Javier Villarroel

Some stochastic properties of a subsystem of Yang–Mills classical mechanics are studied which do not fulfill automatically Gauss’ law, a possibility not considered before. However, significant differences with other subsystems already studied are not found for this system.

The Hamiltonian formalism and the action-angle variables for a generalized version of the Davey-Stewartson system is developed. Special cases include the usual Davey-Stewartson II system and the delta limit of the Davey-Stewartson I equations.

The method of solution to the (2+1)-dimensional Toda equation is discussed. This equation reduces to the well-known Toda lattice in 1+1 dimensions and via an appropriate asymptotic reduction to the 2+1 Kadomtsev-Petviashvili equation in the continuous limit. The solution exhibits a number of interesting aspects depending on the choice of parameters such as linear instability with a bounded growth rate and ill-posedness. In the ill-posed case we can relate the solution to a boundary value problem and give a formal construction of the solution. For one choice of signs, an analogue of the Sommerfeld radiation condition is used to identify a unique solution. In general, the method of solution of the Toda ``molecule'' equation requires an implementation of the dbar technique to cases where the associated eigenfunctions exhibit both smooth regions of nonholomorphicity and a discontinuity across a curve, which in this problem is the unit circle.

A method to obtain a new class of discrete eigenfunctions and associated real, nonsingular, decaying, ``reflectionless'' potentials to the time dependent Schrödinger equation is presented. Using the inverse scattering transform, related solutions of the Kadomtsev-Petviashvili equation are found. The eigenfunctions have poles of order m, m>1 in the complex plane and are also characterized by an index, or ``charge,'' which is obtained as a constraint in the theory.