Lax Integrability and the Peakon Problem for the Modified Camassa-Holm Equation
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.
- Publication:
-
Communications in Mathematical Physics
- Pub Date:
- February 2018
- DOI:
- 10.1007/s00220-017-3076-6
- arXiv:
- arXiv:1705.06451
- Bibcode:
- 2018CMaPh.358..295C
- Keywords:
-
- Nonlinear Sciences - Exactly Solvable and Integrable Systems;
- Mathematical Physics;
- Mathematics - Analysis of PDEs;
- Mathematics - Classical Analysis and ODEs;
- 35D30;
- 35Q51;
- 34K29;
- 37J35;
- 35Q53;
- 34B05;
- 41A21
- E-Print:
- 42 pages