Two Wave Interactions in a Fermi– Pasta–Ulam Model

Abstract

The work is devoted to the dynamic properties of the solutions of boundary value problems associated with the classical system of Fermi – Pasta – Ulam (FPU). We study this problem in infinite-dimensional case, when a countable number of roots of characteristic equations tend to an imaginary axis. Under these conditions, we built a special non-linear partial differential equation, which plays the role of a quasinormal form, i.e, it defines the dynamics of the original boundary value problem with the initial conditions in a sufficiently small neighborhood of the equilibrium state. The modified Korteweg de Vries (KdV) equation and the Korteweg de Vries Burgers (KdVB) one are quasinormal forms depending on the parameter values. Under some additional assumptions, we apply the procedure of renormalization to the obtained boundary value problems. This procedure leads to an infinite-dimensional system of ordinary differential equations. We describe a method of folding this system in the special boundary value problem, which is an analogue of the normal form. The main result is that the analytical methods of nonlinear dynamics explored the interaction of waves moving in different directions, in the problem of the FPU. It was shown that waves influence on each other is asymptotically small and does not change the shape of waves, contributing only a shift in their speed, which does not change over time.

Keywords

• fermi–pasta–ulam model
• generalized kdv equation
• quasinormal form
• boundary value problem