Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 0707.0141

Soliton turbulences in the complex Ginzburg-Landau equation

Hidetsugu Sakaguchi
View original source ↗Read source document ↗Export metadata ↓

Abstract

We study spatio-temporal chaos in the complex Ginzburg-Landau equation in parameter regions of weak amplification and viscosity. Turbulent states involving many soliton-like pulses appear in the parameter range, because the complex Ginzburg-Landau equation is close to the nonlinear Schrödinger equation. We find that the distributions of amplitude and wavenumber of pulses depend only on the ratio of the two parameters of the amplification and the viscosity. This implies that a one-parameter family of soliton turbulence states characterized by different distributions of the soliton parameters exists continuously around the completely integrable system.

Explore related subjects

nlin.PSnlin.CD

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Hidetsugu Sakaguchi. 2007-07-02. Soliton turbulences in the complex Ginzburg-Landau equation. https://doi.org/10.1103/physreve.76.017205

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Routes to chaos in a mass-conserving two-species reaction-diffusion model

Mass-conserving reaction-diffusion systems with two species correspond to a seemingly simple case where pattern formation occurs under the influence of a conservation law. Here, we first revisit their linear stability behavior and point out that generically two instabilities can occur: a stationary large-scale mass-conserving (Cahn-Hilliard) instability and an instability that combines features of a stationary large-scale non-mass-conserving (Allen-Cahn) instability and a oscillatory large-scale mass-conserving (conserved-Hopf) instability. We term it an Allen-Cahn-Hopf instability. Second, we investigate the nonlinear dynamics for a specific model related to the formation of cell polarization where only a Cahn-Hilliard instability can occur, i.e., all primary bifurcations are stationary. We analyze how secondary and further bifurcations subsequently give rise to various oscillatory states. The emerging rich spectrum of spatiotemporal behavior includes several period-doubling cascades related to different forms of spatial and temporal symmetry breaking. Beside regular states, three types of low-dimensional spatiotemporal chaos occur and involve transitions like fusion and an outer crises. Our results demonstrate the importance of nonlinear interactions in the dynamics of mass-conserving reaction-diffusion systems, and show that even a simple two-species system with primary bifurcations of Cahn-Hilliard type can show complex spatiotemporal behavior.

nlin.PS↗
arXiv2026

Duck hunting with quantum mechanics

We bridge two sides of singular perturbation theory: the classical theory of slow-fast systems and the semi-classical approach to quantum mechanical systems. For a specific but physically important class of dynamical systems, we show that purely classical and exotic objects, so-called canard solutions, are shadows of instantons in the corresponding quantum system. We demonstrate that canard solutions exist in a domain of parameter space whose boundaries are determined by an instanton action. We illustrate our statements analytically for the relevant example, the overdamped Josephson junction, and confirm them numerically. For the Josephson junction, the canard window is the exponentially narrow gap between consecutive Shapiro steps.

nlin.PS↗