scholarly journals The Bifurcations of Traveling Wave Solutions of the Kundu Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Yating Yi ◽  
Zhengrong Liu
Keyword(s):  
Dynamical Systems ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Bifurcation Method ◽  
Wave Solutions

We use the bifurcation method of dynamical systems to study the bifurcations of traveling wave solutions for the Kundu equation. Various explicit traveling wave solutions and their bifurcations are obtained. Via some special phase orbits, we obtain some new explicit traveling wave solutions. Our work extends some previous results.

scholarly journals Traveling Wave Solutions for the Generalized Zakharov Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Ming Song ◽  
Zhengrong Liu
Keyword(s):  
Dynamical Systems ◽  
Solitary Wave ◽  
Traveling Wave ◽  
Blow Up ◽  
Traveling Wave Solutions ◽  
Periodic Wave ◽  
Solitary Wave Solutions ◽  
Bifurcation Method ◽  
Periodic Wave Solutions ◽  
Wave Solutions

We use the bifurcation method of dynamical systems to study the traveling wave solutions for the generalized Zakharov equations. A number of traveling wave solutions are obtained. Those solutions contain explicit periodic wave solutions, periodic blow-up wave solutions, unbounded wave solutions, kink profile solitary wave solutions, and solitary wave solutions. Relations of the traveling wave solutions are given. Some previous results are extended.

scholarly journals Application of Bifurcation Method to the Generalized Zakharov Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-8 ◽  
Author(s):  
Ming Song
Keyword(s):  
Dynamical Systems ◽  
Solitary Wave ◽  
Traveling Wave ◽  
Blow Up ◽  
Traveling Wave Solutions ◽  
Solitary Wave Solutions ◽  
Bifurcation Method ◽  
Wave Solutions

We use the bifurcation method of dynamical systems to study the traveling wave solutions for the generalized Zakharov equations. A number of traveling wave solutions are obtained. Those solutions contain explicit periodic blow-up wave solutions and solitary wave solutions.

scholarly journals Further Results on the Traveling Wave Solutions for an Integrable Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Chaohong Pan ◽  
Zhengrong Liu
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Equation ◽  
Traveling Wave ◽  
Kdv Equation ◽  
Traveling Wave Solutions ◽  
Integrable Equation ◽  
Bifurcation Method ◽  
Generalized Kdv Equation ◽  
Wave Solutions ◽  
Relationship Of

The objective of this paper is to extend some results of pioneers for the nonlinear equationmt=(1/2)(1/mk)xxx−(1/2)(1/mk)xintroduced by Qiao. The equivalent relationship of the traveling wave solutions between the integrable equation and the generalized KdV equation is revealed. Moreover, whenk=−(p/q)  (p≠qandp,q∈ℤ+), we obtain some explicit traveling wave solutions by the bifurcation method of dynamical systems.

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 Exact Traveling Wave Solutions and Bifurcation of a Generalized (3+1)-Dimensional Time-Fractional Camassa-Holm-Kadomtsev-Petviashvili Equation

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Zhigang Liu ◽  
Kelei Zhang ◽  
Mengyuan Li
Keyword(s):  
Traveling Wave ◽  
Dynamical Behavior ◽  
Traveling Wave Solutions ◽  
Phase Portraits ◽  
Wave System ◽  
Bifurcation Method ◽  
Exact Traveling Wave Solutions ◽  
Fractional Complex Transform ◽  
Wave Solutions ◽  
Petviashvili Equation

In this paper, we study the (3+1)-dimensional time-fractional Camassa-Holm-Kadomtsev-Petviashvili equation with a conformable fractional derivative. By the fractional complex transform and the bifurcation method for dynamical systems, we investigate the dynamical behavior and bifurcation of solutions of the traveling wave system and seek all possible exact traveling wave solutions of the equation. Furthermore, the phase portraits of the dynamical system and the remarkable features of the solutions are demonstrated via interesting figures.

Exact Cuspon and Compactons of the Novikov Equation

2014 ◽  
Vol 24 (03) ◽  
pp. 1450037 ◽  
Author(s):  
Jibin Li
Keyword(s):  
Dynamical Systems ◽  
Qualitative Analysis ◽  
Traveling Wave ◽  
Wave Solution ◽  
Traveling Wave Solutions ◽  
Phase Portraits ◽  
Breaking Wave ◽  
Wave Solutions ◽  
Novikov Equation

In this paper, we apply the method of dynamical systems to the traveling wave solutions of the Novikov equation. Through qualitative analysis, we obtain bifurcations of phase portraits of the traveling system and exact cuspon wave solution, as well as a family of breaking wave solutions (compactons). We find that the corresponding traveling system of Novikov equation has no one-peakon solution.

On the existence of traveling wave solutions and upper and lower bounds for some Fisher–Kolmogorov type equations

2014 ◽  
Vol 07 (05) ◽  
pp. 1450050 ◽  
Author(s):  
Juan Belmonte-Beitia
Keyword(s):  
Mathematical Biology ◽  
Dynamical Systems ◽  
Lower Bounds ◽  
Traveling Waves ◽  
Traveling Wave ◽  
Systems Approach ◽  
Traveling Wave Solutions ◽  
Upper And Lower Bounds ◽  
Kolmogorov Type ◽  
Wave Solutions

In this paper, we use a dynamical systems approach to prove the existence of traveling waves solutions for the Fisher–Kolmogorov density-dependent equation. Moreover, we prove the existence of upper and lower bounds for these traveling wave solutions found previously. Finally, we present a particular example which has several applications in the mathematical biology field.

scholarly journals The Traveling Wave Solutions and Their Bifurcations for the BBM-LikeB(m,n)Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-17 ◽  
Author(s):  
Shaoyong Li ◽  
Zhengrong Liu
Keyword(s):  
Traveling Wave ◽  
Wave Solution ◽  
Blow Up ◽  
Traveling Wave Solutions ◽  
Periodic Wave ◽  
Periodic Wave Solution ◽  
Simulation Approach ◽  
Bifurcation Method ◽  
Periodic Wave Solutions ◽  
Wave Solutions

We investigate the traveling wave solutions and their bifurcations for the BBM-likeB(m,n)equationsut+αux+β(um)x−γ(un)xxt=0by using bifurcation method and numerical simulation approach of dynamical systems. Firstly, for BBM-likeB(3,2)equation, we obtain some precise expressions of traveling wave solutions, which include periodic blow-up and periodic wave solution, peakon and periodic peakon wave solution, and solitary wave and blow-up solution. Furthermore, we reveal the relationships among these solutions theoretically. Secondly, for BBM-likeB(4,2)equation, we construct two periodic wave solutions and two blow-up solutions. In order to confirm the correctness of these solutions, we also check them by software Mathematica.

scholarly journals Bounded Traveling Wave Solutions and Their Relations for the Generalized HD Type Equation

2016 ◽  
Vol 2016 ◽  
pp. 1-15
Author(s):  
Qing Meng ◽  
Bin He
Keyword(s):  
Solitary Wave ◽  
Traveling Waves ◽  
Traveling Wave ◽  
Sufficient Conditions ◽  
Traveling Wave Solutions ◽  
Type Equation ◽  
Periodic Wave ◽  
Point Of View ◽  
Bifurcation Method ◽  
Wave Solutions

The generalized HD type equation is studied by using the bifurcation method of dynamical systems. From a dynamic point of view, the existence of different kinds of traveling waves which include periodic loop soliton, periodic cusp wave, smooth periodic wave, loop soliton, cuspon, smooth solitary wave, and kink-like wave is proved and the sufficient conditions to guarantee the existence of the above solutions in different regions of the parametric space are given. Also, all possible exact parametric representations of the bounded waves are presented and their relations are stated.

scholarly journals Approximate Damped Oscillatory Solutions for Generalized KdV-Burgers Equation and Their Error Estimates

2011 ◽  
Vol 2011 ◽  
pp. 1-26 ◽  
Author(s):  
Weiguo Zhang ◽  
Xiang Li
Keyword(s):  
Dynamical Systems ◽  
Error Estimates ◽  
Traveling Wave ◽  
Wave Solution ◽  
Burgers Equation ◽  
Traveling Wave Solutions ◽  
Approximate Solutions ◽  
Traveling Wave Solution ◽  
Oscillatory Solutions ◽  
Wave Solutions

We focus on studying approximate solutions of damped oscillatory solutions of generalized KdV-Burgers equation and their error estimates. The theory of planar dynamical systems is employed to make qualitative analysis to the dynamical systems which traveling wave solutions of this equation correspond to. We investigate the relations between the behaviors of bounded traveling wave solutions and dissipation coefficient, and give two critical valuesλ1andλ2which can characterize the scale of dissipation effect, for right and left-traveling wave solution, respectively. We obtain that for the right-traveling wave solution if dissipation coefficientα≥λ1, it appears as a monotone kink profile solitary wave solution; that if0<α<λ1, it appears as a damped oscillatory solution. This is similar for the left-traveling wave solution. According to the evolution relations of orbits in the global phase portraits which the damped oscillatory solutions correspond to, we obtain their approximate damped oscillatory solutions by undetermined coefficients method. By the idea of homogenization principle, we give the error estimates for these approximate solutions by establishing the integral equations reflecting the relations between exact and approximate solutions. The errors are infinitesimal decreasing in the exponential form.

Sign in / Sign up

Export Citation Format

Share Document