scholarly journals EXISTENCE OF SOLITARY WAVES AND PERIODIC WAVES TO A PERTURBED GENERALIZED KDV EQUATION

2014 ◽  
Vol 19 (4) ◽  
pp. 537-555 ◽  
Author(s):  
Weifang Yan ◽  
Zhengrong Liu ◽  
Yong Liang
Keyword(s):  
Perturbation Theory ◽  
Solitary Waves ◽  
Wave Speed ◽  
Kdv Equation ◽  
Upper And Lower Bounds ◽  
Singular Perturbation Theory ◽  
Geometric Singular Perturbation Theory ◽  
Periodic Waves ◽  
Regular Perturbation ◽  
Generalized Kdv Equation

In this paper, the existence of solitary waves and periodic waves to a perturbed generalized KdV equation is established by applying the geometric singular perturbation theory and the regular perturbation analysis for a Hamiltonian system. Moreover, upper and lower bounds of the limit wave speed are obtained. Some previous results are extended.

scholarly journals Dynamics of solitary waves and periodic waves for a generalized KP-MEW-Burgers equation with damping

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Zengji Du ◽  
Xiaojie Lin ◽  
Yulin Ren
Keyword(s):  
Lower Bounds ◽  
Solitary Waves ◽  
Wave Speed ◽  
Burgers Equation ◽  
Systems Approach ◽  
Upper And Lower Bounds ◽  
Singular Perturbation Theory ◽  
Geometric Singular Perturbation Theory ◽  
Periodic Waves ◽  
Small Positive Parameter

<p style='text-indent:20px;'>This paper discusses the existence of solitary waves and periodic waves for a generalized (2+1)-dimensional Kadomtsev-Petviashvili modified equal width-Burgers (KP-MEW-Burgers) equation with small damping and a weak local delay convolution kernel by using the dynamical systems approach, specifically based on geometric singular perturbation theory and invariant manifold theory. Moreover, the monotonicity of the wave speed is proved by analyzing the ratio of Abelian integrals. The upper and lower bounds of the limit wave speed are given. In addition, the upper and lower bounds and monotonicity of the period <inline-formula><tex-math id="M1">\begin{document}$ T $\end{document}</tex-math></inline-formula> of traveling wave when the small positive parameter <inline-formula><tex-math id="M2">\begin{document}$ \tau\rightarrow 0 $\end{document}</tex-math></inline-formula> are also obtained. Perhaps this paper is the first discussion on the solitary waves and periodic waves for the delayed KP-MEW-Burgers equations and the Abelian integral theory may be the first application to the study of the (2+1)-dimensional equation.</p>

Periodic solutions for equations with distributed delays

2006 ◽  
Vol 136 (6) ◽  
pp. 1317-1325 ◽  
Author(s):  
Guojian Lin ◽  
Rong Yuan
Keyword(s):  
Perturbation Theory ◽  
Singular Perturbation ◽  
Periodic Solutions ◽  
General Theorem ◽  
Linear Chain ◽  
Distributed Delays ◽  
Singular Perturbation Theory ◽  
Geometric Singular Perturbation Theory ◽  
Geometric Singular Perturbation ◽  
Existence Of Periodic Solutions

A general theorem about the existence of periodic solutions for equations with distributed delays is obtained by using the linear chain trick and geometric singular perturbation theory. Two examples are given to illustrate the application of the general the general therom.

Interaction of solitary waves for the generalized KdV equation

2012 ◽  
Vol 17 (8) ◽  
pp. 3204-3218 ◽  
Author(s):  
Martin G. Garcia Alvarado ◽  
Georgii A. Omel’yanov
Keyword(s):  
Solitary Waves ◽  
Kdv Equation ◽  
Generalized Kdv Equation

Canard explosion, homoclinic and heteroclinic orbits in singularly perturbed generalist predator–prey systems

2020 ◽  
pp. 2150003
Author(s):  
Ali Atabaigi
Keyword(s):  
Perturbation Theory ◽  
Singular Perturbation ◽  
Limit Cycles ◽  
Heteroclinic Orbits ◽  
Singular Perturbation Theory ◽  
Geometric Singular Perturbation Theory ◽  
Generalist Predator ◽  
Normal Form Theory ◽  
Predator Prey ◽  
Geometric Singular Perturbation

This paper studies the dynamics of the generalist predator–prey systems modeled in [E. Alexandra, F. Lutscher and G. Seo, Bistability and limit cycles in generalist predator–prey dynamics, Ecol. Complex. 14 (2013) 48–55]. When prey reproduces much faster than predator, by combining the normal form theory of slow-fast systems, the geometric singular perturbation theory and the results near non-hyperbolic points developed by Krupa and Szmolyan [Relaxation oscillation and canard explosion, J. Differential Equations 174(2) (2001) 312–368; Extending geometric singular perturbation theory to nonhyperbolic points—fold and canard points in two dimensions, SIAM J. Math. Anal. 33(2) (2001) 286–314], we provide a detailed mathematical analysis to show the existence of homoclinic orbits, heteroclinic orbits and canard limit cycles and relaxation oscillations bifurcating from the singular homoclinic cycles. Moreover, on global stability of the unique positive equilibrium, we provide some new results. Numerical simulations are also carried out to support the theoretical results.

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