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School-Seminar "Interaction of Mathematics and Physics: New Perspectives" for graduate students and young researchers
August 29, 2012 15:00–16:30, Moscow, Moscow State Institute of Electronics and Mathematics, Steklov Mathematical Institute

Propagation and interaction of solution singularities for nonlinear equations

Vladimir Danilov

Moscow State Institute of Electronics and Mathematics — Higher School of Economics
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Vladimir Danilov
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Abstract: Nonlinear solitary waves which are solutions to nonlinear problems with a small parameter can be considered as a regularizations of certain generalized functions. Singularities appear as a weak limits or a weak asymptotics with respect to the small parameter. Exploiting this viewpoint we are going to discuss the following topics:
- well-posed definition to a weak solution for nonlinear equations with a small parameter admitting limit passage (examples: KdV equation, phase field system);
- propagation and interaction of solution singularities for regularized conservation laws (examples: delta-shock type solutions, continuity equation in a discontinuous velocity field );
- propagation and interaction of narrow solitons for integrable and nonintegrable models of KdV type;
- constructing of global in time WKB-type asymptotic solutions for parabolic equations using Hamilton trajectories only;
- forward and backward in time asymptotic solutions to Cauchy problem for the equations mentioned above.
One of the key points of our consideration is the notion of finitely generated asymptotic algebras of generalized functions inspired by the well-known algebras of generalized functions introduced by V. P. Maslov and by J.-F. Colombeau.

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