Dynamical systems in the theory of nonlinear waves with allowance for nonlocal interactions (the Whitham–Benjamin equation)

1994 ◽  
Vol 4 (2) ◽  
pp. 377-384
Author(s):  
V. M. Eleonsky ◽  
V. G. Korolev ◽  
N. E. Kulagin ◽  
L. P. Shil’nikov
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Waves ◽  
Nonlocal Interactions ◽  
Benjamin Equation

SOME COMMON EXPRESSIONS AND NEW BIFURCATION PHENOMENA FOR NONLINEAR WAVES IN A GENERALIZED mKdV EQUATION

2013 ◽  
Vol 23 (03) ◽  
pp. 1330007 ◽  
Author(s):  
RUI LIU ◽  
WEIFANG YAN
Keyword(s):  
Dynamical Systems ◽  
Solitary Waves ◽  
Nonlinear Waves ◽  
Blow Up ◽  
Mkdv Equation ◽  
Bifurcation Method ◽  
The Third ◽  
Bifurcation Phenomena ◽  
Kink Waves ◽  
The Common

Using the bifurcation method of dynamical systems, we study nonlinear waves in the generalized mKdV equation ut + a(1 + bu2)u2ux + uxxx = 0. (i) We obtain four types of new expressions. The first type is composed of four common expressions of the symmetric solitary waves, the kink waves and the blow-up waves. The second type includes four common expressions of the anti-symmetric solitary waves, the kink waves and the blow-up waves. The third type is made of two trigonometric expressions of periodic-blow-up waves. The fourth type is composed of two fractional expressions of 1-blow-up waves. (ii) We point out that there are two sets of kink waves which are called tall-kink waves and low-kink waves, respectively. (iii) We reveal two kinds of new bifurcation phenomena. The first phenomenon is that the low-kink waves can be bifurcated from four types of nonlinear waves, the symmetric solitary waves, blow-up waves, tall-kink waves and anti-symmetric solitary waves. The second phenomenon is that the 1-blow-up waves can be bifurcated from the periodic-blow-up waves. We also show that the common expressions include many results given by pioneers.

Bifurcations of Nonlinear Waves in the Generalized KdV–mKdV-Like Equation

2021 ◽  
Vol 31 (14) ◽  
Author(s):  
Yiren Chen ◽  
Wensheng Chen
Keyword(s):  
Dynamical Systems ◽  
Solitary Waves ◽  
Nonlinear Waves ◽  
Blow Up ◽  
Analytic Method ◽  
Periodic Waves ◽  
Bifurcation Phenomena ◽  
Kink Waves ◽  
Explicit Expressions

Using bifurcation analytic method of dynamical systems, we investigate the nonlinear waves and their bifurcations of the generalized KdV–mKdV-like equation. We obtain the following results : (i) Three types of new explicit expressions of nonlinear waves are obtained. They are trigonometric expressions, exp-function expressions, and hyperbolic expressions. (ii) Under different parameteric conditions, these expressions represent different waves, such as solitary waves, kink waves, 1-blow-up waves, 2-blow-up waves, smooth periodic waves and periodic blow-up waves. (iii) Two kinds of new interesting bifurcation phenomena are revealed. The first phenomenon is that the single-sided periodic blow-up waves can bifurcate from double-sided periodic blow-up waves. The second phenomenon is that the double-sided 1-blow-up waves can bifurcate from 2-blow-up waves. Furthermore, we show that the new expressions encompass many existing results.

scholarly journals Two-dimensional surface waves in magnetohydrodynamics

2019 ◽  
Vol 85 (4) ◽  
Author(s):  
Matthew Hunt
Keyword(s):  
Nonlinear Waves ◽  
Water Wave ◽  
Wave Model ◽  
Model Description ◽  
Kp Equation ◽  
Seminal Paper ◽  
Weakly Nonlinear ◽  
Higher Dimensional ◽  
Benjamin Equation ◽  
Dimensional Surface

The study of nonlinear waves in water has a long history beginning with the seminal paper by Korteweg & de Vries (Phil. Mag., vol. 39, 1895, p. 240) and more recently for magnetohydrodynamics Danov & Ruderman (Fluid Dyn., vol. 18, 1983, pp. 751–756). The appearance of a Hilbert transform in the nonlinear equation for magnetohydrodynamics (MHD) distinguishes it from the water wave model description. In this paper, we are interested in examining weakly nonlinear interfacial waves in $2+1$ dimensions. First, we determine the wave solution in the linear case. Next, we derive the corresponding generalisation for the Kadomtsev–Petviashvili (KP) equation with the inclusion of an equilibrium magnetic field. The derived governing equation is a generalisation of the Benjamin–Ono (BO) equation called the Benjamin equation first derived in Benjamin (J. Fluid Mech., vol. 245, 1992, pp. 401–411) and in the higher-dimensional context in Kim & Akylas (J. Fluid Mech., vol. 557, 2006, pp. 237–256).

scholarly journals New Traveling Wave Solutions and Interesting Bifurcation Phenomena of Generalized KdV-mKdV-Like Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Yiren Chen ◽  
Shaoyong Li
Keyword(s):  
Dynamical Systems ◽  
Solitary Waves ◽  
Nonlinear Waves ◽  
Traveling Wave ◽  
Blow Up ◽  
Traveling Wave Solutions ◽  
Bifurcation Method ◽  
Bifurcation Phenomenon ◽  
Wave Solutions ◽  
Bifurcation Phenomena

Using the bifurcation method of dynamical systems, we investigate the nonlinear waves and their limit properties for the generalized KdV-mKdV-like equation. We obtain the following results: (i) three types of new explicit expressions of nonlinear waves are obtained. (ii) Under different parameter conditions, we point out these expressions represent different waves, such as the solitary waves, the 1-blow-up waves, and the 2-blow-up waves. (iii) We revealed a kind of new interesting bifurcation phenomenon. The phenomenon is that the 1-blow-up waves can be bifurcated from 2-blow-up waves. Also, we gain other interesting bifurcation phenomena. We also show that our expressions include existing results.

Asymptotic methods for stochastic dynamical systems with small non-Gaussian Lévy noise

2014 ◽  
Vol 15 (01) ◽  
pp. 1550004 ◽  
Author(s):  
Huijie Qiao ◽  
Jinqiao Duan
Keyword(s):  
Dynamical Systems ◽  
Differential Equations ◽  
Numerical Simulations ◽  
Asymptotic Methods ◽  
Lévy Noise ◽  
Stochastic Dynamical Systems ◽  
Escape Probability ◽  
Nonlocal Interactions ◽  
Escape Probabilities ◽  
Non Gaussian

The goal of the paper is to analytically examine escape probabilities for dynamical systems driven by symmetric α-stable Lévy motions. Since escape probabilities are solutions of a type of integro-differential equations (i.e. differential equations with nonlocal interactions), asymptotic methods are offered to solve these equations to obtain escape probabilities when noises are sufficiently small. Three examples are presented to illustrate the asymptotic methods, and asymptotic escape probability is compared with numerical simulations.

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