Spectral theory for periodic self-adjoint Dirac operators and inverse scattering transform for the defocusing non-linear Schrödinger equation with periodic boundary conditions
Abstract
The inverse spectral theory for a self-adjoint one-dimensional Dirac operator associated periodic potentials is formulated via a Riemann–Hilbert problem (RHP) approach. The resulting formalism is also used to solve the initial value problem for the non-linear Schrödinger (NLS) equation. A uniqueness theorem for the solutions of the RHP is established, which provides a new method for obtaining the potential from the spectral data. Two additional, scalar RHPs are also formulated that provide conditions for the periodicity in space and time of the solution generated by arbitrary sets of spectral data. The formalism applies for both finite-genus and infinite-genus potentials. The formalism also shows that only a single set of Dirichlet eigenvalues is needed in order to uniquely reconstruct the potential of the Dirac operator and the corresponding solution of the defocusing NLS equation, in contrast with the representation of the solution of the NLS equation via the trace formulae in the finite-genus formalism, in which two different sets of Dirichlet eigenvalues are used.