arXiv subjects

Toan T. Huynh

Publications and source records attributed to Toan T. Huynh.

11 recordsLinked to original sources

arXiv2024

A Finite Difference Scheme for (2+1)D Cubic-Quintic Nonlinear Schrödinger Equations with Nonlinear Damping

Solitons of the purely cubic nonlinear Schrödinger equation in a space dimension of $n \geq 2$ suffer critical and supercritical collapses. These solitons can be stabilized in a cubic-quintic nonlinear medium. In this paper, we analyze the Crank-Nicolson finite difference scheme for the (2+1)D cubic-quintic nonlinear Schrödinger equation with cubic damping. We show that both the discrete solution, in the discrete $L^2$-norm, and discrete energy are bounded. By using appropriate settings and estimations, the existence and the uniqueness of the numerical solution are proved. In addition, the error estimations are established in terms of second order for both space and time in discrete $L^2$-norm and $H^1$-norm. Numerical simulations for the (2+1)D cubic-quintic nonlinear Schrödinger equation with cubic damping are conducted to validate the convergence.

math.NA↗

arXiv2024

Highly controllable stabilization and switching of multiple colliding soliton sequences with generic Ginzburg-Landau gain-loss

We investigate propagation of J soliton sequences in a nonlinear optical waveguide array with generic weak Ginzburg-Landau (GL) gain-loss and nearest-neighbor (NN) interaction. The propagation is described by a system of J perturbed coupled nonlinear Schrödinger (NLS) equations. The NN interaction property leads to the elimination of collisional three-pulse interaction effects, which prevented the observation of stable multisequence soliton propagation with J>2 sequences in the presence of generic GL gain-loss in all previous studies. We show that the dynamics of soliton amplitudes can be described by a generalized J-dimensional Lotka-Volterra (LV) model. Stability and bifurcation analysis for the equilibrium points of the LV model, which is augmented by an application of the Lyapunov function method, is used to develop setups that lead to robust and scalable transmission stabilization and switching for a general J value. The predictions of the LV model are confirmed by extensive numerical simulations with the perturbed coupled-NLS model with J=3, 4, and 5 soliton sequences. Furthermore, soliton stability and the agreement between the LV model's predictions and the simulations are independent of J. Therefore, our study provides the first demonstration of robust control of multiple colliding sequences of NLS solitons in the presence of generic weak GL gain-loss with an arbitrary number of sequences. Due to the robustness and scalability of the results, they can have important applications in stabilization and switching of broadband soliton-based optical waveguide transmission.

nlin.PS↗

arXiv2023

Analysis of a Crank-Nicolson finite difference scheme for (2+1)D perturbed nonlinear Schrödinger equations with saturable nonlinearity

We analyze a Crank-Nicolson finite difference discretization for the perturbed (2+1)D nonlinear Schrödinger equation with saturable nonlinearity and a perturbation of cubic loss. We show the boundedness, the existence and uniqueness of a numerical solution. We establish the error bound to prove the convergence of the numerical solution. Moreover, we find that the convergence rate is at the second order in both time step and spatial mesh size under a mild assumption. The numerical scheme is validated by the extensive simulations of the (2+1)D saturable nonlinear Schrödinger model with cubic loss. The simulations for travelling solitons are implemented by using an accelerated imaginary-time evolution scheme and the Crank-Nicolson finite difference method.

math.NA↗

arXiv2022

Fast two-pulse collisions in linear diffusion-advection systems with weak quadratic loss in spatial dimension 2

We investigate the dynamics of fast two-pulse collisions in linear diffusion-advection systems with weak quadratic loss in spatial dimension 2. We introduce a two-dimensional perturbation method, which generalizes the perturbation method used for studying two-pulse collisions in spatial dimension 1. We then use the generalized perturbation method to show that a fast collision in spatial dimension 2 leads to a change in the pulse shape in the direction transverse to the advection velocity vector. Moreover, we show that in the important case of a separable initial condition, the longitudinal part in the expression for the amplitude shift has a simple universal form, while the transverse part does not. Additionally, we show that anisotropy in the initial condition leads to a complex dependence of the amplitude shift on the orientation angle between the pulses. Our perturbation theory predictions are in very good agreement with results of extensive numerical simulations with the weakly perturbed diffusion-advection model. Thus, our study significantly enhances and generalizes the results of previous works on fast collisions in diffusion-advection systems, which were limited to spatial dimension 1.

nlin.PS↗

arXiv2022

Strong effects of fast collisions between pulsed optical beams in a linear medium with weak cubic loss

We investigate fast collisions between pulsed optical beams in a linear medium with weak cubic loss that arises due to nondegenerate two-photon absorption. We introduce a perturbation method with two small parameters and use it to obtain general formulas for the collision-induced changes in the pulsed-beam's shape and amplitude. Moreover, we use the method to design and characterize collision setups that lead to strong localized and nonlocalized intensity reduction effects. The values of the collision-induced changes in the pulsed-beam's shape in both setups are larger by one to two orders of magnitude compared with the values obtained in previous studies of fast two-pulse collisions. Furthermore, we show that for nonlocalized setups, the graph of the collision-induced amplitude shift vs the difference between the first-order dispersion coefficients for the two pulsed-beams has two local minima. This finding represents the first observation of a deviation of the graph from the common funnel shape that was obtained in all previous studies of fast two-pulse collisions in the presence of weak nonlinear loss. The predictions of our perturbation theory are in good agreement with results of numerical simulations with the perturbed linear propagation model, despite the strong collision-induced effects. Our results can be useful for multisequence optical communication links and for reshaping of pulsed optical beams.

nlin.PS↗

arXiv2021

Collision-induced amplitude dynamics of fast 2D solitons in saturable nonlinear media with weak nonlinear loss

We study the amplitude dynamics of two-dimensional (2D) solitons in a fast collision described by the coupled nonlinear Schrödinger equations with a saturable nonlinearity and weak nonlinear loss. We extend the perturbative technique for calculating the collision-induced dynamics of two one-dimensional (1D) solitons to derive the theoretical expression for the collision-induced amplitude dynamics in a fast collision of two 2D solitons. Our perturbative approach is based on two major steps. The first step is the standard adiabatic perturbation for the calculations on the energy balance of perturbed solitons and the second step, which is the crucial one, is for the analysis of the collision-induced change in the envelope of the perturbed 2D soliton. Furthermore, we also present the dependence of the collision-induced amplitude shift on the angle of the two 2D colliding-solitons. In addition, we show that the current perturbative technique can be simply applied to study the collision-induced amplitude shift in a fast collision of two perturbed 1D solitons. Our analytic calculations are confirmed by numerical simulations with the corresponding coupled nonlinear Schrödinger equations in the presence of the cubic loss and in the presence of the quintic loss.

nlin.PS↗

arXiv2021

Fast two-beam collisions in a linear optical medium with weak cubic loss in spatial dimension higher than 1

We study the dynamics of fast two-beam collisions in linear optical media with weak cubic loss in spatial dimension higher than 1. For this purpose, we extend the perturbation theory that was developed for analyzing two-pulse collisions in spatial dimension 1 to spatial dimension 2. We use the extended two-dimensional version of the perturbation theory to show that the collision leads to a change in the beam shapes in the direction transverse to the relative velocity vector. Furthermore, we show that in the important case of a separable initial condition for both beams, the longitudinal part in the expression for the amplitude shift is universal, while the transverse part is not universal. Additionally, we demonstrate that the same behavior holds for collisions between pulsed optical beams in spatial dimension 3. We check these predictions of the perturbation theory along with other predictions concerning the effects on the collision of partial beam overlap and anisotropy in the initial condition by extensive numerical simulations with the weakly perturbed linear propagation model in spatial dimensions 2 and 3. The agreement between the perturbation theory and the simulations is very good. Therefore, our study significantly extends and generalizes the results of previous works, which were limited to spatial dimension 1.

physics.optics↗

arXiv2019

Frequency shifting for solitons based on transformations on the Fourier domain and applications

We develop the theoretical procedures for shifting the frequency of a single soliton and of a sequence of solitons of the nonlinear Schrödinger equation. The procedures are based on simple transformations of the soliton pattern in the Fourier domain and on the shape-preserving property of solitons. These theoretical frequency shifting procedures are verified by numerical simulations with the nonlinear Schrödinger equation using the split-step Fourier method. In order to demonstrate the use of the frequency shifting procedures, two important applications are presented: (1) stabilization of the propagation of solitons in waveguides with frequency dependent linear gain-loss; (2) induction of repeated soliton collisions in waveguides with weak cubic loss. The results of numerical simulations with the nonlinear Schrödinger model are in very good agreement with the theoretical predictions.

nlin.PS↗

arXiv2018

Universality of the amplitude shift in fast two-pulse collisions in weakly perturbed linear physical systems

We demonstrate that the amplitude shifts in fast two-pulse collisions in perturbed linear physical systems with weak nonlinear dissipation exhibit universal soliton-like behavior. The behavior is demonstrated for linear optical waveguides with weak cubic loss and for systems described by linear diffusion-advection models with weak quadratic loss. We show that in both systems, the expressions for the collision-induced amplitude shifts due to the nonlinear loss have the same form as the expression obtained for a fast collision between two solitons of the nonlinear Schrödinger equation in the presence of weak cubic loss. Furthermore, we show that the expressions for the amplitude shifts are universal in the sense that they are independent of the exact details of the initial pulse shapes. We demonstrate the universal soliton-like behavior of the collision-induced amplitude shifts by carrying out numerical simulations with the two perturbed coupled linear evolution models with three different initial conditions corresponding to pulses with exponentially decreasing tails, pulses with power-law decreasing tails, and pulses that are initially nonsmooth and that develop significant tails during the collision. In all six cases we observe very good agreement between the analytic predictions for the amplitude shifts and the results of the numerical simulations.

nlin.PS↗

arXiv2017

Soliton-like behavior in fast two-pulse collisions in weakly perturbed linear physical systems

We demonstrate that pulses of linear physical systems, weakly perturbed by nonlinear dissipation, exhibit soliton-like behavior in fast collisions. The behavior is demonstrated for linear waveguides with weak cubic loss and for systems described by linear diffusion-advection models with weak quadratic loss. We show that in both systems, the expressions for the collision-induced amplitude shifts due to the nonlinear loss have the same form as the expression for the amplitude shift in a fast collision between two optical solitons in the presence of weak cubic loss. Our analytic predictions are confirmed by numerical simulations with the corresponding coupled linear evolution models with weak nonlinear loss. These results open the way for studying dynamics of fast collisions between pulses of weakly perturbed linear physical systems in an arbitrary spatial dimension.

nlin.PS↗

arXiv2017

Stable scalable control of soliton propagation in broadband nonlinear optical waveguides

We develop a method for achieving scalable transmission stabilization and switching of $N$ colliding soliton sequences in optical waveguides with broadband delayed Raman response and narrowband nonlinear gain-loss. We show that dynamics of soliton amplitudes in $N$-sequence transmission is described by a generalized $N$-dimensional predator-prey model. Stability and bifurcation analysis for the predator-prey model are used to obtain simple conditions on the physical parameters for robust transmission stabilization as well as on-off and off-on switching of $M$ out of $N$ soliton sequences. Numerical simulations for single-waveguide transmission with a system of $N$ coupled nonlinear Schrödinger equations with $2 \le N \le 4$ show excellent agreement with the predator-prey model's predictions and stable propagation over significantly larger distances compared with other broadband nonlinear single-waveguide systems. Moreover, stable on-off and off-on switching of multiple soliton sequences and stable multiple transmission switching events are demonstrated by the simulations. We discuss the reasons for the robustness and scalability of transmission stabilization and switching in waveguides with broadband delayed Raman response and narrowband nonlinear gain-loss, and explain their advantages compared with other broadband nonlinear waveguides.

nlin.PS↗