On the KP-II Limit of Two-Dimensional FPU Lattices

Abstract

We study a two-dimensional Fermi-Pasta-Ulam lattice in the long-amplitude, small-wavelength limit. The one-dimensional lattice has been thoroughly studied in this limit, where it has been established that the dynamics of the lattice is well-approximated by the Korteweg–De Vries (KdV) equation for timescales of the order ε^−3. Further it has been shown that solitary wave solutions of the FPU lattice in the one dimensional case are well approximated by solitary wave solutions of the KdV equation. A two-dimensional analogue of the KdV equation, the Kadomtsev–Petviashvili (KP-II) equation, is known to be a good approximation of certain two-dimensional FPU lattices for similar timescales, although no proof exists. In this thesis we present a rigorous justification that the KP-II equation is the long-amplitude, small-wavelength limit of a two-dimensional FPU model we introduce, analogous to the one-dimensional FPU system with quadratic nonlinearity. We also prove that the cubic KP-II equation is the limit of a model analogous to a one-dimensional FPU system with cubic nonlinearity. Further we study whether stability of line solitons in the KP-II equation extends to stability of one-dimensional FPU solitary waves in the two-dimensional lattices.