Solutions and connections of nonlocal derivative nonlinear Schrödinger equations

Shi, Ying; Shen, Shoufeng; Zhao, Song‐lin

doi:10.1007/s11071-018-4627-x

Cited by 40 publications

(22 citation statements)

References 34 publications

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“…Such a method has proved effectively in presenting solutions to many nonlocal integrable equations, cf. [24,25,[35][36][37][38].…”

Section: Introductionmentioning

confidence: 99%

Solutions to integrable space-time shifted nonlocal equations

Liu,

Wang,

Zhang

2021

Preprint

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In this paper we present a reduction technique based on bilinearization and double Wronskians (or double Casoratians) to obtain explicit multi-soliton solutions for the integrable space-time shifted nonlocal equations introduced very recently by Ablowitz and Musslimani in [Phys. Lett. A, 2021]. Examples include the space-time shifted nonlocal nonlinear Schrödinger and modified Kortewegde Vries hierarchies and the semi-discrete nonlinear Schrödinger equation. It is shown that these nonlocal integrable equations with or without space-time shift(s) reduction share same distributions of eigenvalues but the space-time shift(s) brings new constraints to phase terms in solutions.

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“…Such a method has proved effectively in presenting solutions to many nonlocal integrable equations, cf. [24,25,[35][36][37][38].…”

Section: Introductionmentioning

confidence: 99%

Solutions to integrable space-time shifted nonlocal equations

Liu,

Wang,

Zhang

2021

Preprint

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“…We have derived the classical and nonlocal GP equation (13) from the second order nonisospectral AKNS system (9) that allows both classical and nonlocal reductions. Double Wronskian solutions of the general GP equation ( 13) are obtained by employing a reduction technique that has been proved efficient in generating solutions for both classical and nonlocal systems [26,34,[40][41][42]. We mainly analyzed dynamics of the obtained solutions of the classical GP equation (2).…”

Section: Discussionmentioning

confidence: 99%

“…If B N and H N are corresponding Jordan form as in (42), φ takes the form (52) but where θ 1 and ρ 1 are given by (54).…”

Section: Case By Casementioning

confidence: 99%

New dynamics of the classical and nonlocal Gross-Pitaevskii equation with a parabolic potential

Liu,

Wu,

Zhang

2020

Preprint

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Solutions of the classical and nonlocal Gross-Pitaevskii (GP) equation with a parabolic potential and a gain term are derived by using a second order nonisospectral Ablowitz-Kaup-Newell-Segur system and reduction technique of double Wronskians. Solutions of the classical GP equation show typical space-time localized characteristics. An interesting dynamics, solitons carrying an oscillating wave, are found with mathematical analysis and illustrations. Solutions of some nonlocal cases are also illustrated.

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“…The first example is the nonlocal NLS equation, which was introduced by Ablowitz and Musslimani from the particular reductions of the Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy [7]. Since their seminal works, hierarchies of nonlocal integrable equations and their exact solutions were proposed and studied [8][9][10][11][12][13][14][15][16][17][18][19][20][21]. Very recently, Ablowitz and Mussliman considered a new nonlocal reduction of the AKNS hierarchy called space-shifted nonlocal reduction [22], and proposed the following space-shifted nonlocal NLS equation:…”

Section: Introductionmentioning

confidence: 99%

Bright–dark solitons in the space-shifted nonlocal coupled nonlinear Schrödinger equation

Ren

Rao

2022

Nonlinear Dyn

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Multiple bright-dark soliton solutions in terms of determinants for the space-shifted nonlocal coupled nonlinear Schrödinger (CNLS) equation are constructed by using the bilinear (Kadomtsev-Petviashvili) KP hierarchy reduction method. It is found that the bright-dark two-soliton only occur elastic collisions. Upon their amplitudes, the bright two solitons only admit one pattern whose amplitude are equal, and the dark two solitons have three different non-degenerated patterns and two different degenerated patterns. The bright-dark four-soliton is the superposition of the two-soliton pairs and can generated bound-state solitons. The multiple double-pole bright-dark soliton solutions are generated through the long wave limit of the obtained bright-dark soliton solutions, and their collision dynamics are also investigated.

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Solutions and connections of nonlocal derivative nonlinear Schrödinger equations

Cited by 40 publications

References 34 publications

Solutions to integrable space-time shifted nonlocal equations

Solutions to integrable space-time shifted nonlocal equations

New dynamics of the classical and nonlocal Gross-Pitaevskii equation with a parabolic potential

Bright–dark solitons in the space-shifted nonlocal coupled nonlinear Schrödinger equation

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