Egorova I. Inverse problem method and nondecreasing solutions of nonlinear evolutionary equations

The scattering theory is developed in the thesis for the one-dimensional Schroedinger operator and for the Jacobi operator on the whole axis with asymptotically finite-gap, steplike coefficients in the class of perturbations with given smoothness and finite moments. The Cauchy problem is solved for the Toda hierarchy and for the KdV and the modified KdV equations with asymptotically finite-gap, steplike initial data. The KdV and the defocusing NS equations are integrated in a class of almost periodic initial data, that have the spectrum of the Cantor type. It is proved, that the solutions are uniformly almost periodic functions with respect to the time variable.