Higher Order Asymptotics of the Modified Non-Linear Schrödinger Equation
نویسنده
چکیده
Using the matrix Riemann-Hilbert factorisation approach for non-linear evolution systems which take the form of Lax-pair isospectral deformations, the higher order asymptotics as t→±∞ (x/t∼O(1)) of the solution to the Cauchy problem for the modified non-linear Schrödinger equation, i∂tu+ 1 2 ∂ x u+ |u|u+is∂x(|u|u) = 0, s ∈ R>0, which is a model for non-linear pulse propagation in optical fibres in the subpicosecond time scale, are obtained: also derived are analogous results for two gauge-equivalent nonlinear evolution equations; in particular, the derivative non-linear Schrödinger equation, i∂tq+∂ 2 xq−i∂x(|q|q)=0. AMS subject classifications. 35Q15, 35Q55, 58F07, 78A60 PACS. 02.30.Jr, 42.81.Dp, 42.65.Tg, 02.30.Mv Abbreviated title. Higher Order Asymptotics of the MNLSE
منابع مشابه
Leading Order Temporal Asymptotics of the Modified Non-Linear Schrödinger Equation: Solitonless Sector∗
Using the matrix Riemann-Hilbert (RH) factorisation approach for non-linear evolution equations (NLEEs) integrable in the sense of the inverse scattering method (ISM), we obtain, in the solitonless sector, the leading order asymptotics as t → ±∞ of the solution to the Cauchy initial-value problem for the modified non-linear Schrödinger equation (MNLSE), i∂tu + 1 2 ∂ xu + |u|2u + is∂x(|u|u) = 0,...
متن کاملLong-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...
متن کاملComparison of The LBM With the Modified Local Crank-Nicolson Method Solution of Transient Two-Dimensional Non-Linear Burgers Equation
Burgers equation is a simplified form of the Navier-Stokes equation that represents the non-linear features of it. In this paper, the transient two-dimensional non-linear Burgers equation is solved using the Lattice Boltzmann Method (LBM). The results are compared with the Modified Local Crank-Nicolson method (MLCN) and exact solutions. The LBM has been emerged as a new numerical method for sol...
متن کاملLong-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
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...
متن کاملAsymptotics of Solutions to the Modified Nonlinear Schrödinger Equation: Solitons on a Non-Vanishing Continuous Background
Using the matrix Riemann-Hilbert factorization approach for nonlinear evolution systems which take the form of Lax-pair isospectral deformations and whose corresponding Lax operators contain both discrete and continuous spectra, the leading-order asymptotics as t → ±∞ of the solution to the Cauchy problem for the modified nonlinear Schrödinger equation, i∂tu+ 1 2 ∂ xu+ |u|2u+ is∂x(|u|u) = 0, s ...
متن کامل