Periodic Cusp Wave Research Articles

Dynamical behavior of traveling waves in a generalized VP-mVP equation with non-homogeneous power law nonlinearity

<abstract><p>In this paper, we investigate the dynamical behavior of traveling waves for a generalized Vakhnenko-Parkes-modified Vakhnenko-Parkes (VP-mVP) equation with non-homogeneous power law nonlinearity. By the dynamical systems approach and the singular traveling wave theory, the existence of all possible bounded traveling wave solutions is discussed, including smooth solutions (solitary wave solutions, periodic wave solutions and breaking wave solutions) and non-smooth solutions (solitary cusp wave solutions and periodic cusp wave solutions). We not only obtain all the explicit parametric conditions for the existence of 5 kinds of bounded traveling wave solutions, but also give their exact explicit expressions. Moreover, we qualitatively analyze the dynamical behavior of these traveling waves by using the bifurcation of phase portraits under different parameter conditions, and strictly prove the evolution of different traveling waves with their exact expressions.</p></abstract>

The new optical behaviour of the LPD model with Kerr law and parabolic law of nonlinearity

In this paper, the newly obtained optical analytical solutions of the Lakshmanan–Porsezian–Daniel model with two forms of nonlinearity is investigated. These are Kerr law and parabolic law of nonlinearity. The novel analytical method namely, the modified Khater method is used to achieve the newly derived solutions. The attained solutions are bright, dark, anti-kink, u-shaped periodic cusp wave, singular solution and evolving shape of anti-kink solutions. The analysed characteristics of the reported solutions with different forms of nonlinearity are the most compelling parts of the proposed method. This ensures the efficacy and constructiveness of the method. The novel aspect of this study is the newly achieved solutions that can be used in predicting the performance of the information carrier.

Nonlinear singular traveling waves in a slightly compressible thermo-hyperelastic cylindrical shell

Nonlinear dynamic researches of rubber-like materials have always been the focus of attention. However, it is usually based on the completely incompressible hyperelastic constitutive model. Moreover, the temperature effect is ignored. Here, the constitutive relation of a slightly compressible thermo-hyperelastic material is adopted, and strongly nonlinear traveling waves in a thermo-hyperelastic cylindrical shell are investigated. Under the assumption of axisymmetric deformations, the coupled partial differential equations describing the radial and axial motions of the shell are derived. Based on the bifurcation theory of dynamical systems, the qualitative properties of the system are discussed in detail. The influences of inner and outer boundary temperatures on the presence of bounded traveling waves are investigated. Furthermore, the effects of compressibility and material parameters on the presence of periodic waves, solitary waves, non-smooth solitary cusp waves, and periodic cusp waves are studied, and the corresponding critical bifurcation values are determined. Finally, numerical validations of the analytical results are carried out. Due to these interesting phenomena, the necessity of studying the inner and outer boundary temperatures and compressibility for nonlinear traveling waves in a cylindrical shell is demonstrated.

Abundant Dynamical Behaviors of Bounded Traveling Wave Solutions to Generalized θ-Equation

We study existence and dynamics of bounded traveling wave solutions to generalized $$\theta $$ -equation from the perspective of dynamical systems. We obtain bifurcation of traveling wave solutions for the equation, prove the existence of several types of bounded traveling wave solutions, including solitary wave solutions, periodic wave solutions, peakons, periodic cusp waves, compactons and kink-like (antikink-like) waves, and derive some of their exact expressions. Most importantly, we confirm abundant dynamical behaviors of the traveling wave solutions to the equation, which are summarized as follows: (1) We confirm that three types of orbits give rise to solitary wave solutions, that is, the homoclinic orbit passing the singular point, the composed homoclinic orbit which is comprised of three heteroclinic orbits of the associated system, and the composed homoclinic orbit which is comprised of two heteroclinic orbits and tangent to the singular line at the singular point of the associated system. (2) We confirm that four types of orbits correspond to periodic wave solutions, that is, the periodic orbit surrounding a center, the periodic orbit surrounding two connected homoclinic orbits, the composed periodic orbit which is comprised of two heteroclinic orbits of the associated system, and the homoclinic orbit of the associated system which is tangent to the singular line at the singular point of the associated system. (3) We confirm that two types of orbits correspond to periodic cusp waves, that is, the semiellipse orbit surrounding a center, and the semiellipse-like orbit surrounding two connected homoclinic orbits. (4) We confirm that two families of periodic orbits, which surround two connected homoclinic orbits and are comprised of two heteroclinic orbits of associated system, respectively, and the composed homoclinic orbit, which is comprised of two heteroclinic orbits and tangent to the singular line at the singular point of associated system, have envelope.

Some Interesting Traveling Waves in a Transversely Isotropic Incompressible Hyperelastic Semi-infinite Rod

In this paper, the bounded traveling wave solutions are examined for a system of partial differential equations with certain boundary conditions. Interestingly, this system can be used to describe the axially symmetric motion of a semi-infinite incompressible hyperelastic cylindrical rod. With the aid of traveling wave transformations, the partial differential equations can be reduced to a traveling wave equation. The implicit analytical expressions determining the traveling waves are derived. Significantly, the influences of material parameters on the qualitative properties are discussed in detail by using the phase portraits of the traveling wave system. Smooth solutions such as the periodic traveling wave solutions and the solitary wave solutions with the peak form are shown. In particular, some interesting singular traveling wave solutions including the solitary cusp wave solutions and the periodic cusp wave solutions with the peak form are obtained. Numerical examples for all these waves are given.

Novel solitary wave solution in shallow water and ion acoustic plasma waves in-terms of two nonlinear models via MSE method

By using modified simple equation method, we study the generalized RLW equation and symmetric RLW equation, the subsistence of solitary wave, periodic cusp wave, periodic bell wave solutions are obtained. We establish some conditions on the parameters for which the obtained solutions are dark or bright soliton. The proficiency of the methods for constructing exact solutions has been established. Finally, the variety of structure and graphical representation makes the dynamics of the equations visible and provides the mathematical foundation in shallow water, plasma and ion acoustic plasma waves.

PLANAR BIFURCATION METHOD OF DYNAMICAL SYSTEM FOR INVESTIGATING DIFFERENT KINDS OF BOUNDED TRAVELLING WAVE SOLUTIONS OF A GENERALIZED CAMASSA-HOLM EQUATION

In this study, by using planar bifurcation method of dynamical system, we study a generalized Camassa-Holm (gCH) equation. As results, under different parameter conditions, many bounded travelling wave solutions such as periodic waves, periodic cusp waves, solitary waves, peakons, loops and kink waves are given. The dynamic properties of these exact solutions are investigated.

Dynamical behaviours and exact travelling wave solutions of modified generalized Vakhnenko equation

By using the bifurcation theory of planar dynamical systems and the qualitative theory of differential equations, we studied the dynamical behaviours and exact travelling wave solutions of the modified generalized Vakhnenko equation (mGVE). As a result, we obtained all possible bifurcation parametric sets and many explicit formulas of smooth and non-smooth travelling waves such as cusped solitons, loop solitons, periodic cusp waves, pseudopeakon solitons, smooth periodic waves and smooth solitons. Moreover, we provided some numerical simulations of these solutions.

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

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.

Three kinds of periodic wave solutions and their limit forms for a modified KdV-type equation

A modified KdV-type equation is studied by using the bifurcation theory of dynamical system. By investigating the dynamical behavior with phase space analysis, all possible explicit exact traveling wave solutions including peakon solutions, kink and anti-kink wave solutions, blow-up wave solutions, smooth periodic wave solutions, periodic cusp wave solutions, and periodic blow-up wave solutions are obtained. When the first integral varies, we also show the convergence of the periodic wave solutions, such as the smooth periodic wave solutions converge to the kink and anti-kink wave solutions, the periodic cusp wave solutions converge to the peakon solution, the periodic blow-up wave solutions converge to the blow-up wave solution, the blow-up wave solutions converge to the blow-up wave solution, and the periodic blow-up wave solutions converge to the periodic blow-up wave solution.

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

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.

Exact Solutions and Bifurcations of a Modulated Equation in a Discrete Nonlinear Electrical Transmission Line (III)

In this paper, we consider a modulated equation in a discrete nonlinear electrical transmission line. This model is an integrable planar dynamical system having three singular straight lines. By using the theory of singular systems to investigate the dynamical behavior for this system, we obtain bifurcations of phase portraits under different parameter conditions. Corresponding to some special level curves, we derive exact explicit parametric representations of solutions (including smooth solitary wave solutions, peakons, compactons, periodic cusp wave solutions) under different parameter conditions.

Bifurcation and Traveling Wave Solutions for the Fokas Equation

This paper is devoted to discussing bifurcation and traveling wave solutions for the Fokas equation. By investigating the dynamical behavior with phase space analysis, we may derive all possible exact traveling wave solutions, including compactons, cuspons, periodic cusp wave solutions, and smooth solitary wave solutions.

Exact traveling wave solutions and bifurcations of the dual Ito equation

In this paper, we consider the traveling wave solutions for the dual Ito equation. The corresponding traveling wave systems are two singular planar dynamical systems with one and two singular straight lines, respectively. On the basis of the theory of the singular traveling wave systems, we obtain the bifurcations of phase portraits and all possible explicit exact parametric representations of solutions (including solitary wave solutions, periodic cusp wave solutions, smooth periodic wave solutions, compactons, etc.) under different parameter conditions.

Bifurcations and nonlinear wave solutions for the generalized two-component integrable Dullin–Gottwald–Holm system

In this paper, we study the bifurcations and nonlinear wave solutions for a generalized two-component integrable Dullin–Gottwald–Holm system. Through the bifurcations of phase portraits, we not only show the existence of several types of nonlinear wave solutions, including solitary waves, peakons, periodic cusp waves, periodic waves, compacton-like waves and kink-like waves, but also obtain their implicit expressions. Additionally, the numerical simulations of the nonlinear wave solutions are given to show the correctness of our results.

Single Peak Soliton and Periodic Cusp Wave of the Generalized Schrodinger–Boussinesq Equations* *Supported by National Natural Science Foundation of China under Grant Nos. 11361017, 11161013 and Natural Science Foundation of Guangxi under Grant Nos. 2012GXNSFAA053003, 2013GXNSFAA019010, and Program for Innovative Research Team of Guilin University of Electronic Technology

In this paper, we study peakon, cuspon, smooth soliton and periodic cusp wave of the generalized Schrödinger–Boussinesq equations. Based on the method of dynamical systems, the generalized Schrödinger–Boussinesq equations are shown to have new the parametric representations of peakon, cuspon, smooth soliton and periodic cusp wave solutions. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given.

Bifurcations and Exact Solutions of a Modulated Equation in a Discrete Nonlinear Electrical Transmission Line (II)

In this paper, we consider a model which is the modulated equation in a discrete nonlinear electrical transmission line. This model is an integrable planar dynamical system having three singular straight lines. By using the theory of singular systems and investigating the dynamical behavior, we obtain bifurcations of the phase portraits of the system under different parameter conditions. Corresponding to some special level curves, we derive possible exact explicit parametric representations of solutions (including smooth solitary wave and periodic wave solutions, periodic cusp wave solutions) under different parameter conditions.

Exact Solutions and Bifurcations of a Modulated Equation in a Discrete Nonlinear Electrical Transmission Line (I)

In this paper, we consider a model which is a modulated equation in a discrete nonlinear electrical transmission line. By investigating the dynamical behavior and bifurcations of solutions of the planar dynamical systems, we derive all explicit exact parametric representations of solutions (including smooth solitary wave solutions, smooth periodic wave solutions, peakons, compactons, periodic cusp wave solutions, etc.) under different parameter conditions.

PEAKON, PSEUDO-PEAKON, LOOP, AND PERIODIC CUSP WAVE SOLUTIONS OF A THREE-DIMENSIONAL 3DKP(2, 2) EQUATION WITH NONLINEAR DISPERSION

In this paper, we study the three-dimensional Kadomtsev-Petviashvili equation (3DKP(m, n)) with nonlinear dispersion for <i>m</i>=<i>n</i>=2. By using the bifurcation theory of dynamical systems, we study the dynamical behavior and obtain peakon, pseudo-peakon, loop and periodic cusp wave solutions of the three-dimensional 3DKP(2, 2) equation. The parameter expressions of peakon, pseudo-peakon, loop and periodic cusp wave solutions are obtained and numerical graph are provided for those peakon, pseudo-peakon, loop and periodic cusp wave solutions.

Bifurcations and exact traveling wave solutions for a class of nonlinear fourth-order partial differential equations

In this paper, we study the bifurcations and exact traveling wave solutions for a class of nonlinear fourth-order partial differential equations. Based on the method of bifurcation theory of dynamical systems, the nonlinear fourth-order partial differential equations are shown the existence and implicit expressions of peakon, loop, periodic cusp wave, unbounded wave and periodic wave solutions. In particular, the numerical simulations of the bounded wave solutions are given to show the correctness of our results.