scholarly journals Traveling Waves in the Underdamped Frenkel–Kontorova Model

2018 ◽  
Vol 2018 ◽  
pp. 1-9
Author(s):  
Hengyan Li ◽  
Shaowei Liu
Keyword(s):  
Boundary Condition ◽  
Traveling Waves ◽  
Traveling Wave ◽  
Periodic Boundary Condition ◽  
Periodic Boundary ◽  
Traveling Wave Solutions ◽  
Iteration Scheme ◽  
Moser Iteration ◽  
Wave Solutions ◽  
Kontorova Model

This paper studies a damped Frenkel–Kontorova model with periodic boundary condition. By using Nash–Moser iteration scheme, we prove that such model has a family of smooth traveling wave solutions.

scholarly journals Existence of Global Solution and Traveling Wave of the Modified Short-Wave Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Hyungjin Huh
Keyword(s):  
Boundary Condition ◽  
Wave Equation ◽  
Traveling Wave ◽  
Periodic Boundary Condition ◽  
Periodic Boundary ◽  
Traveling Wave Solutions ◽  
Finite Energy ◽  
Short Wave ◽  
Existence Of Solution ◽  
Wave Solutions

The modified short-wave equation is considered under periodic boundary condition. We prove the global existence of solution with finite energy. We also find traveling wave solutions which is the form of elliptic function.

Numerical Study of Periodic Traveling Wave Solutions for the Predator–Prey Model with Landscape Features

2015 ◽  
Vol 25 (09) ◽  
pp. 1550117 ◽  
Author(s):  
Ana Yun ◽  
Jaemin Shin ◽  
Yibao Li ◽  
Seunggyu Lee ◽  
Junseok Kim
Keyword(s):  
Boundary Condition ◽  
Traveling Waves ◽  
Traveling Wave ◽  
Dirichlet Boundary ◽  
Numerical Technique ◽  
Traveling Wave Solutions ◽  
Predator Prey ◽  
Landscape Features ◽  
Wave Solutions ◽  
Periodic Traveling Wave

We numerically investigate periodic traveling wave solutions for a diffusive predator–prey system with landscape features. The landscape features are modeled through the homogeneous Dirichlet boundary condition which is imposed at the edge of the obstacle domain. To effectively treat the Dirichlet boundary condition, we employ a robust and accurate numerical technique by using a boundary control function. We also propose a robust algorithm for calculating the numerical periodicity of the traveling wave solution. In numerical experiments, we show that periodic traveling waves which move out and away from the obstacle are effectively generated. We explain the formation of the traveling waves by comparing the wavelengths. The spatial asynchrony has been shown in quantitative detail for various obstacles. Furthermore, we apply our numerical technique to the complicated real landscape features.

TRAVELING WAVES IN A REACTION-DIFFUSION PREDATOR-PREY SYSTEM WITH NONLOCAL DELAYS

2012 ◽  
Vol 05 (05) ◽  
pp. 1250043
Author(s):  
ZHE LI ◽  
RUI XU
Keyword(s):  
Traveling Waves ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Reaction Diffusion ◽  
Iteration Scheme ◽  
Monotonicity Condition ◽  
Predator Prey ◽  
Wave Solutions ◽  
Prey System ◽  
Theoretical Results

This paper is concerned with the existence of traveling wave solutions in a reaction-diffusion predator-prey system with nonlocal delays. By introducing a partially exponential quasi-monotonicity condition and a new cross iteration scheme, we reduce the existence of traveling wave solutions to the existence of a pair of upper-lower solutions. By constructing a desirable pair of upper-lower solutions, we establish the existence of traveling wave solutions. Finally, some numerical examples are carried out to illustrate the theoretical results.

STABILITY OF STEADY STATE AND TRAVELING WAVES SOLUTIONS IN COUPLED MAP LATTICES

2008 ◽  
Vol 18 (01) ◽  
pp. 219-225 ◽  
Author(s):  
DANIEL TURZÍK ◽  
MIROSLAVA DUBCOVÁ
Keyword(s):  
Steady State ◽  
Essential Spectrum ◽  
Traveling Waves ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Linear Operators ◽  
Coupled Map Lattices ◽  
Basic Tool ◽  
Wave Solutions ◽  
The Stability

We determine the essential spectrum of certain types of linear operators which arise in the study of the stability of steady state or traveling wave solutions in coupled map lattices. The basic tool is the Gelfand transformation which enables us to determine the essential spectrum completely.

On the existence of traveling wave solutions and upper and lower bounds for some Fisher–Kolmogorov type equations

2014 ◽  
Vol 07 (05) ◽  
pp. 1450050 ◽  
Author(s):  
Juan Belmonte-Beitia
Keyword(s):  
Mathematical Biology ◽  
Dynamical Systems ◽  
Lower Bounds ◽  
Traveling Waves ◽  
Traveling Wave ◽  
Systems Approach ◽  
Traveling Wave Solutions ◽  
Upper And Lower Bounds ◽  
Kolmogorov Type ◽  
Wave Solutions

In this paper, we use a dynamical systems approach to prove the existence of traveling waves solutions for the Fisher–Kolmogorov density-dependent equation. Moreover, we prove the existence of upper and lower bounds for these traveling wave solutions found previously. Finally, we present a particular example which has several applications in the mathematical biology field.

scholarly journals Bounded Traveling Wave Solutions and Their Relations for the Generalized HD Type Equation

2016 ◽  
Vol 2016 ◽  
pp. 1-15
Author(s):  
Qing Meng ◽  
Bin He
Keyword(s):  
Solitary Wave ◽  
Traveling Waves ◽  
Traveling Wave ◽  
Sufficient Conditions ◽  
Traveling Wave Solutions ◽  
Type Equation ◽  
Periodic Wave ◽  
Point Of View ◽  
Bifurcation Method ◽  
Wave Solutions

The generalized HD type equation is studied by using the bifurcation method of dynamical systems. From a dynamic point of view, the existence of different kinds of traveling waves which include periodic loop soliton, periodic cusp wave, smooth periodic wave, loop soliton, cuspon, smooth solitary wave, and kink-like wave is proved and the sufficient conditions to guarantee the existence of the above solutions in different regions of the parametric space are given. Also, all possible exact parametric representations of the bounded waves are presented and their relations are stated.

EXACT SOLUTIONS AND THEIR DYNAMICS OF TRAVELING WAVES IN THREE TYPICAL NONLINEAR WAVE EQUATIONS

2009 ◽  
Vol 19 (07) ◽  
pp. 2249-2266 ◽  
Author(s):  
JIBIN LI ◽  
YI ZHANG ◽  
GUANRONG CHEN
Keyword(s):  
Nonlinear Wave ◽  
Traveling Waves ◽  
Traveling Wave ◽  
Wave Equations ◽  
Visual Illusion ◽  
Soliton Solutions ◽  
Traveling Wave Solutions ◽  
Dynamical Analysis ◽  
Nonlinear Wave Equations ◽  
Wave Solutions

It was reported in the literature that some nonlinear wave equations have the so-called loop- and inverted-loop-soliton solutions, as well as the so-called loop-periodic solutions. Are these true mathematical solutions or just numerical artifacts? To answer the question, this article investigates all traveling wave solutions in the parameter space for three typical nonlinear wave equations from a theoretical viewpoint of dynamical systems. Dynamical analysis shows that all these loop- and inverted-loop-solutions are merely visual illusion of numerical artifacts. To reveal the nature of such special phenomena, this article also offers the mathematical parametric representations of these traveling wave solutions precisely in analytic forms.

scholarly journals Orbital Stability of Solitary Traveling Waves of Moderate Amplitude

2015 ◽  
Vol 2015 ◽  
pp. 1-7
Author(s):  
Zhengyong Ouyang
Keyword(s):  
Traveling Waves ◽  
Traveling Wave ◽  
Water Regime ◽  
Orbital Stability ◽  
Traveling Wave Solutions ◽  
Stability Theorem ◽  
Hamiltonian Form ◽  
Free Surface Waves ◽  
Wave Solutions ◽  
Invariants Of Motion

We consider the orbital stability of solitary traveling wave solutions of an equation describing the free surface waves of moderate amplitude in the shallow water regime. Firstly, we rewrite this equation in Hamiltonian form and construct two invariants of motion. Then using the abstract stability theorem of solitary waves proposed by Grillakis et al. (1987), we prove that the solitary traveling waves of the equation under consideration are orbital stable.

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.

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