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The numerous scientific feedbacks that the FitzHugh-Rinzel system (FHR) is having in various scientific fields, lead to further studies on the determination of its explicit solutions. Indeed, such a study can help to get a better…
Some classes of the so called "travelling wave" solutions of Einstein and Einstein - Maxwell equations in General Relativity and of dynamical equations for massless bosonic fields in string gravity in four and higher dimensions are…
In this paper we consider a scalar parabolic equation on a star graph; the model is quite general but what we have in mind is the description of traffic flows at a crossroad. In particular, we do not necessarily require the continuity of…
The recent theoretical discovery of families of travelling wave solutions in pipe flow at Reynolds numbers lower than the transitional range naturally raises the question of their relevance to the turbulent transition process. Here a series…
We consider the stability of position control of traveling waves in reaction-diffusion system as proposed in {[}J. L\"ober, H. Engel, arXiv:1304.2327{]}. Instead of analyzing the controlled reaction-diffusion system, stability is studied on…
Reaction-diffusion equations appear in biology and chemistry, and combine linear diffusion with different kind of reaction terms. Some of them are remarkable from the mathematical point of view, since they admit families of travelling waves…
We study travelling wave solutions of a 1D continuum model for collective cell migration in which cells are characterised by position and polarity. Four different types of travelling wave solutions are identified which represent…
We study travelling-wave solutions for a reaction-diffusion system arising as a model for host-tissue degradation by bacteria. This system consists of a parabolic equation coupled with an ordinary differential equation. For large values of…
We study a non-linear and non-local evolution equation for curves obtained as the sharp interface limit of a phase-field model for crawling motion of eukaryotic cells on a substrate. We establish uniqueness of solutions to the sharp…
We examine the applicability of the weak wave turbulence theory in explaining experimental scaling results obtained for the diffusion and relative diffusion of particles moving on turbulent surface waves. For capillary waves our theoretical…
We consider the Euler-Poisson system for ions where the electrons are given by a Maxwell-Boltzmann distribution, and we investigate the existence of one-dimensional periodic traveling waves. More precisely, we first establish the existence…
We investigate the connection between the existence of an explicit travelling wave solution and the travelling wave with minimal speed in a scalar monostable reaction-diffusion equation.
We derive parametric travelling-wave solutions of non-linear QCD equations. They describe the evolution towards saturation in the geometric scaling region. The method, based on an expansion in the inverse of the wave velocity, leads to a…
A modified method of functional constraints is used to construct the exact solutions of nonlinear equations of reaction-diffusion type with delay and which are associated with variable coefficients. This study considers a most generalized…
We study traveling wave solutions for a nonlinear Schr\"odinger system with quadratic interaction. For the non mass resonance case, the system has no Galilean symmetry, which is of particular interest in this paper. We construct traveling…
The current paper is devoted to the investigation of wave propagation phenomenon in reaction-diffusion equations with ignition type nonlinearity in time heterogeneous and random media. It is proven that such equations in time heterogeneous…
We prove existence, uniqueness, and stability of transition fronts (generalized traveling waves) for reaction-diffusion equations in cylindrical domains with general inhomogeneous ignition reactions. We also show uniform convergence of…
In this paper, we study the existence and uniqueness of traveling wave solution for the accelerated Frenkel-Kontorova model. This model consists in a system of ODE that describes the motion particles in interaction. The most important…
We prove the existence of a continuous family of positive and generally non-monotone travelling fronts in delayed reaction-diffusion equations $u_t(t,x) = \Delta u(t,x)- u(t,x) + g(u(t-h,x)) (*)$, when $g \in C^2(R_+,R_+)$ has exactly two…
In this work, a systematic study, examining the propagation of periodic and solitary wave along the magnetic field in a cold collision-free plasma, is presented. Employing the quasi-neutral approximation and the conservation of momentum…