Attractors of Hamiltonian nonlinear partial differential equations

TL;DR

This paper surveys the theory of attractors in nonlinear Hamiltonian PDEs, discussing stability, solitons, and quantum phenomena, and proposes a new hypothesis on attractors in G-invariant systems.

Contribution

It provides a comprehensive survey of attractor theory in Hamiltonian PDEs and introduces a new hypothesis linking attractors to fundamental quantum phenomena.

Findings

Results on global attraction to stationary states and solitons

Numerical simulations illustrating attractor behavior

A new hypothesis connecting attractors to quantum phenomena

Abstract

We survey the theory of attractors of nonlinear Hamiltonian partial differential equations since its appearance in 1990. These are results on global attraction to stationary states, to solitons and to stationary orbits, on adiabatic effective dynamics of solitons and their asymptotic stability. Results of numerical simulations are also given. Based on these results, we propose a new general hypothesis on attractors of $G$-invariant nonlinear Hamiltonian partial differential equations. The obtained results suggest a novel dynamical interpretation of basic quantum phenomena: Bohr's transitions between quantum stationary states, wave-particle duality, and probabilistic interpretation.