The lifespan of small solutions to cubic derivative nonlinear Schr\"odinger equations in one space dimension

TL;DR

This paper establishes explicit lower bounds on the lifespan of solutions to cubic derivative nonlinear Schrödinger equations in one dimension, refining previous results by considering gauge-invariant nonlinearities.

Contribution

It provides a detailed, explicit lower bound estimate for the lifespan of solutions, extending prior work to include gauge-invariant nonlinearities.

Findings

Explicit lower bound formulas for solution lifespan

Refinement of previous lifespan estimates

Extension to gauge-invariant nonlinearities

Abstract

Consider the initial value problem for cubic derivative nonlinear Schr\"odinger equations in one space dimension. We provide a detailed lower bound estimate for the lifespan of the solution, which can be computed explicitly from the initial data and the nonlinear term. This is an extension and a refinement of the previous work by one of the authors where the gauge-invariant nonlinearity was treated.