Asymptotics and Uniqueness of Travelling Waves for Non-Monotone Delayed Systems on 2D Lattices

2013 ◽  
Vol 56 (3) ◽  
pp. 659-672 ◽  
Author(s):  
Zhi-Xian Yu ◽  
Ming Mei
Keyword(s):  
Asymptotic Behavior ◽  
Traveling Waves ◽  
Travelling Waves ◽  
A Priori ◽  
Travelling Wave ◽  
Delayed Systems

Abstract.We establish asymptotics and uniqueness (up to translation) of travelling waves for delayed 2D lattice equations with non-monotone birth functions. First, with the help of Ikehara’s Theorem, the a priori asymptotic behavior of travelling wave is exactly derived. Then, based on the obtained asymptotic behavior, the uniqueness of the traveling waves is proved. These results complement earlier results in the literature.

scholarly journals Travelling waves for diffusive and strongly competitive systems: Relative motility and invasion speed

2015 ◽  
Vol 26 (4) ◽  
pp. 521-534 ◽  
Author(s):  
LÉO GIRARDIN ◽  
GRÉGOIRE NADIN
Keyword(s):  
Interspecific Competition ◽  
Travelling Waves ◽  
A Priori ◽  
Travelling Wave ◽  
Volterra System ◽  
Multiplicative Constant ◽  
Competitive Systems ◽  
Invading Species ◽  
Homogeneous Domains

Our interest here is to find the invader in a two species, diffusive and competitive Lotka–Volterra system in the particular case of travelling wave solutions. We investigate the role of diffusion in homogeneous domains. We might expect a priori two different cases: strong interspecific competition and weak interspecific competition. In this paper, we study the first one and obtain a clear conclusion: the invading species is, up to a fixed multiplicative constant, the more diffusive one.

scholarly journals On travelling wave solutions of a model of a liquid film flowing down a fibre

2021 ◽  
pp. 1-30
Author(s):  
HANGJIE JI ◽  
ROMAN TARANETS ◽  
MARINA CHUGUNOVA
Keyword(s):  
Thin Film ◽  
Weak Solutions ◽  
A Priori Estimates ◽  
Travelling Waves ◽  
A Priori ◽  
Travelling Wave ◽  
Model Parameters ◽  
Priori Estimates ◽  
Thin Film Model ◽  
Long Time

Abstract Existence of non-negative weak solutions is shown for a full curvature thin-film model of a liquid thin film flowing down a vertical fibre. The proof is based on the application of a priori estimates derived for energy-entropy functionals. Long-time behaviour of these weak solutions is analysed and, under some additional constraints for the model parameters and initial values, convergence towards a travelling wave solution is obtained. Numerical studies of energy minimisers and travelling waves are presented to illustrate analytical results.

scholarly journals Stability of periodic waves for the fractional KdV and NLS equations

2020 ◽  
pp. 1-33
Author(s):  
Sevdzhan Hakkaev ◽  
Atanas G. Stefanov
Keyword(s):  
Travelling Waves ◽  
A Priori ◽  
Travelling Wave ◽  
Technical Condition ◽  
Travelling Wave Solution ◽  
Periodic Waves ◽  
The Cauchy Problem ◽  
The Waves ◽  
Well Posed ◽  
Two Parameter

We consider the focussing fractional periodic Korteweg–deVries (fKdV) and fractional periodic non-linear Schrödinger equations (fNLS) equations, with L2 sub-critical dispersion. In particular, this covers the case of the periodic KdV and Benjamin-Ono models. We construct two parameter family of bell-shaped travelling waves for KdV (standing waves for NLS), which are constrained minimizers of the Hamiltonian. We show in particular that for each $\lambda \gt 0$ , there is a travelling wave solution to fKdV and fNLS $\phi : \|\phi \|_{L^2[-T,T]}^2=\lambda $ , which is non-degenerate. We also show that the waves are spectrally stable and orbitally stable, provided the Cauchy problem is locally well-posed in Hα/2[ − T, T] and a natural technical condition. This is done rigorously, without any a priori assumptions on the smoothness of the waves or the Lagrange multipliers.

Travelling Wave Control of Rotating Discs and Analysis of its Spillover Effect

1988 ◽  
Vol 202 (2) ◽  
pp. 119-127 ◽  
Author(s):  
C-S Kim ◽  
C-W Lee
Keyword(s):  
Travelling Waves ◽  
Spillover Effect ◽  
Polynomial Equation ◽  
Travelling Wave ◽  
Rotating Disc ◽  
System A ◽  
Finite Dimensional ◽  
Control Scheme ◽  
Actuator System ◽  
Wave Control

A modal control scheme for rotating disc systems is developed based upon the finite-dimensional sub-system model including a few lower backward travelling waves important to the disc response. For the single discrete sensor and actuator system, a polynomial equation, which determines the closed-loop system poles, is derived and the spillover effect is analysed, providing a sufficient condition for stability. Finally, simulation studies are performed to show the effectiveness of the travelling wave control scheme proposed.

TRAVELLING WAVES AND OSCILLATIONS IN SAL’NIKOV’S COMBUSTION REACTION IN A COMPRESSIBLE GAS

2015 ◽  
Vol 56 (3) ◽  
pp. 233-247 ◽  
Author(s):  
RHYS A. PAUL ◽  
LAWRENCE K. FORBES
Keyword(s):  
Asymptotic Solution ◽  
Multiple Scales ◽  
Travelling Waves ◽  
Method Of Lines ◽  
Travelling Wave ◽  
Reaction Scheme ◽  
Compressible Viscous Gas ◽  
The Method Of Lines ◽  
Temperature Waves ◽  
Parameter Values

We consider a two-step Sal’nikov reaction scheme occurring within a compressible viscous gas. The first step of the reaction may be either endothermic or exothermic, while the second step is strictly exothermic. Energy may also be lost from the system due to Newtonian cooling. An asymptotic solution for temperature perturbations of small amplitude is presented using the methods of strained coordinates and multiple scales, and a travelling wave solution with a sech-squared profile is derived. The method of lines is then used to approximate the full system with a set of ordinary differential equations, which are integrated numerically to track accurately the evolution of the reaction front. This numerical method is used to verify the asymptotic solution and investigate behaviours under different conditions. Using this method, temperature waves progressing as pulsatile fronts are detected at appropriate parameter values.

Nonlinear travelling internal waves with piecewise-linear shear profiles

2018 ◽  
Vol 856 ◽  
pp. 984-1013 ◽  
Author(s):  
K. L. Oliveras ◽  
C. W. Curtis
Keyword(s):  
Piecewise Linear ◽  
Travelling Waves ◽  
Tangential Velocity ◽  
Travelling Wave ◽  
Numerical Continuation ◽  
Travelling Wave Solutions ◽  
Water Wave Problem ◽  
Two Fluids ◽  
Wave Solutions ◽  
Linear Shear

In this work, we study the nonlinear travelling waves in density stratified fluids with piecewise-linear shear currents. Beginning with the formulation of the water-wave problem due to Ablowitz et al. (J. Fluid Mech., vol. 562, 2006, pp. 313–343), we extend the work of Ashton & Fokas (J. Fluid Mech., vol. 689, 2011, pp. 129–148) and Haut & Ablowitz (J. Fluid Mech., vol. 631, 2009, pp. 375–396) to examine the interface between two fluids of differing densities and varying linear shear. We derive a systems of equations depending only on variables at the interface, and numerically solve for periodic travelling wave solutions using numerical continuation. Here, we consider only branches which bifurcate from solutions where there is no slip in the tangential velocity at the interface for the trivial flow. The spectral stability of these solutions is then determined using a numerical Fourier–Floquet technique. We find that the strength of the linear shear in each fluid impacts the stability of the corresponding travelling wave solutions. Specifically, opposing shears may amplify or suppress instabilities.

scholarly journals Chaotic period doubling

2009 ◽  
Vol 29 (2) ◽  
pp. 381-418 ◽  
Author(s):  
V. V. M. S. CHANDRAMOULI ◽  
M. MARTENS ◽  
W. DE MELO ◽  
C. P. TRESSER
Keyword(s):  
Fixed Point ◽  
Asymptotic Behavior ◽  
Chaotic Dynamics ◽  
A Priori ◽  
Period Doubling ◽  
Small Scale ◽  
Unimodal Maps ◽  
One Dimensional ◽  
Renormalization Operator ◽  
Renormalization Theory

AbstractThe period doubling renormalization operator was introduced by Feigenbaum and by Coullet and Tresser in the 1970s to study the asymptotic small-scale geometry of the attractor of one-dimensional systems that are at the transition from simple to chaotic dynamics. This geometry turns out not to depend on the choice of the map under rather mild smoothness conditions. The existence of a unique renormalization fixed point that is also hyperbolic among generic smooth-enough maps plays a crucial role in the corresponding renormalization theory. The uniqueness and hyperbolicity of the renormalization fixed point were first shown in the holomorphic context, by means that generalize to other renormalization operators. It was then proved that, in the space ofC2+αunimodal maps, forα>0, the period doubling renormalization fixed point is hyperbolic as well. In this paper we study what happens when one approaches from below the minimal smoothness thresholds for the uniqueness and for the hyperbolicity of the period doubling renormalization generic fixed point. Indeed, our main result states that in the space ofC2unimodal maps the analytic fixed point is not hyperbolic and that the same remains true when adding enough smoothness to geta prioribounds. In this smoother class, calledC2+∣⋅∣, the failure of hyperbolicity is tamer than inC2. Things get much worse with just a bit less smoothness thanC2, as then even the uniqueness is lost and other asymptotic behavior becomes possible. We show that the period doubling renormalization operator acting on the space ofC1+Lipunimodal maps has infinite topological entropy.

Existence of bistable waves for a nonlocal and nonmonotone reaction-diffusion equation

2019 ◽  
Vol 150 (2) ◽  
pp. 721-739
Author(s):  
Sergei Trofimchuk ◽  
Vitaly Volpert
Keyword(s):  
Diffusion Equation ◽  
A Priori Estimates ◽  
Travelling Waves ◽  
A Priori ◽  
Unbounded Domains ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Nonlocal Nonlinearity ◽  
Estimates Of Solutions ◽  
Priori Estimates

AbstractReaction-diffusion equation with a bistable nonlocal nonlinearity is considered in the case where the reaction term is not quasi-monotone. For this equation, the existence of travelling waves is proved by the Leray-Schauder method based on the topological degree for elliptic operators in unbounded domains and a priori estimates of solutions in properly chosen weighted spaces.

scholarly journals Nonuniform Continuity of the Osmosis K(2, 2) Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Aiyong Chen ◽  
Yong Ding ◽  
Wentao Huang
Keyword(s):  
Differential Equations ◽  
Small Amplitude ◽  
Travelling Waves ◽  
Travelling Wave ◽  
Qualitative Theory ◽  
Travelling Wave Solutions ◽  
Solution Map ◽  
Wave Solutions ◽  
Periodic Travelling Wave ◽  
Uniformly Continuous

The qualitative theory of differential equations is applied to the osmosis K(2, 2) equation. The parametric conditions of existence of the smooth periodic travelling wave solutions are given. We show that the solution map is not uniformly continuous by using the theory of Himonas and Misiolek. The proof relies on a construction of smooth periodic travelling waves with small amplitude.

scholarly journals FKPP equation with impulses on unbounded domain

2017 ◽  
Vol 1 ◽  
pp. 1 ◽  
Author(s):  
Valaire Yatat ◽  
Yves Dumont
Keyword(s):  
Travelling Waves ◽  
Reaction Diffusion ◽  
Bare Soil ◽  
Travelling Wave ◽  
Reaction Diffusion Equations ◽  
Spreading Speed ◽  
Systems Dynamics ◽  
Travelling Wave Solutions ◽  
Minimal Speed ◽  
Wave Solutions

This paper deals with the problem of travelling wave solutions in a scalar impulsive FKPP-like equation. It is a first step of a more general study that aims to address existence of travelling wave solutions for systems of impulsive reaction-diffusion equations that model ecological systems dynamics such as fire-prone savannas. Using results on scalar recursion equations, we show existence of populated vs. extinction travelling waves invasion and compute an explicit expression of their spreading speed (characterized as the minimal speed of such travelling waves). In particular, we find that the spreading speed explicitly depends on the time between two successive impulses. In addition, we carry out a comparison with the case of time-continuous events. We also show that depending on the time between two successive impulses, the spreading speed with pulse events could be lower, equal or greater than the spreading speed in the case of time-continuous events. Finally, we apply our results to a model of fire-prone grasslands and show that pulse fires event may slow down the grassland vs. bare soil invasion speed.

Sign in / Sign up

Export Citation Format

Share Document