scholarly journals Bifurcation Approach to Analysis of Travelling Waves in Nonlocal Hydrodynamic-Type Models

2014 ◽  
Vol 2014 ◽  
pp. 1-12
Author(s):  
Jianping Shi ◽  
Jibin Li
Keyword(s):  
Solitary Wave ◽  
Wave Solution ◽  
Solitary Wave Solution ◽  
Dynamical Behavior ◽  
Type Systems ◽  
Travelling Wave ◽  
Hydrodynamic Type ◽  
Phase Portraits ◽  
Breaking Wave ◽  
Wave Solutions

The paper considers the nonlocal hydrodynamic-type systems which are two-dimensional travelling wave systems with a five-parameter group. We apply the method of dynamical systems to investigate the bifurcations of phase portraits depending on the parameters of systems and analyze the dynamical behavior of the travelling wave solutions. The existence of peakons, compactons, and periodic cusp wave solutions is discussed. When the parameternequals 2, namely, let the isochoric Gruneisen coefficient equal 1, some exact analytical solutions such as smooth bright solitary wave solution, smooth and nonsmooth dark solitary wave solution, and periodic wave solutions, as well as uncountably infinitely many breaking wave solutions, are obtained.

Qualitative analysis and solutions of bounded travelling waves for the fluidized-bed modelling equation

2010 ◽  
Vol 140 (2) ◽  
pp. 241-257
Author(s):  
Weiguo Zhang ◽  
Lanyun Bian ◽  
Yan Zhao
Keyword(s):  
Solitary Wave ◽  
Wave Solution ◽  
Solitary Wave Solution ◽  
Travelling Wave ◽  
Oscillatory Solution ◽  
Phase Portraits ◽  
Travelling Wave Solutions ◽  
Oscillatory Solutions ◽  
Wave Solutions ◽  
Damped Oscillatory Solution

We apply the theory of planar dynamical systems to carry out a qualitative analysis for the planar dynamical system corresponding to the fluidized-bed modelling equation. We obtain the global phase portraits of this system under various parameter conditions and the existence conditions of bounded travelling-wave solutions of this equation. According to the discussion on relationships between the behaviours of bounded travelling-wave solutions and the dissipation coefficients ε and δ, we find a critical value λ0 for arbitrary travelling-wave velocity υ. This equation has a unique damped oscillatory solution as ∥ε + δυ∥ < λ0 and ∥ε + δυ∥ ≠ 0, while it has a unique monotone kink profile solitary-wave solution as ∥ε + δυ∥ > λ0. By means of the undetermined coefficients method, we obtain the exact bell profile solitary-wave solution and monotone kink profile solitary-wave solution. Meanwhile, we obtain the approximate damped oscillatory solution. We point out the positions of these solutions in the global phase portraits. Finally, based on integral equations that reflect the relationships between the approximate damped oscillatory solutions and the implicit exact damped oscillatory solutions, error estimates for the approximate damped oscillatory solutions are presented.

A New Type of Solitary Wave Solution of the mKdV Equation Under Singular Perturbations

2020 ◽  
Vol 30 (11) ◽  
pp. 2050162
Author(s):  
Lijun Zhang ◽  
Maoan Han ◽  
Mingji Zhang ◽  
Chaudry Masood Khalique
Keyword(s):  
Solitary Wave ◽  
Singular Perturbations ◽  
Wave Solution ◽  
Solitary Wave Solution ◽  
Mkdv Equation ◽  
Geometric Singular Perturbation Theory ◽  
Solitary Wave Solutions ◽  
Wave Speeds ◽  
Wave Solutions ◽  
New Type

In this work, we examine the solitary wave solutions of the mKdV equation with small singular perturbations. Our analysis is a combination of geometric singular perturbation theory and Melnikov’s method. Our result shows that two families of solitary wave solutions of mKdV equation, having arbitrary positive wave speeds and infinite boundary limits, persist for selected wave speeds after small singular perturbations. More importantly, a new type of solitary wave solution possessing both valley and peak, named as breather in physics, which corresponds to a big homoclinic loop of the associated dynamical system is observed. It reveals an exotic phenomenon and exhibits rich dynamics of the perturbed nonlinear wave equation. Numerical simulations are performed to further detect the wave speeds of the persistent solitary waves and the nontrivial one with both valley and peak.

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.

THE EXPLICIT NONLINEAR WAVE SOLUTIONS AND THEIR BIFURCATIONS OF THE GENERALIZED CAMASSA–HOLM EQUATION

2011 ◽  
Vol 21 (11) ◽  
pp. 3119-3136 ◽  
Author(s):  
ZHENGRONG LIU ◽  
YONG LIANG
Keyword(s):  
Solitary Wave ◽  
Nonlinear Wave ◽  
Wave Solution ◽  
Wave Speed ◽  
Solitary Wave Solution ◽  
Wave Solutions ◽  
Holm Equation ◽  
Bifurcation Phenomena ◽  
Singular Wave ◽  
Nonlinear Wave Solutions

In this paper, we study the explicit nonlinear wave solutions and their bifurcations of the generalized Camassa–Holm equation [Formula: see text]Not only are the precise expressions of the explicit nonlinear wave solutions obtained, but some interesting bifurcation phenomena are revealed.Firstly, it is verified that k = 3/8 is a bifurcation parametric value for several types of explicit nonlinear wave solutions.When k < 3/8, there are five types of explicit nonlinear wave solutions, which are(i) hyperbolic peakon wave solution,(ii) fractional peakon wave solution,(iii) fractional singular wave solution,(iv) hyperbolic singular wave solution,(v) hyperbolic smooth solitary wave solution.When k = 3/8, there are two types of explicit nonlinear wave solutions, which are fractional peakon wave solution and fractional singular wave solution.When k > 3/8, there is not any type of explicit nonlinear wave solutions.Secondly, it is shown that there are some bifurcation wave speed values such that the peakon wave and the anti-peakon wave appear alternately.Thirdly, it is displayed that there are other bifurcation wave speed values such that the hyperbolic peakon wave solution becomes the fractional peakon wave solution, and the hyperbolic singular wave solution becomes the fractional singular wave solution.

Characteristics of the breathers, rogue waves and soliton waves in a (2 + 1)-dimensional generalized Nizhnik–Novikov–Veselov equation

2019 ◽  
Vol 33 (03) ◽  
pp. 1950014 ◽  
Author(s):  
Xiu-Bin Wang ◽  
Bo Han
Keyword(s):  
Solitary Wave ◽  
Wave Solution ◽  
Rogue Wave ◽  
Solitary Wave Solution ◽  
Dynamical Behavior ◽  
Rogue Waves ◽  
Bilinear Forms ◽  
Wave Fields ◽  
Rogue Wave Solution ◽  
Bell’S Polynomials

In this work, a (2 + 1)-dimensional generalized Nizhnik–Novikov–Veselov (GNNV) equation, which can be reduced to several integrable equations, is under investigation. By virtue of Bell’s polynomials, an effective and straightforward way is presented to succinctly construct its two bilinear forms. Furthermore, based on the bilinear formalism and the extended homoclinic test, the breather wave solution, rogue-wave solution and solitary-wave solution of the equation are well constructed. The results can be used to enrich the dynamical behavior of the (2 + 1)-dimensional nonlinear wave fields.

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.

THE STABILITY OF THE SOLITARY WAVE SOLUTIONS TO THE GENERALIZED COMPOUND KDV EQUATION

2011 ◽  
Vol 04 (03) ◽  
pp. 475-480
Author(s):  
Xiaohua Liu ◽  
Weiguo Zhang
Keyword(s):  
Solitary Wave ◽  
Variational Method ◽  
Wave Solution ◽  
Lyapunov Functional ◽  
Solitary Wave Solution ◽  
Kdv Equation ◽  
Solitary Wave Solutions ◽  
Wave Solutions ◽  
The Stability ◽  
Lyapunov Sense

Using variational method, we investigate that the solitary wave solution u(x - ct) to the Generalized Compound Kdv Equation with two nonlinear terms is stable in the Lyapunov sense when 0 < p < 2 holds. The result is new. There shows a new method to consider the extremum of Lyapunov functional.

EXPLICIT PERIODIC WAVE SOLUTIONS AND THEIR BIFURCATIONS FOR GENERALIZED CAMASSA–HOLM EQUATION

2010 ◽  
Vol 20 (08) ◽  
pp. 2507-2519 ◽  
Author(s):  
ZHENGRONG LIU ◽  
HAO TANG
Keyword(s):  
Solitary Wave ◽  
Wave Solution ◽  
Blow Up ◽  
Solitary Wave Solution ◽  
Periodic Wave ◽  
Periodic Wave Solution ◽  
Periodic Wave Solutions ◽  
Bifurcation Phenomenon ◽  
Wave Solutions ◽  
Holm Equation

In this paper, through qualitative analysis and integration, we study the explicit periodic wave solutions and their bifurcations for the generalized Camassa–Holm equation [Formula: see text] When the parameter k satisfies k < 3/8 and the constant wave speed c satisfies [Formula: see text], we obtain two types of explicit periodic wave solutions, elliptic smooth periodic wave solution and elliptic periodic blow-up solutions. These solutions include a bifurcation parameter α which has four bifurcation values αi(i = 1, 2, 3, 4). When α tends to the bifurcation values, the elliptic periodic wave solutions become three types of other solutions, the hyperbolic smooth solitary wave solution, the hyperbolic blow-up solution and the trigonometric periodic blow-up solution. Especially, a new bifurcation phenomenon is found, that is, the periodic blow-up solution can become a smooth solitary wave solution when α varies. When k > 3/8, we guess that there is no other explicit solution except the explicit periodic blow-up solution.

scholarly journals Exact Solitary Wave Solution in the ZK-BBM Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Juan Zhao ◽  
Wei Li
Keyword(s):  
Solitary Wave ◽  
Wave Solution ◽  
Solitary Wave Solution ◽  
Homoclinic Orbits ◽  
Traveling Wave Solution ◽  
Phase Portraits ◽  
Dynamical System Theory ◽  
Ode System ◽  
Nonlinear Ode ◽  
Bbm Equation

The traveling wave solution for the ZK-BBM equation is considered, which is governed by a nonlinear ODE system. The bifurcation structure of fixed points and bifurcation phase portraits with respect to the wave speed c are analyzed by using the dynamical system theory. Furthermore, the exact solutions of the homoclinic orbits for the nonlinear ODE system are obtained which corresponds to the solitary wave solution curve of the ZK-BBM equation.

Cross-kink wave, solitary, dark, and periodic wave solutions by bilinear and He’s variational direct methods for the KP–BBM equation

2021 ◽  
Author(s):  
Baolin Feng ◽  
Jalil Manafian ◽  
Onur Alp Ilhan ◽  
Amitha Manmohan Rao ◽  
Anand H. Agadi
Keyword(s):  
Solitary Wave ◽  
Traveling Wave ◽  
Wave Solution ◽  
Solitary Wave Solution ◽  
Traveling Wave Solutions ◽  
Periodic Wave ◽  
Periodic Wave Solution ◽  
Wave Solutions ◽  
Ito Equation ◽  
Bbm Equation

This paper deals with cross-kink waves in the (2+1)-dimensional KP–BBM equation in the incompressible fluid. Based on Hirota’s bilinear technique, cross-kink solutions related to KP–BBM equation are constructed. Taking the special reduction, the exact expression of different types of solutions comprising exponential, trigonometric and hyperbolic functions is obtained. Moreover, He’s variational direct method (HVDM) based on the variational theory and Ritz-like method is employed to construct the abundant traveling wave solutions of the (2+1)-dimensional generalized Hirota–Satsuma–Ito equation. These traveling wave solutions include kinky dark solitary wave solution, dark solitary wave solution, bright solitary wave solution, periodic wave solution and so on, which are all depending on the initial hypothesis for the Ritz-like method. In continuation, the modulation instability is engaged to discuss the stability of the obtained solutions. Moreover, the rational [Formula: see text] method on the generalized Hirota–Satsuma–Ito equation is investigated. The applicability and effectiveness of the acquired solutions are presented through the numerical results in the form of 3D and 2D graphs. A variety of interactions are illustrated analytically and graphically. The influence of parameters on propagation is analyzed and summarized. The results and phenomena obtained in this paper enrich the dynamic behavior of the evolution of nonlinear waves.

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