Related papers: Localized patterns in the Gierer Meinhardt model o…
We study the discrete Gierer-Meinhardt model of reaction-diffusion on three different types of networks: regular, random and scale-free. The model dynamics lead to the formation of stationary Turing patterns in the steady state in certain…
We analyze the existence and stability of localized solutions in the one-dimensional discrete nonlinear Schr\"{o}dinger (DNLS) equation with a combination of competing self-focusing cubic and defocusing quintic onsite nonlinearities. We…
Quadratic scale-invariant gravity non minimally coupled to a scalar field provides a competitive model for inflation, characterized by the transition from an unstable to a stable fixed point, both characterized by constant scalar field…
We numerically investigate the formation of soliton pairs (bound states) in mode-locked fiber ring lasers. In the distributed model (complex cubic-quintic Ginzburg-Landau equation) we observe a discrete family of soliton pairs with…
We propose and analyze a structure-preserving space-time variational discretization method for the Cahn-Hilliard-Navier-Stokes system. Uniqueness and stability for the discrete problem is established in the presence of concentration…
We address the issue of mobility of localized modes in two-dimensional nonlinear Schr\"odinger lattices with saturable nonlinearity. This describes e.g. discrete spatial solitons in a tight-binding approximation of two-dimensional optical…
We present three examples of delayed bifurcations for spike solutions of reaction-diffusion systems. The delay effect results as the system passes slowly from a stable to an unstable regime, and was previously analysed in the context of…
We introduce a system of two linearly coupled discrete nonlinear Schr\"{o}dinger equations (DNLSEs), with the coupling constant subject to a rapid temporal modulation. The model can be realized in bimodal Bose-Einstein condensates (BEC).…
I show stable, localized, single and multi-spot patterns of three classes - stationary, moving, and rotating - that exist within a limited range of parameter values in the two-dimensional Gray-Scott reaction-diffusion model with ${\sigma} =…
We study, analytically and numerically, the dynamical behavior of the solutions of the complex Ginzburg-Landau equation with diffraction but without diffusion, which governs the spatial evolution of the field in an active nonlinear laser…
In this paper we characterise the global stability, global boundedness and recurrence of solutions of a scalar nonlinear stochastic differential equation. The differential equation is a perturbed version of a globally stable autonomous…
We explore a mechanism of pattern formation arising in processes described by a system of a single reaction-diffusion equation coupled with ordinary differential equations. Such systems of equations arise from the modeling of interactions…
We have found several families of vortex soliton solutions in two-dimensional discrete dissipative systems governed by the cubic-quintic complex Ginzburg-Landau equation. There are symmetric and asymmetric solutions, and some of them have…
In this paper we investigate the stability properties of the so-called gBBKS and GeCo methods, which belong to the class of nonstandard schemes and preserve the positivity as well as all linear invariants of the underlying system of…
We show that spinor systems with scalar self-interaction, such as the Dirac--Klein--Gordon system with Yukawa coupling or the Soler model, generically have bi-frequency solitary wave solutions. We develop the approach to stability…
We develop an unconditionally energy-stable tensor-product space-time discretization framework for the solution of a linear kinetic transport equation in one space dimension. The kinetic equation is a simplified model of radiative transfer…
Reaction-diffusion equations coupled to ordinary differential equations (ODEs) may exhibit spatially low-regular stationary solutions. This work provides a comprehensive theory of asymptotic stability of bounded, discontinuous or…
In this paper, we prove that the slowly-rotating Kerr-de Sitter family of black holes are linearly stable as a family of solutions to the Einstein vacuum equations with $\Lambda>0$ in harmonic (wave) gauge. This article is part of a series…
We consider a prototypical dynamical lattice model, namely, the discrete nonlinear Schroedinger equation on nonsquare lattice geometries. We present a systematic classification of the solutions that arise in principal six-lattice-site and…
Localized patterns are spatially confined structures that arise in lattice dynamical systems and play an important role in physics, biology, and materials science. While their existence and bifurcation structure are well-understood, the…