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 formulae 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 $\overline{\partial }$ nonlinear steepest descent analysis to the associated Riemann–Hilbert problem.