scholarly journals Solitons, Peakons, and Periodic Cuspons of a Generalized Degasperis-Procesi Equation

2009 ◽  
Vol 2009 ◽  
pp. 1-13 ◽  
Author(s):  
Jiangbo Zhou ◽  
Lixin Tian
Keyword(s):  
Dynamical Systems ◽  
Bifurcation Theory ◽  
Soliton Solutions ◽  
Travelling Wave ◽  
Travelling Wave Solutions ◽  
Wave Solutions ◽  
The Relationship ◽  
Planar Dynamical Systems ◽  
Explicit Expressions

We employ the bifurcation theory of planar dynamical systems to investigate the exact travelling wave solutions of a generalized Degasperis-Procesi equationut−uxxt+4uux+γ(u−uxx)x=3uxuxx+uuxxx. The implicit expression of smooth soliton solutions is given. The explicit expressions of peaked soliton solutions and periodic cuspon solutions are also obtained. Further, we show the relationship among the smooth soliton solutions, the peaked soliton solutions, and the periodic cuspon solutions. The physical relevance of the found solutions and the reason why these solutions can exist in this equation are also given.

scholarly journals Periodic Loop Solutions and Their Limit Forms for the Kudryashov-Sinelshchikov Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Bin He ◽  
Qing Meng ◽  
Jinhua Zhang ◽  
Yao Long
Keyword(s):  
Dynamical Systems ◽  
Soliton Solutions ◽  
Periodic Wave ◽  
Travelling Wave ◽  
Phase Portraits ◽  
Travelling Wave Solutions ◽  
Bifurcation Method ◽  
Periodic Wave Solutions ◽  
Wave Solutions

The Kudryashov-Sinelshchikov equation is studied by using the bifurcation method of dynamical systems and the method of phase portraits analysis. We show that the limit forms of periodic loop solutions contain loop soliton solutions, smooth periodic wave solutions, and periodic cusp wave solutions. Also, some new exact travelling wave solutions are presented through some special phase orbits.

Bright soliton solutions, power series solutions and travelling wave solutions of a (3+1)-dimensional modified Korteweg–de Vries–Kadomtsev–Petviashvili equation

2018 ◽  
Vol 32 (06) ◽  
pp. 1850082
Author(s):  
Ding Guo ◽  
Shou-Fu Tian ◽  
Li Zou ◽  
Tian-Tian Zhang
Keyword(s):  
Power Series ◽  
Soliton Solutions ◽  
Travelling Wave ◽  
Bright Soliton ◽  
Travelling Wave Solutions ◽  
Series Solutions ◽  
Wave Solutions ◽  
Petviashvili Equation ◽  
Power Series Solutions ◽  
De Vries

In this paper, we consider the (3[Formula: see text]+[Formula: see text]1)-dimensional modified Korteweg–de Vries–Kadomtsev–Petviashvili (mKdV-KP) equation, which can be used to describe the nonlinear waves in plasma physics and fluid dynamics. By using solitary wave ansatz in the form of sech[Formula: see text] function and a direct integrating way, we construct the exact bright soliton solutions and the travelling wave solutions of the equation, respectively. Moreover, we obtain its power series solutions with the convergence analysis. It is hoped that our results can provide the richer dynamical behavior of the KdV-type and KP-type equations.

scholarly journals Travelling-Wave Solutions for Wave Equations with Two Exponential Nonlinearities

2018 ◽  
Vol 73 (10) ◽  
pp. 883-892
Author(s):  
Stefan C. Mancas ◽  
Haret C. Rosu ◽  
Maximino Pérez-Maldonado
Keyword(s):  
Dynamical Systems ◽  
Elliptic Equations ◽  
Hypergeometric Functions ◽  
Wave Equations ◽  
Constant Term ◽  
Travelling Wave ◽  
Travelling Wave Solutions ◽  
Simple Method ◽  
Solutions Of Wave Equations ◽  
Wave Solutions

AbstractWe use a simple method that leads to the integrals involved in obtaining the travelling-wave solutions of wave equations with one and two exponential nonlinearities. When the constant term in the integrand is zero, implicit solutions in terms of hypergeometric functions are obtained, while when that term is nonzero, all the basic travelling-wave solutions of Liouville, Tzitzéica, and their variants, as as well sine/sinh-Gordon equations with important applications in the phenomenology of nonlinear physics and dynamical systems are found through a detailed study of the corresponding elliptic equations.

Three Kind of Triangle Rational Functions and Applications for Discrete Nonlinear Equations

2011 ◽  
Vol 317-319 ◽  
pp. 2168-2171
Author(s):  
Xiu Rong Guo ◽  
Zheng Tao Liu ◽  
Mei Guo
Keyword(s):  
Nonlinear Equations ◽  
Difference Equations ◽  
Soliton Solutions ◽  
Rational Functions ◽  
Travelling Wave ◽  
Travelling Wave Solutions ◽  
Wave Solutions ◽  
Periodic Travelling Wave ◽  
Formal Solutions

In order to efficiently search for new soliton solutions to differential-difference equations (DDEs), three kinds of triangle rational functions are first introduced. Then a kind of formal solutions of DDEs are presented which are expressed by a unified nonlinear combination of the three kinds of triangle rational functions. As illustrative examples, the periodic travelling-wave solutions of the discrete modified KdV(mKdV) equations are obtained.

New Soliton Solutions of Chaffee-Infante Equations Using the Exp-Function Method

2010 ◽  
Vol 65 (3) ◽  
pp. 197-202 ◽  
Author(s):  
Rathinasamy Sakthivel ◽  
Changbum Chun
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Symbolic Computation ◽  
Function Method ◽  
Nonlinear Partial Differential Equations ◽  
Soliton Solutions ◽  
Mathematical Physics ◽  
Travelling Wave ◽  
Travelling Wave Solutions ◽  
Wave Solutions

In this paper, the exp-function method is applied by using symbolic computation to construct a variety of new generalized solitonary solutions for the Chaffee-Infante equation with distinct physical structures. The results reveal that the exp-function method is suited for finding travelling wave solutions of nonlinear partial differential equations arising in mathematical physics

scholarly journals New exact soliton solutions, bifurcation and multistability behaviors of traveling waves for the (3+1)-dimensional modified Zakharov-Kuznetsov equation with higher order dispersion

2021 ◽  
Author(s):  
Asit Saha ◽  
Battal Gazi Karakoç ◽  
Khalid K. Ali
Keyword(s):  
Traveling Wave ◽  
Soliton Solutions ◽  
Travelling Wave ◽  
Higher Order ◽  
Physical Parameters ◽  
Travelling Wave Solutions ◽  
Wave Solutions ◽  
Higher Order Dispersion ◽  
Order Dispersion ◽  
Bifurcation Behavior

The goal of the present paper is to obtain and analyze new exact travelling wave solutions and bifurcation behavior of modified Zakharov-Kuznetsov (mZK) equation with higher order dispersion term. For this purpose, first and second simple methods are used to build soliton solutions of travelling wave solutions. Furthermore, bifurcation behavior of traveling waves including new type of quasiperiodic and multi-periodic traveling wave motions have been examined depending on the physical parameters. Multistability for the nonlinear mZK equation has been investigated depending on fixed values of physical parameters with various initial conditions. The suggested methods for the analytical solutions are powerful and benefical tools to obtain the exact travelling wave solutions of nonlinear evolution equations (NLEEs). Two and three-dimensional plots are also provided to illustrate the new solutions. Bifurcation and multistability behaviors of traveling wave solution of the nonlinear mZK equation with higher order dispersion will add some value in the literature of mathematical and plasma physics.

scholarly journals Method for Studying the Multisoliton Solutions of the Korteweg-de Vries Type Equations

2015 ◽  
Vol 2015 ◽  
pp. 1-9
Author(s):  
Yuriy Turbal ◽  
Andriy Bomba ◽  
Mariana Turbal
Keyword(s):  
Soliton Solutions ◽  
Travelling Wave ◽  
Travelling Wave Solutions ◽  
New Approach ◽  
Wave Solutions ◽  
De Vries ◽  
Multisoliton Solutions

We present a new approach to find travelling wave solutions for the Korteweg-de Vries type equations, which allows extending the class of known soliton solutions. Also we propose method for studying the multisoliton solutions of the Korteweg-de Vries type equations.

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.

Bell-shaped soliton solutions and travelling wave solutions of the fifth-order nonlinear modified Kawahara equation

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Ali Başhan
Keyword(s):  
Soliton Solutions ◽  
Travelling Wave ◽  
Test Problems ◽  
Simple Type ◽  
Order Partial Differential Equation ◽  
Travelling Wave Solutions ◽  
Kawahara Equation ◽  
Third Order ◽  
Wave Solutions ◽  
Fifth Order

AbstractThe main aim of this work is to investigate numerical solutions of the two different types of the fifth-order modified Kawahara equation namely bell-shaped soliton solutions and travelling wave solutions that occur thereby the different type of the Korteweg–de Vries equation. For this approach, we have used an effective and simple type of finite difference method namely Crank-Nicolson scheme for time integration and third-order modified cubic B-spline-based differential quadrature method for space integration. We preferred the third-order modified cubic B-splines to solve the fifth-order partial differential equation because of by using low energy, less algebraic process and produce better results than earlier works. To display the efficiency and accuracy of the present fresh approach famous test problems namely bell-shaped single soliton that has negative amplitude and travelling wave solutions that have the both of the positive and negative amplitudes are solved and the error norms L2 and L∞ are calculated and compared with earlier works. Comparison of the error norms show that present fresh approach obtained superior results than earlier works by using same parameters. At the same time, two lowest invariants of the test problems during the simulations are calculated and reported. Besides those, relative changes of invariants are computed and reported.

scholarly journals On the Solutions of the b-Family of Novikov Equation

Symmetry ◽  
2021 ◽  
Vol 13 (10) ◽  
pp. 1765
Author(s):  
Tingting Wang ◽  
Xuanxuan Han ◽  
Yibin Lu
Keyword(s):  
Explicit Formula ◽  
Soliton Solutions ◽  
Travelling Wave ◽  
Travelling Wave Solutions ◽  
Wave Solutions ◽  
Novikov Equation

In this paper, we study the symmetric travelling wave solutions of the b-family of the Novikov equation. We show that the b-family of the Novikov equation can provide symmetric travelling wave solutions, such as peakon, kink and smooth soliton solutions. In particular, the single peakon, two-peakon, stationary kink, anti-kink, two-kink, two-anti-kink, bell-shape soliton and hat-shape soliton solutions are presented in an explicit formula.

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