N. Hristov and D. Pelinovsky
Justification of the KP-II approximation in dynamics of two-dimensional FPU systems
Z. Angew. Math. Phys. 73 (2022) 213 (26 pages)
Abstract:
Dynamics of the Fermi-Pasta-Ulam (FPU) system on a two-dimensional
square lattice is considered in the limit of small-amplitude long-scale waves with slow
transverse modulations. In the absence of transverse modulations, dynamics of such
waves, even at an oblique angle with respect to the square lattice, is known to be
described by the Korteweg-de Vries (KdV) equation. For the three basic directions
(horizontal, vertical, and diagonal), we prove that the modulated waves are well
described by the Kadomtsev-Petviashvili (KP-II) equation. The result was expected long
ago but proving rigorous bounds on the approximation error turns out to be complicated
due to the nonlocal terms of the KP-II equation and the vector structure of the
FPU systems on two-dimensional lattices. We have obtained these error bounds by
extending the local well-posedness result for the KP-II equation in Sobolev spaces and by
controlling the error terms with energy estimates. The bounds are useful in the analysis
of transverse stability of solitary and periodic waves in two-dimensional FPU systems
due to many results available for the KP-II equation.
Keywords:
FPU lattice, KdV equation, KP-II equation, justification of amplitude equations, stability of nonlinear waves.