On Asymptotic Completeness of Scattering in the Nonlinear Lamb System, II

TL;DR

This paper proves asymptotic completeness for the nonlinear Lamb system around hyperbolic stationary states, using a Banach space inverse function theorem, and provides counterexamples for nonhyperbolic states.

Contribution

It establishes asymptotic completeness in the nonlinear Lamb system for hyperbolic states and constructs specific trajectories using advanced mathematical techniques.

Findings

Asymptotic completeness holds for hyperbolic stationary states.

Counterexamples demonstrate nonexistence of trajectories for nonhyperbolic states.

The proof employs the inverse function theorem in Banach spaces.

Abstract

We establish the asymptotic completeness in the nonlinear Lamb system for hyperbolic stationary states. For the proof we construct a trajectory of a reduced equation (which is a nonlinear nonautonomous ODE) converging to a hyperbolic stationary point using the Inverse Function Theorem in a Banach space. We give the counterexamples showing nonexistence of such trajectories for nonhyperbolic stationary points.