General soliton solution to a nonlocal nonlinear Schrödinger equation with zero and nonzero boundary conditions

Feng, Bao‐Feng; Luo, Xu‐Dan; Ablowitz, Mark J.; Musslimani, Ziad H.

doi:10.1088/1361-6544/aae031

Cited by 163 publications

(74 citation statements)

References 92 publications

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“…By checking qualitative features of the evolution, we note that further decrease of ∆x and/or ∆t did not produce any conspicuous effect. The VAs (5), (7), and (14) are also integrated using the fourth-order Runge-Kutta method. The varying-in-time variables are then inserted back into the ansatz to obtain the spatial profile of the VA. 7…”

Section: Variational Methods For Soliton Collisionsmentioning

confidence: 99%

“…Various soliton solutions of the nonlocal NLS equation (1) have been obtained, exploiting the integrability of the equations, such as dark and antidark solitons [13,14], standing waves in terms of elliptic functions [15], and rational solutions both in the focusing and defocusing nonlinearity [16,17]. Note, however, that the study of soliton dynamics, especially interactions between solitons, has been done only using Darboux transformation [13,16,18,19] or via a combination of Hirota's bilinear method and the Kadomtsev-Petviashvili (KP) hierarchy [14], which are limited to specific combinations of parameter values and initial conditions. A general study of interactions of many bright, dark and antidark solitons has been presented recently, where again the same transformation method was used [20].…”

Section: Introductionmentioning

confidence: 99%

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Variational approximations of soliton dynamics in the Ablowitz-Musslimani nonlinear Schrödinger equation

2019

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We study the integrable nonlocal nonlinear Schrödinger equation proposed by Ablowitz and Musslimani, that is considered as a particular example of equations with parity-time (PT ) symmetric self-induced potential. We consider dynamics (including collisions) of moving solitons. Analytically we develop a collective coordinate approach based on variational methods and examine its applicability in the system. We show numerically that a single moving soliton can pass the origin and decay or be trapped at the origin and blows up at a finite time. Using a standard soliton ansatz, the variational approximation can capture the dynamics well, including the finite-time blow up, even though the ansatz is relatively far from the actual blowing-up soliton solution. In the case of two solitons moving towards each other, we show that there can be a mass transfer between them, in addition to wave scattering. We also demonstrate that defocusing nonlinearity can support bright solitons.

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Section: Variational Methods For Soliton Collisionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Variational approximations of soliton dynamics in the Ablowitz-Musslimani nonlinear Schrödinger equation

2019

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“…5.2 q(x, y, t) = kp(ε 1 x, ε 2 y, ε 3 t), ε 2 1 = ε 2 2 = ε 2 3 = 1, k is a real constant When we apply this reduction to the AKNS(-3) system (11) and (12), it reduces consistently to the nonlocal complex AKNS(-3) equation,…”

Section: Q(x Y T) = Kp(x Y T) K Is a Real Constantmentioning

confidence: 99%

“…Let us take ε = 1. Hence one-soliton solutions of the AKNS(-3) system given by (11) and (12) and the AKNS(-4) system given by (18) and (19)…”

Section: Introductionmentioning

confidence: 99%

Special Smarandache curves in R31

GÜRSES¹,

BEKTAS²,

Yuce³

et al. 2016

Communications Faculty of Science University of Ankara Series A1Mathematics and Statistics

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“…Following the introduction of this equation, its properties have been extensively investigated [6,[8][9][10][11][12][13][14][15][16][17][18][19]. In addition, other nonlocal integrable equations have been reported [20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38].…”

Section: Introductionmentioning

confidence: 99%

Physically significant nonlocal nonlinear Schrödinger equation and its soliton solutions

Yang

2018

Phys. Rev. E

109

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A new integrable nonlocal nonlinear Schrödinger (NLS) equation with clear physical motivations is proposed. This equation is obtained from a special reduction of the Manakov system, and it describes Manakov solutions whose two components are related by a parity symmetry. Since the Manakov system governs wave propagation in a wide variety of physical systems, this new nonlocal equation has clear physical meanings. Solitons and multi-solitons in this nonlocal equation are also investigated in the framework of Riemann-Hilbert formulations. Surprisingly, symmetry relations of discrete scattering data for this equation are found to be very complicated, where constraints between eigenvectors in the scattering data depend on the number and locations of the underlying discrete eigenvalues in a very complex manner. As a consequence, general N -solitons are difficult to obtain in the Riemann-Hilbert framework. However, one-and two-solitons are derived, and their dynamics investigated. It is found that two-solitons are generally not a nonlinear superposition of one-solitons, and they exhibit interesting dynamics such as meandering and sudden position shifts. As a generalization, other integrable and physically meaningful nonlocal equations are also proposed, which include NLS equations of reverse-time and reverse-space-time types as well as nonlocal Manakov equations of reverse-space, reverse-time and reverse-space-time types.

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General soliton solution to a nonlocal nonlinear Schrödinger equation with zero and nonzero boundary conditions

Cited by 163 publications

References 92 publications

Variational approximations of soliton dynamics in the Ablowitz-Musslimani nonlinear Schrödinger equation

Variational approximations of soliton dynamics in the Ablowitz-Musslimani nonlinear Schrödinger equation

Special Smarandache curves in R31

Physically significant nonlocal nonlinear Schrödinger equation and its soliton solutions

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