Toda Soliton Limits on General Backgrounds

Renger, Walter

doi:10.1006/jdeq.1998.3505

Cited by 4 publications

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“…Within the ISM framework, the asymptotic analysis of the solution to the Cauchy problem for the D f NLSE with finite-density initial data is divided into two steps: (1) the analysis of the solitonless (pure radiative, or continuous) component of the solution; and (2) the inclusion of the N -dark soliton component via the application of a "dressing" procedure to the solitonless background [33,34,35,36,37]. 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].…”

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confidence: 99%

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

Vartanian¹

2003

Recent Developments in Integrable Systems and Riemann-Hilbert Problems

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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].

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