Dispersive blow-up for the fifth order Korteweg-de Vries equation on the line

TL;DR

This paper proves dispersive blow-up for the fifth order Korteweg-de Vries equation by establishing local well-posedness in Bourgain spaces and constructing specific initial data.

Contribution

It introduces a novel dispersive blow-up result for the fifth order KdV equation using Bourgain space analysis and nonlinear regularity properties.

Findings

Dispersive blow-up occurs for certain initial data.

Local well-posedness in Bourgain spaces is established.

Constructive method for initial data leading to blow-up.

Abstract

In this work we establish a dispersive blow-up result for the initial value problem (IVP) for the fifth order Korteweg-de Vries equation \begin{align*} \left. \begin{array}{rlr} u_t+\partial_x^5 u+u\partial_x u&\hspace{-2mm}=0,&\quad x\in\mathbb R,\; t>0,\\ u(x,0)&\hspace{-2mm}=u_0(x),& \end{array} \right\} \end{align*} To achieve this, we prove a local well-posedness result in Bourgain spaces of the type $X^{s,b}$ for appropriate values of $s$ and $b$, along with a regularity property for the nonlinear part of that solution. This property enables the construction of initial data that leads to the dispersive blow-up phenomenon.