Damping to prevent the blow-up of the Korteweg-de Vries equation

TL;DR

This paper investigates how damping mechanisms can prevent blow-up in solutions of the generalized damped Korteweg-de Vries equation, providing theoretical conditions and numerical constructions for effective damping.

Contribution

It establishes conditions on damping operators to prevent blow-up in the generalized damped KdV equation for p ≥ 4 and demonstrates these through numerical examples.

Findings

Conditions on damping operators to prevent blow-up for p ≥ 4

Local well-posedness results for the generalized damped KdV

Numerical sequences of damping constructed to prevent blow-up

Abstract

We study the behavior of the solution of a generalized damped KdV equation $u_t + u_x + u_{xxx} + u^p u_x + \mathscr{L}_{\gamma}(u)= 0$. We first state results on the local well-posedness. Then when $p \geq 4$, conditions on $\mathscr{L}_{\gamma}$ are given to prevent the blow-up of the solution. Finally, we numerically build such sequences of damping.