## 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.

Original language | English |
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Article number | e22 |

Pages (from-to) | 1-55 |

Number of pages | 55 |

Journal | Forum of Mathematics, Sigma |

Volume | 6 |

Early online date | 14 Nov 2018 |

DOIs | |

Publication status | Published - 2018 |