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Existence theorem for solitary waves on lattices. (English) Zbl 0807.35121

Summary: We give an existence theorem for localized travelling wave solutions on one-dimensional lattices with Hamiltonian \[ H = \sum_{n \in \mathbb{Z}} \left( {1 \over 2} p^ 2_ n + V(q_{n+1} - q_ n) \right), \] where \(V(\cdot)\) is the potential energy due to nearest-neighbour interactions. Until now, apart from rare integrable lattices like the Toda lattice \[ V(\varphi) = ab^{-1} (e^{-b \varphi} + b \varphi - 1), \] 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 \[ V(\varphi) = {1 \over 2} \varphi^ 2 + {1 \over 3} a \varphi^ 3, \quad V(\varphi) = {1 \over 2} \varphi^ 2 + {1 \over 4} b \varphi^ 4, \] thereby shedding new light on the recurrence phenomena in these systems observed first by Fermi, Pasta and Ulam 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).

MSC:

35Q51 Soliton equations
81T25 Quantum field theory on lattices
49J40 Variational inequalities
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