A Unified Integral Equation Approach to Conservation Laws for Nonlinear Schrödinger Equations

Abstract

We present a unified framework for the rigorous derivation of conservation laws and related identities for nonlinear Schrödinger 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.