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.
Highlights
We study the irregular solutions to the boundary value problem (4) and (6)
It is shown that various partial differential equations arise while describing the leading approximations of solutions in different domains of the phase space of the boundary value problem (3)
Asymptotic representations of the irregular solutions studied above contain superposition of functions depending on the following: the ‘slow’ time τ = εt, the ‘medium’ time t, and ‘fast’ time ε−1t
Summary
The transition from the Equation (3) to a special nonlinear partial differential equation was made to study regular solutions with a certain degree of accuracy, with respect to the parameter ε. We are interested in the study of influence of this value on the asymptotics of the solutions. We study the irregular solutions to the boundary value problem (4) and (6). The structure of such solutions consists of the superposition of functions that depend smoothly (regularly) on the parameter ε as well as the functions that depend smoothly on the parameter ε−1. Mathematics 2021, 9, 2872 periodic and continuous with respect to x initial conditions u(0, x) φ1(x),
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