ORBITAL STABILITY OF PEAKONS FOR A MODIFIED CAMASSA-HOLM EQUATION WITH HIGHER-ORDER NONLINEARITY

. In this paper, we consider the orbital stability of peakons for a modiﬁed Camassa-Holm equation with higher-order nonlinearity, which admits the single peakons and multi-peakons. We ﬁrstly show the existence of the single peakon and prove two useful conservation laws. Then by constructing certain Lyapunov functionals, we give the proof of stability result of peakons in the energy space H 1 ( R )-norm.


Introduction.
Recently, the great interest in the well-known Camassa-Holm (CH) equation [2] has inspired the search for various CH-type equations with cubic or higher-order nonlinearity. One of the most concerned is the following modified CH equation: which was derived by Fuchssteiner [9] and Olver and Rosenau [16] using the method of tri-Hamiltonian duality to the bi-Hamiltonian representation of the modified KdV equation. Subsequently, Eq. (1) was shown to arise from the asymptotic theory of surface water waves by Fokas [7]. Moreover, Qiao [17] deduced it from the twodimensional Euler equations, where u(t, x) denotes the fluid velocity and hence y(t, x) represents its potential density. Hence the modified CH equation is also called FORQ equation in some literature.
The modified CH equation is completely integrable [16]. It has a bi-Hamiltonian structure and also admits a Lax pair [19], and hence can be solved by the inverse scattering transform method. Compared with the classical CH equation, Eq. (1) admits not only peaked solitary waves (peakons), but also possesses cusp solitons (cuspons) and weak kink solutions (u, u x , u t are continuous, but u xx has a jump at its peak point) [19,24]. It has also significant differences from the CH equation about the dynamics of the two-peakons and peakon-kink solutions [18,24]. The so called white solitons and dark ones of Eq. (1) were presented in [22] and [12], respectively. In [1], the authors applied the geometric and analytic approaches to give a geometric interpretation of the variable y(t, x) and construct an infinitedimensional Lie algebra of symmetries. The Cauchy problem of Eq. (1) in Besov spaces and the blow-up scenario were studied in [8]. The nonuniform dependence 5506 XINGXING LIU on the initial data for Eq. (1) was established in [11]. In [10], the authors considered the formulation of the singularities of solutions and showed that some solutions with certain initial date would blow up in finite time. Then the blow-up phenomena were systematically investigated in [3,15]. There exist single peakon of the form [10] u(t, x) = ϕ c (t, x) = 3c 2 e −|x−ct| , c > 0, and the multi-peakons where p i (t) and q i (t) satisfy the following system: j=1 p i p j e −|qj −qi| + 4 1≤k<i,i<j≤N p k p j e −|q k −qj | . It is noticed that, the peakons replicate a feature that is characteristic for the Stokes wave of greatest height which is an exact solution of the governing equations for water waves [4,5,23]. On the other hand, the multi-peakon solutions of Eq. (1) in comparison with the CH equation have only constant amplitudes. Later, the orbital stability of single peakon and the train of peakons were proven in [20] and [14], respectively.
In this paper, we consider the following nonlinear equation with higher-order nonlinearity: Indeed, Eq. (2) is the case n = 3 of the following generalized modified CH equation y t + (u 2 − u 2 x ) n y x = 0, 1 ≤ n ∈ N + , which is proposed by Recio and Anco [21] in the study of the family of nonlinear dispersive wave equations involving two arbitrary functions. It is known that one of the main remarkable features of the CH equation (with quadratic nonlinearity) and the modified CH equation (1) (with cubic nonlinearity) is the existence of stable peakons. Interestingly, Eq. (2) (with higher-order nonlinearity) studied here can also represent single peakon and multi-peakons [21]. As far as we know, no attempt has been made here to prove the stability of peakons for Eq. (2) up to now. Thus it is of great interest to identify whether the single peakon of Eq. (2) has the similar stable result as the CH and modified CH equations.
To pursue the above goal, we briefly recall the impressive proof of stability of peakons for the CH equation provided by Constantin and Strauss [6]. The CH equation has two useful invariants The idea is mainly based on expanding the conserved energyẼ(u) around the peakon ϕ c . Then, one can find that the minimum of the H 1 -norm between the solution u(t, x) and the peakon ϕ c (· − ξ(t)) is precisely controlled by an error term |M − max x∈R ϕ c | (M max x∈R u(x)). As we know, a crucial ingredient to estimate this error term is the following polynomial inequality betweenẼ(u) andF (u) The above inequality can be obtained from the observation of the following two integral formulas, whereg is an auxiliary function (see Lemma 2 in [6]) andh u. After analyzing the root structure of the above polynomial inequality, one can easily get the desired result.
In order to apply the above approach to Eq. (2), we need to overcome several difficulties. Since the conservation law E(u) of Eq. (2) is the same asẼ(u), we also expect orbital stability of peakons for Eq. (2) in the sense of the energy space H 1 -norm. While the other conservation law F (u) is much more complicated than the cases of the CH and modified CH equations. Hence we here require more subtle computation on the introduction of the higher-order conservation F (u) and new auxiliary function h(x) (see Lemma 3.3), which is quite different from the case of simply takingh(x) = u(x) for the CH equation. On the other hand, to get the key inequality related to the conservation laws E(u) and F (u) as the CH and modified CH equations, we have to show h ≤ 64 35 (max x∈R u) 6 . However, since h is of the form u 6 ∓ 58 x due to its higher-order nonlinear structure, we here treat it firstly by applying the Cauchy-Schwarz inequality to the term ∓u 5 u x ± u 3 u 3 x , and then the term ∓u 5 u x − u 4 u 2 x . Lastly we reach to an obviously correct inequality (34) by the delicate combination of the terms of h. Moreover, it is noted to point out that the reason why we do not study the case n = 2 of the generalized modified CH equation, even though the existence of peakons and the conservation laws E(u), F (u) hold for this case. As mentioned above, the current approach requires principally one ingredient: the precise polynomial inequality between the conservation laws. However, when n = 2 in the generalized modified CH equation, the auxiliary function h(x) is the form of u 4 ∓ 6 5 u 3 u x − 2 5 u 2 u 2 x ± 2 5 uu 3 x + 1 5 u 4 x , which cannot generally be bounded by 8 5 u 4 due to the positive term 1 5 u 4 x . Thus the choice of the generalized modified CH equation when n is a positive odd integer seems to be the best adapted to our approach used here.
The remainder of this paper is organized as follows. In Section 2, we mainly demonstrate that the existence of peakons which can be understood as weak solutions for Eq. (2), and then prove two conserved quantities which are crucial in the proof of the stability theorem. In Section 3, the orbital stability of peakons for Eq.
(2) is proved based on several useful lemmas. In Section 4, we give the proof of last inequality of (36) in Lemma 3.3.
Notation. In the following, the symbols is used to express the corresponding inequality that includes a universal constant. For example, f g denotes that there exists a constant C > 0 such that f ≤ Cg.

2.
Preliminaries. As shown in [25], one can easily obtain the following local wellposedness result by solving a transport equation satisfied by the momentum y instead of u.
Then there exists a time T > 0 such that the Cauchy problem (2) has a unique strong solution Next we restate Eq. (2) in a more convenient form. Note that Eq. (2) is equivalent to the following one: Applying the operator (1 − ∂ 2 x ) −1 to the both sides of (3), we can obtain In order to understand the meaning of a peakon solution to Eq. (2), we can apply Eq. (4) to define the notion of weak solutions for Eq. (2).
is called a weak solution to Eq. (2). If u is a weak solution on [0, T ) for every T > 0, then it is called a global weak solution.
In the following theorem, we discuss the existence result of the single peakons for Eq. (2). Theorem 2.3. For any a = 0, the peaked functions of the form is a global weak solution to Eq. (2) in the sense of Definition 2.2.
Proof. For any test function φ(·) ∈ C ∞ c (R), integrating by parts, we infer Thus, for all t ≥ 0, we have in the sense of distribution S (R). Letting ϕ 0,c (x) ϕ c (0, x), then we get The same computation as in (6), for all t ≥ 0, yields, Combining (6)- (8) and using integration by parts, for any test function Using the definition of ϕ c and c = 16 35 a 6 , we have, for x > ct, and for x ≤ ct, On the other hand, by Definition 2.2, we derive We calculate from (6) that, which together with (12) leads to For x > ct, we can split the right hand side of (14) into the following three parts, A direct calculation for each one of the terms I i , 1 ≤ i ≤ 3, yields Inserting the above equalities I 1 -I 3 into (15), we find that for x > ct, While for x ≤ ct, we also split the right hand side of (14) into three parts, We now directly compute each one of the terms II i , 1 ≤ i ≤ 3, as follows, Inserting the above equalities II 1 -II 3 into (17), we find that for x ≤ ct, Hence, by (10)-(11), (16) and (18), we deduce that for all (t, x) ∈ (0, +∞) × R, Thanks to (9), (13) and (19), we conclude that for every test function φ(t, x) ∈ C ∞ c ([0, ∞) × R). This completes the proof of Theorem 2.3.
In the following lemma, we give two useful conservation laws which are crucial in our development.

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Lemma 2.4. If the initial data u 0 ∈ H s (R) with s > 5 2 , then the following two functionals are invariants for Eq. (2).

Proof. A direct computation yields
Similarly, one obtains

STABILITY OF PEAKONS FOR A MODIFIED CAMASSA-HOLM EQUATION
Combining the above five equalities, we have This completes the proof of Lemma 2.4.
The proof of the following lemma is very similar to Lemmas 2.8-2.9 in [13], thus we omit it here.
does not change sign, then y(t, x) will not change sign for all t ∈ [0, T ). It follows that if y 0 ≥ 0, then the corresponding solution u(t, x) of Eq. (2) is positive for (t, x) ∈ [0, T ) × R. Furthermore, if y 0 ≥ 0, then the corresponding solution u(t, x) of Eq. (2) satisfies 3. Stability of peakons. In this section, we prove the orbital stability of peakons for Eq. (2). Our main theorem reads, Theorem 3.1. The peaked soliton ϕ c (t, x) defined in (5) traveling with the speed c > 0 is orbitally stable in the following sense. If u 0 (x) ∈ H s (R), for some s > 5 2 , y 0 (x) = (1 − ∂ 2 x )u 0 = 0 is a nonnegative Radon measure of finite total mass, and where a = 6 35c 16 by (5) where T > 0 is the maximal existence time, ξ(t) ∈ R is the maximum point of function u(t, ·) and the constant A(c, u 0 H s ) > 0 depends only on wave speed c > 0 and the norm u 0 H s .
We break the proof of Theorem 3.1 into several lemmas. Note that the assumptions on the initial profile in Theorem 3.1 guarantee the existence of the unique local positive solution Eq. (2) by Lemma 2.1 and Lemma 2.5. It is obvious that ϕ c (x) = aϕ(x) = ae −|x| ∈ H 1 (R) has the peak at x = 0, and hence max R ϕ c = ϕ c (0) = a. By a simple computation, we have E(ϕ c ) = 2a 2 and F (ϕ c ) = 32 35 a 8 . Here a = 6 35c 16 given by (5).
Lemma 3.2. For every u ∈ H 1 (R) and ξ ∈ R, we have Proof. Using (5) and integration by parts, as [6], we calculate where we used the fact that E(ϕ c ) = 2a 2 . This completes the proof of Lemma 3.2.
Next, we derive a crucial inequality between the two conserved quantities E(u) and F (u).
Proof. Let M be taken at x = ξ, and define the same function as in [6] g Then we have Next we introduce the other function as Integrating by parts, we calculate By Lemma 2.5, the solution u of Eq. (2) satisfies To see this, it suffices to show that that is, By the Cauchy-Schwarz inequality and (26), we deduce Thanks to (29), to prove (28), we only need to show that In a similar manner, Owing to (31), to prove (30), it suffices to show that Using (29) again, to prove (32), we only need to show that 11 2 Since , to prove (33), it suffices to show that Obviously, (34) holds. This proves our claim (27). Combining (24)- (25) and (27), we obtain which gives the desired inequality (23). This completes the proof of Lemma 3.3.
where the constant A(c, u H s ) > 0 depends only on wave speed c > 0 and the norm u H s .

Proof. A direct computation yields
under the assumption 0 < ε < 3 − 2 √ 2 a. For the estimation |F (u) − F (ϕ c )|, we have where we give the proof of the last inequality of (36) in Appendix for convenience. This completes the proof of Lemma 3.4.
Appendix. In this section, we complete the proof of the last inequality of (36) in Lemma 3.4.
Using the relation (24), we have for all v ∈ H 1 (R),

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For the term J 1 , by (35) and (43), we have Similarly, for the term J 2 , we have For the term J 3 , by the Hölder inequality, we obtain For K, a direct use of Young's inequality yields, Since u ∈ H s (R) ⊂ H 2 (R), s > 5 2 , by the Gagliardo-Nirenberg inequality, we have Since a = 6 35c 16 , then we have In a similar manner to treat J 4 and J 6 , using the fact that ∂ x ϕ c 6 L 6 = 1 3 a 6 and ∂ x ϕ c 10 L 10 = 1 5 a 10 , we obtain For the terms J 5 and J 7 , a direct use of the Hölder inequality yields,