Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1210.0938

On transverse stability of discrete line solitons

Dmitry E. Pelinovsky·Jianke Yang
View original source ↗Read source document ↗Export metadata ↓

Abstract

We obtain sharp criteria for transverse stability and instability of line solitons in the discrete nonlinear Schrödinger equations on one- and two-dimensional lattices near the anti-continuum limit. On a two-dimensional lattice, the fundamental line soliton is proved to be transversely stable (unstable) when it bifurcates from the $X$ ($Γ$) point of the dispersion surface. On a one-dimensional (stripe) lattice, the fundamental line soliton is proved to be transversely unstable for both signs of transverse dispersion. If this transverse dispersion has the opposite sign to the discrete dispersion, the instability is caused by a resonance between isolated eigenvalues of negative energy and the continuous spectrum of positive energy. These results hold for both focusing and defocusing nonlinearities via a staggering transformation. When the line soliton is transversely unstable, asymptotic expressions for unstable eigenvalues are also derived. These analytical results are compared with numerical results, and perfect agreement is obtained.

Explore related subjects

nlin.PS

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Dmitry E. Pelinovsky, Jianke Yang. 2012-10-02. On transverse stability of discrete line solitons. https://doi.org/10.1016/j.physd.2013.03.011

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↗