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SPATIAL HAMILTONIAN IDENTITIES FOR NONLOCALLY COUPLED SYSTEMS

Part of: Nonlinear integral equations Infinite-dimensional dissipative dynamical systems Representations of solutions Infinite-dimensional Hamiltonian systems Miscellaneous topics - Partial differential equations

Published online by Cambridge University Press:  14 November 2018

BENTE BAKKER  and
ARND SCHEEL
BENTE BAKKER
Affiliation:
Department of Mathematics, VU Universiteit Amsterdam, De Boelelaan 1081, 1081 HV Amsterdam, The Netherlands; bente.h.bakker@gmail.com
ARND SCHEEL
Affiliation:
School of Mathematics, University of Minnesota, 206 Church St. SE, Minneapolis, 55455, USA; scheel@umn.edu

Abstract

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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.

MSC classification

Primary: 35S30: Fourier integral operators 45G15: Systems of nonlinear integral equations
Secondary: 35C07: Traveling wave solutions 37K05: Hamiltonian structures, symmetries, variational principles, conservation laws 37L10: Normal forms, center manifold theory, bifurcation theory 37L45: Hyperbolicity; Lyapunov functions
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 6 , 2018 , e22
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2018

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