Danilov200711

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One of the most important properties of linear problems is the existence of linear superposition principle. It is used in different ways and is a foundation of all linear theories. In the case of linear problems the superposition principle is in fact the basis of global in time approximating (asymptotic) solutions constructions-these are classical Maslov canonical operator and Fourier integral operators introduced by L.Hormander. These constructions have discovered the deep connection between symplectic geometry objects and PDE's.

Another situation is for nonlinear problems. For these problems a superposition principle is known only for particular cases (linearization using Baklund transformation, equations integrated by inverse scattering problem method). The problem of the nonlinear waves interaction decryption in general case seems now as unsolvable. But for the particular case of nonlinear solitary waves (solitons, kinks) the situation looks more optimistically. Here one can construct approximating superposition description based on the algebraic nature of the weak (in D' sense) asymptotics of these solutions. Firstly it was noted by V.Maslov at the beginning of 70th. This approach was developed by V.Shelkovich, G.Omel'yanov and the author. In the framework of this approach the following problems were solved:

-the scenario of solitons interaction in nonintegrable KdV type equations, -the problem of free boundaries confluence in phase field model, -the problem of kinks interaction in nonintegrable Sine-Gordon type models.

Besides the problems of shock and singular(delta) shock waves generation also were solved. The last ones are directly connected with geometry: the reason of their apparence is the characteristics intersection (singularity of projection mapping) just like in linear theory. All said above relates to special (constructive) approach to NPDE's investigations that is based on explicit (in a sense)formulas construction. This approach can give information about very fine solution properties. Here there is very interesting interaction with numerical investigations. From one hand side one can try to explane some effects discovered numerically and from another hand side it is possible to try to determine quantitative values of the phenomena observed by theoretical investigations. In the talk we are planning to give a review of the topics mentioned above.

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