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

Guo, Boling; Liu, Nan

doi:10.1002/mma.5698

Cited by 10 publications

(9 citation statements)

References 35 publications

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“…The GIE was decomposed into two systems of solvable ordinary differential equations and the solutions were derived in terms of the Riemann theta functions (Dai and Fan 2004). The GI-type derivative nonlinear Schrödinger equation, with fifth degree nonlinearity was studied for initial value problem (Guo and Liu 2019). The DT was used to derive a variety of soliton solutions of the nonlocal nonlocal nonlinear GIE (Li et al 2021).…”

Section: Introductionmentioning

confidence: 99%

Approximate-analytic optical soliton solutions of a modified-Gerdjikov–Ivanov equation: modulation instability

Abdel‐Gawad

2023

Opt Quant Electron

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The Gerdjikov–Ivanov equation (GIE) occupied a remarkable area of research in the literature. In the present work, a modified GIE (MGIE) is considered which is new and was not studied in the literature. Also, the modified-unified method (MUM) is used to obtain approximate analytic solutions (AASs) of MGIE. Up to our knowledge, no AASs for non-integrable complex field equation were found up to now. Thus the AASs found, here, are novel. The UM addresses finding the exact solutions to integrable equations. In this sense as no exact solution for MGIE exists, consequently, it is not integrable. So, here, approximate analytic optical soliton solutions are invoked. The UM stands for expressing the solution of nonlinear evolution equations in polynomial and rational forms in an auxiliary function (AF) with an appropriate auxiliary equation. For finding exact solutions by the UM, the coefficients of the AF, with all powers, are set equal to zero, For a non-integrable equation, only approximate solutions are affordable. In this case, we are led to utilizing the MUM. Herein, non-zero coefficients (residue terms (RTs)) are considered as errors, which are space and time-independent. It is worth mentioning that, this is in contrast to the errors found by the different numerical methods, where they are space and time-dependent. Further, in the present case, the maximum error is controlled via an adequate choice of the parameters in the RTs. These solutions are displayed in graphs. Breather soliton, chirped soliton and M-shape soliton, among others, are observed. Furthermore, modulation instability (MI) is studied and it is found MI triggers when the coefficient of the nonlinear dispersion exceeds a critical value.

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Section: Introductionmentioning

confidence: 99%

Approximate-analytic optical soliton solutions of a modified-Gerdjikov–Ivanov equation: modulation instability

Abdel‐Gawad

2023

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“…This system is an important and integrable model in mathematics and physics, which was proposed by Gerdjikov and Ivanov 11 . Since its discovery, many researches have taken this system as research object and then lots of results have been obtained 12–21 …”

Section: Introductionmentioning

confidence: 99%

Inverse scattering transform for the two‐component Gerdjikov–Ivanov equation with nonzero boundary conditions: Dark‐dark solitons

Zhu

2023

Stud Appl Math

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The two-component Gerdjikov-Ivanov equation with nonzero boundary conditions is studied by the inverse scattering transform. A fundamental set of analytic eigenfunctions is obtained with the aid of the associated adjoint problem. Three symmetry conditions are discussed to curb the scattering data. The behavior of the Jost functions and the scattering matrix at the branch points is discussed. The inverse scattering problem is formulated by a matrix Riemann-Hilbert problem. The trace formula in terms of the scattering data and the so-called asymptotic phase difference for the potential are obtained. The solitons classification is described in detail. When the discrete eigenvalues lie on the circle, the dark-dark soliton is obtained for the first time in this work. And the discrete eigenvalues off the circle generate the dark-bright, bright-bright, breather-breather, M(-type)-W(-type) solitons, and their interactions.

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“…Biondini and his cooperators have studied the soliton solutions and the long-time asymptotics for the focusing NLS equation with NZBCs in [42] and [43], respectively. After that, long-time asymptotics of the focusing Kundu-Eckhaus equation with NZBCs were studied in [44], long-time dynamics of the Gerdjikov-Ivanov type derivative nonlinear Schrödinger equation with NZBCs were studied in [45], long-time dynamics of the Hirota equation with NZBCs were studied in [46], and long-time dynamics of the modified Landau-Lifshitz equation with NZBCs were studied in [47]. Besides, the long-time asymptotic behavior of nonlocal integrable NLS solutions with NZBCs were studied in [48].…”

Section: Introductionmentioning

confidence: 99%

Asymptotic stage of modulation instability for a fourth-order dispersive nonlinear Schrödinger equation with nonzero boundary conditions

Peng¹,

Chen²

2022

Preprint

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In this work, we consider the long-time asymptotics for the Cauchy problem of a fourth-order dispersive nonlinear Schrödinger equation with nonzero boundary conditions at infinity. Firstly, in order to construct the basic Riemann-Hilbert problem associated with nonzero boundary conditions, we analysis direct scattering problem. Then we deform the corresponding matrix Riemann-Hilbert problem to explicitly solving models via using the nonlinear steepest descent method and employing the g-function mechanism to eliminate the exponential growths of the jump matrices. Finally, we obtain the asymptotic stage of modulation instability for the fourth-order dispersive nonlinear Schrödinger equation.

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

Cited by 10 publications

References 35 publications

Approximate-analytic optical soliton solutions of a modified-Gerdjikov–Ivanov equation: modulation instability

Approximate-analytic optical soliton solutions of a modified-Gerdjikov–Ivanov equation: modulation instability

Inverse scattering transform for the two‐component Gerdjikov–Ivanov equation with nonzero boundary conditions: Dark‐dark solitons

Asymptotic stage of modulation instability for a fourth-order dispersive nonlinear Schrödinger equation with nonzero boundary conditions

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