Solving the Physical Hierarchy of Nonlinear Water Wave Equations via a Nearby Lax Integrable Hierarchy with Hamiltonian Perturbations

The goal of this talk is the study of the non-integrable, physical hierarchy of equations in 2+1 dimensions: Nonlinear Schroedinger, Dysthe, Trulsen-Dysthe and Zakharov. These equations describe, with increasing order, the dynamics of water waves in two dimensions. I demonstrate a procedure to determine a nearby hierarchy of these equations, which is Lax Integrable and the Lax pairs and the Its-Matveev formulae are given. I am then able to show that the solutions of each of these equations can be reduced to a quasiperiodic Fourier series with coherent structure basis functions. These include Stokes waves, envelope solitons and breather packets. In order to return to the original hierarchy of the Nonlinear Schroedinger, Dysthe, Trulsen-Dysthe and Zakharov equations I make a Hamiltonian perturbation of the Lax integrable hierarchy. I then apply a theorem of Kuksin, and a further theorem of Baker and Mumford, to write the algebraic geometric solutions. These steps provide me with an approach for the hyperfast numerical integration of these equations for physics and engineering purposes, and for the analysis of recorded data in one and two dimensions through a procedure I refer to as nonlinear Fourier analysis.