Blow-up criteria for the 1D NLS with combined nonlinearities

TL;DR

This paper establishes new finite-time blow-up criteria for 1D nonlinear Schrödinger equations with combined nonlinearities, including negative and positive energy conditions, expanding understanding of solution behaviors.

Contribution

It introduces two novel blow-up criteria for the 1D NLS with combined nonlinearities, covering both negative and positive energy cases, and includes examples of initial data satisfying these criteria.

Findings

Negative energy criterion for triple combined nonlinearity

Positive energy criterion for double nonlinearity with defocusing-focusing coefficients

Examples of initial data satisfying the criteria

Abstract

In this paper we develop two different types of criteria for the finite time blow-up solutions to the combined nonlinear Schr\"odinger equation in 1D. The first one is a negative energy criterion developed for triple combined nonlinearity and then for a (convergent) infinite sum of combined nonlinearities, including an exponential nonlinearity. The second one is a positive energy criterion for the double nonlinearity with the defocusing-focusing coefficients. We also provide examples of initial data that satisfy either of the criteria.