What question did this study set out to answer?

The study aims to improve the understanding of local well-posedness in 1-D semilinear Schrödinger equations with quadratic nonlinearities.

April 3, 2026Open Access

On the Well-posedness of the 1-D Quadratic Semilinear Schrödinger Equations

Key Points

The study aims to improve the understanding of local well-posedness in 1-D semilinear Schrödinger equations with quadratic nonlinearities.
Analyzed the initial value problem in specific function spaces
Focused on Sobolev spaces H^s with s > -1/4 for nonlinearities u
Considered Sobolev spaces H^s with s > -3/4 for nonlinearities u^2
Developed new estimates for Fourier restriction norms with homogeneous weights
Established improved local well-posedness estimates compared to previous findings
Demonstrated less restrictive conditions for initial data behavior at t=0 in phase space

Abstract

This paper deals with a slight improvement on the results of the 1-D semilinear Schrdinger equations with quadratic nonlinearities.We study the local well-posedness of the initial value problem in particular function spaces containing the Sobolev spaces H s with s > -1/4 for the nonlinearity u, and with s > -3/4 for u 2 or 2 , in which the local well-posedness was proved by Kenig, Ponce and Vega.Our improvement lies in the estimate of the Fourier restriction norm with a homogeneous weight || s .It makes the behavior of the initial data at = 0 in the phase space less restrictive.

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