Weak asymptotic method for describing the propagation and interactions of solitary waves

In the report, we will talk about some approach to the construction of formulas that describe the propagation and interaction of solitary waves such as solitons, kinks and delta shock waves. Such solutions arise in various problems: from the ones listed above to, for example, phase transition problems in the phase field model or well-known methods of front- and wave-tracking. Because we are talking about solutions with a given structure, and the problem contains, one way or another, a small parameter, the problem arises of correctly determining a generalized solution that allows a limit transition to a non-smooth “real” generalized solution of the limiting problem. In some situations, for example, in the low viscosity method, this solution can be constructed according to the classical scheme, and for the (scalar) KdV equation, such a definition consists of two integral identities. This approach was invented by me in collaboration with V.M. Shelkovich and in some ways similar to the approach of J.-F. Colombo, but better suited for the study of equations.