Dynamical Systems and Nonlinear Waves in Plasmas

2021 ◽  
Author(s):  
Asit Saha ◽  
Santo Banerjee
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Waves

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 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.

Nonlinear Waves, Solitons and Chaos

2000 ◽  
Author(s):  
Eryk Infeld ◽  
George Rowlands
Keyword(s):  
Nonlinear Waves
Sign in / Sign up

Export Citation Format

Share Document