Long‐Time Asymptotics for the Focusing Nonlinear Schrödinger Equation with Nonzero Boundary Conditions at Infinity and Asymptotic Stage of Modulational Instability

Biondini, Gino; Mantzavinos, Dionyssios

doi:10.1002/cpa.21701

Cited by 109 publications

(164 citation statements)

References 81 publications

(162 reference statements)

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“…Finally, one can write m (4) in the form

m^{false(4 false)} false(x, t; z false) = m^{e r r} false(x, t; z false) m^{m o d} false(x, t; z false),

where m mod ( x , t ; z ) solves the model problem:

m^{m o d +} false(x, t; z false) = m^{m o d -} false(x, t; z false) J^{m o d}, z \in γ \cup \overset{γ}{true}

with constant jump matrix

J^{m o d} = (\begin{array}{ll} 0 & \frac{2}{{\overset{q}{true}}_{-}} \\ - \frac{{\overset{q}{true}}_{-}}{2} & 0 \end{array}),

and

m^{e r r} false(x, t; z false) = I + O false(t^{- \frac{1}{2}} false) .

The last estimate can be justified by considering the parametrix associated with the RH problem for m (4) ( x , t ; z ), see. () The error of order O ( t −1/2 ) comes from the contribution of the jump near z − .…”

Section: The Long‐time Asymptoticsmentioning

confidence: 99%

“…In this section, we will use the approach proposed in Biondini and Mantzavinos to formulate the main RH problem, which allows us to give a representation of the solution for the Equation .…”

Section: The Riemann‐hilbert Problemmentioning

confidence: 99%

“…In this section, we compute the long‐time asymptotic behavior of the solution q ( x , t ) of the GI‐type derivative NLS Equation , as obtained by Equation by using the Deift‐Zhou nonlinear steepest descent method to perform the asymptotic analysis of oscillating RH problems. () The key point of this method is the choice of contour deformations, which depends crucially on the sign structure of the quantity Re(i θ ). We note that

θ false(ξ; z false) = \sqrt{z^{2} + \frac{1}{4} q_{0}^{4}} false(ξ - 2 z false) .

Thus, we get

\frac{normald θ false(ξ; z false)}{normald z} = \frac{- 4 z^{2} + ξ z - \frac{1}{2} q_{0}^{4}}{λ},

which implies that θ has two stationary points

z_{\pm} = \frac{1}{8} ((), ξ \pm \sqrt{ξ^{2} - 8 q_{0}^{2}}) .

Letting z = z 1 +i z 2 , we have

θ^{2} (ξ; z) = (z_{1}^{2} - z_{2}^{2} + \frac{1}{4} q_{0}^{4} + 2 i z_{1} z_{2}) (4 z_{1}^{2} - 4 z_{2}^{2} + ξ^{2} - 4 ξ z_{1} + i (8 z_{1} z_{2} - 4 ξ z_{2})) .

Hence, according to

{Im θ (ξ; z) = 0} = {Im θ^{2} (ξ; z) = 0} \cap

…”

Section: The Long‐time Asymptoticsmentioning

confidence: 99%

“…This implies that it is not possible anymore to use the previous factorizations and deformations to lift the contours off the real z ‐axis in such a way that the corresponding jump matrices remain bounded as t → ∞ . Therefore, developing and extending the ideas used in,() we introduce a new g ‐function g ( z ) appropriate for the region under consideration, which has a new real stationary point

z_{0} \in R^{-}

.…”

Section: The Long‐time Asymptoticsmentioning

confidence: 99%

“…Our present work was motivated by the long‐time asymptotic analysis for the focusing NLS equation developed in Biondini and Mantzavinos . Our goal here is to compute the long‐time asymptotics for the GI‐type derivative NLS Equation with the initial value condition satisfied .…”

Section: Introductionmentioning

confidence: 99%

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The Gerdjikov‐Ivanov–type derivative nonlinear Schrödinger equation: Long‐time dynamics of nonzero boundary conditions

Guo

Liu

2019

Math Methods in App Sciences

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We consider the Gerdjikov‐Ivanov–type derivative nonlinear Schrödinger equation iqt+qxx−iq2q¯x+12|qfalse|4−q04q=0 on the line. The initial value q(x,0) is given and satisfies the symmetric, nonzero boundary conditions at infinity, that is, q(x,0)→q± as x→±∞, and |q±|=q0>0. The goal of this paper is to study the asymptotic behavior of the solution of this initial value problem as t→∞. The main tool is the asymptotic analysis of an associated matrix Riemann‐Hilbert problem by using the steepest descent method and the so‐called g‐function mechanism. We show that the solution q(x,t) of this initial value problem has a different asymptotic behavior in different regions of the xt‐plane. In the regions x<−22q02t and x>22q02t, the solution takes the form of a plane wave. In the region −22q02t

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“…Finally, one can write m (4) in the form

m^{false(4 false)} false(x, t; z false) = m^{e r r} false(x, t; z false) m^{m o d} false(x, t; z false),

where m mod ( x , t ; z ) solves the model problem:

m^{m o d +} false(x, t; z false) = m^{m o d -} false(x, t; z false) J^{m o d}, z \in γ \cup \overset{γ}{true}

with constant jump matrix

J^{m o d} = (\begin{array}{ll} 0 & \frac{2}{{\overset{q}{true}}_{-}} \\ - \frac{{\overset{q}{true}}_{-}}{2} & 0 \end{array}),

and

m^{e r r} false(x, t; z false) = I + O false(t^{- \frac{1}{2}} false) .

Section: The Long‐time Asymptoticsmentioning

confidence: 99%

Section: The Riemann‐hilbert Problemmentioning

confidence: 99%

θ false(ξ; z false) = \sqrt{z^{2} + \frac{1}{4} q_{0}^{4}} false(ξ - 2 z false) .

Thus, we get

\frac{normald θ false(ξ; z false)}{normald z} = \frac{- 4 z^{2} + ξ z - \frac{1}{2} q_{0}^{4}}{λ},

which implies that θ has two stationary points

z_{\pm} = \frac{1}{8} ((), ξ \pm \sqrt{ξ^{2} - 8 q_{0}^{2}}) .

Letting z = z 1 +i z 2 , we have

θ^{2} (ξ; z) = (z_{1}^{2} - z_{2}^{2} + \frac{1}{4} q_{0}^{4} + 2 i z_{1} z_{2}) (4 z_{1}^{2} - 4 z_{2}^{2} + ξ^{2} - 4 ξ z_{1} + i (8 z_{1} z_{2} - 4 ξ z_{2})) .

Hence, according to

{Im θ (ξ; z) = 0} = {Im θ^{2} (ξ; z) = 0} \cap

…”

Section: The Long‐time Asymptoticsmentioning

confidence: 99%

z_{0} \in R^{-}

.…”

Section: The Long‐time Asymptoticsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The Gerdjikov‐Ivanov–type derivative nonlinear Schrödinger equation: Long‐time dynamics of nonzero boundary conditions

Guo

Liu

2019

Math Methods in App Sciences

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A Robust Inverse Scattering Transform for the Focusing Nonlinear Schrödinger Equation

Bilman

Miller

2019

Comm Pure Appl Math

146

169

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We propose a modification of the standard inverse scattering transform for the focusing nonlinear Schrödinger equation (also other equations by natural generalization) formulated with nonzero boundary conditions at infinity. The purpose is to deal with arbitrary‐order poles and potentially severe spectral singularities in a simple and unified way. As an application, we use the modified transform to place the Peregrine solution and related higher‐order “rogue wave” solutions in an inverse‐scattering context for the first time. This allows one to directly study properties of these solutions such as their dynamical or structural stability, or their asymptotic behavior in the limit of high order. The modified transform method also allows rogue waves to be generated on top of other structures by elementary Darboux transformations rather than the generalized Darboux transformations in the literature or other related limit processes. © 2019 Wiley Periodicals, Inc.

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Rational solutions of multi‐component nonlinear Schrödinger equation and complex modified KdV equation

Wang

Erdélyi

2022

Math Methods in App Sciences

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In this paper, the crucial condition to achieve rational solutions of the multi‐component nonlinear Schrödinger equation is proposed by introducing two nilpotent Lax matrices. Taking the series multisections of the vector eigenfunction as a set of fundamental eigenfunctions, an explicit formula of the nth‐order rational solution is obtained by the degenerate Darboux transformation, which is used to generate some new patterns of rogue waves. A conjecture about the degree of the nth‐order rogue waves is summarized. This conjecture also holds for rogue waves of the multi‐component complex modified Korteweg–de Vries equation. Finally, the semi‐rational solutions of the Manakov system are discussed.

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Long‐Time Asymptotics for the Focusing Nonlinear Schrödinger Equation with Nonzero Boundary Conditions at Infinity and Asymptotic Stage of Modulational Instability

Cited by 109 publications

References 81 publications

The Gerdjikov‐Ivanov–type derivative nonlinear Schrödinger equation: Long‐time dynamics of nonzero boundary conditions

The Gerdjikov‐Ivanov–type derivative nonlinear Schrödinger equation: Long‐time dynamics of nonzero boundary conditions

A Robust Inverse Scattering Transform for the Focusing Nonlinear Schrödinger Equation

Rational solutions of multi‐component nonlinear Schrödinger equation and complex modified KdV equation

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