期刊
COMMUNICATIONS IN MATHEMATICAL PHYSICS
卷 358, 期 1, 页码 295-341出版社
SPRINGER
DOI: 10.1007/s00220-017-3076-6
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资金
- NSERC [163953]
- Institute of Computational Mathematics, AMSS (CAS)
- Natural Sciences and Engineering Research Council of Canada (NSERC)
- National Natural Science Foundation of China [11731014, 11701550]
- Department of Mathematics and Statistics of the University of Saskatchewan, PIMS postdoctoral fellowship
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 Pad, 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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