Inverse Scattering for Spectral Problems with Algebraic Potentials

Inverse scattering transform was designed in seventies to solve the initial-value problem for nonlinear partial differential equations (PDEs) by means of solving linear spectral problems (for Lax operators). The theorems on existence and uniqueness of solutions of nonlinear PDEs were proven with the help of scattering formalism and advanced complex analysis. Decomposition formulas (orthogonality and completeness) for linear spaces of the Lax operators were found in compliance with standard spectral theory in a Hilbert space.

Nonlinear evolution equations in higher dimensions often exhibit steady-state localized solutions called algebraic solitons or lumps. The multidimensional solitons appear as algebraic non-integrable potentials in the spectral problem. The linear space associated with such potentials expands beyond the Hilbert space and has sign-indefinite inner products, resonant pole singularities and other delicate analytical properties. Fundamental examples of such non-Hilbert spectral problems are the time-dependent Schrodinger equation and the Dirac system in two dimensions.

Two problems of advanced complex analysis can be used to construct a complete decomposition of linear spaces in multidimensional inverse scattering. The first is the Riemann-Hilbert problem of reconstructing a meromorphic function in a plane from its given jump at a real axis. The other is the D-bar formalism defining a non-analytic function from its D-bar derivative.

The decomposition formalism for linear spaces associated with multidimensional algebraic potentials enables further applications such as bifurcation theory for embedded eigenvalues, edge bifurcations, soliton perturbation theory, decay estimates.