scholarly journals Bifurcation Analysis and Different Kinds of Exact Travelling Wave Solutions of a Generalized Two-Component Hunter-Saxton System

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Qing Meng ◽  
Bin He
Keyword(s):  
Solitary Wave ◽  
Blow Up ◽  
Travelling Waves ◽  
Sufficient Conditions ◽  
Periodic Wave ◽  
Travelling Wave ◽  
Point Of View ◽  
Travelling Wave Solutions ◽  
Wave Solutions ◽  
Two Component

This paper focuses on a generalized two-component Hunter-Saxton system. From a dynamic point of view, the existence of different kinds of periodic wave, solitary wave, and blow-up 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, some exact parametric representations of the travelling waves are presented.

scholarly journals Bifurcations and Parametric Representations of Peakon, Solitary Wave and Smooth Periodic Wave Solutions for a Two-Component Degasperis-Procesi Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-14
Author(s):  
Bin He ◽  
Qing Meng ◽  
Jinhua Zhang
Keyword(s):  
Dynamical Systems ◽  
Solitary Wave ◽  
Periodic Wave ◽  
Travelling Wave ◽  
Phase Portraits ◽  
Travelling Wave Solutions ◽  
Bifurcation Method ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
Two Component

By using the bifurcation method of dynamical systems and the method of phase portraits analysis, we consider a two-component Degasperis-Procesi equation:mt=-3mux-mxu+kρρx,  ρt=-ρxu+2ρux,the existence of the peakon, solitary wave and smooth periodic wave is proved, and exact parametric representations of above travelling wave solutions are obtained in different parameter regions.

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.

A Variety of Exact Periodic Wave and Solitary Wave Solutions for the Coupled Higgs Equation

2012 ◽  
Vol 67 (10-11) ◽  
pp. 545-549 ◽  
Author(s):  
Houria Trikia ◽  
Abdul-Majid Wazwazb
Keyword(s):  
Solitary Wave ◽  
Soliton Solutions ◽  
Higgs Field ◽  
Periodic Wave ◽  
Travelling Wave ◽  
Jacobi Elliptic Function ◽  
Travelling Wave Solutions ◽  
Solitary Wave Solutions ◽  
Wave Solutions ◽  
Jacobi Elliptic Function Expansion

In this work, the coupled Higgs field equation is studied. The extended Jacobi elliptic function expansion methods are efficiently employed to construct the exact periodic solutions of this model. As a result, many exact travelling wave solutions are obtained which include new shock wave solutions or kink-shaped soliton solutions, solitary wave solutions or bell-shaped soliton solutions, and combined solitary wave solutions are formally obtained.

Traveling Wave Solutions of a Two-Component Dullin–Gottwald–Holm System

2016 ◽  
Vol 12 (3) ◽  
Author(s):  
Jiyu Zhong ◽  
Shengfu Deng
Keyword(s):  
Solitary Wave ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Periodic Wave ◽  
Phase Portraits ◽  
Wave System ◽  
Solitary Wave Solutions ◽  
Wave Solutions ◽  
Two Component ◽  
Planar Systems

In this paper, we investigate the traveling wave solutions of a two-component Dullin–Gottwald–Holm (DGH) system. By qualitative analysis methods of planar systems, we investigate completely the topological behavior of the solutions of the traveling wave system, which is derived from the two-component Dullin–Gottwald–Holm system, and show the corresponding phase portraits. We prove the topological types of degenerate equilibria by the technique of desingularization. According to the dynamical behaviors of the solutions, we give all the bounded exact traveling wave solutions of the system, including solitary wave solutions, periodic wave solutions, cusp solitary wave solutions, periodic cusp wave solutions, compactonlike wave solutions, and kinklike and antikinklike wave solutions. Furthermore, to verify the correctness of our results, we simulate these bounded wave solutions using the software maple version 18.

Nonlinear travelling internal waves with piecewise-linear shear profiles

2018 ◽  
Vol 856 ◽  
pp. 984-1013 ◽  
Author(s):  
K. L. Oliveras ◽  
C. W. Curtis
Keyword(s):  
Piecewise Linear ◽  
Travelling Waves ◽  
Tangential Velocity ◽  
Travelling Wave ◽  
Numerical Continuation ◽  
Travelling Wave Solutions ◽  
Water Wave Problem ◽  
Two Fluids ◽  
Wave Solutions ◽  
Linear Shear

In this work, we study the nonlinear travelling waves in density stratified fluids with piecewise-linear shear currents. Beginning with the formulation of the water-wave problem due to Ablowitz et al. (J. Fluid Mech., vol. 562, 2006, pp. 313–343), we extend the work of Ashton & Fokas (J. Fluid Mech., vol. 689, 2011, pp. 129–148) and Haut & Ablowitz (J. Fluid Mech., vol. 631, 2009, pp. 375–396) to examine the interface between two fluids of differing densities and varying linear shear. We derive a systems of equations depending only on variables at the interface, and numerically solve for periodic travelling wave solutions using numerical continuation. Here, we consider only branches which bifurcate from solutions where there is no slip in the tangential velocity at the interface for the trivial flow. The spectral stability of these solutions is then determined using a numerical Fourier–Floquet technique. We find that the strength of the linear shear in each fluid impacts the stability of the corresponding travelling wave solutions. Specifically, opposing shears may amplify or suppress instabilities.

scholarly journals New Exact Solitary Wave Solutions of a Coupled Nonlinear Wave Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
XiaoHua Liu ◽  
CaiXia He
Keyword(s):  
Wave Equation ◽  
Solitary Wave ◽  
Nonlinear Wave ◽  
Nonlinear Wave Equation ◽  
Sufficient Conditions ◽  
Periodic Wave ◽  
Solitary Wave Solutions ◽  
Wave Solutions ◽  
Undetermined Coefficient ◽  
Planar Dynamical Systems

By using the theory of planar dynamical systems to a coupled nonlinear wave equation, the existence of bell-shaped solitary wave solutions, kink-shaped solitary wave solutions, and periodic wave solutions is obtained. Under the different parametric values, various sufficient conditions to guarantee the existence of the above solutions are given. With the help of three different undetermined coefficient methods, we investigated the new exact explicit expression of all three bell-shaped solitary wave solutions and one kink solitary wave solutions with nonzero asymptotic value for a coupled nonlinear wave equation. The solutions cannot be deduced from the former references.

scholarly journals Nonuniform Continuity of the Osmosis K(2, 2) Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Aiyong Chen ◽  
Yong Ding ◽  
Wentao Huang
Keyword(s):  
Differential Equations ◽  
Small Amplitude ◽  
Travelling Waves ◽  
Travelling Wave ◽  
Qualitative Theory ◽  
Travelling Wave Solutions ◽  
Solution Map ◽  
Wave Solutions ◽  
Periodic Travelling Wave ◽  
Uniformly Continuous

The qualitative theory of differential equations is applied to the osmosis K(2, 2) equation. The parametric conditions of existence of the smooth periodic travelling wave solutions are given. We show that the solution map is not uniformly continuous by using the theory of Himonas and Misiolek. The proof relies on a construction of smooth periodic travelling waves with small amplitude.

scholarly journals FKPP equation with impulses on unbounded domain

2017 ◽  
Vol 1 ◽  
pp. 1 ◽  
Author(s):  
Valaire Yatat ◽  
Yves Dumont
Keyword(s):  
Travelling Waves ◽  
Reaction Diffusion ◽  
Bare Soil ◽  
Travelling Wave ◽  
Reaction Diffusion Equations ◽  
Spreading Speed ◽  
Systems Dynamics ◽  
Travelling Wave Solutions ◽  
Minimal Speed ◽  
Wave Solutions

This paper deals with the problem of travelling wave solutions in a scalar impulsive FKPP-like equation. It is a first step of a more general study that aims to address existence of travelling wave solutions for systems of impulsive reaction-diffusion equations that model ecological systems dynamics such as fire-prone savannas. Using results on scalar recursion equations, we show existence of populated vs. extinction travelling waves invasion and compute an explicit expression of their spreading speed (characterized as the minimal speed of such travelling waves). In particular, we find that the spreading speed explicitly depends on the time between two successive impulses. In addition, we carry out a comparison with the case of time-continuous events. We also show that depending on the time between two successive impulses, the spreading speed with pulse events could be lower, equal or greater than the spreading speed in the case of time-continuous events. Finally, we apply our results to a model of fire-prone grasslands and show that pulse fires event may slow down the grassland vs. bare soil invasion speed.

scholarly journals Periodic Wave Solutions and Their Limits for the Generalized KP-BBM Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-25 ◽  
Author(s):  
Ming Song ◽  
Zhengrong Liu
Keyword(s):  
Dynamical Systems ◽  
Solitary Wave ◽  
Blow Up ◽  
Periodic Wave ◽  
Solitary Wave Solutions ◽  
Bifurcation Method ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
Kink Wave ◽  
Bbm Equation

We use the bifurcation method of dynamical systems to study the periodic wave solutions and their limits for the generalized KP-BBM equation. A number of explicit periodic wave solutions are obtained. These solutions contain smooth periodic wave solutions and periodic blow-up solutions. Their limits contain periodic wave solutions, kink wave solutions, unbounded wave solutions, blow-up wave solutions, and solitary wave solutions.

Instability modulation properties of the (2 + 1)-dimensional Kundu–Mukherjee–Naskar model in travelling wave solutions

2021 ◽  
pp. 2150217
Author(s):  
Haci Mehmet Baskonus ◽  
Juan Luis García Guirao ◽  
Ajay Kumar ◽  
Fernando S. Vidal Causanilles ◽  
German Rodriguez Bermudez
Keyword(s):  
Function Method ◽  
Periodic Wave ◽  
Travelling Wave ◽  
Travelling Wave Solutions ◽  
Validity Of Results ◽  
Periodic Wave Solutions ◽  
Wave Solutions

This paper focuses on the instability modulation and new travelling wave solutions of the (2 + 1)-dimensional Kundu–Mukherjee–Naskar equation via the tanh function method. Dark, mixed dark–bright, complex solitons and periodic wave solutions are archived. Strain conditions for the validity of results are also reported. Instability modulation properties of the governing model are also extracted. Various wave simulations in 2D, 3D and contour graphs under the strain conditions are presented.

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