New Irregular Solutions in the Spatially Distributed Fermi–Pasta–Ulam Problem

Abstract

For the spatially-distributed Fermi–Pasta–Ulam (FPU) equation, irregular solutions are studied that contain components rapidly oscillating in the spatial variable, with different asymptotically large modes. The main result of this paper is the construction of families of special nonlinear systems of the Schrödinger type—quasinormal forms—whose nonlocal dynamics determines the local behavior of solutions to the original problem, as t→∞. On their basis, results are obtained on the asymptotics in the main solution of the FPU equation and on the interaction of waves moving in opposite directions. The problem of “perturbing” the number of N elements of a chain is considered. In this case, instead of the differential operator, with respect to one spatial variable, a special differential operator, with respect to two spatial variables appears. This leads to a complication of the structure of an irregular solution.

Asymptotics of Regular and Irregular Solutions in Chains of Coupled van der Pol Equations

“…To do this, we introduce some notation. By D, J and J 0 , we denote [24] operators defined on continuously differentiable functions v(x, y) of two variables x and y, acting according to the rules…”

Nonlinear Dynamics

“…The paper "New Irregular Solutions in the Spatially Distributed Fermi-Pasta-Ulam Problem" [5] by Kashchenko and Tolbey presents the construction of families of special nonlinear systems of the Schrödinger-type quasinormal forms, whose nonlocal dynamics determines the local behavior of solutions to the spatially distributed Fermi-Pasta-Ulam equation. Irregular solutions are studied that contain components rapidly oscillating in the spatial variable, with different asymptotically large modes.…”