Techniques and instabilities for 1+1 and 2+1 dimensional integrable partial differential equations.

Abstract

We study the semiclassical limit of the 1+1 dimensional initial value problems in the focusing nonlinear Schrodinger (NLS) hierarchy, and establish a rigorous connection of odd flows in this hierarchy to all the members of the Korteweg-de Vries (KdV) hierarchy in the same limit. We also demonstrate numerically that the microscopic oscillations differ in the limit. These initial value problems are closely related to the spectrum of the AKNS operator, which we use to study this limit. To examine this spectrum numerically, we introduce a new method for the calculation of the AKNS Floquet spectrum, and argue that the canonical auxiliary spectrum associated with the NLS flows is intrinsically sensitive to variations of the potential. We then consider the semiclassical limit of the 2+1 dimensional Kadomtsev-Petviashvilli-II (KP-II) equation. We present numerical simulations which suggest the semiclassical limit exists. Like the NLS flows, the KP-II flow has an invariant spectral set, known as the Floquet multiplier curve or Heat curve. To study the KP-II flow, we develop a numerical method for calculating the branches of this curve for small potentials. The numerical integration of these initial value problems employ several new temporal integration techniques. We develop preconditioning methods, spectral multi-grid methods, and asymptotic corrections for these initial value problems. Dissipative techniques which are appropriate for conservative initial value problems are developed, as well as variable timestep methods for conservative partial and ordinary differential equations.