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We prove existence and uniqueness of travelling waves for a reaction-diffusion system coupling a classical reaction-diffusion equation in a strip with a diffusion equation on a line. To do this we use a continuation method which leads to…
By a refinement of the technique used by Johnson and Zumbrun to show stability under localized perturbations, we show that spectral stability implies nonlinear modulational stability of periodic traveling-wave solutions of reaction…
We investigate the linearized hydrodynamic equations of interacting self-propelled particles in two dimensional space. It is found that the small perturbations of density and polarization fields satisfy the hyperbolic partial differential…
Generalizing similar results for viscous shock and relaxation waves, we establish sharp pointwise Green function bounds and linearized and nonlinear stability for traveling wave solutions of an abstract viscous combustion model including…
The diffusion in two dimensions of non-interacting active particles that follow an arbitrary motility pattern is considered for analysis. Accordingly, the transport equation is generalized to take into account an arbitrary distribution of…
We study traveling wave solutions to bistable differential equations on infinite $k$-ary trees. These graphs generalize the notion of classical square infinite lattices and our results complement those for bistable lattice equations on…
A general system of n ordinary differential equations coupled with one reaction-diffusion equation, considered in a bounded N-dimensional domain, with no-flux boundary condition is studied in a context of pattern formation. Such initial…
We develop a complete stability theory for two-dimensional periodic traveling waves of reaction-diffusion systems. More precisely, we identify a diffusive spectral stability assumption, prove that it implies nonlinear stability and provide…
In the present work, we study coherent structures in a one-dimensional discrete nonlinear Schr\"odinger lattice in which the coupling between waveguides is periodically modulated. Numerical experiments with single-site initial conditions…
The structure of spiral waves is investigated in super-excitable reaction-diffusion systems where the local dynamics exhibits multi-looped phase space trajectories. It is shown that such systems support stable spiral waves with broken…
In this paper, we focus on the existence of propagation fronts, solutions to non-local dispersion reaction models. Our aim is to provide a unified proof of this existence in a very broad framework using simple real analysis tools. In…
It has been reported that traveling waves propagate periodically and stably in sub-excitable systems driven by noise [Phys. Rev. Lett. \textbf{88}, 138301 (2002)]. As a further investigation, here we observe different types of traveling…
An analysis of traveling wave solutions of pure cross-diffusion systems, i.e., systems that lack reaction and self-diffusion terms, is presented. Using the qualitative theory of phase plane analysis the conditions for existence of different…
We investigate dynamics near Turing patterns in reaction-diffusion systems posed on the real line. Linear analysis predicts diffusive decay of small perturbations. We construct a "normal form" coordinate system near such Turing patterns…
We extend the result on the stability of travelling waves for stochastic Nagumo equations in [St] to general bistable reaction-diffusion equations with both additive and multiplicative noise, using a variational approach based on functional…
Mass-conserving reaction-diffusion (MCRD) systems are widely used to model phase separation and pattern formation in cell polarity, biomolecular condensates, and ecological systems. Numerical simulations and formal asymptotic analysis…
This paper is concerned with quasilinear parabolic reaction-diffusion-advection systems on extended domains. Frameworks for well-posedness in Hilbert spaces and spaces of continuous functions are presented, based on known results using…
We consider the problem of wave propagation for a 2-D rectilinear optical waveguide which presents some perturbation. We construct a mathematical framework to study such a problem and prove the existence of a solution for the case of small…
We analyze the pattern forming ability and pattern stability for a one-dimensional non-linear transport-diffusion equation on the circle. We show that the trivial steady state is stable when diffusion is sufficiently strong. In the limit…
A key feature of $(1+1)$-dimensional nonlinear wave equations is that they admit left or right traveling waves, under appropriate algebraic conditions on the nonlinearities. In this paper, we prove global stability of such traveling wave…