Well-posedness and long time behavior for the Schr\"odinger-Korteweg-de Vries interactions on the half-Line

TL;DR

This paper investigates the well-posedness and long-term behavior of solutions to the Schr"odinger-Korteweg-de Vries system on half-lines, establishing conditions for global existence and describing blow-up rates using a virial identity.

Contribution

It provides new results on global solutions for positive coupling and blow-up behavior for negative coupling in the Schr"odinger-KdV system on half-lines, including energy space analysis.

Findings

Global existence for positive coupling in energy space

Finite-time blow-up conditions for negative coupling

Super-linear blow-up rate in weighted norms

Abstract

The initial-boundary value problem for the Schr\"odinger-Korteweg-de Vries system is considered on the left and right half-line for a wide class of initial-boundary data, including the energy regularity $H^1(\R^{\pm})\times H^1(\R^{\pm})$ for initial data. Assuming homogeneous boundary conditions it is shown for positive coupling interactions that local solutions can be extended globally in time for initial data in the energy space; furthermore, for negative coupling interactions it was proved, for a certain class of regular initial data, the following result: if the respective solution does not exhibits finite time blow-up in $H^1(\R^-)\times H^1(\R^-)$, then the norm of the weighted space $L^2\big(\R^-,\, |x|dx\big)\times L^2\big(\R^-,\, |x|dx\big)$ blows-up at infinity time with \textit{super-linear rate}, this is obtained by using a satisfactory algebraic manipulation of a new global virial type identity associated to the system .