Higher order asymptotics of the modified non-linear schrödinger equation

Abstract

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,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∂ t q+∂ 2 x q−i∂ x (|q| 2 q) = 0.

Asymptotics of Solutions to the Modified Nonlinear Schrödinger Equation: Solitons on a Nonvanishing Continuous Background

“…Up to the leading (O(t −1 )) term, the asymptotic expansion was proved in [7]. The O(t −1 ) term constitutes the leading-order contribution from the firstorder stationary phase point at λ = 0: the complete proof of this asymptotic expansion can be found in [22].…”

“…(21),(22), and (24),||G(·;λ 0 )|| L ∞ (C\∪ ℵ∈{0,±λ 0 } N (ℵ;ǫ 0 );M 2 (C)) < ∞, G(λ; ·) ∈ S(R >M ;M 2 (C)), G(λ;λ 0 ) ∼ O C(λ 0 ) λas λ → ∞ with C(λ 0 ) ∈ S(R >M ;M 2 (C)), and satisfies the following involutions, χ c (−λ) = σ 3 χ c (λ)σ 3 and χ c (λ) = σ 1 χ c (λ)σ 1 .…”

Higher order asymptotics for the Hirota equation via Deift–Zhou higher order theory

“…The result (2) (2) into the Hirota equation (1) as the methodology in [12] (for details, see [24]). The calculation is complicated and trivial, we don't explicitly deduce here.…”

“…The aim of this paper is to establish the full asymptotic expansion for the above solutions in the so-called similarity region of such solutions using the high order asymptotic method due to Deift and Zhou [23] (see also [24] …”

Long-Time Asymptotic Behavior for the Discrete Defocusing mKdV Equation

“…Since Deift and Zhou developed nonlinear steepest descent method in 1993 [1], it has been successfully applied to analyze the long-time asymptotic behavior of a wide variety of continuous integrable systems, such as the mKdV equation, the NLS equation, the sine-Gordon equation, the KdV equation, and the Cammasa-Holm equation [2][3][4][5][6]. However, there still has been little work on the long-time behavior of the discrete integrable systems, except for the Toda lattice and discrete NLS equation [7,8].…”