Nonstandard bilinearization of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:mi mathvariant="script">PT</mml:mi></mml:math>-invariant nonlocal nonlinear Schrödinger equation: Bright soliton solutions

Stalin, S.; Senthilvelan, M.; Lakshmanan, M.

doi:10.1016/j.physleta.2017.05.026

Cited by 33 publications

(36 citation statements)

References 12 publications

(42 reference statements)

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“…For the case S-symmetric nonlocal NLS equation (1.8) we are at variance with Stalin et al's results [19] (see Remark 2 and Remark 3 in Sections 4.1 and 4.2 respectively). They claim that they produce soliton solutions of the nonlocal NLS equation (S-symmetric) but it seems that they are solving the NLS system of equations (1.1) and (1.2) rather than solving nonlocal NLS equation (1.8), because they ignore the constraint equations satisfied by the parameters of the one-soliton solutions.…”

Section: Introductioncontrasting

confidence: 89%

“…Ablowitz and Musslimani have found many other nonlocal integrable equations such as nonlocal modified Korteweg-de Vries equation, nonlocal Davey-Stewartson equation, nonlocal sine-Gordon equation, and nonlocal (2 + 1)-dimensional three-wave interaction equations [2]- [4]. After the work of Ablowitz and Musslimani there is an increasing interest in obtaining the nonlocal reductions of systems of integrable equations and their properties [5]- [19].…”

Section: Introductionmentioning

confidence: 99%

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Nonlocal nonlinear Schrödinger equations and their soliton solutions

Gürses

Pekcan

2018

Journal of Mathematical Physics

149

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We study standard and nonlocal nonlinear Schrödinger (NLS) equations obtained from the coupled NLS system of equations (Ablowitz-Kaup-Newell-Segur (AKNS) equations) by using standard and nonlocal reductions respectively. By using the Hirota bilinear method we first find soliton solutions of the coupled NLS system of equations then using the reduction formulas we find the soliton solutions of the standard and nonlocal NLS equations. We give examples for particular values of the parameters and plot the function |q(t, x)| 2 for the standard and nonlocal NLS equations.

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Section: Introductioncontrasting

confidence: 89%

Section: Introductionmentioning

confidence: 99%

Nonlocal nonlinear Schrödinger equations and their soliton solutions

Gürses

Pekcan

2018

Journal of Mathematical Physics

149

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“…In this way we have obtained a set of three bilinear equations For vanishing deformation parameter δ → 0 the equations (4.2) and (4.4) constitute the bilinearisation for the nonlocal NLSE. As our equation differ from the ones recently proposed for that model in [35] we will comment below on some solutions related to that specific case. The local equations presented in the previous section are obtained forf * → f ,g → g, h → g * as in this case the two equations in (4.4) combine into the one equation (3.4).…”

Section: Hirota's Direct Methodsmentioning

confidence: 72%

Integrable nonlocal Hirota equations

Cen

Correa

Fring

2019

Journal of Mathematical Physics

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We construct several new integrable systems corresponding to nonlocal versions of the Hirota equation, which is a particular example of higher order nonlinear Schrödinger equations. The integrability of the new models is established by providing their explicit forms of Lax pairs or zero curvature conditions. The two compatibility equations arising in this construction are found to be related to each other either by a parity transformation P, by a time reversal T or a PT -transformation possibly combined with a conjugation. We construct explicit multi-soliton solutions for these models by employing Hirota's direct method as well as Darboux-Crum transformations. The nonlocal nature of these models allows for a modification of these solution procedures as the new systems also possess new types of solutions with different parameter dependence and different qualitative behaviour. The multi-soliton solutions are of varied type, being for instance nonlocal in space, nonlocal in time of time crystal type, regular with local structures either in time/space or of rogues wave type. Integrable nonlocal Hirota equationswith constants ε, α, β, γ ∈ R. Besides the NLSE for ε = 0, four cases are known to be integrable. When the ratio of the constants are taken to be α : β : γ = 0 : 1 : 1 or α : β : γ = 0 : 1 : 0 one obtains the derivative NLSE of type I [8] and II [9], respectively, which are in fact related to each other by a dependent variable transformation [10]. For α : β : γ = 1 : 6 : 3 one obtains the Sasa-Satsuma equation [11] and for α : β : γ = 1 : 6 : 0 the Hirota equation [12]. Variations of the latter are the subject of this manuscript.We notice that the additional term in the HNLSE when compared to the NLSE, i.e. (1.1) for ε = 0, shares the same PT -symmetry with the NLSE, as it is invariant with respect to PT :Hence HNLSEs may also be viewed as PT -symmetric extensions of the NLSE. Similarly as for many other PT -symmetric nonlinear integrable systems [13], various other PT -symmetric generalizations have been proposed and investigated by adding terms to the original equation, e.g. [14,15,16]. A further option, that will be important here, was explored by Ablowitz and Musslimani [17,18] who identified a new class of nonlinear integrable systems closely related to the NLSE by exploiting a hitherto unexplored PTsymmetry present in the zero curvature condition. Exploring this option below for the Hirota equation will lead us to new integrable systems with nonlocal properties.Our manuscript is organized as follows: In section 2 we discuss the zero curvature condition or AKNS-equation for the new class of integrable systems. The solutions to these systems involve fields at different points in space or time and reduce in certain limits to the standard Hirota equation, so that we refer to them as nonlocal Hirota equations. The equations possess two types of solutions of qualitatively different behaviour and parameter dependence. We identify the origin for this novel feature within the context of Hirota's direct method as w...

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“…To explore general soliton solutions, we adopt the non-standard bilinearization procedure developed for the scalar NNLS equation [14]. Using this procedure, we bilinearize both the nonlocal Manakov equation and the following a pair of coupled equations that arise in the zero curvature condition [43], that is…”

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confidence: 99%

Degenerate soliton solutions and their dynamics in the nonlocal Manakov system: I symmetry preserving and symmetry breaking solutions

2018

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In this paper, we construct degenerate soliton solutions (which preserve PT -symmetry/break PT -symmetry) to the nonlocal Manakov system through a nonstandard bilinear procedure. Here by degenerate we mean the solitons that are present in both the modes which propagate with same velocity. The degenerate nonlocal soliton solution is constructed after briefly indicating the form of nondegenerate onesoliton solution. To derive these soliton solutions, we simultaneously solve the nonlocal Manakov equation and a pair of coupled equations that arise from the zero curvature condition. The later consideration yields general soliton solution which agrees with the solutions that are already reported in the literature under certain specific parametric choice. We also discuss the salient features associated with the obtained degenerate soliton solutions.Keywords nonlocal Manakov equation · Hirota's bilinear method · Soliton solutions 1 IntroductionIn the context of PT -symmetric classical optics [1,2], recently a nonlocal nonlinear Schrödinger (NNLS) equation, namely iq t (x, t) + q xx (x, t) + 2σq(x, t)q * (−x, t)q(x, t) = 0, σ = ±1.(1)

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

Cited by 33 publications

References 12 publications

Nonlocal nonlinear Schrödinger equations and their soliton solutions

Nonlocal nonlinear Schrödinger equations and their soliton solutions

Integrable nonlocal Hirota equations

Degenerate soliton solutions and their dynamics in the nonlocal Manakov system: I symmetry preserving and symmetry breaking solutions

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