On the focusing non-linear Schrödinger equation with non-zero boundary conditions and double poles

The inverse scattering transform for the focusing non-linear Schrödinger (NLS) equation with non-zero boundary conditions at infinity and double zeros of the analytic scattering coefficients is presented. The direct problem is discussed, including a detailed analysis of the discrete spectrum in the presence of such double zeros. Such cases lead to double poles for the meromorphic matrices appearing in the inverse problem. Explicit formulae for the coefficients of the singular part of the Laurent expansions of the meromorphic functions at the discrete spectrum are also derived, and the inverse problem is reduced to a standard set of linear algebraic-integral equations. The general solution of the NLS equation in the case of an arbitrary finite number of double zeros is given, and an explicit formula for soliton solutions arising in the case of a single quartet of purely imaginary double zeros is presented. The long-Time asymptotic behaviour of such a solution is also studied, and the solution is shown to describe the interaction of two solitons with same amplitude and velocity parameters, which diverge from each other logarithmically as in the case of zero boundary conditions.