Abstract
Peakons are special weak solutions of a class of nonlinear partial differential equations modelling non-linear phenomena such as the breakdown of regularity and the onset of shocks. We show that the natural concept of weak solutions in the case of the modified Camassa–Holm equation studied in this paper is dictated by the distributional compatibility of its Lax pair and, as a result, it differs from the one proposed and used in the literature based on the concept of weak solutions used for equations of the Burgers type. Subsequently, we give a complete construction of peakon solutions satisfying the modified Camassa–Holm equation in the sense of distributions; our approach is based on solving certain inverse boundary value problem, the solution of which hinges on a combination of classical techniques of analysis involving Stieltjes’ continued fractions and multi-point Padé approximations. We propose sufficient conditions needed to ensure the global existence of peakon solutions and analyze the large time asymptotic behaviour whose special features include a formation of pairs of peakons that share asymptotic speeds, as well as Toda-like sorting property.
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Xiangke Chang was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC), the Department of Mathematics and Statistics of the University of Saskatchewan, PIMS postdoctoral fellowship, the Institute of Computational Mathematics, AMSS (CAS), and the National Natural Science Foundation of China (#11701550; #11731014). Jacek Szmigielski was supported in part by NSERC #163953.
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Chang, X., Szmigielski, J. Lax Integrability and the Peakon Problem for the Modified Camassa–Holm Equation. Commun. Math. Phys. 358, 295–341 (2018). https://doi.org/10.1007/s00220-017-3076-6
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DOI: https://doi.org/10.1007/s00220-017-3076-6