scholarly journals A variational problem associated with the minimal speed of traveling waves for spatially periodic KPP type equations

2019 ◽  
Vol 119 (3) ◽  
pp. 654-680 ◽  
Author(s):  
Ryunosuke Mori ◽  
Dongyuan Xiao
Keyword(s):  
Variational Problem ◽  
Traveling Waves ◽  
Minimal Speed ◽  
Spatially Periodic

scholarly journals Hyperbolic traveling waves driven by growth

2014 ◽  
Vol 24 (06) ◽  
pp. 1165-1195 ◽  
Author(s):  
Emeric Bouin ◽  
Vincent Calvez ◽  
Grégoire Nadin
Keyword(s):  
Nonlinear Stability ◽  
Traveling Waves ◽  
Hyperbolic Model ◽  
Kpp Equation ◽  
Transition Parameter ◽  
Minimal Speed ◽  
Traveling Fronts ◽  
Traveling Front ◽  
Maximal Speed ◽  
Existence And Stability

We perform the analysis of a hyperbolic model which is the analog of the Fisher-KPP equation. This model accounts for particles that move at maximal speed ϵ-1 (ϵ > 0), and proliferate according to a reaction term of monostable type. We study the existence and stability of traveling fronts. We exhibit a transition depending on the parameter ϵ: for small ϵ the behavior is essentially the same as for the diffusive Fisher-KPP equation. However, for large ϵ the traveling front with minimal speed is discontinuous and travels at the maximal speed ϵ-1. The traveling fronts with minimal speed are linearly stable in weighted L2 spaces. We also prove local nonlinear stability of the traveling front with minimal speed when ϵ is smaller than the transition parameter.

Strongly Nonlinear Spatially Periodic Traveling Waves in Granular Dimer Chains

2012 ◽  
Author(s):  
K. R. Jayaprakash ◽  
Alexander F. Vakakis ◽  
Yuli Starosvetsky
Keyword(s):  
Traveling Waves ◽  
Periodic Boundary Conditions ◽  
Mass Ratio ◽  
Efficient Energy ◽  
Periodic Traveling Waves ◽  
Strongly Nonlinear ◽  
Spatial Periodicity ◽  
Spatially Periodic ◽  
New Formulation ◽  
Primary Pulse

In the present work we study the dynamics of spatially periodic traveling waves in granular 1:1 (each bead is followed and preceded by a bead of different mass and/or stiffness) dimer chain with no pre-compression. The dynamics of a 1:1 dimer chain is governed by a single parameter, the mass ratio of the two beads forming each dimer pair of the chain. In particular, we demonstrate numerically the formation of special families of traveling waves with spatially periodic waveforms that are realized in semi-infinite dimer chains with the application of an arbitrary impulse. These traveling waves were first observed in the form of oscillatory tails in the trail of the propagating primary pulse. The energy radiated by the propagating primary pulse manifests in the form of traveling waves of varying spatial periodicity depending on the mass ratio. These traveling waves depend only on the mass ratio and are rescalable with respect to any arbitrary applied energy. The dynamics of these families of traveling waves is systematically studied by considering finite dimer chains (termed the ‘reduced systems’) subject to periodic boundary conditions. We demonstrate that these waves may exhibit interesting bifurcations or loss of stability as the system parameter varies. In turn, these bifurcations and stability exchanges in infinite dimer chains are correlated to previous studies of pulse attenuation in finite dimer chains through efficient energy radiation from the propagating pulse to the far field, mainly in the form of traveling waves. Based on these results a new formulation of attenuation and propagation zones (stop and pass bands) in semi-infinite granular dimer chains is proposed.

scholarly journals Monotone traveling waves for delayed neural field equations

2016 ◽  
Vol 26 (10) ◽  
pp. 1919-1954 ◽  
Author(s):  
Jian Fang ◽  
Grégory Faye
Keyword(s):  
Traveling Waves ◽  
Traveling Wave ◽  
Level Sets ◽  
Single Layer ◽  
Traveling Wave Solutions ◽  
Neural Field ◽  
Field Equations ◽  
Minimal Speed ◽  
Wave Solutions ◽  
Neural Field Equations

We study the existence of traveling wave solutions and spreading properties for single-layer delayed neural field equations. We focus on the case where the kinetic dynamics are of monostable type and characterize the invasion speeds as a function of the asymptotic decay of the connectivity kernel. More precisely, we show that for exponentially bounded kernels the minimal speed of traveling waves exists and coincides with the spreading speed, which further can be explicitly characterized under a KPP type condition. We also investigate the case of algebraically decaying kernels where we prove the non-existence of traveling wave solutions and show the level sets of the solutions eventually locate in-between two exponential functions of time. The uniqueness of traveling waves modulo translation is also obtained.

scholarly journals Existence of Periodic Traveling Waves in the Klausmeier Model of Desertification and its Modification: A Comparative Study

2019 ◽  
Vol 38 ◽  
pp. 27-46
Author(s):  
Md AS Howlader ◽  
Md Ariful Islam Arif ◽  
LS Andallah ◽  
M Osman Gani
Keyword(s):  
Traveling Waves ◽  
Arid Ecosystems ◽  
Vegetation Patterns ◽  
Transport Parameter ◽  
Variable Model ◽  
Periodic Traveling Waves ◽  
Self Organized ◽  
Spatially Periodic ◽  
Semi Arid ◽  
Banded Vegetation

Self-organized and spatially periodic banded vegetation patterns have been observed in many semi-arid ecosystems. In order to understand the mechanism of these patterns, we consider a system of reaction-advection-diffusion equations in a two-variable model of desertification. This work deals with the investigation of the existence of periodic traveling waves in a one-parameter family of solutions. In addition, we investigate the existence of periodic traveling waves as a function of water transport parameter in the model. GANIT J. Bangladesh Math. Soc.Vol. 38 (2018) 27-46

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