A Nonlinear Adiabatic Theorem for Coherent States

TL;DR

This paper proves that in a semi-classical nonlinear Schrödinger setting, coherent states maintain their mode separation and obey a superposition principle, with nonlinear effects being critically balanced.

Contribution

It introduces a nonlinear adiabatic theorem for coherent states in a matrix-valued potential, highlighting mode preservation and superposition in nonlinear regimes.

Findings

Mode separation is preserved during nonlinear evolution.

A nonlinear superposition principle for adiabatic wave packets is established.

The analysis handles critical nonlinear effects in the semi-classical limit.

Abstract

We consider the propagation of wave packets for a one-dimensional nonlinear Schrodinger equation with a matrix-valued potential, in the semi-classical limit. For an initial coherent state polarized along some eigenvector, we prove that the nonlinear evolution preserves the separation of modes, in a scaling such that nonlinear effects are critical (the envelope equation is nonlinear). The proof relies on a fine geometric analysis of the role of spectral projectors, which is compatible with the treatment of nonlinearities. We also prove a nonlinear superposition principle for these adiabatic wave packets.