On symmetry preserving and symmetry broken bright, dark and antidark soliton solutions of nonlocal nonlinear Schrödinger equation

Priya, N. Vishnu; Senthilvelan, M.; Rangarajan, Govindan; Lakshmanan, M.

doi:10.1016/j.physleta.2018.10.011

“…Nonlocal integrable systems are very much important relative to physical and mathematical viewpoints because these equations admit different types of unique solutions [22][23][24][25][26][27][28][29][30]. Different types of soliton solutions of nonlocal reverse space, reverse time, and reverse space-time NLS-type equations have been investigated in recent years [31][32][33][34][35].…”

Section: Introductionmentioning

confidence: 99%

On multi-hump solutions of reverse space-time nonlocal nonlinear Schrödinger equation

Hanif

¹

,

Saleem

²

2023

Phys. Scr.

1

0

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In this article multi-soliton solutions of reverse space-time nonlocal nonlinear Schrödinger (NLS) equation have been constructed. Darboux transformation is applied to the associated linear eigenvalue problem for the generalized NLS equation and we obtain a determinant formula for multi-soliton solutions. Under suitable reduction conditions and appropriate choice of spectral parameters, the generalized expression of first-order nontrivial solution gives some novel solutions such as double-hump and flat-top soliton solutions for reverse space-time nonlocal NLS equation. The dynamics and interaction of double-hump soliton solutions are studied in detailand it is indicated that these solutions undergo collisions without any energy redistribution. For higher-order double-hump solutions, the relative velocities of solitons play a crucial role to have humps and also induce nonlinear interference in the collision zone. The dynamics of individual decaying and growing unstable and stable double-humps as well as their interactions are explained and illustrated.

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“…In 2013, Ablowitz and Musslimani first proposed the continuous PT-symmetry nonlocal nonlinear Schrödinger (NNLS) equation from a new symmetry reduction of the well-known AKNS system, which is an integrable system admitting the Lax pair and an infinite number of conservation laws, and has been solved by the inverse scattering transform method. [15] The PT-symmetric NNLS equation, which can keep lossless propagation due to the balanced gain and loss, [16] admits remarkable new characteristics that are not observed by comparison with its classical standard NLS equation and has received great attention, see Refs. [15][16][17][18][19][20][21].…”

Section: Introductionmentioning

confidence: 99%

“…[15] The PT-symmetric NNLS equation, which can keep lossless propagation due to the balanced gain and loss, [16] admits remarkable new characteristics that are not observed by comparison with its classical standard NLS equation and has received great attention, see Refs. [15][16][17][18][19][20][21]. Moreover, in Refs.…”

Section: Introductionmentioning

confidence: 99%

“…Some methods for solving local equations have been extended to nonlocal equations such as the inverse scattering transform method, [15,24,30] Hirota's bilinear method, [25,26,31,32,[42][43][44] the function expansion method, [27] the Riemann-Hilbert method, [28] and the DT method. [9,12,16,29,33,[35][36][37][38][39][40][41][45][46][47][48][49][50][51] Among them, DT is a simple algebraic technique to generate a family of infinite nontrivial solutions of a linear equation from a known trivial solution which is widely used in soliton theory. [9,45,46] It needs to be noted that the N-fold DT technique is a very effective method to derive N-soliton solutions compared to iterative DT because there is no iterative process.…”

Section: Introductionmentioning

confidence: 99%

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Soliton interactions and asymptotic state analysis in a discrete nonlocal nonlinear self-dual network equation of reverse-space type*

Yuan¹,

Wen²

2021

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We propose a reverse-space nonlocal nonlinear self-dual network equation under special symmetry reduction, which may have potential applications in electric circuits. Nonlocal infinitely many conservation laws are constructed based on its Lax pair. Nonlocal discrete generalized (m, N – m)-fold Darboux transformation is extended and applied to solve this system. As an application of the method, we obtain multi-soliton solutions in zero seed background via the nonlocal discrete N-fold Darboux transformation and rational solutions from nonzero-seed background via the nonlocal discrete generalized (1, N – 1)-fold Darboux transformation, respectively. By using the asymptotic and graphic analysis, structures of one-, two-, three- and four-soliton solutions are shown and discussed graphically. We find that single component field in this nonlocal system displays unstable soliton structure whereas the combined potential terms exhibit stable soliton structures. It is shown that the soliton structures are quite different between discrete local and nonlocal systems. Results given in this paper may be helpful for understanding the electrical signals propagation.

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“…Here instead we follow Yang's suggestion [15] and set ψ 2 (x, t) = ψ 1 (−x, t) in equation (1.3). The NNLSE, its variants and soliton solutions, have been studied in a variety of physical contexts [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30]. Specifically, the NNLSE finds applications in the context of self-induced potentials in classical optics, coupled waveguides and photonic lattices.…”

Section: Introductionmentioning

confidence: 99%

Stability of trapped solutions of a nonlinear Schrödinger equation with a nonlocal nonlinear self-interaction potential

Charalampidis¹,

Cooper²,

Khare³

et al. 2021

J. Phys. A: Math. Theor.

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This work focuses on the study of the stability of trapped soliton-like solutions of a (1 + 1)-dimensional nonlinear Schrödinger equation (NLSE) in a nonlocal, nonlinear, self-interaction potential of the form [ | ψ ( x , t ) | 2 + | ψ ( − x , t ) | 2 ] κ where κ is an arbitrary nonlinearity parameter. Although the system with κ = 1 (i.e. fully integrable case) was first reported by Yang (2018 Phys. Rev. E 98 042202), in the present work, we extend this model to the one in which κ is arbitrary. This allows us to compare the stability properties of the now trapped solutions to previously found solutions of the more usual NLSE with κ ≠ 1 which are moving soliton solutions. We show that there is a simple, one-component, nonlocal Lagrangian and corresponding action governing the dynamics of the system. Using a collective coordinate method derived from the action as well as assuming the validity of Derrick’s theorem, we find that these trapped solutions are stable for 0 < κ < 2 and unstable when κ > 2. At the critical value of κ, i.e. κ = 2, the solution can either collapse or blowup linearly in time when q 0 = 0, where q 0 is the center of the initial density ρ(x, t = 0) = ψ ⋆ ψ of the solution. For q 0 ≠ 0 the displaced solution collapses. When κ > 2 initial small displacements from the origin also lead to collapse of the wave function. This phenomenon is not seen in the usual NLSE.

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On symmetry preserving and symmetry broken bright, dark and antidark soliton solutions of nonlocal nonlinear Schrödinger equation

Cited by 25 publications

References 14 publications

On multi-hump solutions of reverse space-time nonlocal nonlinear Schrödinger equation

On multi-hump solutions of reverse space-time nonlocal nonlinear Schrödinger equation

Soliton interactions and asymptotic state analysis in a discrete nonlocal nonlinear self-dual network equation of reverse-space type*

Stability of trapped solutions of a nonlinear Schrödinger equation with a nonlocal nonlinear self-interaction potential

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