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We complete the description, initiated in [ 6 ], of a free boundary traveling at constant speed in a half plane, where the propagation is controlled by a line having a large diffusion on its own. The main result of this work is that the free boundary is asymptotic to a line at infinity, whose angle to the horizontal is dictated by the velocity of the wave on the line. This helps understanding some counter-intuitive simulations of [ 7 ].

We consider a one-dimensional reaction–diffusion equation of Fisher–Kolmogoroff–Petrovsky–Piscounoff type. We investigate the effect of the interaction between the nonlinear diffusion coefficient and the reaction term on the existence and non-existence of travelling waves. Our diffusion coefficient is allowed to be degenerate or singular at both equilibrium points, 0 and 1, while the reaction term need not be differentiable. These facts influence the existence and qualitative properties of travelling waves in a substantial way.

International audience; In this paper, we use the exponential transform to give a unified formal upper bound for the asymptotic rate of spread of a population propagating in a one dimensional habitat. We show through examples how this upper bound can be obtained directly for discrete and continuous time models. This upper bound has the form min s>0 ln(ρ(s))/s and coincides with the speeds of several models found in the literature. 1. Introduction. Under favourable biological conditions, a well or newly established population in a one dimensional continuous habitat (e.g. coastline [24], river [27], a preferred direction in a multidimensional space), can spread at left (in the region x ≤ 0), at right (in the region x ≥ 0) or in both directions causing direct and indirect effects on the ecosystem [44, 7, 10, 14]. One important measure quantifying this spread is the asymptotic speed of propagation of the expanding population in the environment, generally called the invasion speed when dealing with the spread of invasive species [13]. This asymptotic quantity has been studied for a large class of biological models like reaction-diffusion [9, 1, 4, 19, 35], integrodifferential [30, 28], integrodifference [18, 42] and discrete-time recursion [48, 25] equations. A mathematical formulation of the asymptotic speed of spread (in short, spreading speed) appeared in the work of Aronson and Weinberger [1] on the Fisher-KPP reaction-diffusion equation under specific conditions. They showed that the solutions N (t, x) (population density at time t and position x) travel in a one dimensional habitat of carrying capacity unity with a spreading speed c * > 0 characterized by the behaviour of N (t, ξ + ct) for large t, where ξ and c are real numbers. More precisely, for some α ∈ (0, 1] and every ξ, it holds

The quasi-linear model of viscoelasticity is a constitutive law widely used to investigate the time dependent behaviour of soft tissues and bio-materials. For this model, we study the shearing motion and discuss the existence of kink-type wave solutions. In particular, we derive a nonlinear second-order ordinary differential equation which allows to widen the class of solutions given by Samsonov (1995). When the stress relaxation function is a Prony series, kink-wave solutions can exist for strongly elliptic strain energy functions, except for the Mooney–Rivlin model. We provide numerical simulations for the Yeoh model.

We study dynamical properties of a complex equation with spatially-distributed parameters. Families of special parabolic equations that de¯ne the behavior of initial problem solutions were built.

Layer II stellate cells in the medial enthorinal cortex (MEC) express hyperpolarisation-activated cyclic-nucleotide-gated (HCN) channels that allow for rebound spiking via an \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$I_{\text{h}}$\end{document}Ih current in response to hyperpolarising synaptic input. A computational modelling study by Hasselmo (Philos. Trans. R. Soc. Lond. B, Biol. Sci. 369:20120523, 2013) showed that an inhibitory network of such cells can support periodic travelling waves with a period that is controlled by the dynamics of the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$I_{\text{h}}$\end{document}Ih current. Hasselmo has suggested that these waves can underlie the generation of grid cells, and that the known difference in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$I_{\text{h}}$\end{document}Ih resonance frequency along the dorsal to ventral axis can explain the observed size and spacing between grid cell firing fields. Here we develop a biophysical spiking model within a framework that allows for analytical tractability. We combine the simplicity of integrate-and-fire neurons with a piecewise linear caricature of the gating dynamics for HCN channels to develop a spiking neural field model of MEC. Using techniques primarily drawn from the field of nonsmooth dynamical systems we show how to construct periodic travelling waves, and in particular the dispersion curve that determines how wave speed varies as a function of period. This exhibits a wide range of long wavelength solutions, reinforcing the idea that rebound spiking is a candidate mechanism for generating grid cell firing patterns. Importantly we develop a wave stability analysis to show how the maximum allowed period is controlled by the dynamical properties of the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$I_{\text{h}}$\end{document}Ih current. Our theoretical work is validated by numerical simulations of the spiking model in both one and two dimensions. Electronic Supplementary Material The online version of this article (doi:10.1186/s13408-017-0051-7) contains supplementary material.

This article studies existence and stability of travelling wave of unstirred Gray-Scott system in biological pattern formation which models an isothermal chemical reaction $ A + 2B \rightarrow 3B $ involving two chemical species, a reactant A and an auto-catalyst B, and a linear decay $ B \rightarrow C $ , where C is an inert product. Our result shows a new and very distinctive feature of Gray-Scott type of models in generating rich and structurally different travelling pulses than related models in the literature. In particular, the existence of multiple travelling waves which have distinctive number of local maxima is proved. Furthermore, the stability of travelling wave of the reaction–diffusion system of isothermal diffusion system $ A + 2 B \rightarrow 3 B $ , is also studied.

We study some properties of the monotone solutions of the boundary value problem ( P ( u ′ ) ) ′ - cu ′ + f ( u ) = 0 , u ( - ∞ ) = 0 , u ( + ∞ ) = 1 , where f is a continuous function, positive in ( 0 , 1 ) and taking the value zero at 0 and 1, and P may be an increasing homeomorphism of [ 0 , 1 ) or [ 0 , + ∞ ) onto [ 0 , + ∞ ) . This problem arises when we look for travelling waves for the reaction diffusion equation ∂ u ∂ t = ∂ ∂ x P ∂ u ∂ x + f ( u ) with the parameter c representing the wave speed. A possible model for the nonlinear diffusion is the relativistic curvature operator P ( v ) = v 1 - v 2 . The same ideas apply when P is given by the one-dimensional p -Laplacian P ( v ) = v p - 2 v . In this case, an advection term is also considered. We show that, as for the classical Fisher–Kolmogorov–Petrovski–Piskounov equations, there is an interval of admissible speeds [ c ∗ , + ∞ ) and we give characterisations of the critical speed c ∗ . We also present some examples of exact solutions.

Within the framework of the Maxwell-Cattaneo relaxation model extended to reaction-diffusion systems with nonlinear advection, travelling wave (TW) solutions are analytically investigated by studying a normalized reaction-telegraph equation in the case of the reaction and advection terms described by quadratic functions. The problem involves two governing parameters: (i) a ratio ${\ensuremath{\varphi}}^{2}$ of the relaxation time in the Maxwell-Cattaneo model to the characteristic time scale of the reaction term, and (ii) the normalized magnitude $N$ of the advection term. By linearizing the equation at the leading edge of the TW, (i) necessary conditions for the existence of TW solutions that are smooth in the entire interval of $\ensuremath{-}\ensuremath{\infty}l\ensuremath{\zeta}l\ensuremath{\infty}$ are obtained, (ii) the smooth TW speed is shown to be less than the maximal speed ${\ensuremath{\varphi}}^{\ensuremath{-}1}$ of the propagation of a substance, (iii) the lowest TW speed as a function of $\ensuremath{\varphi}$ and $N$ is determined. If the necessary condition of $Ng\ensuremath{\varphi}\ensuremath{-}{\ensuremath{\varphi}}^{\ensuremath{-}1}$ does not hold, e.g., if the magnitude $N$ of the nonlinear advection is insufficiently high in the case of ${\ensuremath{\varphi}}^{2}g1$, then, the studied equation admits piecewise smooth TW solutions with sharp leading fronts that propagate at the maximal speed ${\ensuremath{\varphi}}^{\ensuremath{-}1}$, with the substance concentration or its spatial derivative jumping at the front. An increase in $N$ can make the solution smooth in the entire spatial domain. Moreover, an explicit TW solution to the considered equation is found provided that $Ng\ensuremath{\varphi}$. Subsequently, by invoking a principle of the maximal decay rate of TW solution at its leading edge, relevant TW solutions are selected in a domain of $(\ensuremath{\varphi},N)$ that admits the smooth TWs. Application of this principle to the studied problem yields transition from pulled (propagation speed is controlled by the TW leading edge) to pushed (propagation speed is controlled by the entire TW structure) TW solutions at $N={N}_{\mathrm{cr}}=\sqrt{1+{\ensuremath{\varphi}}^{2}}$, with the pulled (pushed) TW being relevant at smaller (larger) $N$. An increase in the normalized relaxation time ${\ensuremath{\varphi}}^{2}$ results in increasing ${N}_{\mathrm{cr}}$, thus promoting the pulled TW solutions. The domains of $(\ensuremath{\varphi},N)$ that admit either the smooth or piecewise smooth TWs are not overlapped and, therefore, the selection problem does not arise for these two types of solutions. All the aforementioned results and, in particular, the maximal-decay-rate principle or appearance of the piecewise smooth TW solutions, are validated by numerically solving the initial boundary value problem for the reaction-telegraph equation with natural initial conditions localized to a bounded spatial region.

Mathematical models of bacterial populations are often written as systems of partial differential equations for the densities of bacteria and concentrations of extracellular (signal) chemicals. This approach has been employed since the seminal work of Keller and Segel in the 1970s (Keller and Segel, J. Theor. Biol. 30:235-248, 1971). The system has been shown to permit travelling wave solutions which correspond to travelling band formation in bacterial colonies, yet only under specific criteria, such as a singularity in the chemotactic sensitivity function as the signal approaches zero. Such a singularity generates infinite macroscopic velocities which are biologically unrealistic. In this paper, we formulate a model that takes into consideration relevant details of the intracellular processes while avoiding the singularity in the chemotactic sensitivity. We prove the global existence of solutions and then show the existence of travelling wave solutions both numerically and analytically. © 2010 Society for Mathematical Biology.

International audience; We investigate numerically the two dimensional travelling waves of the Nonlinear Schrödinger Equation for a general nonlinearity and with nonzero condition at infinity. In particular, we are interested in the energy-momentum diagrams. We propose a numerical strategy based on the variational structure of the equation. The key point is to characterize the saddle points of the action as minimizers of another functional, that allows us to use a gradient flow. We combine this approach with a continuation method in speed in order to obtain the full range of velocities. Through various examples, we show that even though the nonlinearity has the same behaviour as the well-known Gross-Pitaevskii nonlinearity, the qualitative properties of the travelling waves may be extremely different. For instance, we observe cusps, a modified (KP-I) asymptotic in the transonic limit, various multiplicity results and ''one dimensional spreading'' phenomena.

In an earlier work (Berestycki et al., 2013), we introduced a parabolic system to describe biological invasions in the presence of a line (the “road”) with a specific diffusion, included in a domain (the “field”) subject to Fisher–KPP propagation. We determined the asymptotic spreading speed in the direction of the road, w ∗ , in terms of the various parameters. The new result we establish here is the existence of travelling fronts for this system for any speed c ⩾ w ∗ . In addition, we show that no front can travel with a speed c w ∗ . We further extend the results to a new model where there is Fisher–KPP growth on the road, embedded in an elsewhere unfavourable plane. This is relevant to discuss certain invasions that have been observed along roads. We establish sharp conditions for extinction/invasion and describe the invasion dynamics.

International audience; We consider a lattice equation modelling one-dimensional metamaterials formed by a discrete array of nonlinear resonators. We focus on periodic travelling waves due to the presence of a periodic force. The existence and uniqueness results of periodic travelling waves of the system are presented. Our analytical results are found to be in good agreement with direct numerical computations.

Yazar adının diğer kullanılan biçimi: Mahir Hasanov This paper is devoted to multi-parameter eigenvalue problems for one-dimensional perturbed p-Laplacians, modelling travelling waves for a class of nonlinear evolution PDE. Dispersion relations between the eigen-parameters, the existence of eigenfunctions and positive eigenfunctions, variational principles for eigenvalues and constructing solutions in the analytical and implicit forms are the main subject of this paper. We use both variational and analytical methods.

Scalar one-dimensional conservation laws with a nonlocal diffusion term corresponding to a Riesz–Feller differential operator are considered. Solvability results for the Cauchy problem in L ∞ are adapted from the case of a fractional derivative with homogeneous symbol. The main interest of this work is the investigation of smooth shock profiles. In the case of a genuinely nonlinear smooth flux function we prove the existence of such travelling waves, which are monotone and satisfy the standard entropy condition. Moreover, the dynamic nonlinear stability of the travelling waves under small perturbations is proven, similarly to the case of the standard diffusive regularisation, by constructing a Lyapunov functional.

The paper deals with a problem of interaction between hydrodynamics and mechanics of nonlinear elastic bodies. The existence question for two-dimensional symmetric steady waves travelling on the surface of a deep ocean beneath a heavy elastic membrane is analyzed as a problem in bifurcation theory. The behaviour of the two-dimensional cross-section of the membrane is modelled as a thin (unshearable), heavy, hyperelastic Cosserat rod, following Antman's elasticity theory, and the fluid beneath is supposed to be in steady 2D irrotational motion under gravity. Assuming that gravity and the density of the undeformed membrane are prescribed, the free parameters of the problem are the speed of the wave and drift velocity of the membrane. The analysis relies upon a conformal formulation of the hydro-elastic problem developed in previous papers; the basic tool for the study of the bifurcation picture is the implicit function theorem, under some non-resonance assumptions. The most interesting part of the final result is the existence of a symmetry-breaking 'third sheet' of solutions, which bifurcates from primary sheets, and is a hydro-elastic analogue of the phenomenon known as 'Wilton ripples' in the surface tension case., Comment: 23 pages, 3 figures

The Korteweg-de Vries and the Korteweg-de Vries-Burgers equations are considered. Using the travelling wave the general solutions of these equations are presented. "New travelling wave solutions" of the KdV and the KdV-Burgers equations by Wazzan [Wazzan L Commun Nonlinear Sci Numer Simulat 2009:14:443-50] are analyzed. We demonstrate that all his solutions are not new and are transformed to known solutions. (C) 2008 Elsevier B.V. All rights reserved.

In this paper we consider approximate travelling wave solutions to the Korteweg-de Vries equation. The heat-balance integral method is first applied to the problem, using two different quartic approximating functions, and then the refined integral method is investigated. We examine two types of solution, chosen by matching the wave speed to that of the exact solution and by imposing the same area. The first set of solutions is generally better with an error that is fixed in time. The second set of solutions has an error that grows with time. This is shown to be due to slight discrepancies in the wave speed.

We study the time-asymptotic behaviour of a solution of the Cauchy initial-value problem for a conservation law with non-linear divergent viscosity. We shall prove that, when a bounded measurable initial function has limits at +/-infinity, a solution of the cauchy initial-value problem converges uniformly to a system of waves consisting of travelling waves and rarefaction waves, where the phase shifts of the travelling waves are allowed to depend oil time. The rate of convergence is estimated under additional conditions oil the initial function.

For a simple one dimensional lattice version of a travelling wave equation, we obtain an exact relation between the initial condition and the position of the front at any later time. This exact relation takes the form of an inverse problem: given the times $t_n$ at which the travelling wave reaches the positions $n$, one can deduce the initial profile. We show, by means of complex analysis, that a number of known properties of travelling wave equations in the Fisher-KPP class can be recovered, in particular Bramson's shifts of the positions. We also recover and generalize Ebert-van Saarloos' corrections depending on the initial condition., Comment: For version 2: some typos + clarification of (87)

International audience; We introduce a precise definition of algebraic travelling wave solution of n-th order partial differential equations and prove that the only algebraic travelling waves solutions for the celebrated Fisher–Kolmogorov equation are the ones found in 1979 by Ablowitz and Zeppetella. This question is equivalent to study when an associated one-parameter family of planar ordinary differential systems has invariant algebraic curves.

An infinite-dimensional Evans function theory is developed for the elliptic eigenvalue problem associated with the stability of travelling solitary waves in a channel. Also, a bundle is constructed over the complex domain, so that its first Chern number gives the number of eigenvalues inside the domain.

The existence of travelling wave with given end points for parabolic system of nonlinear equations is proven. The nonlinear term depends also on a · x − ct where x is the multidimensional space variable, t —time, c —the speed of the wave and a —the direction of travel.

In this paper, a cellular neural network (CNN) model of FitzHugh–Nagumo equation is introduced. Dynamical behavior of this model is investigated using harmonic balance method. For the CNN model of FitzHugh–Nagumo equation, propagation of solitary waves have been proved.

We give short proofs of two classical results about the position of the extremal particle in a branching Brownian motion, one concerning the median position and another the almost sure behaviour., Published in at http://dx.doi.org/10.1214/12-AOP753 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

International audience; The paper is devoted to a reaction-diffusion equation in an infinite two-dimensional strip with nonlinear boundary conditions. The existence of travelling waves is proved in the bistable case by the Leray-Schauder method. It is based on a topological degree for elliptic problems in unbounded domains and on a priori estimates of solutions.

We are interested in the existence of travelling waves for the Benjamin-Bona-Mahony equation on a network. First we construct an explicit wave, defined in $\mathbb{R}$. Then, we use this wave to derive some conditions on the coefficients appearing in the equations and on the geometry of the network to ensure the existence of travelling waves on the network., Comment: 18 pages, 7 figures

We study positive solutions $y(u)$ for the first order differential equation $$y'=q(c\,{y}^{\frac{1}{p}}-f(u))$$ where $c>0$ is a parameter, $p>1$ and $q>1$ are conjugate numbers and $f$ is a continuous function in $[0,1]$ such that $f(0)=0=f(1)$. We shall be particularly concerned with positive solutions $y(u)$ such that $y(0)=0=y(1)$. Our motivation lies in the fact that this problem provides a model for the existence of travelling wave solutions for analogues of the FKPP equation in one space dimension, where diffusion is represented by the $p$-Laplacian operator. We obtain a theory of admissible velocities and some other features that generalize classical and recent results, established for $p=2$.

This paper considers some one dimensional reaction diffusion equations driven by a one dimensional multiplicative white noise. The existence of a stochastic travelling wave solution is established, as well as a sufficient condition to be in its domain of attraction. The arguments use stochastic ordering techniques.

In this paper we make a full analysis of the symmetry reductions of a beam equation by using the classical Lie method of infinitesimals and the nonclassical method. We consider travelling wave reductions depending on the form of an arbitrary function. We have found several new classes of solutions that have not been considered before: solutions expressed in terms of Jacobi elliptic functions, Wadati solitons and compactons. Several classes of coherent structures are displayed by some of the solutions: kinks, solitons, two humps compactons., 17 páginas

We provide a rigorous mathematical derivation of the convergence in the long-wave transonic limit of the minimizing travelling waves for the two-dimensional Gross-Pitaevskii equation towards ground states for the Kadomtsev-Petviashvili equation (KP I)., Final version published in Dynamics of Partial Differential Equations, 5, 3, 2008, p. 241-280

This is a study of two-dimensional steady periodic travelling waves on the surface of an infinitely deep irrotational ocean, when the top streamline is in contact with a membrane which has a nonlinear response to stretching and bending, and the pressure in the air above is constant. It is not supposed that the waves have small amplitude. The problem of existence of such waves is addressed using methods from the calculus of variations. The analysis involves the Hilbert transform and a Riemann-Hilbert formulation., Comment: 44 pages, 1 figure

The generalized Davey-Stewartson (GDS) equations, as derived by Babaoglu & Erbay (2004, Int. J. Non-Linear Mech., 39, 941-949), is a system of three coupled equations in (2 + 1) dimensions modelling wave propagation in an infinite elastic medium. The physical parameters (gamma, m(1), m(2), lambda and n) of the system allow one to classify the equations as elliptic-elliptic-elliptic (EEE), elliptic-elliptic-hyperbolic (EEH), elliptic-hyperbolic-hyperbolic (EHH), hyperbolic-elliptic-elliptic (HEE), hyperbolic-hyperbolic-hyperbolic (HHH) and hyperbolic-elliptic-hyperbolic (HEH) (Babaoglu et al., 2004, preprint). In this note, we only consider the EEE and HEE cases and seek travelling wave solutions to GDS systems. By deriving Pohozaev-type identities we establish some necessary conditions on the parameters for the existence of travelling waves, when solutions satisfy some integrability conditions. Using the explicit solutions given in Babaoglu & Erbay (2004) we also show that the parameter constraints must be weaker in the absence of such integrability conditions. Publisher's Version Q3 WOS:000226983100003

We propose a model for transport of solutes in a porous medium participating in a dissolution/precipitation reaction, in general not in equilibrium. For an unbounded spatial domain, travelling wave solutions exists, if and only if the charge distribution is constant and we deal with a dissolution situation. The travelling wave in fact exhibits a sharp dissolution front. The wave is given in a nearly explicit manner. Also for the limit cases of equilibrium reaction or no dispersion, travelling waves are established under the same conditions, but with different qualitative properties.

We consider the one-dimensional Kolmogorov equation driven by a particular space-time white noise term and show that there exist stochastic wavelike solutions which travel with a linear limiting speed.

Let $R_t$ be the position of the rightmost particle at time $t$ in a time-homogeneous one-dimensional branching diffusion process. Let $\gamma(\alpha,t)$ be the $\alpha$th quantile of $R_t$ under $P^0$, where $P^x$ denotes the probability measure of the branching diffusion process starting with a single particle at position $x$. We show that $\gamma(\alpha,t)$ is a limiting quantile of $R_t$ under $P^x$ in the sense that $\lim_{t \rightarrow\infty}P^x\{R_t \leq \gamma(\alpha,t)\}$ exists for all $\alpha \in (0,1)$ and all $x \in \mathbb{R}$. If the underlying diffusion is recurrent, we show that, after an appropriate rescaling of space, the $P^x$ distribution of $R_t - t$ converges weakly to a nontrivial limiting distribution $w_x$.