Long-Time Asymptotics of Solutions to the Cauchy Problem for the Defocusing Non-Linear Schrödinger Equation with Finite-Density Initial Data. II. Dark Solitons on Continua

2002/10/22 by A. H. Vartanian, Vartanian, A. H. · 1 citation
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Physical sciences #Nonlinear Photonic Systems #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.nlin/0210050

openalex publication_date 2002/10/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 (D_fNLSE), \mi ∂_tu + ∂_x²u - 2(| u |² - 1) u = 0, with finite-density initial data u(x,0) =_{x → ± ∞} exp (\tfrac\mi (1 ∓ 1) θ2)(1 + o(1)), θ∈ [0,2π). The D_fNLSE dark soliton position shifts in the presence of the continuum are also obtained.

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