Spatial Hamiltonian identities for nonlocally coupled systems

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.Comment: 39 pages, 1 figur

Eigenvalues of Curvature, Lyapunov exponents and Harder-Narasimhan filtrations

Inspired by Katz-Mazur theorem on crystalline cohomology and by Eskin-Kontsevich-Zorich's numerical experiments, we conjecture that the polygon of Lyapunov spectrum lies above (or on) the Harder-Narasimhan polygon of the Hodge bundle over any Teichm\"uller curve. We also discuss the connections between the two polygons and the integral of eigenvalues of the curvature of the Hodge bundle by using Atiyah-Bott, Forni and M\"oller's works. We obtain several applications to Teichm\"uller dynamics conditional to the conjecture.Comment: 37 pages. We rewrite this paper without changing the mathematics content. arXiv admin note: text overlap with arXiv:1112.5872, arXiv:1204.1707 by other author

Periodic solutions of nonlinear wave equations with general nonlinearities

We prove the existence of small amplitude periodic solutions, with strongly irrational frequency \om close to one, for completely resonant nonlinear wave equations. We provide multiplicity results for both monotone and nonmonotone nonlinearities. For \om close to one we prove the existence of a large number N_\om of 2 \pi \slash \om -periodic in time solutions $u_1, ..., u_n, ..., u_N$: N_\om \to + \infty as \om \to 1 . The minimal period of the $n$-th solution $u_n$ is proved to be 2 \pi \slash n \om . The proofs are based on a Lyapunov-Schmidt reduction and variational arguments.Comment: 29 page

Periodic solutions of fully nonlinear autonomous equations of Benjamin-Ono type

We prove the existence of time-periodic, small amplitude solutions of autonomous quasilinear or fully nonlinear completely resonant pseudo-PDEs of Benjamin-Ono type in Sobolev class. The result holds for frequencies in a Cantor set that has asymptotically full measure as the amplitude goes to zero. At the first order of amplitude, the solutions are the superposition of an arbitrarily large number of waves that travel with different velocities (multimodal solutions). The equation can be considered as a Hamiltonian, reversible system plus a non-Hamiltonian (but still reversible) perturbation that contains derivatives of the highest order. The main difficulties of the problem are: an infinite-dimensional bifurcation equation, and small divisors in the linearized operator, where also the highest order derivatives have nonconstant coefficients. The main technical step of the proof is the reduction of the linearized operator to constant coefficients up to a regularizing rest, by means of changes of variables and conjugation with simple linear pseudo-differential operators, in the spirit of the method of Iooss, Plotnikov and Toland for standing water waves (ARMA 2005). Other ingredients are a suitable Nash-Moser iteration in Sobolev spaces, and Lyapunov-Schmidt decomposition. (Version 2: small change in Section 2).Comment: 47 page