Orbital Stability of Peakons and Multi-peakons for a Generalized Cubic–Quintic Camassa–Holm Type Equation

The peakons and mulit-peakons for a generalized cubic–quintic Camassa–Holm type equation have been obtained by Weng et al. (Monatsh Math, 2022. https://doi.org/10.1007/s00605-022-01699-w). In this paper, by constructing certain Lyapunov functionals, we prove that the peakons were orbitally stable in the energy space. Furthermore, using energy argument and combining the method of the orbital stability of peakons with monotonicity of the local energy norm, we also prove that the sum of N sufficiently decoupled peakons is orbitally stable in the energy space.


Proof of Theorem 1.1
In the section, we will get the orbital stability of peakons for Eq. (1.1). Using (1.18) and (1.19), we obtain and We next consider the expansion of the conservation law E(u) around the peakon in the H 1 (ℝ)-norm and show several important lemmas.
(2.34) P 0 (M) = P(M) + 4k 1 15 We define the following neighborhood of size of the superposition of N peakons of speed c i , with spatial shifts z i that satisfied > 0 , To complete the proof of Theorem 1.2, it is sufficient to show that there exist A > 0, L 0 > 0, and 0 > 0 such that for all L > L 0 and 0 < < 0 , if u 0 satisifies m 0 ≥ 0 , (1.9) and (1.10), and if for some 0 < t * < T, Hence, we only need to prove (3.4) under the assumption (3.3) for some L > L 0 and 0 < < 0 , with A, L 0 , and 0 to be specified later.

Control of the Distance Between the Peakons
In this subsection, we use the modulation theory to prove Lemma 3.1, which is an important result for proving Theorem 1.2.
Lemma 3.1 Let the initial data u 0 satisfy the assumptions given in Theorem 1.2.
(3.12) Proof We refer to the proof of [10,25]. From the implicit function theorem, most of the conclusions of this lemma have been proved by [10,25]. In fact, we just need to prove (3.10). Now, we show that the speed of x j stays close to c j . Notice that Differentiating (3.8) with respect to t, we infer that Therefore, we have x) into (1.14) and using the following equation of R i (t): where and Hence, we deduce that v(t, x) satisfies on [0, t * ]: where (3.14) and Using the L 2 (ℝ)-scalar product with x R j , and integrating by parts, we deduce that for t ∈ [0, t * ], From Appendix 1 and the decay of x R j , we obtain that which yields (3.10). Therefore, we complete the proof of Lemma 3.1. ◻

Monotonicity Property
In this subsection, we will show the almost monotonicity of functionals that are very close to the energy at the right of the (i − 1) th bump of u, i = 2, … , N . In order to get the almost monotonicity property, we first need to prove the following Lemma 3.2.
, s > 5∕2 and T > 0 be the maximal time of existence of the corresponding strong solution u(t, x) with initial data u 0 (x) . Then for any smooth function w(x) and all t ∈ [0, T) , the following identity holds: x ) −1 f and taking the derivative to (1.14) with respect to x, we infer that Integrating by parts yields (3.20) (3.25) where Ψ j,K (t, x) = Ψ K (x − y j (t)) with y j (t) defined in Lemma 3.1.
(3.29)  Using (3.10), then for 0 < < 0 and L > L 0 , we obtain that By (1.17), we deduce that Thus, we infer from (3.34) that We estimate each item on the right hand side of (3.40) in Appendix 2. From the detailed calculations of Appendix 2, we obtain that Using the Gronwall argument on [0, t] with t ≤ t * , we show that for any t ∈ [0, t * ], This completes the proof of Lemma 3.3. ◻

Global Identity and Localized Estimate
In this subsection, we establish the global identity and the localized estimate similar to that in [10]. For Z = (z 1 , … , z N ) , we denote where c i = a 2 i 8 15 k 1 a 2 i + 2 3 k 2 . R z i (x) has the peak at x = z i and max x∈ℝ R z i = R z i (z i ) = a i . From (1.18) and (1.19), we deduce that Proof See reference [10,25] for proof of this lemma. We omit the process. ◻ In order to get the following localized estimate, we define the weight function . Taking L > 0 and L∕K > 0 large enough, using the exponentially asymptotic behavior of Φ i , we deduce that and We define the following localized version of (1.18) and (1.19) as In the next lemma, we take K = √ L∕8 to derive the appropriate estimates.

Lemma 3.5 Given N real numbers
Define the interval J i as in (3.12). Suppose that, for any fixed positive function u ∈ H s (ℝ) with s > 5∕2 , and each i = 1, … , N , there exists i ∈ J i such that Then, for each i = 1, … , N , we have Proof Let i = 1, … , N be fixed and take i ∈ J i satisfying (3.50). We define and We infer that In a similar manner, By the construction of Φ i and the exponential decay of Ψ , taking K = √ L∕8 , we deduce that with a constant C > 0, (3.63)

End the Proof of Theorem 1.2
In this subsection, using the method of estimation in [25], we will finish the proof of Theorem 1.2. In the next lemma, we estimate the differences between the local maximum of the solution u(t, x) and the maximum of each single peakon.
(3.69) P i (y) = 16k 1 45 (3.70) To complete the proof of Theorem 1.2, in view of (3.3), it suffices to prove that there exists a constant C > 0 independent of A such that at time t * , there exist To this end, we need to take in (3.45) (3.14), that is, Combining (3.11) and (3.13), we infer that

Appendix 1
In this Appendix 1, we complete the estimates of ∫ ℝ Q 1 x R j dx and ∫ ℝ Q 2 x R j dx for (3.18). We denote ) and the exponential decay of R i , we deduce all x ∈ ℝ that (3.75) � . (3.79) Similarly, we can obtain estimates for the other terms of ∫ ℝ Q 1 x R j dx . The estimate of ∫ ℝ Q 2 x R j dx can refer to [25], and we omit the process here. Therefore, we deduce that and This completes the estimates of (3.18).