Dressing method and quadratic bundles related to symmetric spaces. Vanishing boundary conditions (Q2795518)

The author proposed a way to relate to each Hermitian symmetric space a multi-component nonlinear Schrödinger equation. In fact, he consider a multi-component nonlinear Schrödinger equation which generalizes the equation of Fordy. The main result of the study show how one can apply Zakharov-Shabat's dressing method to quadratic bundles of a special type (by means of an operator which is tightly related to a Hermitian symmetric space). This method allows one to obtain reflectionless potentials and thus generate special types of solutions on a trivial background in an algebraic manner. The author proved that there are two different types of solutions: generic soliton type solutions and rational solutions. The soliton type solutions are associated with dressing factors whose poles are generic while the rational solutions correspond to factors whose poles lie on a continuous spectrum. Finally, the author explicitly construct reflectionless potentials and particular solutions of either of the aforementioned types and show how these solutions can be reduced to well-known solutions to the scalar derivative nonlinear Schrödinger equation.