Traveling wave solutions in n-dimensional delayed nonlocal diffusion system with mixed quasimonotonicity

2014 ◽  
Vol 13 (01) ◽  
pp. 23-43
Author(s):  
Weifang Yan ◽  
Rui Liu
Keyword(s):  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Iteration Scheme ◽  
Existence Results ◽  
Nonlocal Diffusion ◽  
Volterra Model ◽  
Wave Solutions ◽  
Delayed System ◽  
Definition Of ◽  
Special Case

This paper is devoted to the study of an n-dimensional delayed system with nonlocal diffusion and mixed quasimonotonicity. By developing a new definition of upper–lower solutions and a new cross iteration scheme, we establish some existence results of traveling wave solutions. These results are applied to a nonlocal diffusion model which takes the four-species Lotka–Volterra model as its special case.

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.

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.

TRAVELING WAVE SOLUTIONS IN A THREE-SPECIES FOOD-CHAIN MODEL WITH DIFFUSION AND DELAYS

2012 ◽  
Vol 05 (01) ◽  
pp. 1250002 ◽  
Author(s):  
YANKE DU ◽  
RUI XU
Keyword(s):  
Steady State ◽  
Food Chain ◽  
Traveling Wave ◽  
Time Delays ◽  
Traveling Wave Solutions ◽  
Iteration Scheme ◽  
Traveling Wave Solution ◽  
Chain Model ◽  
Wave Solutions ◽  
Food Chain Model

This paper deals with the existence of traveling wave solutions in a three-species food-chain model with spatial diffusion and time delays due to gestation and negative feedback. By using a cross iteration scheme and Schauder's fixed point theorem, we reduce the existence of traveling wave solutions to the existence of a pair of upper-lower solutions. By constructing a pair of upper-lower solutions, we derive the existence of a traveling wave solution connecting the trivial steady state and the positive steady state. Numerical simulations are carried out to illustrate the main results. In particular, our results extend and improve some known results.

Traveling Wave Solutions to the Burgers-αβ Equations

2016 ◽  
Vol 16 (1) ◽  
pp. 147-157 ◽  
Author(s):  
Byungsoo Moon
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Traveling Wave ◽  
Burgers Equation ◽  
Nonlinear Partial Differential Equation ◽  
Traveling Wave Solutions ◽  
Partial Differential ◽  
Decay Condition ◽  
Wave Solutions ◽  
Special Case

AbstractThe Burgers-αβ equation, which was first introduced by Holm and Staley [4], is considered in the special case where ${\nu=0}$ and ${b=3}$. Traveling wave solutions are classified to the Burgers-αβ equation containing four parameters ${b,\alpha,\nu}$, and β, which is a nonintegrable nonlinear partial differential equation that coincides with the usual Burgers equation and viscous b-family of peakon equation, respectively, for two specific choices of the parameter ${\beta=0}$ and ${\beta=1}$. Under the decay condition, it is shown that there are smooth, peaked and cusped traveling wave solutions of the Burgers-αβ equation with ${\nu=0}$ and ${b=3}$ depending on the parameter β. Moreover, all traveling wave solutions without the decay condition are parametrized by the integration constant ${k_{1}\in\mathbb{R}}$. In an appropriate limit ${\beta=1}$, the previously known traveling wave solutions of the Degasperis–Procesi equation are recovered.

Traveling wave solutions of the one-dimensional Boussinesq paradigm equation

2013 ◽  
Author(s):  
V. M. Vassilev ◽  
P. A. Djondjorov ◽  
M. Ts. Hadzhilazova ◽  
I. M. Mladenov
Keyword(s):  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
One Dimensional ◽  
Wave Solutions ◽  
The One

scholarly journals Lie Group Method for Solving the Negative-Order Kadomtsev–Petviashvili Equation (nKP)

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 224
Author(s):  
Ghaylen Laouini ◽  
Amr M. Amin ◽  
Mohamed Moustafa
Keyword(s):  
Lie Group ◽  
Traveling Wave ◽  
Invariant Solution ◽  
Traveling Wave Solutions ◽  
Group Method ◽  
Lie Group Method ◽  
Reduced Equations ◽  
Wave Solutions ◽  
Negative Order ◽  
Petviashvili Equation

A comprehensive study of the negative-order Kadomtsev–Petviashvili (nKP) partial differential equation by Lie group method has been presented. Initially the infinitesimal generators and symmetry reduction, which were obtained by applying the Lie group method on the negative-order Kadomtsev–Petviashvili equation, have been used for constructing the reduced equations. In particular, the traveling wave solutions for the negative-order KP equation have been derived from the reduced equations as an invariant solution. Finally, the extended improved (G′/G) method and the extended tanh method are described and applied in constructing new explicit expressions for the traveling wave solutions. Many new and more general exact solutions are obtained.

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