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
نویسنده
چکیده
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 (DfNLSE), i∂tu+∂ 2 xu−2(|u| −1)u= 0, with finitedensity initial data u(x, 0) =x→±∞ exp( i(1∓1)θ 2 )(1+o(1)), θ ∈ [0, 2π). The DfNLSE dark soliton position shifts in the presence of the continuum are also obtained. 2000 Mathematics Subject Classification. (Primary) 35Q15, 37K40, 35Q55, 37K15: (Secondary) 30E20, 30E25, 81U40 Abbreviated Title. Asymptotics of the DfNLSE Dark Solitons on Continua
منابع مشابه
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)), wher...
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