Rational Solitons in the Parity-Time-Symmetric Nonlocal Nonlinear Schrödinger Model

Li, Min; Xu, Tao; Meng, De-Xin

doi:10.7566/jpsj.85.124001

Cited by 107 publications

(75 citation statements)

References 53 publications

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“…By the Hirota's bilinear method with the perturbation expansion, the periodic line wave solutions defined in (35) with functions f and g are given as…”

Section: The Period and Line Breather Solutions Of The Nonlocal Dsiimentioning

confidence: 99%

“…Next, we proceed to analyze typical dynamics of rogue waves constructed through the polynomials defined in (35) and (41). The fundamental rogue waves in nonlocal DSII equation can be obtained by taking parameters in (41)…”

Section: Rational Solutions Of the Nonlocal Dsii Equationmentioning

confidence: 99%

“…In this subsection, we take a long wave limit of the periodic solutions in (37) partially, and keep the other part of them still in exponential functions, thus semirational solutions expressed in a combination of rational and exponential functions are generated through (35). These semirational solutions demonstrate typical dynamics of rogue waves on a background of periodic line waves.…”

Section: Semirational Solutions Of the Nonlocal Dsii Equationmentioning

confidence: 99%

“…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%

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Rational and Semirational Solutions of the Nonlocal Davey–Stewartson Equations

Rao¹,

Cheng

2017

Stud Appl Math

183

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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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“…By the Hirota's bilinear method with the perturbation expansion, the periodic line wave solutions defined in (35) with functions f and g are given as…”

Section: The Period and Line Breather Solutions Of The Nonlocal Dsiimentioning

confidence: 99%

Section: Rational Solutions Of the Nonlocal Dsii Equationmentioning

confidence: 99%

Section: Semirational Solutions Of the Nonlocal Dsii Equationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Rational and Semirational Solutions of the Nonlocal Davey–Stewartson Equations

Rao¹,

Cheng

2017

Stud Appl Math

183

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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%

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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Elliptic‐ and hyperbolic‐function solutions of the nonlocal reverse‐time and reverse‐space‐time nonlinear Schrödinger equations

Zhang

et al. 2022

Math Methods in App Sciences

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In this paper, we obtain the stationary elliptic-and hyperbolic-function solutions of the nonlocal reverse-time and reverse-space-time nonlinear Schrödinger (NLS) equations based on their connection with the standard Weierstrass elliptic equation. The reverse-time NLS equation possesses the bounded dn-, cn-, sn-, sech-, and tanh-function solutions. Of special interest, the tanh-function solution can display both the dark-and antidark-soliton profiles. The reverse-space-time NLS equation admits the general Jacobian elliptic-function solutions (which are exponentially growing at one infinity or display the periodical oscillation in x), the bounded dn-and cn-function solutions, as well as the K-shifted dn-and snfunction solutions. In addition, the hyperbolic-function solutions may exhibit an exponential growth behavior at one infinity, or show the gray/bright-soliton profiles.

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Rational Solitons in the Parity-Time-Symmetric Nonlocal Nonlinear Schrödinger Model

Cited by 107 publications

References 53 publications

Rational and Semirational Solutions of the Nonlocal Davey–Stewartson Equations

Rational and Semirational Solutions of the Nonlocal Davey–Stewartson Equations

Nonintegrable spatial discrete nonlocal nonlinear schrödinger equation

Elliptic‐ and hyperbolic‐function solutions of the nonlocal reverse‐time and reverse‐space‐time nonlinear Schrödinger equations

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