On the modified fractional Korteweg-de Vries and related equations

TL;DR

This paper studies modified fractional KdV and related equations, analyzing their dispersive properties, shock formation, and blow-up phenomena through rigorous proofs and numerical simulations, highlighting differences between weakly and strongly dispersive cases.

Contribution

It provides the first rigorous analysis of shock formation and blow-up in modified fractional KdV equations, including explicit calculations and numerical insights.

Findings

Shock formation with explicit blow-up time and location in weakly dispersive case

Numerical simulations illustrating dynamics in strongly dispersive case

Extension of shock results to equations with generalized nonlinearities

Abstract

We consider in this paper modified fractional Korteweg-de Vries and related equations (modified Burgers-Hilbert and Whitham). They have the advantage with respect to the usual fractional KdV equation to have a defocusing case with a different dynamics. We will distinguish the weakly dispersive case where the phase velocity is unbounded for low frequencies and tends to zero at infinity and the strongly dispersive case where the phase velocity vanishes at the origin and goes to infinity at infinity. In the former case, the nonlinear hyperbolic effects dominate for large data, leading to the possibility of shock formation though the dispersive effects manifest for small initial data where scattering is possible. In the latter case, finite time blow-up is possible in the focusing case but not the shock formation. In the defocusing case global existence and scattering is expected in the energy subcritical case, while finite time blow-up is expected in the energy supercritical case. We establish rigorously the existence of shocks with blow-up time and location being explicitly computed in the weakly dispersive case, while most of the results on the strongly dispersive case are derived via numerical simulations, for large solutions. Moreover, the shock formation result can be extended to the weakly dispersive equation with some generalized nonlinearity. We will also comment briefly on the BBM versions of those equations.