
Inverse scattering transform is a method to solve the initialvalue problem for nonlinear partial differential
equations by means of linear spectral problems. Existence and uniqueness of solutions of nonlinear equations
are considered with the spectral and complex analysis. Spectral decompositions theorems are proven in Hilbert
and nonHilbert spaces, associated with the selfadjoint and nonselfadjoint Lax operators.
Nonlinear evolution equations often exhibit localized solutions called algebraic solitons or lumps. The algebraic
solitons become nonintegrable potentials in the fundamental spectral problems, such as the timedependent Schrodinger
equation, the Dirac equation and the AblowitzKaupNewellSegur system.
Analytical properties of nonHilbert spectral problems, associated with the algebraically decaying potentials
are studied with the RiemannHilbert problem, the dbar problem, and the Evans function formalism. Bifurcations
of embedded eigenvalues, resonances, and other deformations of the spectral data are considered in view of
applications to the nonlinear integrable systems.

