Alfimov G.L., Eleonsky V.M.

Nonlocal phenomena in nonlinear media
G.L. Alfimov, V.M. Eleonsky
F.V. Lukin Research Institute of Physical Problems,
Moscow, Zelenograd, 103460, Russia

Nowadays the investigation of nonlinear phenomena in various branches of science reveals a list of model nonlinear problems which can be labelled as classical ones. They are, for example, Nonlinear Klein-Gordon equation, Korteveg-de Vries equation and Nonlinear Scrödinger equation. Besides the having of some interesting mathematical properties, these equations are of great importance for applications because they arise simultaneously in numerous branches of science (hydrodynamics, solid state physics, theory of macromolecules etc) as the simplest nonlinear approximations.

At the same time in many cases the next step to construct more realistic model incorporates the nonlocality. From mathematical viewpoint these models are covered by nonlocal generalizations of classical equations. Thus the two questions arise:

• which of the solutions of classical equations (such as solitons etc) survive under the transition to nonlocal generalization?
• which new solutions arise in nonlocal model?

These questions are not trivial ones. For example, the inclusion of nonlocality in the Korteveg-de Vries equation allows to describe sharp crested waves and the phenomenon of the breaking of waves. The incorporation of nonlocality in sine-Gordon model can destroy the traditional 2 -kink solution and can cause the existence of 4 - and 6 -kinks as well as solutions of topological charge 0.

The report presented contains a short survey of these examples of nonlocal problems. The case of nonlocal sine-Gordon equation will be considered in more detail and some new features of this model will be discussed.