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 Some Explicit Expressions and Interesting Bifurcation Phenomena for Nonlinear Waves in Generalized Zakharov Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-19 ◽  
Author(s):  
Shaoyong Li ◽  
Rui Liu
Keyword(s):  
Solitary Waves ◽  
Nonlinear Waves ◽  
Blow Up ◽  
Complex Function ◽  
Periodic Waves ◽  
Bifurcation Method ◽  
Bifurcation Phenomena ◽  
Kink Waves ◽  
Fourth Kind ◽  
Explicit Expressions

Using bifurcation method of dynamical systems, we investigate the nonlinear waves for the generalized Zakharov equationsutt-cs2uxx=β(|E|2)xx,  iEt+αExx-δ1uE+δ2|E|2E+δ3|E|4E=0,whereα,β,δ1,δ2,δ3, andcsare real parameters,E=E(x,t)is a complex function, andu=u(x,t)is a real function. We obtain the following results. (i) Three types of explicit expressions of nonlinear waves are obtained, that is, the fractional expressions, the trigonometric expressions, and the exp-function expressions. (ii) Under different parameter conditions, these expressions represent symmetric and antisymmetric solitary waves, kink and antikink waves, symmetric periodic and periodic-blow-up waves, and 1-blow-up and 2-blow-up waves. We point out that there are two sets of kink waves which are called tall-kink waves and low-kink waves, respectively. (iii) Five kinds of interesting bifurcation phenomena are revealed. The first kind is that the 1-blow-up waves can be bifurcated from the periodic-blow-up and 2-blow-up waves. The second kind is that the 2-blow-up waves can be bifurcated from the periodic-blow-up waves. The third kind is that the symmetric solitary waves can be bifurcated from the symmetric periodic waves. The fourth kind is that the low-kink waves can be bifurcated from four types of nonlinear waves, the symmetric solitary waves, the 1-blow-up waves, the tall-kink waves, and the antisymmetric solitary waves. The fifth kind is that the tall-kink waves can be bifurcated from the symmetric periodic waves. We also show that the exp-function expressions include some results given by pioneers.

scholarly journals Bifurcation Phenomena of Nonlinear Waves in a Generalized Zakharov-Kuznetsov Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-14
Author(s):  
Yun Wu ◽  
Zhengrong Liu
Keyword(s):  
Solitary Waves ◽  
Nonlinear Waves ◽  
Blow Up ◽  
Periodic Waves ◽  
The Third ◽  
Bifurcation Phenomena ◽  
Kink Waves ◽  
Fourth Kind

We study the bifurcation phenomena of nonlinear waves described by a generalized Zakharov-Kuznetsov equationut+au2+bu4ux+γuxxx+δuxyy=0. We reveal four kinds of interesting bifurcation phenomena. The first kind is that the low-kink waves can be bifurcated from the symmetric solitary waves, the 1-blow-up waves, the tall-kink waves, and the antisymmetric solitary waves. The second kind is that the 1-blow-up waves can be bifurcated from the periodic-blow-up waves, the symmetric solitary waves, and the 2-blow-up waves. The third kind is that the periodic-blow-up waves can be bifurcated from the symmetric periodic waves. The fourth kind is that the tall-kink waves can be bifurcated from the symmetric periodic 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.

scholarly journals Some Interesting Bifurcations of Nonlinear Waves for the Generalized Drinfel’d-Sokolov System

2014 ◽  
Vol 2014 ◽  
pp. 1-20
Author(s):  
Huixian Cai ◽  
Chaohong Pan ◽  
Zhengrong Liu
Keyword(s):  
Solitary Waves ◽  
Nonlinear Waves ◽  
Periodic Waves ◽  
Bifurcation Phenomena ◽  
Kink Waves

We study the bifurcations of nonlinear waves for the generalized Drinfel’d-Sokolov systemut+(vm)x=0,vt+a(vn)xxx+buxv+cuvx=0calledD(m,n)system. We reveal some interesting bifurcation phenomena as follows. (1) ForD(2,1)system, the fractional solitary waves can be bifurcated from the trigonometric periodic waves and the elliptic periodic waves, and the kink waves can be bifurcated from the solitary waves and the singular waves. (2) ForD(1,2)system, the compactons can be bifurcated from the solitary waves, and the peakons can be bifurcated from the solitary waves and the singular cusp waves. (3) ForD(2,2)system, the solitary waves can be bifurcated from the smooth periodic waves and the singular periodic waves.

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.

scholarly journals New Types of Nonlinear Waves and Bifurcation Phenomena in Schamel-Korteweg-de Vries Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-18 ◽  
Author(s):  
Yun Wu ◽  
Zhengrong Liu
Keyword(s):  
Solitary Waves ◽  
Nonlinear Waves ◽  
Blow Up ◽  
Vries Equation ◽  
Periodic Wave ◽  
Bifurcation Phenomena ◽  
De Vries ◽  
Kink Waves

We study the nonlinear waves described by Schamel-Korteweg-de Vries equationut+au1/2+buux+δuxxx=0. Two new types of nonlinear waves called compacton-like waves and kink-like waves are displayed. Furthermore, two kinds of new bifurcation phenomena are revealed. The first phenomenon is that the kink waves can be bifurcated from five types of nonlinear waves which are the bell-shape solitary waves, the blow-up waves, the valley-shape solitary waves, the kink-like waves, and the compacton-like waves. The second phenomenon is that the periodic-blow-up wave can be bifurcated from the smooth periodic wave.

scholarly journals The Bounded Traveling Wave Solutions of a (3+1) Dimensions mKdv-ZK Equation

2020 ◽  
Author(s):  
Yuzhong Zhang
Keyword(s):  
Solitary Waves ◽  
Traveling Waves ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Periodic Waves ◽  
Zk Equation ◽  
Wave Solutions ◽  
Fixed Parameter ◽  
Kink Waves ◽  
Parameter Condition

The bounded traveling waves solutions of a (3+1) dimensions mKdv-ZK equation be investigated by using method of dynamical systems. The exact expressions of bounded periodic waves, solitary waves and kink waves are given. Under fixed parameter condition, the planar simulation graphs of the bounded periodic waves, solitary waves and kinks are obtained by using the software Mathematica 7.

scholarly journals New Exact Solutions for the(2+1)-Dimensional Broer-Kaup-Kupershmidt Equations

2010 ◽  
Vol 2010 ◽  
pp. 1-9 ◽  
Author(s):  
Ming Song ◽  
Shaoyong Li ◽  
Jun Cao
Keyword(s):  
Dynamical Systems ◽  
Solitary Wave ◽  
Exact Solutions ◽  
Blow Up ◽  
Qualitative Theory ◽  
Solitary Wave Solutions ◽  
Bifurcation Method ◽  
New Exact Solutions ◽  
Wave Solutions ◽  
Explicit Expressions

We investigate the(2+1)-dimensional Broer-Kaup-Kupershmidt equations. Some explicit expressions of solutions for the equations are obtained by using the bifurcation method and qualitative theory of dynamical systems. These solutions contain kink-shaped solutions, blow-up solutions, periodic blow-up solutions, and solitary wave solutions. Some previous results are extended.

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