The inverse scattering transform for the focusing nonlinear Schrödinger equation with asymmetric boundary conditions

Demontis, Francesco; Prinari, B.; Mee, Cornelis van der; Vitale, F.

doi:10.1063/1.4898768

Cited by 59 publications

(46 citation statements)

References 31 publications

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“…Simply requiring

Q_{n} - Q_{\pm} \in ℓ_{\pm}^{1} = false{f_{n} : Z \mapsto C : {0false \sum}_{j = \mp \infty}^{n} false| f_{j} false| < \infty false}

is not sufficient to guarantee continuity of the eigenfunctions at the branch points

ζ_{o}^{\pm} = i false(r \pm Q_{o} false)

. On the other hand, similarly to what happens in the continuous case, the stronger summability condition for the AL potential also ensures the eigenfunctions are continuous at the branch points. On the other hand, in general the scattering coefficients have simple poles at the branch points.…”

Section: Direct Scattering Problemmentioning

confidence: 98%

“…The inverse scattering transform (IST) as a tool to solve the initial‐value problem for the focusing NLS equations on the whole line (

- \infty < x < \infty

) has been known since the pioneer work by Zakharov and Shabat . In recent years, there has been an ongoing effort to extend the IST for potentials with a nonzero background (NZBG), and specifically to boundary conditions of the form:

q false(x, t false) \to q_{\pm} false(t false) = q_{o} e^{2 i σ q_{o}^{2} t + i θ_{\pm}} x \to \pm \infty .

The defocusing NLS with such nonzero boundary conditions was actually first considered in 1973, and the IST was subsequently investigated and generalized in various works . Many recent papers have dealt with the same problem for the focusing NLS equation, which has turned out to have a richer family of solutions, including breathers and rational solutions such as the celebrated Peregrine and higher order Peregrine solutions .…”

Section: Introductionmentioning

confidence: 99%

“…The defocusing NLS with such nonzero boundary conditions was actually first considered in 1973, 2 and the IST was subsequently investigated and generalized in various works. [3][4][5][6][7][8][9][10][11] Many recent papers have dealt with the same problem for the focusing NLS equation, which has turned out to have a richer family of solutions, including breathers and rational solutions such as the celebrated Peregrine and higher order Peregrine solutions. [12][13][14][15][16][17][18][19][20][21] These solutions are currently studied worldwide, in light of the fact that the nonlinear stage of modulation instability in the governing equation has been proposed as a mechanism for the formation of "rogue" waves, i.e., waves whose amplitude is significantly larger than the background.…”

Section: Introductionmentioning

confidence: 99%

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Inverse scattering transform for the defocusing Ablowitz–Ladik system with arbitrarily large nonzero background

Ortiz

Prinari

2019

Stud Appl Math

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In this work we develop the inverse scattering transform (IST) for the defocusing Ablowitz–Ladik (AL) equation with arbitrarily a large nonzero background at space infinity. The IST was developed in previous works under the assumption that the amplitude of the background Qo satisfies a “small norm” condition 01, and exhibits discrete rogue wave solutions, some of which are regular for all times. Here, we construct the IST for the defocusing AL with Qo>1, analyze the spectrum, and characterize the soliton and rational solutions from a spectral point of view. We formulate the direct and inverse problems by using a suitable uniformization variable, and pose the inverse problem as an RHP across a simple contour in the complex plane of the uniform variable. As a by‐product of the IST, we also obtain explicit soliton solutions, which are the discrete analog of the celebrated Kuznetsov–Ma, Akhmediev, Peregrine solutions, and which mimic the corresponding solutions for the focusing AL equation. Soliton solutions that are the analog of the dark soliton solutions of the defocusing AL equation in the case 0

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“…Simply requiring

Q_{n} - Q_{\pm} \in ℓ_{\pm}^{1} = false{f_{n} : Z \mapsto C : {0false \sum}_{j = \mp \infty}^{n} false| f_{j} false| < \infty false}

is not sufficient to guarantee continuity of the eigenfunctions at the branch points

ζ_{o}^{\pm} = i false(r \pm Q_{o} false)

Section: Direct Scattering Problemmentioning

confidence: 98%

“…The inverse scattering transform (IST) as a tool to solve the initial‐value problem for the focusing NLS equations on the whole line (

- \infty < x < \infty

q false(x, t false) \to q_{\pm} false(t false) = q_{o} e^{2 i σ q_{o}^{2} t + i θ_{\pm}} x \to \pm \infty .

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Inverse scattering transform for the defocusing Ablowitz–Ladik system with arbitrarily large nonzero background

Ortiz

Prinari

2019

Stud Appl Math

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“…The three-component defocusing NLS equation with nonzero boundary conditions was analyzed in Biondini et al 14 by the theory of IST. On the other hand, for the asymmetric nonzero boundary conditions (ie, when the limiting values of the solution at space infinities have different nonzero moduli), the IST for the focusing and defocusing NLS equation were formulated in Demontis et al 15 and Biondini et al, 16 respectively. Our present work was motivated by the long-time asymptotic analysis for the focusing NLS equation developed in Biondini and Mantzavinos.…”

Section: Introductionmentioning

confidence: 99%

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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“…Most research involves solutions vanishing as x → ±∞ [5,6,16,26,31,33]. Recently there has been much interest in solutions nonvanishing as x → ±∞ [8,10,11]. In this article we study the IST for the focusing NLS equation which is discrete in position and continuous in time, obtained by applying forward differencing to the focusing Zakharov-Shabat system…”

Section: Introductionmentioning

confidence: 99%

Inverse Scattering Transform for the Discrete Focusing Nonlinear Schrödinger Equation with Nonvanishing Boundary Conditions

Mee¹

2021

JNMP

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In this article we develop the direct and inverse scattering theory of the Ablowitz-Ladik system with potentials having limits of equal positive modulus at infinity. In particular, we introduce fundamental eigensolutions, Jost solutions, and scattering coefficients, and study their properties. We also discuss the discrete eigenvalues and the corresponding norming constants. We then go on to derive the left Marchenko equations whose solutions solve the inverse scattering problem. We specify the time evolution of the scattering data to solve the initial-value problem of the corresponding integrable discrete nonlinear Schrödinger equation. The one-soliton solution is also discussed.

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The inverse scattering transform for the focusing nonlinear Schrödinger equation with asymmetric boundary conditions

Cited by 59 publications

References 31 publications

Inverse scattering transform for the defocusing Ablowitz–Ladik system with arbitrarily large nonzero background

Inverse scattering transform for the defocusing Ablowitz–Ladik system with arbitrarily large nonzero background

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

Inverse Scattering Transform for the Discrete Focusing Nonlinear Schrödinger Equation with Nonvanishing Boundary Conditions

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