Orbital stability of elliptic periodic peakons for the modified Camassa-Holm equation

The orbital stability of peakons and hyperbolic periodic peakons for the Camassa-Holm equation has been established by Constantin and Strauss in [A. Constantin, W. Strauss, Comm. Pure. Appl. Math. 53 (2000) 603-610] and Lenells in [J. Lenells, Int. Math. Res. Not. 10 (2004) 485-499], respectively. In this paper, we prove the orbital stability of the elliptic periodic peakons for the modified Camassa-Holm equation. By using the invariants of the equation and controlling the extrema of the solution, it is demonstrated that the shapes of these elliptic periodic peakons are stable under small perturbations in the energy space. Throughout the paper, the theory of elliptic functions and elliptic integrals is used in the calculation.

1. Introduction. In [18], Holm and Staley studied an one-dimensional version of an active fluid transport that is described by the following nonlinear equation Equation (1.2) was proposed as a model for the unidirectional propagation of the shallow water waves over a flat bottom [2,3], with u(x, t) representing the water's free surface in nondimentional variables. It can be found using the method of recursion operators as an example of bi-Hamiltonian equation with an infinite number of conserved functionals by Fokas and Fuchssteiner [14]. The CH equation has attracted much attention in the last two decades because of its interesting properties: complete integrability [2,14], geometric formulations and the presence of breaking waves [7,8,9] (i.e. a wave profile remains bounded while its slope becomes unbounded in finite time).

AIYONG CHEN AND XINHUI LU
In [17], Hakkaev, Iliev and Kirchev studied the following equation: where a, b : R → R are smooth functions. It is easy to see that whatever a, b is, the equation for travelling wave solutions u = ϕ(x−ct) has no dissipative terms. Hence, any travelling wave solution of (1.3) is determined from Newton's equation which we can write in the form (ϕ ) 2 = U (ϕ)(see section 2). If a(u) = u 3 and b(u) = u, then (1.3) becomes the modified Camassa-Holm (mCH) equation (1. 4) The orbital stability of negative solitary waves to the mCH equation was proved by Yin, Tian and Fan [30]. Recently, Darós and Arruda [12] investigated the orbital instability of snoidal waves of the mCH equation by using the abstract method of Grillakis, Shatah and Strauss [15]. For the CH equation (1.2), a remarkable property is that it admits the peakons in the following forms u(x, t) = cϕ(x − ct) = ce −|x−ct| . (1.5) The peakons were proved to be orbital stable by Constantin and Strauss in [10]. A variational approach for proving the orbital stability of the peakons was introduced by Constantin and Molient [11]. Using variational approach, the stability of the Camassa-Holm peakons in the dynamics of an integrable shallow-water-type system was investigated by Chen et al. in [4]. Orbital stability of multi-peakon solutions was proved by Dika and Molient in [13]. Liu, Liu and Qu [24] considered the modified Camassa-Holm equation with cubic nonlinearity, which is integrable and admits the single peakons and mult-peakons. Using energy argument and combining the method of the orbital stability of a single peakon with monotonicity of the local energy norm, they proved that the sum of N sufficiently decoupled peakons is orbitally stable in the energy space. Moreover, the orbital stability of the single peakons for the DP equation was proved by Lin and Liu [23]. They developed the approach due to Constantin and Strauss [10] in a delicate way. The approach in [10] was extended in [25] to prove the orbital stability of the peakons for the Novikov equation. Recently, Guo et al. [16] proved the orbital stability of peakons for the generalized modified Camassa-Holm (gmCH) equation. In addition, equation (1.2) has also the periodic peakons where (1. 7) and ϕ(x) is defined for x ∈ [0, 1) and extends periodically to the whole real line. Orbital stability of the periodic peakons for the CH equation was proved by Lenells in [20,21]. Wang and Tian [29] extended Lenell's approach to prove the orbital stability of the periodic peakons for the Novikov equation. The nonlinear partial differential equation µ(u t ) − u xxt = −2µ(u)u x + 2u x u xx + uu xxx , t > 0, x ∈ S 1 = R/Z, (1. 8) where u(x, t) is a real-valued spatially periodic function and µ(u) = S 1 u(x, t)dx denotes its mean, was introduced in [19] as an integrable equation arising in the study of the diffeomorphism group of the circle. It describes the propagation of selfinteracting, weakly nonlinear orientation waves in a massive nematic liquid crystal under the influence of an external magnetic field. It was noted in [22] that the µCH equation also admits periodic peakons: For any c ∈ R, the periodic peaked traveling wave u( and ϕ is extended periodically to the real line, is a solution of (1.8). Chen, Lenells and Liu [5] proved that the periodic peakons of the µCH equation are orbitally stable. Liu, Qu and Zhang [26] further proved that the periodic peakons of the modified µCH equation are orbitally stable. The approach in [5] was further extended in [28] to prove the orbital stability of the periodic peakons for the generalized µCH equation. This work has the interest in to investigate the existence and orbital stability of elliptic periodic peakon solutions of the modified Camassa-Holm (mCH) equation (1.4). It was noted in [27] that the mCH equation also admits peakons and periodic peakons with similar formulas (1.5) and (1.6). By constructing certain Lyapunov functionals, it is demonstrated that the shapes of the peakons and periodic peakons are stable under small perturbations in the energy space [27]. Equation (1.4) has the elliptic periodic peakon (1. 10) where ϕ(x) is given for x ∈ [−T, T ] by Jacobi elliptic function 5 and cn −1 (x, k) represents the inverse function of Jacobi elliptic function cn(x, k). The local well-posedness of the Cauchy problem for equation (1.4) was established by Hakkaev, Iliev and Kirchev in [17].
The periodic peakons of the CH equation and the µCH equation are given by hyperbolic functions and quadratic functions, respectively. According to the previous statement, their orbital stability has been established. Recently, Chen, Deng and Huang [6] proved that the trigonometric periodic peakons for the modified Camassa-Holm equation are orbitally stable. In the present work, we will prove that the elliptic periodic peakons for the modified Camassa-Holm equation are orbitally stable. To the best of our knowledge, this is the first result about the orbital stability of elliptic periodic peakons.
The proof is inspired by [21] where the case of hyperbolic periodic peakons of (1.4) is considered. The approach taken here is similar but there are differences. The main difference is that in [21] the H 1 -norm u − ϕ(· − ξ) 2 H 1 (S) associated with a solution u(x, t) is represented by the difference between conservation law H 1 (u) and H 1 (ϕ) and the difference between max u and max ϕ, whereas here the H 1 -norm However, since this integrals can be controlled by the conservation law H 0 if u(x) − ϕ(x − ξ) preserves sign, we can ensure that they remain small enough for later times. We conjecture that the Theorem 1.1 still holds without the hypothesis that u(x) − ϕ(x − ξ) preserves sign.
2. Elliptic periodic peakons of the mCH equation. In this section, we convert equation (1.4) into a planar dynamical system. By substituting u(x, t) = ϕ(τ ) with τ = x − ct into equation (1.4), then it follows that where ϕ is the derivative with respect to τ . Integrating equation (2.1) once we obtain where g is the integral constant. Letting y = dϕ dτ , then we obtain the following planar dynamic system where d 2 = c 3 − 2c 2 + 4g, d 1 = c 2 − 2c and d 0 = c, and h is an integral constant.
Where α > β > γ and α, β, γ satisfying the relations    α + β + γ = 0, αβ + αγ + βγ = −1, Notice that a periodic peakon corresponds to the heteroclinic orbit (arch curve) defined by H(ϕ, y) = 0 (see Fig. 1). Now, equation (2.6) becomes By using the first equation of system (2.5) to do the integration, we have Therefore, we obtain the exact parameter representation of ϕ as following represents the inverse function of Jacobi elliptic function cn(x, k). We can get the following elliptic periodic peakon where n = 0, 1, 2, · · ·. The profile of elliptic periodic peakon is shown in Fig. 2.  Remark 1. Note that the algebraic curves defined by H(ϕ, y) = 0 consists of a closed curve and a open curve (see Fig.1(b)), the open curve corresponds periodic peakon solution (2.10) and the close curve corresponds smooth periodic wave solution Recently, Darós and Arruda [12] investigated the orbital instability of smooth periodic waves (2.11) of the mCH equation by using the abstract method of Grillakis, Shatah and Strauss [15]. The stability of the periodic peakon seems not to enter the general framework developed in [15], especially because of the non-smoothness of the periodic peakon.
3. Proof of stability. Note that a small change in the shape of a peakon can yield another one with a different speed. The appropriate notion of stability is, therefore, that of orbital stability: a periodic wave with an initial profile close to a peakon remains close to some translate of it for all later times. That is, the shape of the wave remains approximately the same for all times. Equation (1.4) has the conservation laws We will identify S with [−T, T ] and view functions u on S as periodic functions on the real line with period 2T . For an integer n ≥ 1, we let H n (S) be the Sobolev space of all square integrable functions f ∈ L 2 (S) with distributional derivatives ∂ i x f ∈ L 2 (S) for i = 1, ..., n. These Hilbert spaces are endowed with the inner products Moreover, ϕ is smooth on (−T, T ). This gives, as where (3.7) Refer to appendix B for details, by calculation, we can get (3.10) Using the identity ϕ 2 where Refer to appendix B for details, through a series of calculations, we know (3.14) The proof of Theorem 1.1 will proceed through a series of lemmas. First we consider the expansion of the conservation law H 1 around the peakon ϕ in the H 1 (S)-norm. The following lemma suggests that the associated with a solution u(x, t) is not only represented by the difference between conservation law H 1 (u) and H 1 (ϕ) and the difference between max u and max ϕ, but also it is also related to the integrals Lemma 3.1. For every u ∈ H 1 (S) and ξ ∈ R, Proof. We have On account of ϕ(T ) = M ϕ , we can get This proves the lemma.
As Lenells said in [21], for a wave profile u ∈ H 1 (S), the functional H 0 [u] and H 1 [u] represents the momentum and kinetic energy, respectively. Lemma 3.1 says that if a wave u ∈ H 1 (S) has momentum H 0 [u], energy H 1 [u] and height M u close to the elliptic periodic peakon's momentum, energy and height, then the whole shape of u is close to that of the elliptic periodic peakon. Another physically relevant consequence of Lemma 3.1 is that among all waves of fixed momentum and energy, the elliptic periodic peakon has maximal height.
Note that the periodic peakon ϕ satisfies the differential equation  (3.20) and extend it periodically to the real line. We compute where the details calculation of is given in appendix C. In the same way, we compute where the details calculation of We have actually proved the following lemma Lemma 3.2. For any positive u ∈ H 1 (S), define a function Proof. Using M ϕ = 1 and m ϕ = Similarly, referring to Appendix C for details, we can get where ϕ 1 = sn −1 ( (1 − k 2 ) tnϕ 1 dnϕ 1 nc 6 ϕ 1 .
Similarly, we can get J 20 , J 21 , J 22 and J 23 are 0, as well as J 24 tn(ϕ 1 )dn(ϕ 1 )nc 4 (ϕ 1 ) + On the other hand, we have Similarly, we have . (3.38) According to the derivation integral with parameters and referring to appendix B, we get .
where A ≈ 0.5349, equality holds in (3.50) if and only if f = ϕ(· − ξ) for some ξ ∈ R that is, if and only if has the shape of a peakon.
Proof. For x ∈ S, we have (3.51) Then, we get Whereupon, we have following inequality since, The Lemma 3.4 is proved.  Lemma 3.6. Let u ∈ C([0, T ); H 1 (S)) be a solution of (1.4). Given a small neighborhood U of (M ϕ , m ϕ ) in R 2 , there is a δ > 0 such that so F u is a small perturbation of F ϕ . The effect of the perturbation near the point (M ϕ , m ϕ ) can be made arbitrarily small by choosing the ε s small. Lemma 3.3 says that F ϕ (M ϕ , m ϕ ) = 0 and that F ϕ has a critical point with negative definite second derivative at (M ϕ , m ϕ ). By continuity of the second derivative, there is a neighborhood around (M ϕ , m ϕ ) , where F ϕ is concave with curvature bounded away from zero. Therefore, after a small perturbation, the set where F u ≥ 0 near (M ϕ , m ϕ ) will be contained in a neighbor-hood of (M ϕ , m ϕ ). Let U be given as in the statement. Shrinking U if necessary, we infer the existence of a δ > 0 such that for u with it holds that the set where F u ≥ 0 near (M ϕ , m ϕ ) is contained in U , and U is surrounded by a set where F u < 0. Lemma 3.2 and Lemma 3.5 say that M u(t) and m u(t) are continuous functions of t ∈ [0, T ), and F u (M u(t) , m u(t) ) ≥ 0 for t ∈ [0, T ). We conclude that for u satisfying (3.54), we have However, the continuity of the conserved functionals H i : H 1 (S) → R, i = 0, 1, 2, shows that there is a δ > 0 such that (3.54) holds for all u with Moreover, by Lemma 3.4, taking a smaller δ if necessary, we may also assume that (M u(0) , m u(0) ) ∈ U if u(·, 0) − ϕ H 1 (S) < δ. This proves the lemma.
Proof of Theorem 1.1. Let u ∈ C([0, T ); H 1 (S)) be a solution of (1.4) and suppose we are given an ε > 0. Pick a neighborhood U of (M ϕ , m ϕ ) small enough that Choose a δ > 0 as in Lemma 3.6 so that (3.53) holds. Taking a smaller δ if necessary, we may also assume that Using the hypothesis that u(x) − ϕ(x − ξ) preserves sign, from Lemma 3.1, we conclude that and ξ(t) ∈ R is any point where u(ξ(t) + T, t) = M u(t) . This completes the proof of the theorem.
Appendix A. In this Appendix, some basic properties of Jacobian elliptic integrals (see [1]) is collected for the convenience of the reader. we started setting the normal elliptic integral of the first kind where y = sin ϕ and ϕ = amu 1 , however, the normal elliptic integral of the second kind is The parameter k is called the modulus and belongs to the interval (0, 1). The number k is referred to as the complementary modulus and is related to k by k = √ 1 − k 2 . The variable ϕ is called the argument of the normal elliptic integrals. It is usually understood that 0 ≤ y ≤ 1 or 0 ≤ ϕ ≤ π 2 . For y = 1, It is said that the integrals above are complete. In this case, one writes Clearly, we have K(0) = E(0) = π/2 and E(0, k) = 0. We using the inverse function of the elliptic integral of the first kind define the Jacobian Elliptic Functions. This inverse function exists because that which is a strictly increasing function of the variable y 1 , in its algebraic form, this integral has the property of being finite for all values of y 1 . Its inverse function may thus be defined by y 1 = sin ϕ = sn(u, k) and ϕ = am(u, k), or simply y 1 = sn(u), ϕ = amu when not necessary emphasize the modulus k, these may be read sine amplitude u and amplitude u. So, sn is an odd function. The other two basic elliptic functions, the cnoidal and dnoidal functions, are defined in terms of sn by    cn(u, k) = 1 − y 2 1 = 1 − sn 2 (u, k), dn(u, k) = 1 − k 2 y 2 1 = 1 − k 2 sn 2 (u, k).
6. Differentiation of the jacobian inverse functions: d dy sn −1 (y, k) = 1 Appendix B. In this Appendix, we mainly present some of the formulas used in this article to calculate the elliptic function, which are from the references [1]. Formula 1: