In this work we will look at the focusing Davey-Stewartson equation from two different angles, using advanced numerical tools.
As a nonlinear dispersive PDE and a generalisation of the non-linear Schrödinger equation, DS possesses solutions that develop a singularity in finite time. We numerically study the long time behaviour and potential blow-up of solutions to the focusing Davey-Stewartson II equation for various initial data and propose a conjecture describing the blow up rate and solution profiles near the singularity.
Secondly, DS is an integrable system and can be studied as an inverse scat- tering problem. Both the forward and inverse scattering transformation in this case are reduced to a d-bar system which plays the role that Riemann-Hilbert problems play in one dimensional problems. We will present numerical solutions for Schwartzian and compactly supported potentials. In all studied cases we use spectral methods and achieve machine precision.
Based on joint works with Christian Klein and Ken McLaughlin