arXiv subjects

P. S. Vinayagam

Publications and source records attributed to P. S. Vinayagam.

16 recordsLinked to original sources

arXiv2026

PINN-Based Framework for Soliton Solutions of Gross Pitaevskii and Nonlinear Schrodinger Equations

This study presents a data-driven framework for solving nonlinear wave equations, specifically the Gross-Pitaevskii equation (GPE) and the single-component nonlinear Schrodinger equation (NLSE), using Physics-Informed Neural Networks (PINNs). The approach integrates physical constraints directly into the neural network's loss function, enabling efficient training without requiring labelled data. We implement a PINN-based framework for solitons that models a variety of localized wave structures across both equations. Predicted solutions are compared with exact analytical results and show strong agreement with low error. The method effectively captures soliton profiles in both the GPE and NLSE. The accuracy and flexibility of the framework suggest its usefulness for studying nonlinear differential equations relevant to Bose--Einstein condensates and nonlinear optics.

nlin.PS↗

arXiv2024

Robust Dynamics of Rogue waves, Breathers and Mixed Bound State Solutions in Spin-Orbit and Rabi Coupled Condensates

In this paper, based on an integrable model governed by four system parameters, namely, spin-orbit coupling and Rabi coupling which are constants while the other two parameters, namely the harmonic trap and scattering lengths which are time dependent, we investigate the spin orbit-Rabi coupled condensates governed by the two coupled Gross-Pitaevskii(GP) equation. Employing Darboux transformation approach with nontrivial seed solution, we generate rogue waves, breathers, mixed rogue-dark-bright and classical dark-bright solitons. While the addition of SOC contributes to rapid oscillations in the amplitude of the rogue waves, Rabi coupling results in the appearance of stripes in the temporal direction. When the transient trap is switched on, these stripes which remain stable within the confining region, begin to overlap exponentially in the expulsive domain and eventually shrink leading to instability. We have shown that the instability arising in the rogue waves in a transient trap can be completely overcome by manipulating the scattering length through Feshbach resonance. In the case of breathers, the Rabi coupling introduces temporal stripes with single and double mode peaks around the origin while under the influence of the transient trap, the breathers get compressed and tilted. On the other hand, the amplitude of breathers which stays constant between a maxima and minima despite oscillating with time undergoes rapid fluctuations in a given spatial domain under the influence of SOC. In the case of classical dark-bright solitons, we see the flipping of dark solitons to attain positive density much similar to bright solitons under the impact of Rabi coupling. One also witnesses a 45 degree shift in the trajectory of dark and bright solitons when the transient trap is switched on. The width of the solitons widen in the confining trap while they shrink in the expulsive domain.

nlin.PS↗

arXiv2024

Collisional Dynamics of Solitons and Pattern Formation in an Integrable Cross Coupled Nonlinear Schrodinger equation with constant background

We investigate the dynamics arising out of the propagation of light pulses with different polarizations through a condensate (referred to as a constant background field) with cross coupling described by a coupled nonlinear Schrodinger equation(NLSE) type equation. We then employ Gauge and Darboux transformation approach to bring out the rich dynamics arising out of the background field and cross coupling. The collisional dynamics of bright solitons is found to be inelastic. The constant background field is found to facilitate the periodic localization of light pulses during propagation. We have also unearthed breathers, bright-bright, bright-dark and dark-bright solitons of the coupled NLSE. While the amplitude of breathers oscillate with time as predicted, their maximum(or minimum) amplitude is found to remain a constant and the addition of cross coupling only contributes to the rapid fluctuations in its amplitude over a period of time. In addition, the reinforcement of cross coupling in the presence of constant wave field facilitates the interference of light pulses leading to interesting pattern formation among bright-bright, bright-dark and dark-bright solitons. The highlight of the results is that one obtains various localized excitations like breathers, bright and dark solitons by simply manipulating the amplitude of the constant wave field.

nlin.SI↗

arXiv2024

New classes of solutions in the Coupled PT Symmetric Nonlocal Nonlinear Schrodinger Equations with Four Wave Mixing

We investigate generalized nonlocal coupled nonlinear Schroedinger equation containing Self-Phase Modulation, Cross-Phase Modulation and Four-Wave Mixing involving nonlocal interaction. By means of Darboux transformation, we obtained a family of exact breathers and solitons including the Peregrine soliton, Kuznetsov-Ma breather, Akhmediev breather along with all kinds of soliton-soliton and breather-soliton interactions. We analyze and emphasize the impact of the four-wave mixing on the nature and interaction of the solutions. We found that the presence of Four-Wave Mixing converts a two-soliton solution into an Akhmediev breather. In particular, the inclusion of Four-Wave Mixing results in the generation of a new solution which is spatially and temporally periodic called "Soliton (Breather) lattice".

nlin.PS↗

arXiv2018

Enhanced mobility of discrete solitons in anisotropic two-dimensional waveguide arrays with modulated separations

We consider two-dimensional waveguide arrays with anisotropic coupling coefficients. We show using numerical and variational calculations that four stationary soliton types exist: Site-Centered, Bond-Centered, Hybrid-X and Hybrid-Y. For the isotropic case the last two modes become identical and equivalent to the known hybrid soliton. With a variational calculation using a gaussian trial function and six variational parameters corresponding to the soliton's position, width, and velocity components, the four stationary soliton types are reproduced and their equilibrium widths are accounted for accurately for a wide range of anisotropy ratios. We obtained using the variational calculation the Peierls-Nabarro potential and barrier heights for the four soliton types and different anisotropy ratios. We have also obtained a phase diagram showing regions of soliton stability against collapse and subregions of mobility in terms of the initial kick-in speed and anisotropy ratio. The phase diagram shows that 2D solitons become highly mobile for anisotropy ratios larger than some critical values that depend on the initial kick-in speed. This fact was then exploited to design tracks within the 2D waveguide array along which the soliton can be accelerated and routed. We have calculated the actual waveguide separations needed to realist the proposed guided trajectories of 2D solitons.

nlin.PS↗

arXiv2017

Collisional Dynamics of Solitons in the Coupled PT symmetric Nonlocal nonlinear Schrodinger equations

We investigate the focusing coupled PT-symmetric nonlocal nonlinear Schrodinger equation employing Darboux transformation approach. We find a family of exact solutions including pairs of Bright-Bright, Dark-Dark and Bright-Dark solitons in addition to solitary waves. We show that one can convert bright bound state onto a dark bound state in a two-soliton solution by selectively fine tuning the amplitude dependent parameter. We also show that the energy in each mode remains conserved unlike the celebrated Manakov model. We also characterize the behaviour of the soliton solutions in detail. We emphasize that the above phenomenon occurs due to the nonlocality of the model.

nlin.SI↗

arXiv2017

Bright soliton dynamics in Spin Orbit-Rabi coupled Bose-Einstein condensates

We investigate the dynamics of a spin-orbit (SO) coupled BECs in a time dependent harmonic trap and show the dynamical system to be completely integrable by constructing the Lax pair. We then employ gauge transformation approach to witness the rapid oscillations of the condensates for a relatively smaller value of SO coupling in a time independent harmonic trap compared to their counterparts in a transient trap. Keeping track of the evolution of the condensates in a transient trap during its transition from confining to expulsive trap, we notice that they collapse in the expulsive trap. We further show that one can manipulate the scattering length through Feshbach resonance to stretch the lifetime of the confining trap and revive the condensate. Considering a SO coupled state as the initial state, the numerical simulation indicates that the reinforcement of Rabi coupling on SO coupled BECs generates the striped phase of the bright solitons and does not impact the stability of the condensates despite destroying the integrability of the dynamical system.

cond-mat.quant-gas↗

arXiv2016

Manipulation of light in a generalized coupled Nonlinear Schrodinger equation

We investigate a generalized coupled nonlinear Schrodinger (GCNLS) equation containing Self-Phase Modulation (SPM), Cross-Phase Modulation (XPM) and Four Wave Mixing (FWM) describing the propagation of electromagnetic radiation through an optical fibre and generate the associated Lax-pair. We then construct bright solitons employing gauge transformation approach. The collisional dynamics of bright solitons indicates that it is not only possible to manipulate intensity (energy) between the two modes (optical beams), but also within a given mode unlike the Manakov model which does not have the same freedom. The freedom to manipulate intensity (energy) in a given mode or between two modes arises due to a suitable combination of SPM, XPM and FWM.While SPM and XPM are controlled by an arbitrary real parameter each, FWM is governed by two arbitrary complex parameters. The above model may have wider ramifications in nonlinear optics and Bose-Einstein Condensates (BECs).

cond-mat.quant-gas↗

arXiv2015

Engineering Bright Solitons to Enhance the Stability of Two-Component Bose-Einstein Condensates

We consider a system of coupled Gross-Pitaevskii (GP) equations describing a binary quasi-one-dimensional Bose-Einstein condensate (BEC) with intrinsic time-dependent attractive interactions, placed in a time-dependent expulsive parabolic potential, in a special case when the system is integrable (a deformed Manakov's system). Since the nonlinearity in the integrable system which represents binary attractive interactions exponentially decays with time, solitons are also subject to decay. Nevertheless, it is shown that the robustness of bright solitons can be enhanced in this system, making their respective lifetime longer, by matching the time dependence of the interaction strength (adjusted with the help of the Feshbach-resonance management) to the time modulation of the strength of the parabolic potential. The analytical results, and their stability, are corroborated by numerical simulations. In particular, we demonstrate that the addition of random noise does not impact the stability of the solitons.

cond-mat.quant-gas↗

arXiv2015

Persistent bright solitons in sign-indefinite coupled nonlinear Schrodinger equations with a time-dependent harmonic trap

We introduce a model based on a system of coupled nonlinear Schrodinger (NLS) equations with opposite signs infront of the kinetic and gradient terms in the two equations. It also includes time-dependent nonlinearity coefficients and a parabolic expulsive potential. By means of a gauge transformation, we demonstrate that, with a special choice of the time dependence of the trap, the system gives rise to persistent solitons. Exact single and two-soliton analytical solutions and their stability are corroborated by numerical simulations. In particular, the exact solutions exhibit inelastic collisions between solitons.

cond-mat.quant-gas↗

arXiv2015

Generalized gauge transformation approach to construct dark solitons of coupled nonlinear Schrodinger type equations

We harness the freedom in the celebrated gauge transformation approach to generate dark solitons of coupled nonlinear Schrödinger (NLS) type equations. The new approach which is purely algebraic could prove to be very useful, particularly in the construction of vector dark solitons in the fields of nonlinear optics, plasma physics and Bose-Einstein condensates. We have employed this algebraic method to coupled Gross- Pitaevskii (GP) and NLS equations and obtained dark solitons.

nlin.SI↗

arXiv2015

An Analytical Window into the World of Ultracold Atoms

We review the recent developments which had taken place in the domain of quasi one dimensional BECs from the viewpoint of integrability. To start with, we consider the dynamics of scalar BECs in a time independent harmonic trap and observe that the scattering length (SL) can be suitably manipulated either to compress the bright solitons to attain peak matter wave density without causing their explosion or to broaden the width of the condensates without diluting them. When the harmonic trap frequency becomes time dependent, we notice that one can stabilize the condensates in the confining domain while the density of the condensates continue to increase in the expulsive region. We also observe that the trap frequency and the temporal SL can be maneuvered to generate matter wave interference patterns indicating the coherent nature of the atoms in the condensates. We also notice that a small repulsive three body interaction when reinforced with attractive binary interaction can extend the region of stability of the condensates in the quasi-one dimensional regime. On the other hand, the investigation of two component BECs in a time dependent harmonic trap suggests that it is possible to switch matter wave energy from one mode to the other confirming the fact that vector BECs are long lived compared to scalar BECs. The Feshbach resonance management of vector BECs indicates that the two component BECs in a time dependent harmonic trap are more stable compared to the condensates in a time independent trap. The introduction of weak time dependent Rabi coupling rapidly compresses the bright solitons which however can be again stabilized through Feshbach resonance or by finetuning the Rabi coupling while the spatial coupling of vector BECs introduces a phase difference between the condensates which subsequently can be exploited to generate interference pattern in the bright or dark solitons.

cond-mat.quant-gas↗

arXiv2013

Spatiotemporal Binary Interaction and Designer quasi particle condensates

We introduce a new integrable model to investigate the dynamics of two component quasi particle condensates with spatio temporal interaction strengths. We derive the associated Lax-pair of the coupled GP equation and construct matter wave solitons. We show that the spatio temporal binary interaction strengths not only facilitate the stabilization of the condensates, but also enables one to fabricate condensates with desirable densities, geometries and properties leading to the so called "designer quasi particle condensates".

cond-mat.quant-gas↗

arXiv2013

Taming Rogue waves in Vector BECs

Using Gauge transformation method, we generate rogue waves for the two component Bose Einstein Condensates (BECs) governed by the symmetric coupled Gross-Pitaevskii (GP) equations and study their dynamics. We also suggest a mechanism to tame the rogue waves either by manipulating the scattering length through Feshbach resonance or the trapping frequency, a new phenomenon not witnessed in the domain of BEC, we believe that these results may have wider ramifications in the management of rogons.

cond-mat.other↗

arXiv2013

Rotation of the Trajectories of Bright soliton and Realignment of Intensity Distribution in the Coupled Nonlinear Schrodinger Equation

We revisit the collisional dynamics of bright solitons in the coupled Nonlinear Schrodinger equation. We observe that apart from the intensity redistribution in the interaction of bright solitons, one also witnesses a rotation of the trajectories of bright solitons . The angle of rotation can be varied by suitably manipulating the Self-Phase Modulation (SPM) or Cross Phase Modulation (XPM) parameters.The rotation of the trajectories of the bright solitons arises due to the excess energy that is injected into the dynamical system through SPM or XPM. This extra energy not only contributes to the rotation of the trajectories, but also to the realignment of intensity distribution between the two modes. We also notice that the angular separation between the bright solitons can also manouvred suitably. The above results which exclude quantum superposition for the field vectors may have wider ramifications in nonlinear optics, Bose-Einstein condensates, Left Handed (LH) and Right Handed (RH) meta materials.

cond-mat.other↗