A variable coefficient nonlinear Schr\"{o}dinger equation with a four-dimensional symmetry group and blow-up of its solutions

TL;DR

This paper investigates a variable coefficient nonlinear Schrödinger equation with a four-dimensional symmetry group, using symmetry transformations to analyze blow-up behaviors of solutions in various norms, including $L_p$, $L_\infty$, and distributional senses.

Contribution

It introduces a specific variable coefficient nonlinear Schrödinger equation with a rich symmetry group and applies symmetry transformations to study solution blow-ups.

Findings

Symmetry transformations relate known solutions to new ones.

Blow-up phenomena are analyzed in multiple norms.

Solutions exhibit finite-time blow-up behavior.

Abstract

A canonical variable coefficient nonlinear Schr\"{o}dinger equation with a four dimensional symmetry group containing $\SL(2,\mathbb{R})$ group as a subgroup is considered. This typical invariance is then used to transform by a symmetry transformation a known solution that can be derived by truncating its Painlev\'e expansion and study blow-ups of these solutions in the $L_p$-norm for $p>2$, $L_\infty$-norm and in the sense of distributions.