A weak form of the soliton resolution conjecture for high-dimensional fourth-order Schrodinger equations

TL;DR

This paper establishes a weak form of the soliton resolution conjecture for high-dimensional fourth-order Schrödinger equations, demonstrating asymptotic frequency and spatial localization of solutions.

Contribution

It introduces a novel approach combining dispersive properties and phase symmetries to prove localization properties for these equations.

Findings

High-frequency solutions exhibit better dispersive decay.

Solutions show asymptotic spatial localization.

The method relies on phase symmetry analysis.

Abstract

We prove a weak form of the soliton resolution conjecture of bounded solutions of high-dimensional fourth-order Schrodinger equations. The result relies upon two properties to be proved: the asymptotic frequency localization and the asymptotic spatial localization. In order to prove the asymptotic frequency localization we use the fact that the high frequency pieces of the free solution have better dispersive properties than the lower ones. In order to prove the asymptotic spatial localization, we use the symmetries of the phase of the fundamental solution.