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.