Long-time asymptotics for the Fokas–Lenells equation with decaying initial value problem: Without solitons

Jian, Xu; Fan, Engui

doi:10.1016/j.jde.2015.02.046

Cited by 190 publications

(76 citation statements)

References 20 publications

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“…This method has been used to study rigorously the long−time asymptotics of a wide variety of integral systems, such as the mKdV equation [12] and the non-focusing NLS equation [11],the sine-Gordon equation [10], the modified Schrödinger equation [20,21], the KdV equation [15], the Cammasa−Holm equation [9], Fokas-Lenells equation [29], derivative NLS equation [31], short pulse equation [28,30], Sine-Gordon equation [16], Kundu-Eckhaus Equation [33].…”

Section: Introductionmentioning

confidence: 99%

Long-Time Asymptotics for the Nonlocal MKdV Equation*

He¹,

Fan²,

Jian³

2019

Commun. Theor. Phys.

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In this paper, we study the Cauchy problem with decaying initial data for the nonlocal modified Korteweg-de Vries equation (nonlocal mKdV)which can be viewed as a generalization of the local classical mKdV equation. We first formulate the Riemann-Hilbert problem associated with the Cauchy problem of the nonlocal mKdV equation. Then we apply the Deift-Zhou nonlinear steepest-descent method to analyze the long-time asymptotics for the solution of the nonlocal mKdV equation. In contrast with the classical mKdV equation, we find some new and different results on long-time asymptotics for the nonlocal mKdV equation and some additional assumptions about the scattering data are made in our main results.

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Section: Introductionmentioning

confidence: 99%

Long-Time Asymptotics for the Nonlocal MKdV Equation*

He¹,

Fan²,

Jian³

2019

Commun. Theor. Phys.

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“…In addition, Yang proposed inverse scattering transform based on RH problem. Furthermore, the RH method can also be used to study the initial boundary value problem of integrable nonlinear equations and the asymptotic behavior of solutions . It is well known that the nonlinear Schrödinger equation can be used to describe physical phenomena in deep water, optics, and plasma physics.…”

Section: Introductionmentioning

confidence: 99%

“…Furthermore, the RH method can also be used to study the initial boundary value problem of integrable nonlinear equations and the asymptotic behavior of solutions. [11][12][13][14][15][16][17][18][19][20][21] It is well known that the nonlinear Schrödinger equation can be used to describe physical phenomena in deep water, optics, and plasma physics. However, due to the existence of higher-order effects, such as the third-order dispersion (TOD), the self-steepening (SS), and the stimulated Raman scattering (SRS) effects, some higher-order nonlinear Schrödinger equations, such as two components and three components, have been proposed.…”

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confidence: 99%

The N‐coupled higher‐order nonlinear Schrödinger equation: Riemann‐Hilbert problem and multi‐soliton solutions

Yang

Tian

Peng

et al. 2019

Math Methods in App Sciences

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In this work, the Riemann‐Hilbert (RH) problem of the N‐coupled high‐order nonlinear Schrödinger (N‐CHNLS) equations is studied carefully, which controls the propagation of N fields with all high‐order effects such as high‐order dispersion, self‐steepening effect, and Raman scattering in optical fiber. The spectral analysis of the Lax pair associated with a (2N+1)×(2N+1) matrix spectral problem for the N‐CHNLS equations is firstly carried out, from which a kind of RH problem is structured. Then a series of multi‐soliton solutions including breather, bright, and dark solutions for the N‐CHNLS equations can be formulated by the RH problem with the reflection‐less case. In addition, with N=4 taken as an example, the propagation behavior of these solutions and their interactions are presented by selecting appropriate parameters with some graphics.

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“…The nonlinear steepest descent method was first introduced in 1993 by Deift and Zhou [8], it turn out to be very successful for analysing the long-time asymptotics of initial-value problems for a large range of nonlinear integrable evolution equations in a rigorous and transparent form. Numerous new significant results about the asymptotics theory of initialvalue problems for different completely integrable nonlinear equations were obtained based on the analysis of the corresponding Riemann-Hilbert (RH) problems [3,[5][6][7]18,22,23]. After that, Fokas announced a new unified approach [11,12] to construct the matrix RH problems for the analysis of initial-boundary value (IBV) problems for linear and nonlinear integrable systems.…”

Section: Introductionmentioning

confidence: 99%

Long-time asymptotics for the Hirota equation on the half-line

2018

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We consider the Hirota equation on the quarter plane with the initial and boundary values belonging to the Schwartz space. The goal of this paper is to study the long-time behavior of the solution of this initial-boundary value problem based on the asymptotic analysis of an associated matrix Riemann-Hilbert problem. 0 1 , J 2 (x, t, k) = (J 1 J −1 4 J 3 )(x, t, k) = 1 − r(k)e −tΦ(k) −r(k)e tΦ(k) 1 + r(k)r(k) ,

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Long-time asymptotics for the Fokas–Lenells equation with decaying initial value problem: Without solitons

Cited by 190 publications

References 20 publications

Long-Time Asymptotics for the Nonlocal MKdV Equation*

Long-Time Asymptotics for the Nonlocal MKdV Equation*

The N‐coupled higher‐order nonlinear Schrödinger equation: Riemann‐Hilbert problem and multi‐soliton solutions

Long-time asymptotics for the Hirota equation on the half-line

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