Bounds for the nonlinear Schrödinger approximation of the Fermi–Pasta–Ulam system

Abstract

We prove that the evolution of a slowly varying envelope of small amplitude of an underlying oscillating wave packet in the Fermi–Pasta–Ulam (FPU) system can be described approximately by the nonlinear Schrödinger equation. In contrast to other lattice equations for which this question has been addressed in the existing literature, the FPU system possesses a nontrivial quadratic resonance due to the curves of eigenvalues which vanish at the wave number k = 0. The proof of the error estimates is based on normal form transforms and a wave number-dependent scaling of the error function.

Keywords:

• lattice equation
• envelope equation
• multi-scale approximation
• quadratic resonance
• normal form transform
• approximation

AMS Subject Classifications::

• 34C15
• 35C20
• 35Q55
• 35Q70