Perturbation of the nonlinear Schrödinger equation by a localized nonlinearity

2025/08/15 by Chen, Gong, Liu, Jiaqi, Tian, Yuanhong
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2508.11463

Abstract

We revisit the perturbative theory of infinite dimensional integrable systems developed by P. Deift and X. Zhou \citeDZ-2, aiming to provide new and simpler proofs of some key L^∞ bounds and L^p \empha priori estimates. Our proofs emphasizes a further step towards understanding focussing problems and extends the applicability to other integrable models. As a concrete application, we examine the perturbation of the one-dimensional defocussing cubic nonlinear Schrödinger equation by a localized higher-order term. We introduce improved estimates to control the power of the perturbative term and demonstrate that the perturbed equation exhibits the same long-time behavior as the completely integrable nonlinear Schrödinger equation.

Citations

• Long-time asymptotics and stability for the sine-Gordon equation
• Long-time Asymptotics for the NLS equation via dbar methods
• Long-time asymptotics for solutions of the NLS equation with initial data in a weighted Sobolev space
• A priori L^p estimates for solutions of Riemann-Hilbert Problems

Related

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• The Nonlinear Schrodinger equation with a potential in dimension 1
• Remarks on global a priori estimates for the nonlinear Schrödinger equation
• On the proximity between the wave dynamics of the integrable focusing nonlinear Schrödinger equation and its non-integrable generalizations
• Asymptotic stability of ground states in 2D nonlinear Schrödinger equation including subcritical cases