Blow-up solutions to a class of nonlinear coupled Schr\"odinger systems with power-type-growth nonlinearities

TL;DR

This paper investigates nonlinear Schrödinger systems with power-type nonlinearities, establishing well-posedness, existence of ground states, and conditions for blow-up versus global solutions, extending previous quadratic cases to higher-order nonlinearities.

Contribution

It extends the analysis of Schrödinger systems to include nonlinearities of arbitrary power, providing new results on well-posedness, ground states, and blow-up behavior for these systems.

Findings

Proved local and global well-posedness in $L^2$ and $H^1$.

Established existence of ground state solutions.

Analyzed the dichotomy between global existence and finite-time blow-up.

Abstract

In this work we consider a system of nonlinear Schr\"odinger equations whose nonlinearities satisfy a power-type-growth. First, we prove that the Cauchy problem is local and global well-posedness in $L^2$ and $H^1$. Next, we establish the existence of ground state solutions. Then we use these solutions to study the dichotomy of global existence versus blow-up in finite time. Similar results were presented in the reference Noguera N. and Pastor A. 2022 https://doi.org/10.1142/S0219199720500236 for the special case when the growth of the nonlinearities was quadratic. Here we will extend them to systems with nolinearities of order $p$ (cubic, quartic and so on). Finally, we recover some known results for two particular systems, one with quadratic and the other with cubic growth nolinearities.