Nonparaxial elliptic waves and solitary waves in coupled nonlinear Helmholtz equations

Tamilselvan, K.; Kanna, T.; Khare, Avinash

doi:10.1016/j.cnsns.2016.02.040

Cited by 30 publications

(14 citation statements)

References 62 publications

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“…which is similar to the Lamé potential with λ being the eigenvalue of the Lamé equation [76,77], is the order of the Lamé polynomial and sn(x, m) denotes the Jacobian ellipticity sine function with elliptic modulus m, 0 ≤ m ≤ 1. In this work, we consider first order Lamé polynomials sn(x, m), cn(x, m) and dn(x, m) as eigenfunctions with eigenvalues (λ =) m + 1, 1 and m, respectively.…”

Section: (I) Periodic Potentials (Optical Lattice Potential)mentioning

confidence: 99%

Spatially modulated two- and three-component Rabi-coupled Gross–Pitaevskii systems

Kanna¹,

Sheela²,

Mareeswaran³

2019

J. Phys. A: Math. Theor.

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Vector rogue wave (RW) formation and their dynamics in Rabi coupled two-and three-species Bose-Einstein condensates with spatially varying dispersion and nonlinearity are studied. For this purpose, we obtain the RW solution of the two-and three-component inhomogeneous Gross-Pitaevskii (GP) systems with Rabi coupling by introducing suitable rotational and similarity transformations. Then, we investigate the effect of inhomogeneity (spatially varying dispersion, trapping potential and nonlinearity) on vector RWs for two different forms of potential strengths, namely periodic (optical lattice) with specific reference to hyperbolic type potentials and parabolic cylinder potentials. First, we show an interesting oscillating boomeronic behaviour of dark-bright solitons due to Rabi coupling in two component-condensate with constant nonlinearities. Then in the presence of inhomogeneity but in the absence of Rabi coupling we demonstrate creation of new daughter RWs co-existing with dark (bright) soliton part in first (second) component of the two-component GP system. Further, the hyperbolic modulation (sech type) of parameter along with Rabi effect leads to the formation of dromion (two-dimensional localized structure) trains even in the (1+1) dimensional two-component GP system, which is a striking feature of Rabi coupling with spatial modulation. Next, our study on three-component condensate, reveals the fact that the three RWs can be converted into broad based zero background RW appearing on top of a bright soliton by introducing spatial modulation only. Further, by including Rabi coupling we observe beating behaviour of solitons with internal oscillations mostly at the wings. Also, we show that by employing parabolic cylinder modulation with model parameter n, one can produce (n + 1) RWs. Our study reveals the fact that the spatially varying environment leads to the possibility of realization of two dimensional nonlinear structure in (1+1) dimensional setting.Spatially modulated two-and three-component Rabi-coupled Gross-Pitaevskii systems 2 features such RWs. Study of RWs has attracted significant attention not only in the more standard ocean-surface-dynamical problem [1,2], but also in other physical contexts. Indeed, there exists a vast amount of theoretical and experimental works on RWs in various fields ranging from optics [4, 6-9], Bose-Einstein condensates (BECs) [1,10], to plasmas [11], and atmospheric dynamics [12] (see also the recent short review [13]). Recently, high-order RWs [14] were excited successfully in a water wave tank. This suggests that a high-order analytic RW solution is of physical relevance and can be realized experimentally [14,15].These comprehensive theoretical and experimental studies on single component RWs, stimulated an increasing interest in investigating multi-component RWs. Rogue waves of the two-component Manakov system [16] have been studied extensively in the focusing regime [17,18] and in the defocusing regime [19][20][21]. The higher order RW solutions of the Manakov system ha...

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Section: (I) Periodic Potentials (Optical Lattice Potential)mentioning

confidence: 99%

Spatially modulated two- and three-component Rabi-coupled Gross–Pitaevskii systems

Kanna¹,

Sheela²,

Mareeswaran³

2019

J. Phys. A: Math. Theor.

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“…In recent years the multicomponent NLH system has also received considerable interest. Special solitary waves and elliptic waves for a multicomponent NLH system exhibiting distinct dynamical behaviours have been reported in [14].…”

Section: Introductionmentioning

confidence: 95%

Lie symmetry analysis and group invariant solutions of the nonlinear Helmholtz equation

Sakkaravarthi

Johnpillai²,

Devi

et al. 2018

Applied Mathematics and Computation

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We consider the nonlinear Helmholtz (NLH) equation describing the beam propagation in a planar waveguide with Kerr-like nonlinearity under non-paraxial approximation. By applying the Lie symmetry analysis, we determine the Lie point symmetries and the corresponding symmetry reductions in the form of ordinary differential equations (ODEs) with the help of the optimal systems of one-dimensional subalgebras. Our investigation reveals an important fact that in spite of the original NLH equation being non-integrable, its symmetry reductions are of Painlevé integrable. We study the resulting sets of nonlinear ODEs analytically either by constructing the integrals of motion using the modified Prelle-Singer method or by obtaining explicit travelling wave-like solutions including solitary and symbiotic solitary wave solutions. Also, we carry out a detailed numerical analysis of the reduced equations and obtain multi-peak nonlinear wave trains. As a special case of the NLH equation, we also make a comparison between the symmetries of the present NLH system and that of the standard nonlinear Schrödinger equation for which symmetries are long available in the literature.

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“…Exact and approximate solutions to these equations are obtained and are shown to exhibit quasi-soliton behavior based on propagation and collision studies. The elliptic and the hyperbolic solitary wave solutions to the coupled NLH system are obtained in [17] and effect of non-paraxiality on speed, pulse-width and amplitude of various solutions are explored. Coupled nonparaxial (2+1) dimensional nonlinear Schrödinger equation has been considered in [18] and bright-bright, dark-dark and bright-dark soliton solutions are obtained.…”

Section: Introductionmentioning

confidence: 99%

Coupled Helmholtz Equations : Chirped Solitary Waves

Saha,

Roy,

Khare

2021

Preprint

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We investigate the existence and stability properties of the chirped gray and anti-dark solitary waves within the framework of coupled cubic nonlinear Helmholtz equation in the presence of self steepening and self frequency shift. We show that for a particular combination of the self steepening and the self frequency shift, there is not only chirping but also chirp reversal. Specifically, the associated nontrivial phase has two intensity dependent terms, one varies as the reciprocal of the intensity while the other, which depends on non-Kerr nonlinearities, is directly proportional to the intensity. This causes chirp reversal across the solitary wave profile. A different combination of non-Kerr terms leads to chirping but no chirp reversal.The influence of nonparaxial parameter on physical quantities such as intensity, speed and pulse-width of the solitary waves is studied too. It is found that the speed of the solitary waves can be tuned by altering the nonparaxial parameter. Stable propagation of these nonparaxial solitary waves is achieved by an appropriate choice of parameters.

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Nonparaxial elliptic waves and solitary waves in coupled nonlinear Helmholtz equations

Cited by 30 publications

References 62 publications

Spatially modulated two- and three-component Rabi-coupled Gross–Pitaevskii systems

Spatially modulated two- and three-component Rabi-coupled Gross–Pitaevskii systems

Lie symmetry analysis and group invariant solutions of the nonlinear Helmholtz equation

Coupled Helmholtz Equations : Chirped Solitary Waves

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