Long-time asymptotics of solutions to the Cauchy problem for the defocusing non-linear Schrödinger equation with finite-density initial data. I. Solitonless sector

Abstract

For Lax-pair isospectral deformations whose associated spectrum, for given initial data, consists of the disjoint union of a finitely denumerable discrete spectrum (solitons) and a continuous spectrum (continuum), the matrix Riemann-Hilbert problem approach is used to derive the leading-order asymptotics as |t| → ∞ (x/t ∼ O(1)) of solutions (u = u(x, t)) to the Cauchy problem for the defocusing non-linear Schrödinger equation 2π). The D f NLSE dark soliton position shifts in the presence of the continuum are also obtained.36, 37]. The complete details of the asymptotic analysis that constitutes stage (1) of the two-step asymptotic paradigm above, which is quite technical and whose results are essential in order to obtain those of the present paper, can be found in [38]: this paper addresses stage (2) of the above programme via the matrix Riemann-Hilbert problem (RHP) approach [7,12,39,40,41,42,43,44,45,46,47]. It is important to note that, to the best of the author's knowledge as at the time of the presents, the first to obtain asymptotics of solutions to the Cauchy problem for the D f NLSE with finite-density initial data in the solitonless sector were Its and Ustinov [48,49].This paper is organized as follows. In Section 2, the necessary facts from the direct and inverse scattering analysis for the D f NLSE with finite-density initial data are given, the (matrix) RHP analysed asymptotically as |t| → ∞ (x/t ∼ O(1)) is stated, and the results of this paper are summarised in Theorems 2.2.1-2.2.4 (and Corollaries 2.2.1 and 2.2.2). In Section 3, an augmented RHP, which is equivalent to the original one stated in Section 2, is formulated, and it is shown that, as t → +∞, modulo exponentially small terms, the solution of the augmented RHP converges to the solution of an explicitly solvable, model RHP. In Section 4, the model RHP is solved asymptotically as t → +∞, from which the asymptotics of u(x, t) andx ±∞ (|u(x ′ , t)| 2 − 1) dx ′ are derived, and, in Appendix A, the-analogous-asymptotic analysis is succinctly reworked for the case when t → −∞. In Appendices B and C, respectively, formulae which are necessary in order to obtain the remaining asymptotic results of this paper are presented, and a panoramic view of the matrix RH theory in the L 2 -Sobolev space is given [44,45,46,50].

On the Asymptotic Stability of $${N}$$ N -Soliton Solutions of the Defocusing Nonlinear Schrödinger Equation

“…The situation is somewhat better understood in the integrable setting where the inverse scattering transform (IST) gives one much stronger control on the behavior of solutions than purely analytic techniques [20,29,41,14,16,12]. Even among the integrable evolutions, most results concern initial data with sufficient decay at spatial infinity, but there have been some recent studies concerning non vanishing initial data [23,7,30,42,43].…”

“…The long time asymptotic behavior of the defocusing NLS equation with finite density data has been studied previously. In a series of papers [42,43,44] Vartanian computed both the leading and first correction terms in the asymptotic expansion of the solution q(x, t) and 'partial masses'…”

The Three-Component Defocusing Nonlinear Schrödinger Equation with Nonzero Boundary Conditions

“…(ii) The computation of the long-time asymptotic behavior using the Deift-Zhou method [21,22] (see [31,45,46] for the defocusing scalar case with NZBC). This problem is also still open even in the two-component (Manakov) case.…”

Asymptotic stability of the black soliton for the Gross–Pitaevskii equation

“…Finally, it would be of interest to investigate further the global bassin of attraction of solitons or multi-solitons (in dimension 1 or higher), in particular having in mind the so-called 'soliton resolution conjecture'. Some results in that direction have been obtained for the integrable onedimensional Gross-Pitaevskii equation using Deift-Zhou's steepest descent method [16,44,45]. Let us emphasize that our proof of asymptotic stability does not rely on the integrability by means of the inverse scattering transform of the one-dimensional Gross-Pitaevskii equation.…”

“…At least formally, this method provides a description of the long-time dynamics, which is governed by solitons and dispersion. More precisely, the solutions are expected to behave as a chain of solitons plus a dispersive part (see, for example, [44,45]). A first step in order to derive rigorously this long-time description is to establish the stability of single solitons and chains of solitons.…”