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  • SPATIAL HAMILTONIAN IDENTITIES FOR NONLOCALLY COUPLED SYSTEMS

  • Part of
  • BENTE BAKKER, ARND SCHEEL
  • Journal:
    Forum of Mathematics, Sigma / Volume 6 / 2018
    Published online by Cambridge University Press:
    14 November 2018, e22
    • Article
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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.