Dispersive blow-up for a coupled Schr\"odinger-fifth order KdV system

TL;DR

This paper proves a dispersive blow-up phenomenon for a coupled Schr"odinger-fifth order KdV system by establishing local well-posedness in Bourgain spaces and constructing special initial data.

Contribution

It introduces a novel dispersive blow-up result for the coupled system and develops a local well-posedness framework in Bourgain spaces for this problem.

Findings

Dispersive blow-up occurs for the coupled Schr"odinger-KdV system.

Established local well-posedness in Bourgain spaces.

Constructed initial data leading to blow-up phenomenon.

Abstract

In this work we establish a dispersive blow-up result for the initial value problem (IVP) for the coupled Schr\"odinger-fifth order Korteweg-de Vries system \begin{align*} \left. \begin{array}{rl} i u_t+\partial_x^2 u &\hspace{-2mm}=\alpha uv + \gamma |u|^2 u, \quad x\in\mathbb R,\quad t\in\mathbb R,\\ \partial_t v + \partial_x^5 v + \partial_x v^2&\hspace{-2mm}=\epsilon \partial_x |u|^2, \quad x\in\mathbb R,\quad t\in\mathbb R,\\ u(x,0)&\hspace{-2mm}= u_0(x), \quad v(x,0)=v_0(x). \end{array} \right\} \end{align*} To achieve this, we prove a local well-posedness result in Bourgain spaces of the type $X^{s+\beta,b}\times Y^{s,b}$, along with a regularity property for the nonlinear part of the IVP solutions. This property enables the construction of initial data that leads to the dispersive blow-up phenomenon.