Transient asymptotics of the modified Camassa-Holm equation

Abstract

We investigate long time asymptotics of the modified Camassa-Holm equation in three transition zones under a nonzero background. The first transition zone lies between the soliton region and the first oscillatory region, the second one lies between the second oscillatory region and the fast decay region, and possibly, the third one, namely, the collisionless shock region, that bridges the first transition region and the first oscillatory region. Under a low regularity condition on the initial data, we obtain Painlevé-type asymptotic formulas in the first two transition regions, while the transient asymptotics in the third region involves the Jacobi theta function. We establish our results by performing a $\bar{\partial }$ nonlinear steepest descent analysis to the associated Riemann-Hilbert problem.

arXiv comment58 pages, 16 figures. Minor corrections for 2nd version according to anonymous referees' comments. To appear in Journal of the London Mathematical Society

References in corpus (8)

• An integrable shallow water equation with peaked solitons
• The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations
• Long-Time Asymptotics for the Camassa-Holm Equation
• Universality of the break-up profile for the KdV equation in the small dispersion limit using the Riemann-Hilbert approach
• Complex oscillations in the delayed Fitzhugh-Nagumo equation
• A Riemann-Hilbert approach to the modified Camassa-Holm equation with nonzero boundary conditions
• Airy and Painlevé asymptotics for the mKdV equation
• The modified Camassa-Holm equation on a nonzero background: large-time asymptotics for the Cauchy problem

Cited by in corpus (2)

• Painlevé XXXIV asymptotics for the defocusing nonlinear Schrödinger equation with a finite-genus algebro-geometric background
• Painlevé transcendents in the defocusing mKdV equation with non-zero boundary conditions