Soliton wave-speed management: Slowing, stopping, or reversing a solitary wave

Baines, Luke W. S.; Gorder, Robert A. Van

doi:10.1103/physreva.97.063814

Cited by 14 publications

(9 citation statements)

References 53 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Used in conjunction, saturability and XPM parameters could be used to position and reinforce the intensity of specific patterns. Alternative manners of control could involve the inclusion of spatial or temporal coefficients, which, if properly tuned, could be used to control the wave envelopes through dispersion management [90], or the speed of traveling fronts through wavespeed management [91]. Such an approach has previously been applied to the cubic complex GL equation [92], and it may be interesting to consider how such an approach may be used in conjunction with saturability of the media in order to modify the behavior of a nonlinear wave solution to (3)-(4).…”

Section: Discussionmentioning

confidence: 99%

Coupled complex Ginzburg–Landau systems with saturable nonlinearity and asymmetric cross-phase modulation

et al. 2018

Self Cite

View full text Add to dashboard Cite

We formulate and study dynamics from a complex Ginzburg-Landau system with saturable nonlinearity, including asymmetric cross-phase modulation (XPM) parameters. Such equations can model phenomena described by complex Ginzburg-Landau systems under the added assumption of saturable media. When the saturation parameter is set to zero, we recover a general complex cubic Ginzburg-Landau system with XPM. We first derive conditions for the existence of bounded dynamics, approximating the absorbing set for solutions. We use this to then determine conditions for amplitude death of a single wavefunction. We also construct exact plane wave solutions, and determine conditions for their modulational instability. In a degenerate limit where dispersion and nonlinearity balance, we reduce our system to a saturable nonlinear Schrödinger system with XPM parameters, and we demonstrate the existence and behavior of spatially heterogeneous stationary solutions in this limit. Using numerical simulations we verify the aforementioned analytical results, while also demonstrating other interesting emergent features of the dynamics, such as spatiotemporal chaos in the presence of modulational instability. In other regimes, coherent patterns including uniform states or banded structures arise, corresponding to certain stable stationary states. For sufficiently large yet equal XPM parameters, we observe a segregation of wavefunctions into different regions of the spatial domain, while when XPM parameters are large and take different values, one wavefunction may decay to zero in finite time over the spatial domain (in agreement with the amplitude death predicted analytically). We also find a collection of transient features, including transient defects and what appear to be rogue waves, while in two spatial dimensions we observe highly localized pattern formation. While saturation will often regularize the dynamics, such transient dynamics can still be observed -and in some cases even prolonged -as the saturability of the media is increased, as the saturation may act to slow the timescale.[5]. Spatial patterning is also possible, with solutions such as spiral waves being found [6]. Bounds on the attractor for complex cubic GL equations have been considered in [7,8]. Quintic or cubic-quintic GL equations have been studied [9], and have application to areas of nonlinear optics including lasers [10], optical waveguides equipped with a Bragg grating [11], quasi-CW Raman fiber lasers [12], and Bose-Einstein condensates [13,14]. Solutions to non-autonomous forms of the complex GL equation also have been studied [15].Coupled or vector complex cubic GL equations have also attracted interest. For example, the effect of slow real modes in reaction-diffusion systems close to a supercritical Hopf bifurcation was studied in [16] via a system of two coupled GL equations, where it was pointed out that a single GL equation cannot reproduce the dynamics from the reaction-diffusion system, suggesting the necessity of considering such coupled systems in some co...

show abstract

Section: Discussionmentioning

confidence: 99%

Coupled complex Ginzburg–Landau systems with saturable nonlinearity and asymmetric cross-phase modulation

et al. 2018

Self Cite

View full text Add to dashboard Cite

show abstract

“…Some analytical studies also exist, albeit for very specialized forms of the non-autonomous GPE, many involving solitary waves or solitons. Extensions of these solutions which account for time-varying potentials have included [25][26][27][28][29][30][31][32]. Similarity solutions are also seen, albeit under very restrictive potentials [33,34].…”

Section: Introductionmentioning

confidence: 84%

Time-varying Bose–Einstein condensates

Gorder

2021

Proc. R. Soc. A.

Self Cite

View full text Add to dashboard Cite

Bose–Einstein condensates (BECs), a state of matter formed when a low-density gas of bosons is cooled to near absolute zero, continue to motivate novel work in theoretical and experimental physics. Although BECs are most commonly studied in stationary ground states, time-varying BECs arise when some aspect of the physics governing the condensate varies as a function of time. We study the evolution of time-varying BECs under non-autonomous Gross–Pitaevskii equations (GPEs) through a mix of theory and numerical experiments. We separately derive a perturbation theory (in the small-parameter limit) and a variational approximation for non-autonomous GPEs on generic bounded space domains. We then explore various routes to obtain time-varying BECs, starting with the more standard techniques of varying the potential, scattering length, or dispersion, and then moving on to more advanced control mechanisms such as moving the external potential well over time to move or even split the BEC cloud. We also describe how to modify a BEC cloud through evolution of the size or curvature of the space domain. Our results highlight a variety of interesting theoretical routes for studying and controlling time-varying BECs, lending a stronger theoretical formulation for existing experiments and suggesting new directions for future investigation.

show abstract

“…Several studies have considered Gross-Pitaevskii equations with time-dependent potentials or traps [60][61][62], with the modulational instability of such systems studied using a quasi-static or frozen time approach [63,64]. Dispersion management is a process by which the gain or loss of a waveform in a fibre optical cable can be mitigated [41][42][43][44], and involves affixing a nonlinear Schrödinger equation (or a related model) with time-dependent coefficients. Consider a Gross-Pitaevskii equation of the form…”

Section: (D) History Dependence Due To Time-dependent Base Statesmentioning

confidence: 99%

“…Complex Ginzburg-Landau equations arise as amplitude equations in reactiondiffusion systems, and hence amplitude equations for the aforementioned applications should similarly be non-autonomous [36,37]. A variety of non-autonomous nonlinear Schrödinger and complex Ginzburg-Landau equations have been considered in their own right [38][39][40], with nonautonomous terms of particular use in the dispersion management [41,42] of nonlinear waves in nonlinear optics [43,44] and atomic or condensed matter physics [45].…”

Section: Introductionmentioning

confidence: 99%

Turing and Benjamin–Feir instability mechanisms in non-autonomous systems

Gorder

2020

Proc. R. Soc. A.

Self Cite

View full text Add to dashboard Cite

The Turing and Benjamin–Feir instabilities are two of the primary instability mechanisms useful for studying the transition from homogeneous states to heterogeneous spatial or spatio-temporal states in reaction–diffusion systems. We consider the case when the underlying reaction–diffusion system is non-autonomous or has a base state which varies in time, as in this case standard approaches, which rely on temporal eigenvalues, break down. We are able to establish respective criteria for the onset of each instability using comparison principles, obtaining inequalities which involve the in general time-dependent model parameters and their time derivatives. In the autonomous limit where the base state is constant in time, our results exactly recover the respective Turing and Benjamin–Feir conditions known in the literature. Our results make the Turing and Benjamin–Feir analysis amenable for a wide collection of applications, and allow one to better understand instabilities emergent due to a variety of non-autonomous mechanisms, including time-varying diffusion coefficients, time-varying reaction rates, time-dependent transitions between reaction kinetics and base states which change in time (such as heteroclinic connections between unique steady states, or limit cycles), to name a few examples.

show abstract

Soliton wave-speed management: Slowing, stopping, or reversing a solitary wave

Cited by 14 publications

References 53 publications

Coupled complex Ginzburg–Landau systems with saturable nonlinearity and asymmetric cross-phase modulation

Coupled complex Ginzburg–Landau systems with saturable nonlinearity and asymmetric cross-phase modulation

Time-varying Bose–Einstein condensates

Turing and Benjamin–Feir instability mechanisms in non-autonomous systems

Contact Info

Product

Resources

About