Inverse Scattering Transform Method for Lattice Equations

Abstract

The main original contribution of this thesis is the development of a fully discrete inverse scattering transform (IST) method for nonlinear partial difference equations. The equations we solve are nonlinear partial difference equations on a quad-graph, also called lattice equations, which are known to be multidimensionally consistent in N dimensions for arbitrary N. Such equations were discovered by Nijhoff, Quispel and Capel and Adler and later classified by Adler, Bobenko and Suris. The main equation solved by our IST framework is the Q3δ lattice equation. Our approach also solves all of its limiting cases, including H1, known as the lattice potential KdV equation. Our results provide the discrete analogue of the solution of the initial value problem on the real line. We provide a rigorous justification that solves the problem for wide classes of initial data given along initial paths in a multidimensional lattice. Moreover, we show how soliton solutions arise from the IST method and also utilise asymptotics of the eigenfunctions to construct infinitely many conservation laws.