Anomalous Diffusion of Dissipative Solitons in the Cubic-Quintic Complex Ginzburg-Landau Equation in Two Spatial Dimensions

Cisternas, Jaime; Descalzi, Orazio; Albers, Tony; Radons, Günter

doi:10.1103/physrevlett.116.203901

Cited by 20 publications

(21 citation statements)

References 32 publications

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“…Our observations and the perfect zig-zag patterns of localized structures [31] are qualitatively different. The one-dimensional and the two-dimensional behaviors of the soliton (the latter was reported in [32,33]) require separate treatments, since in two dimensions, the soliton can suffer perfectly symmetric explosions that do not induce spatial shifts.…”

Section: Explosions Of Dissipative Solitonsmentioning

confidence: 99%

A new kind of chaotic diffusion: anti-persistent random walks of explosive dissipative solitons

2019

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The solitons that exist in nonlinear dissipative media have properties very different from the ones that exist in conservative media and are modeled by the nonlinear Schrödinger equation. One of the surprising behaviors of dissipative solitons is the occurrence of explosions: sudden transient enlargements of a soliton, which as a result induce spatial shifts. In this work using the complex Ginzburg-Landau equation in one dimension, we address the long-time statistics of these apparently random shifts. We show that the motion of a soliton can be described as an anti-persistent random walk with a corresponding oscillatory decay of the velocity correlation function. We derive two simple statistical models, one in discrete and one in continuous time, which explain the observed behavior. Our statistical analysis benchmarks a future microscopic theory of the origin of this new kind of chaotic diffusion.

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Section: Explosions Of Dissipative Solitonsmentioning

confidence: 99%

A new kind of chaotic diffusion: anti-persistent random walks of explosive dissipative solitons

2019

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“…It is well-known that the competing cubic and quintic nonlinear terms-cubic-quintic model-that have opposite strengths of nonlinearities (the most often used is the case with self-focusing cubic and self-defocusing quintic nonlinear terms), can help to generate and stabilize various solitons [29][30][31][32][33][34][35][36][37][38]. In particular, the quintic nonlinearity may be realized in the background of optics with metal-dielectric nanocomposites by varying the proportion of silver nanoparticles suspended and the host medium [39,40], and in the context of Bose-Einstein condensates composed of a dense atom cloud by considering the influence of three-body interactions (threebody collisions on the account of s-wave low-energy atomatom scattering) whose value and even the sign can be tuned in experiments by utilizing the commonly used Feshbach resonance technique [41].…”

Section: Introductionmentioning

confidence: 99%

One-dimensional gap solitons in quintic and cubic–quintic fractional nonlinear Schrödinger equations with a periodically modulated linear potential

Zeng

2019

Nonlinear Dyn

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Competing nonlinearities, such as the cubic (Kerr) and quintic nonlinear terms whose strengths are of opposite signs (the coefficients in front of the nonlinearities), exist in various physical media (in particular, in optical and matter-wave media). A benign competition between self-focusing cubic and self-defocusing quintic nonlinear nonlinearities (known as cubic-quintic model) plays an important role in creating and stabilizing the self-trapping of D-dimensional localized structures, in the contexts of standard nonlinear Schrödinger equation. We incorporate an external periodic potential (linear lattice) into this model and extend it to the space-fractional scenario that begins to surface in very recent years-the nonlinear fractional Schrödinger equation (NLFSE), therefore obtaining the cubic-quintic or the purely quintic NLFSE, and investigate the propagation and stability properties of self-trapped modes therein. Two types of one-dimensional localized gap modes are found, including the fundamental and dipole-mode gap solitons. Employing the techniques based on the linear-stability analysis and direct numerical simulations, we get the stability regions of all the localized modes; and particularly, the anti-Vakhitov-Kolokolov criterion applies for the stable portions of soliton families generated in the frameworks of quintic-only nonlinearity and competing cubic-quintic nonlinear terms.

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“…We have found 11,13 that there are several regimes with significative differences. First there is a regime for values of µ ≤ −0.40, where all explosions are perfectly symmetric and the location of the soliton remains stationary.…”

Section: Effective Diffusion Of Solitonsmentioning

confidence: 74%

“…This article is a continuation of Ref. 13 and has the following three objectives: (i) Describe the transition between no diffusion (soliton exploding in-place) and normal diffusion as a single parameter is varied. (ii) Understand the role of anomalous diffusion in the transition.…”

Section: Introductionmentioning

confidence: 99%

Normal and anomalous random walks of 2-d solitons

Cisternas

Albers

Radons

2018

Chaos: An Interdisciplinary Journal of Nonlinear Science

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Solitons, which describe the propagation of concentrated beams of light through nonlinear media, can exhibit a variety of behaviors as a result of the intrinsic dissipation, diffraction, and the nonlinear effects. One of these phenomena, modeled by the complex Ginzburg-Landau equation, is chaotic explosions, transient enlargements of the soliton that may induce random transversal displacements, which in the long run lead to a random walk of the soliton center. As we show in this work, the transition from nonmoving to moving solitons is not a simple bifurcation but includes a sequence of normal and anomalous random walks. We analyze their statistics with the distribution of generalized diffusivities, a novel approach that has been used successfully for characterizing anomalous diffusion.

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Anomalous Diffusion of Dissipative Solitons in the Cubic-Quintic Complex Ginzburg-Landau Equation in Two Spatial Dimensions

Cited by 20 publications

References 32 publications

A new kind of chaotic diffusion: anti-persistent random walks of explosive dissipative solitons

A new kind of chaotic diffusion: anti-persistent random walks of explosive dissipative solitons

One-dimensional gap solitons in quintic and cubic–quintic fractional nonlinear Schrödinger equations with a periodically modulated linear potential

Normal and anomalous random walks of 2-d solitons

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