Spatial Hamiltonian identities for nonlocally coupled systems

TL;DR

This paper develops a Hamiltonian formalism for a broad class of nonlocal integro-differential systems, revealing conserved quantities and Lyapunov functions relevant to pattern formation.

Contribution

It introduces a novel Hamiltonian calculus for nonlocal systems, extending Noether's theorem and identifying conserved quantities and Lyapunov functions.

Findings

Derived Hamiltonian formalism for nonlocal equations

Identified new conserved quantities and Lyapunov functions

Applied framework to neural field and phase separation models

Abstract

We consider a broad class of systems of nonlinear integro-differential equations posed on the real line that arise as Euler-Lagrange equations to energies involving nonlinear nonlocal interactions. Although these equations are not readily cast as dynamical systems, we develop a calculus that yields a natural Hamiltonian formalism. In particular, we formulate Noether's theorem in this context, identify a degenerate symplectic structure, and derive Hamiltonian differential equations on finite-dimensional center manifolds when those exist. Our formalism yields new natural conserved quantities. For Euler-Lagrange equations arising as traveling-wave equations in gradient flows, we identify Lyapunov functions. We provide several applications to pattern-forming systems including neural field and phase separation problems.