Solitons, Inverse Scattering, and Lattice Waves

Dr. Steve Cohn
Associate Professor
Department of Mathematics
University of Nebraska-Lincoln

Date:  Tuesday, February 24, 2004
Time:  3:30 p.m.
Place:  105 Othmer Hall

Solitons are nonlinear, solitary waves that interact elastically. They were first described in 1844, in a beautiful passage by the British engineer John Scott Russell. In 1895 it was seen that Scott Russell's wave could be derived as traveling-wave solution to the Korteweg-deVries equation. The inverse scattering transform (IST) was developed in the 1960's and 70's to treat nonlinear evolution equations on the real line. It first appeared in a 1967 paper by Gardener, Greene, Kruskal and Miura [GGKM], in which the authors derived explicit solutions to the initial value problem for the Korteweg-deVries equation. in this paper the IST looks more like an ingenious trick than a method. In a 1968 paper, Lax showed that there was indeed a method to the IST. In the following years, the IST was used to solve nonlinear partial differential equations like the sine-Gordon and the cubic Schrodinger, as well as a discrete, particle-chain model known as the Toda lattice.

One remarkable feature of the IST is that it gives an explicit characterization of soliton solutions. In [GGKM] for example, the authors use it to recover and generalize the wave form observed by Scott Russell.

I'll talk about the mathematics of IST and solitons, and about applications and possible generalizations of the lattice particle-chain model. Much of the talk is based on joint work with Henk Viljoen, Chemical Engineering at UNL.