Two-Wave Interactions in the Fermi–Pasta–Ulam Model

Abstract

This work is devoted to investigating the dynamic properties of the solutions to the boundaryvalue problems associated with the classic Fermi–Pasta–Ulam (FPU) system. We analyze these problems for an infinite-dimensional case where a countable number of roots of characteristic equations tend to an imaginary axis. Under these conditions, we build a special nonlinear partial differential equation that acts as a quasi-normal form, i.e., determines the dynamics of the original boundary-value problem with the initial conditions in a sufficiently small neighborhood of the equilibrium state. The modified Korteweg–deVries (KdV) equation and the Korteweg–de Vries–Burgers (KdVB) equation act as quasi-normal forms depending on the parameter values. Under some additional assumptions, we apply the renormalization procedure to the boundary-value problems obtained. This procedure leads to an infinite-dimensional system of ordinary differential equations. We describe a method for folding this system into a special boundary- value problem, which is an analog of the normal form. The main contribution of this work is investigating the interaction of the waves moving in different directions in the FPU problem by using analytical methods of nonlinear dynamics. It is shown that the mutual influence of the waves is asymptotically small, does not affect their shape, and contributes only to a shift in their speeds, which does not change over time.