A Unified Integral Equation Approach to Conservation Laws for Nonlinear Schr\"odinger Equations

TL;DR

This paper introduces a unified integral equation framework for deriving conservation laws and identities in nonlinear Schrödinger equations, leveraging space-time integrability and avoiding regularization.

Contribution

It provides a systematic derivation of multiple conservation laws from a single integral identity using the Duhamel form and Strichartz estimates.

Findings

Derived conservation of charge, energy, and momentum.

Established pseudo-conformal conservation law.

Unified approach simplifies derivation of virial identities.

Abstract

We present a unified framework for the rigorous derivation of conservation laws and related identities for nonlinear Schr\"odinger equations with power-type nonlinearities. This approach treats the equation in its Duhamel form and uses the space-time integrability provided by Strichartz estimates, without relying on smooth approximations or regularization procedures. It was first introduced by the third author in [20] and subsequently developed in [7, 13]. In this paper, we establish a single integral identity from which all of the laws and identities considered here follow systematically. These include the conservation of charge (mass), energy, and momentum, the pseudo-conformal conservation law, and virial-type identities.