Long-Time Asymptotics of Solutions to the Cauchy Problem for the Defocusing Non-Linear Schrödinger Equation with Finite Density Initial Data. I. Solitonless Sector

The methodology of the Riemann-Hilbert (RH) factorisation approach for Lax-pair isospectral deformations is used to derive, in the solitonless sector, the leading-order asymptotics as t→±∞ (x/t∼O(1)) of solutions to the Cauchy problem for the defocusing non-linear Schrödinger equation (DfNLSE), i∂tu+∂ 2 xu−2(|u|−1)u=0, with finite density initial data u(x, 0)=x→±∞ exp( i(1∓1)θ 2 )(1+o(1)), where, as a consequence of a compatibility condition related to the solvability of the RH problem (RHP) associated with the DfNLSE, θ (∈ [0, 2π)) is a real-valued functional of a subset of the complete set of scattering data of the underlying linear auxiliary eigenvalue problem. A limiting case of these asymptotics, related to the RHP for the Painlevé II equation, or one of its special reductions, is also identified. 2000 Mathematics Subject Classification. (Primary) 35Q15, 37K40, 35Q55, 37K15: (Secondary) 30E20, 30E25, 81U40, 78A60 Abbreviated Title. Asymptotics of the Defocusing NLSE