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Lax Integrability and the Peakon Problem for the Modified Camassa-Holm Equation
Chang, Xiangke1,2,3; Szmigielski, Jacek3
AbstractPeakons 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.
Funding ProjectNSERC[163953] ; Institute of Computational Mathematics, AMSS (CAS) ; Natural Sciences and Engineering Research Council of Canada (NSERC) ; National Natural Science Foundation of China[11731014] ; National Natural Science Foundation of China[11701550] ; Department of Mathematics and Statistics of the University of Saskatchewan, PIMS postdoctoral fellowship
WOS Research AreaPhysics
WOS SubjectPhysics, Mathematical
WOS IDWOS:000426904200009
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Document Type期刊论文
Corresponding AuthorChang, Xiangke
Affiliation1.Chinese Acad Sci, Acad Math & Syst Sci, ICMSEC, LSEC, POB 2719, Beijing 100190, Peoples R China
2.Univ Chinese Acad Sci, Sch Math Sci, Beijing 100049, Peoples R China
3.Univ Saskatchewan, Dept Math & Stat, 106 Wiggins Rd, Saskatoon, SK S7N 5E6, Canada
Recommended Citation
GB/T 7714
Chang, Xiangke,Szmigielski, Jacek. Lax Integrability and the Peakon Problem for the Modified Camassa-Holm Equation[J]. COMMUNICATIONS IN MATHEMATICAL PHYSICS,2018,358(1):295-341.
APA Chang, Xiangke,&Szmigielski, Jacek.(2018).Lax Integrability and the Peakon Problem for the Modified Camassa-Holm Equation.COMMUNICATIONS IN MATHEMATICAL PHYSICS,358(1),295-341.
MLA Chang, Xiangke,et al."Lax Integrability and the Peakon Problem for the Modified Camassa-Holm Equation".COMMUNICATIONS IN MATHEMATICAL PHYSICS 358.1(2018):295-341.
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