Title:The inverse scattering method for the Novikov-Veselov equation

Abstract:
The inverse scattering method for the Novikov-Veselov equation. The 
nonlinear Novikov-Veselov (NV) equation is the most natural 2+1 
dimensional generalization of the celebrated 1+1 dimensional nonlinear
Korteweg-deVries (KdV) equation. The KdV equation is known to model 
various nonlinear physical phenomena including soliton waves in water 
canals and nonlinear propagation of light in optical cables. Thus the NV 
equation is expected to be important for the description of some (yet 
unknown) physical phenomena. Inverse scattering transform (IST) solution 
for the non-periodic NV equation has been discussed in the literature 
[Boiti, Leon, Manna and Pempinelli 1987; Tsai 1994], but only formally. 
In the first part of the talk, rigorous mapping properties are proved 
for the IST scheme related to the NV equation. There is no smallness 
assumption on the initial data, but it has to be of "conductivity type". 
However, the crucial question of rigorous solvability of the NV equation 
by the IST method remains open even after the presented theorems. The 
second part of the talk presents numerical evidence for solvability. 
More precisely, we compute the solution in two ways: by directly solving 
the nonlinear evolution equation, and by implementing the IST scheme 
numerically. The results agree to significant precision and provide 
intuition for further theoretical work; in particular the evolving 
solution seems to stay "conductivity type", suggesting the existence of a so
far unknown nonlinear evolution equation for conductivities. The results 
in this talk are done in collaboration with David Isaacson,
Kim Knudsen, Matti Lassas, Jennifer Mueller and Andreas Stahel.