Existence theorem for solitary waves on lattices

Abstract

In this article we give an existence theorem for localized travelling wave solutions on one-dimensional lattices with Hamiltonian where V( -) is the potential energy due to nearest-neighbour interactions. Until now, apart from rare integrable lattices like the Toda lattice V(φ) -ab~l(e~b^ + bφ -1), the only evidence for existence of such solutions has been numerical. Our result in particular recovers existence of solitary waves in the Toda lattice, establishes for the first time existence of solitary waves in the (nonintegrable) cubic and quartic lattices V(φ) = ^ φ 2 + | α</> 3 , V(φ) ~\Φ L ^\ bφ 4 , thereby confirming the numerical findings in [1] and shedding new light on the recurrence phenomena in these systems observed first by Fermi, Pasta and Ulam [2], and shows that contrary to widespread belief, the presence of exact solitary waves is not a peculiarity of integrable systems, but "generic" in this class of nonlinear lattices. The approach presented here is new and quite general, and should also be applicable to other forms of lattice equations: the travelling waves are sought as minimisers of a naturally associated variational problem (obtained via Hamilton's principle), and existence of minimisers is then established using modern methods in the calculus of variations (the concentration-compactness principle of P.-L. Lions [3]).

On the solitary wave pulse in a chain of beads

“…Conversely, any solution of (1) with sufficiently rapid decay must satisfy (6). This applies to the solutions of MacKay and Ji and Hong, since these are continuous and square-integrable on R. The reason is that the theory of [5] implies r(x) = Q(x + 1) − Q(x) where ∞ −∞ Q (s) 2 ds < ∞, and so…”

“…For p = 3 2 , MacKay [6] applied the theory of Friesecke and Wattis [5] to prove the existence of a constant-velocity pulse solution r n = r(n − ct) in this one-dimensional chain. Essentially the same argument works also for p > 1, as shown by Ji and Hong [7].…”

The dynamical behaviour near solitary waves in Hamiltonian lattice differential equations

“…We recall that such solutions are globally defined and approach a specific steady state, say zero, as τ → ±∞. Although the travelling wave equation (2) is ill-posed, existence proofs of solitary waves for broad classes of lattice differential equations relying on variational methods [7,1], center manifold reductions [20,21] or perturbation methods [8] are known.…”

“…However, such an assumption is often violated in the framework of Hamiltonian LDEs like the Fermi-Pasta-Ulam lattice and the Klein Gordon lattice [16,20,21]. Moreover, most existence results of solitary waves such as the result of Friesecke and Wattis [7] rely on variational methods and refer to equations where the hyperbolicity assumption is violated. It is therefore desirable to extent Lin's method also to the context of lattice differential equations, where hyperbolicity of the steady state may be absent as in the Fermi-Pasta-Ulam lattice.…”

“…Surprisingly, their numerical simulation yielded an unexpected result: At least at low energy, the energy of the system remained confined among the initial modes instead of spreading to all modes. Concerning exact dynamics in the Fermi-Pasta-Ulam lattice it was proved only recently by Friesecke and Wattis [7] that (4) exhibits solitary wave solutions, i.e. localized coherent modes.…”