Degenerate and non-degenerate solutions of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mi mathvariant="script">PT</mml:mi></mml:math>-symmetric nonlocal integrable discrete nonlinear Schrödinger equation

Hanif, Y.; Saleem, U.

doi:10.1016/j.physleta.2020.126834

Cited by 11 publications

(16 citation statements)

References 40 publications

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“…For detail study of degenerate and nondegenerate solitons solutions of local and nonlocal continuous and discrete NLS equation see references [44,45]. While to obtain second-order solution of nonlocal reverse spacetime NLS equation (2.7) we use reduction λ 4 = − λ 3 and λ 2 = − λ 1 in equation (4.14) along with…”

Section: Two-soliton Solutionmentioning

confidence: 99%

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

Hanif

Saleem

2023

Phys. Scr.

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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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Section: Two-soliton Solutionmentioning

confidence: 99%

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

Hanif

Saleem

2023

Phys. Scr.

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“…44 Meanwhile, wide classes of explicit solutions for both 𝜖 = 1 and 𝜖 = −1 cases were constructed via various analytical methods. [41][42][43][44][45][46] In the theory of integrable systems, the DT is a special gauge transformation that acts on the Lax pair associated to an integrable model. Although the DT of Equation (3) was studied in the previous literature, [40][41][42][43] there still exists the following deficiency: (i) Refs.…”

Section: Introductionmentioning

confidence: 99%

“…[41][42][43][44][45][46] In the theory of integrable systems, the DT is a special gauge transformation that acts on the Lax pair associated to an integrable model. Although the DT of Equation (3) was studied in the previous literature, [40][41][42][43] there still exists the following deficiency: (i) Refs. [42,43] established the DT of Ablowitz-Ladik scattering problem without the potential constraint, so that some additional parameter conditions must be imposed on the obtained solutions to satisfy the equation; (ii) Ref.…”

Section: Introductionmentioning

confidence: 99%

“…Although the DT of Equation (3) was studied in the previous literature, [40][41][42][43] there still exists the following deficiency: (i) Refs. [42,43] established the DT of Ablowitz-Ladik scattering problem without the potential constraint, so that some additional parameter conditions must be imposed on the obtained solutions to satisfy the equation; (ii) Ref. [40] constructed the onefold DT of Equation (3) but did not provide a rigorous proof.…”

Section: Introductionmentioning

confidence: 99%

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N‐fold Darboux transformation of the discrete PT‐symmetric nonlinear Schrödinger equation and new soliton solutions over the nonzero background

Xu,

An,

et al. 2024

Stud Appl Math

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For the discrete PT‐symmetric nonlinear Schrödinger (dPTNLS) equation, this paper gives a rigorous proof of the N‐fold Darboux transformation (DT) and especially verifies the PT‐symmetric relation between transformed potentials in the Lax pair. Meanwhile, some determinant identities are developed in completing the proof. When the tanh‐function solution is directly selected as a seed for the focusing case, the onefold DT yields a three‐soliton solution that exhibits the solitonic behavior with a wide range of parameter regimes. Moreover, it is shown that the solution contains three pairs of asymptotic solitons, and that each asymptotic soliton can display both the dark and antidark soliton profiles or vanish as . It indicates that the focusing dPTNLS equation admits a rich variety of soliton interactions over the nonzero background, behaving like those in the continuous counterpart.

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“…The integrability [10,11], the Cauchy problem [12], the inverse scattering transform [13], and exact solutions, such as breathers, periodic, and rational solutions [14], general rogue waves [15], multiple bright soliton [16], higher order rational solutions [17], and N-soliton solutions [18] of (1) have been derived. Moreover, other nonlocal integrable systems have also been investigated like nonlocal modified Korteweg-de Vries equation [19,20], nonlocal KP equation [21], nonlocal vector nonlinear nonlinear Schrödinger equation [22,23], nonlocal discrete nonlinear Schrödinger equation [24][25][26], nonlocal Davey-Stewartson I equation [27], etc. Although much advance has been made in nonlocal systems, there are very few studies on nonlocal equations with variable coefficients.…”

Section: Introductionmentioning

confidence: 99%

The Soliton Solutions and Long-Time Asymptotic Analysis for an Integrable Variable Coefficient Nonlocal Nonlinear Schrödinger Equation

Chen

Xin

Zhang

2021

Advances in Mathematical Physics

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An integrable variable coefficient nonlocal nonlinear Schrödinger equation (NNLS) is studied; by employing the Hirota’s bilinear method, the bilinear form is obtained, and the N -soliton solutions are constructed. In addition, some singular solutions and period solutions of the addressed equation with specific coefficients are shown. Finally, under certain conditions, the asymptotic behavior of the two-soliton solution is analyzed to prove that the collision of the two-soliton is elastic.

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Degenerate and non-degenerate solutions ofPT-symmetric nonlocal integrable discrete nonlinear Schrödinger equation

Cited by 11 publications

References 40 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

N‐fold Darboux transformation of the discrete PT‐symmetric nonlinear Schrödinger equation and new soliton solutions over the nonzero background

The Soliton Solutions and Long-Time Asymptotic Analysis for an Integrable Variable Coefficient Nonlocal Nonlinear Schrödinger Equation

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