Dynamics of higher-order rational solitons for the nonlocal nonlinear Schrödinger equation with the self-induced parity-time-symmetric potential

Wen, Xiao-Yong; Yan, Zhenya; Yang, Yunqing

doi:10.1063/1.4954767

Cited by 143 publications

(85 citation statements)

References 53 publications

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“…Different kinds of localized solutions have also been reported in the literature such as periodic and hyperbolic soliton solutions [9], dark and anti-dark soliton solutions [10], breather solutions [11] and rational soliton solutions [12] The two parameter family of breathing one soliton solution which is reported in [4] can be extracted from (11) by restricting the wave numbers k 1 andk 1 to be pure imaginary, that is k 1 = i2η 1 andk 1 = i2η 1 and considering …”

Section: One Bright Soliton Solutionmentioning

confidence: 99%

Nonstandard bilinearization ofPT-invariant nonlocal nonlinear Schrödinger equation: Bright soliton solutions

2017

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In this paper, we succeed to bilinearize the PT -invariant nonlocal nonlinear Schrödinger (NNLS) equation through a nonstandard procedure and present more general bright soliton solutions. We achieve this by bilinearizing both the NNLS equation and its associated parity transformed complex conjugate equation in a novel way. The obtained one and two soliton solutions are invariant under combined space and time reversal transformations and are more general than the known ones. Further, by considering the two-soliton solution we bring out certain novel interaction properties of the PT -invariant multi-soliton solutions.

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Section: One Bright Soliton Solutionmentioning

confidence: 99%

Nonstandard bilinearization ofPT-invariant nonlocal nonlinear Schrödinger equation: Bright soliton solutions

2017

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“…Note that two types of rogue wave solutions for the nonlocal DSI equation were obtained formally in [22], and Darboux transformations and global explicit solutions for the x-nonlocal DSI equation were given in [23]. Besides, a lot of work was done after that for these equations and other nonlocal equations [19,20,[23][24][25][26][27][28][29][30][31][32][33][34][35].…”

Section: Introductionmentioning

confidence: 99%

Rational and Semirational Solutions of the Nonlocal Davey–Stewartson Equations

Rao¹,

Cheng

2017

Stud Appl Math

159

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In this paper, the partially party-time (PT ) symmetric nonlocal Davey-Stewartson (DS) equations with respect to x is called x-nonlocal DS equations, while a fully PT symmetric nonlocal DSII equation is called nonlocal DSII equation. Three kinds of solutions, namely, breather, rational, and semirational solutions for these nonlocal DS equations are derived by employing the bilinear method. For the x-nonlocal DS equations, the usual (2 + 1)-dimensional breathers are periodic in x direction and localized in y direction. Nonsingular rational solutions are lumps, and semirational solutions are composed of lumps, breathers, and periodic line waves. For the nonlocal DSII equation, line breathers are periodic in both x and y directions with parallels in profile, but localized in time. Nonsingular rational solutions are (2 + 1)-dimensional line rogue waves, which arise from a constant background and disappear into the same constant background, and this process only lasts for a short period of time. Semirational solutions describe interactions of line rogue waves and periodic line waves.

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“…where − N 2 ≤ n ≤ N 2 . We use a six-order self-adaptive Runge-Kutta method [45], to simulate the evolution of the Cauchy problems (34) and (35) by setting L = 2 √ 2π, ǫ = 0.1, a = 1 and µ = 2π L , respectively.…”

Section: Numerical Solution Of Cauchy Problem (9)mentioning

confidence: 99%

“…In this performance, we first would like to check that for evolution of the solution to the Cauchy problems, what can be obtained when the space step h changes for fixed initial data. Then, we discuss the influence of initial data on the solutions to the Cauchy problem for fixed step h. We also give a special observation for the difference among Cauchy problems (34), (35) and the following Cauchy problem of discrete NLS equation,…”

Section: Numerical Solution Of Cauchy Problem (9)mentioning

confidence: 99%

Nonintegrable spatial discrete nonlocal nonlinear schrödinger equation

Zhu

2019

Chaos: An Interdisciplinary Journal of Nonlinear Science

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Integrable and nonintegrable discrete nonlinear Schrödinger equations (NLS) are significant models to describe many phenomena in physics. Recently, Ablowitz and Musslimani introduced a class of reverse space, reverse time and reverse space-time nonlocal integrable equations, including nonlocal NLS, nonlocal sine-Gordon equation and nonlocal Davey-Stewartson equation etc. And, the integrable nonlocal discrete NLS has been exactly solved by inverse scattering transform. In this paper, we study a nonintegrable discrete nonlocal NLS which is direct discretization version of the reverse space nonlocal NLS. By applying discrete Fourier transform and modified Neumann iteration, we present its stationary solutions numerically. The linear stability of the stationary solutions is examined. Finally, we study the Cauchy problem for nonlocal NLS equation numerically and find some different and new properties on the numerical solutions comparing with the numerical solutions of the Cauchy problem for NLS equation.problem of this equation is solved under special initial values by inverse scattering transform (IST). But for general initial values, the solutions of Cauchy problem of nonlocal NLS equation can not be obtained by using IST. In our study, we investigate a nonintegrable space discrete nonlocal NLS equation. We obtain its stationary solitary wave solutions by discrete Fourier transform. The linear stability of these stationary solitary wave solutions are examined. Then, numerical simulations for Cauchy problem of nonlocal NLS equation with periodic boundary conditions are performed, in which we use the integrable scheme and the nonintegrable scheme as a spatial discrete model of nonlocal NLS equation. We find some different and new properties on the numerical solutions for Cauchy problem of nonlocal NLS equation comparing with the numerical solutions of the Cauchy problem for the NLS equation.

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Dynamics of higher-order rational solitons for the nonlocal nonlinear Schrödinger equation with the self-induced parity-time-symmetric potential

Cited by 143 publications

References 53 publications

Nonstandard bilinearization ofPT-invariant nonlocal nonlinear Schrödinger equation: Bright soliton solutions

Nonstandard bilinearization ofPT-invariant nonlocal nonlinear Schrödinger equation: Bright soliton solutions

Rational and Semirational Solutions of the Nonlocal Davey–Stewartson Equations

Nonintegrable spatial discrete nonlocal nonlinear schrödinger equation

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