Length scale competition in soliton bearing equations: A collective coordinate approach

Sara Cuenda; Angel Sanchez

doi:10.1063/1.1876632

Home/Archive/arXiv

arXiv · nlin/0411037

Length scale competition in soliton bearing equations: A collective coordinate approach

Sara Cuenda·Angel Sanchez

View original source ↗Read source document ↗Export metadata ↓

Abstract

We study the phenomenon of length scale competition, an instability of solitons and other coherent structures that takes place when their size is of the same order of some characteristic scale of the system in which they propagate. Working on the framework of nonlinear Klein-Gordon models as a paradigmatic example, we show that this instability can be understood by means of a collective coordinate approach in terms of soliton position and width. As a consequence, we provide a quantitative, natural explanation of the phenomenon in much simpler terms than any previous treatment of the problem. Our technique allows to study the existence of length scale competition in most soliton bearing nonlinear models and can be extended to coherent structures with more degrees of freedom, such as breathers.

Explore related subjects

nlin.PS cond-mat.other math.AP physics.class-ph

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Sara Cuenda, Angel Sanchez. 2004-11-15. Length scale competition in soliton bearing equations: A collective coordinate approach. https://doi.org/10.1063/1.1876632

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

KEEP EXPLORING

Related papers

arXiv2026

Is there an accelerating nonspreading wave packet in the Schrödinger equation with higher even-order dispersions?

Probably not. For Airy-type initial data, although nearly all lobes accelerate, the principal lobe remains essentially stationary. Moreover, the overall wave packet gradually disperses.

nlin.PS↗

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↗