Nth-order smooth positon and breather-positon solutions of a generalized nonlinear Schrödinger equation

Priya, N. Vishnu; Monisha, S. Irudaya; Senthilvelan, M.; Rangarajan, Govindan

doi:10.1140/epjp/s13360-022-02861-x

Cited by 11 publications

(5 citation statements)

References 44 publications

(72 reference statements)

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“…where α 1 is the eigenvalue of the spectral parameter, α * 1 is the complex conjugate of α 1 , and η and τ are provided in Equation ( 4). The above-mentioned second-order smooth positon solution (8) can be obtained from the material presented in [58] by considering ν = 0. Substituting this solution into (7) along with the suitable form of R(t), we obtain the second-order matter-wave smooth positon solution of (1).…”

Section: Second-order Matter-wave Smooth Positonsmentioning

confidence: 99%

“…It is important to note that the third-order smooth positon solution can be deduced from the research in [58] by setting ν = 0. By substituting this third-order smooth positon solution of the ccNLS Equation ( 13), along with the appropriate form of the modulated parameter R(t), into Equation (7), we delve into an analysis of the underlying characteristics of the GP Equation (1).…”

Section: Third-order Matter-wave Smooth Positonsmentioning

confidence: 99%

“…Following these advancements, endeavors have been made to construct smooth positon solutions for various nonlinear evolution equations, including the focusing mKdV equation [51], the complex mKdV equation [52], the derivative NLS equation [53,54], the NLS-Maxwell-Bloch equation [55], the higher-order Chen-Lee-Liu equation [56], and the Gerdjikov-Ivanov equation [57]. More recently, smooth positons and breather positons have been derived for the generalized NLS equation with higher-order nonlinearity along with higher-order solutions for an extended NLS equation featuring cubic and quartic nonlinearity [58,59]. Inspired by these advancements in the field of positons, our research aims to construct smooth positon solutions within the GP equation, incorporating time-varying nonlinearity and trap potentials.…”

Section: Introductionmentioning

confidence: 99%

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Controlling Matter-Wave Smooth Positons in Bose–Einstein Condensates

Manikandan,

Serikbayev,

Vijayasree

et al. 2023

Symmetry

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In this investigation, we explore the existence and intriguing features of matter-wave smooth positons in a non-autonomous one-dimensional Bose–Einstein condensate (BEC) system with attractive interatomic interactions. We focus on the Gross–Pitaevskii (GP) equation/nonlinear Schrödinger-type equation with time-modulated nonlinearity and trap potential, which govern nonlinear wave propagation in the BEC. Our approach involves constructing second- and third-order matter-wave smooth positons using a similarity transformation technique. We also identify the constraints on the time-modulated system parameters that give rise to these nonlinear localized profiles. This study considers three distinct forms of modulated nonlinearities: (i) kink-like, (ii) localized or sech-like, and (iii) periodic. By varying the parameters associated with the nonlinearity strengths, we observe a rich variety of captivating behaviors in the matter-wave smooth positon profiles. These behaviors include stretching, curving, oscillating, breathing, collapsing, amplification, and suppression. Our comprehensive studies shed light on the intricate density profile of matter-wave smooth positons in BECs, providing valuable insights into their controllable behavior and characteristics in the presence of time-modulated nonlinearity and trap potential effects.

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Section: Second-order Matter-wave Smooth Positonsmentioning

confidence: 99%

Section: Third-order Matter-wave Smooth Positonsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Controlling Matter-Wave Smooth Positons in Bose–Einstein Condensates

Manikandan,

Serikbayev,

Vijayasree

et al. 2023

Symmetry

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“…The singular and smooth positon solutions have been constructed for a class of integrable equations [21,22,23,24,25,26,27,28,29,30,31,32]. Breather-positons (b-p) are equal amplitude breathers (localized periodic waves on constant background) and they travel with equal speed [33,34,35,36,37,38].…”

Section: Introductionmentioning

confidence: 99%

Higher order smooth positon and breather positon solutions of an extended nonlinear Schrödinger equation with the cubic and quartic nonlinearity

Monisha¹,

Priya²,

Senthilvelan³

et al. 2022

Preprint

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We construct certain higher order smooth positon and breather positon solutions of an extended nonlinear Schrödinger equation with the cubic and quartic nonlinearity. We utilize the generalized Darboux transformation method to construct the aforementioned solutions. The three well-known equations, namely nonlinear Schrödinger equation, Hirota equation, and generalized nonlinear Schrödinger equation, are sub-cases of the considered extended nonlinear Schrödinger equation. The solutions which we construct are more general. We analyze how the positon and breather positon solutions of the constituent equations get modified by the higher order nonlinear and dispersion terms. Our results show that the width and direction of the smooth positon and breather-positon solutions are highly sensitive to higher-order effects. Further, we carryout an asymptotic analysis to predict the behaviour of positons. We observe that during collision positons exhibit a time-dependent phase shift. We also present the exact expression of time-dependent phase shift of positons. Finally, we show that this time-dependent phase shift is directly proportional to the higher order nonlinear and dispersion parameters.

show abstract

“…The singular and smooth positon solutions have been constructed for a class of integrable equations [21][22][23][24][25][26][27][28][29][30][31][32]. Breather-positons (b-p) are equal amplitude breathers (localized periodic waves on constant background) and they travel with equal speed [33][34][35][36][37][38].…”

Section: Introductionmentioning

confidence: 99%