Abstract
In this paper, we study the Cauchy problem for an integrable multi-component ($2N$-component) peakon system which is involved in an arbitrary polynomial function. Based on a generalized Ovsyannikov type theorem, we first prove the existence anduniqueness of solutions for the system in the Gevrey-Sobolev spaces with the lower bound of the lifespan. Then we show the continuity of the data-to-solution map for the system.Furthermore, by introducing afamily of continuous diffeomorphisms of a line and utilizing thefine structure of the system, we demonstrate the system exhibits unique continuation.
Highlights
In 2013, Xia and Qiao proposed the following integrable 2N -component Camassa-Holm system (2N Camassa-Holm equation (CH)) [49]mj,t =x + mjH + 1 (N +1)2Ni=1[mi(uj nj,t =x − njH− ujx)(vi + vix) + mj(vi + vix)], −1 (N +1)2 mj = uj − ujxx, nj = vj − vjxx, Ni=1[ni(ui − uix)(vj 1 ≤ j ≤ N.+ vjx) + nj(ui − uix)(vi + vix)], (1.1)where H is an arbitrary smooth function of uj, vj, 1 ≤ j ≤ N, and their derivatives
Luo and Yin [33] investigated the Gevrey regularity and analyticity for a class of Camassa-Holm type systems including a three-component Camassa-Holm system derived by Geng and Xue [26], a two-component shallow water system proposed by Constantin and Ivanov [21], and a modified two-component Camassa-Holm system found by Holm, Naraigh and Tronci [29]
We show that the 2N -component Camassa-Holm system (2N CH) system exhibits the unique continuation if N = 1
Summary
In 2013, Xia and Qiao proposed the following integrable 2N -component Camassa-Holm system (2N CH) [49]. Barostichi, Himonas and Petronilho [4] established the wellposedness for a class of nonlocal evolution equations in spaces of analytic functions They proved a Cauchy-Kovalevsky theorem for a so-called generalized Camassa-Holm equation in [3]. Luo and Yin [33] investigated the Gevrey regularity and analyticity for a class of Camassa-Holm type systems including a three-component Camassa-Holm system derived by Geng and Xue [26], a two-component shallow water system proposed by Constantin and Ivanov [21], and a modified two-component Camassa-Holm system found by Holm, Naraigh and Tronci [29]. The unique continuation for the SQQ system (1.5) is illustrated
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