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Description

We study nonlinear systems of hyperbolic PDE's and difference equations on multidimensional lattices describing wave propagation. We show that a wide class of such systems, including nonlinear Schrodinger and Maxwell equations, Fermi-Pasta-Ulam model and many other not completely integrable systems, satisfy a principle of "approximate linear modal superposition". The essence of the principle is that the nonlinear evolution of a wave composed of generic wavepackets (defined as almost monochromatic waves) reduces with very high accuracy to independent nonlinear evolution of the involved wavepackets. Such an independence of wavepackets in the course of evolution persists for times long enough to observe fully developed nonlinear phenomena. An essential common property of systems obeying the principle of approximated superposition is the absolute continuity of the spectrum of underlying linear component. Such systems are not covered by either the classical complete integrability theory, including Birkhoff separation of variables theorem, or the KAM theory, nor by the Lax pairs method. In particular, our approach provides a simple justification for numerically observed effect of almost non interaction of solitons passing through each other without any recourse to the complete integrability. The mathematical framework developed for establishing the approximate linear superposition principle includes the theory of analytic functions of infinite-dimensional variable and the asymptotic theory of oscillatory integrals. Joint work with A. Babine