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Lev Ostrovsky, Colloquium

September 22, 2016 @ 4:00 pm - 5:00 pm

Tea at 3:30 pm

Title: Perturbation theory of solitons and their behavior as particles

Abstract: The term “soliton” is now widely known even among the non-specialists. Solitary waves on water surface were first observed by J. Scott Russell in 1834. In 1967 M. Kruskal and N. Zabusky have numerically shown that solitary waves can nonlinearly interact in such a way that after the interaction they remain unchanged, with only an additional time delay. Such a behavior is characteristic of elastic collision of material particles, thus the term “soliton” was introduced. Another breakthrough was the discovery of new exact methods of solving a certain class of equations, especially the so-called inverse scattering method. However, exact description of soliton dynamics is possible almost exclusively for fully integrable equations, and even for them it is often difficult to interpret the formal result for more than two solitons. Thus, different perturbation schemes for solitons were developed since 1970s.

Here, some modern results of perturbation theory for solitons and fronts (kinks) are briefly described. The method is based on expansion of the solution in powers of a small parameter (e.g., small dissipation, slow variations of a medium in space and time, interaction with neighboring solitons, etc.) and preventing systematic increase of the perturbation by the slow variation of parameters of the basic solution (such as its amplitude). Among the physically sound results is the effect of earth rotation on a soliton, and dynamics of interacting solitons with close amplitudes in integrable and non-integrable systems. The latter is reduced to interaction of classical particles creating a field potential decreasing with distance. This interaction can be repulsing, attracting, or result in a multisoliton state. Also interaction of kinks in flat-top solitons in described. Under this approach, the soliton ensembles form “envelope solitons” of Toda type, and ensembles of kinks, by a double Toda system.
The theory is illustrated by examples of nonlinear waves in electromagnetic systems and nonlinear internal waves in the ocean.

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Date:
September 22, 2016
Time:
4:00 pm - 5:00 pm
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