integrable when s < 3/2, and f 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.