Nonlocal integrals and conservation laws in the theory of nonlinear solitons

It is natural to investigate properties of solutions to nonstationary linear equations of mathematical physics by means of time-invariant spaces of linear functionals. In the framework of this approach, there appear nonlinear (nonstationary) partial differential equations (dual to original ones) admitting nontrivial groups of self-similarities. The superposition principle in the space of solutions to an original equation can be reproduced for the dual equation in the form of convolutions of kernels of linear functionals. The corresponding construction is applied to the Schrödinger equation on the line, where the ideas of quantum mechanics allows one to understand this new approach.