scholarly journals Exact Solutions to a Generalized Bogoyavlensky-Konopelchenko Equation via Maple Symbolic Computations

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-6 ◽  
Author(s):  
Shou-Ting Chen ◽  
Wen-Xiu Ma
Keyword(s):  
Nonlinear Equation ◽  
Traveling Wave ◽  
Computer Algebra System ◽  
Traveling Wave Solutions ◽  
Explicit Solutions ◽  
Symbolic Computations ◽  
Wave Solutions ◽  
Hirota Bilinear Form ◽  
Hirota Bilinear ◽  
Bilinear Equation

We aim to construct exact and explicit solutions to a generalized Bogoyavlensky-Konopelchenko equation through the Maple computer algebra system. The considered nonlinear equation is transformed into a Hirota bilinear form, and symbolic computations are made for solving both the nonlinear equation and the corresponding bilinear equation. A few classes of exact and explicit solutions are generated from different ansätze on solution forms, including traveling wave solutions, two-wave solutions, and polynomial solutions.

scholarly journals Traveling Wave Solutions and Infinite-Dimensional Linear Spaces of Multiwave Solutions to Jimbo-Miwa Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Lijun Zhang ◽  
C. M. Khalique
Keyword(s):  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Wave System ◽  
Infinite Dimensional ◽  
Wave Solutions ◽  
Hirota Bilinear Equation ◽  
Periodic Traveling Wave ◽  
Multiwave Solutions ◽  
Hirota Bilinear ◽  
Bilinear Equation

The traveling wave solutions and multiwave solutions to (3 + 1)-dimensional Jimbo-Miwa equation are investigated in this paper. As a result, besides the exact bounded solitary wave solutions, we obtain the existence of two families of bounded periodic traveling wave solutions and their implicit formulas by analysis of phase portrait of the corresponding traveling wave system. We derive the exact 2-wave solutions and two families of arbitrary finiteN-wave solutions by studying the linear space of its Hirota bilinear equation, which confirms that the (3 + 1)-dimensional Jimbo-Miwa equation admits multiwave solutions of any order and is completely integrable.

scholarly journals Solutions of the Bullough–Dodd Model of Scalar Field through Jacobi-Type Equations

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1529
Author(s):  
Rodica Cimpoiasu ◽  
Radu Constantinescu ◽  
Alina Streche Pauna
Keyword(s):  
Scalar Field ◽  
Elliptic Equation ◽  
Nonlinear Equation ◽  
Homogeneous Space ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Auxiliary Equation ◽  
Wave Solutions ◽  
General Elliptic ◽  
Refined Version

A technique based on multiple auxiliary equations is used to investigate the traveling wave solutions of the Bullough–Dodd (BD) model of the scalar field. We place the model in a flat and homogeneous space, considering a symmetry reduction to a 2D-nonlinear equation. It is solved through this refined version of the auxiliary equation technique, and multiparametric solutions are found. The key idea is that the general elliptic equation, considered here as an auxiliary equation, degenerates under some special conditions into subequations involving fewer parameters. Using these subequations, we successfully construct, in a unitary way, a series of solutions for the BD equation, part of them not yet reported. The technique of multiple auxiliary equations could be employed to handle several other types of nonlinear equations, from QFT and from various other scientific areas.

Explicit solutions of the (2 + 1)-dimensional Hirota–Maccari system arising in nonlinear optics

2019 ◽  
Vol 33 (30) ◽  
pp. 1950360 ◽  
Author(s):  
Nauman Raza ◽  
Adil Jhangeer ◽  
Hadi Rezazadeh ◽  
Ahmet Bekir
Keyword(s):  
Nonlinear Optics ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Expansion Method ◽  
Explicit Solutions ◽  
Mathematical Tool ◽  
Exact Traveling Wave Solutions ◽  
Wave Solutions ◽  
Set Up ◽  
Projective Riccati Equation Method

In this paper, new exact traveling wave solutions of the [Formula: see text]-dimensional Hirota–Maccari system arising in nonlinear optics are successfully obtained by using two methods, namely, Improved [Formula: see text]-expansion method and general projective Riccati equation method. The considered methods have been successfully implemented to find exact traveling wave solutions for nonlinear evaluation equations (NLEE) coming for describing nonlinear optics. The results obtained by these methods are straightforward and concise mathematical tool to set up the exact solutions of NLEE.

scholarly journals Explicit solutions of Fisher's equation with three zeros

1990 ◽  
Vol 13 (3) ◽  
pp. 617-620 ◽  
Author(s):  
M. F. K. Abur-Robb
Keyword(s):  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
The Other ◽  
Wave Form ◽  
Explicit Solutions ◽  
Wave Speeds ◽  
Fisher’S Equation ◽  
Wave Solutions ◽  
Fisher's Equation ◽  
Permanent Wave

Explicit traveling wave solutions of Fisher's equation with three simple zerosut=uxx+u(1−u)(u−a),a∈(0,1), are obtained for the wave speedsC=±2(12−a)suggested by pure analytic considerations. Two types of solutions are obtained: one type is of a permanent wave form whereas the other is not.

Symmetry Reductions and Exact Solutions to the Kudryashov–Sinelshchikov Equation

2016 ◽  
Vol 71 (11) ◽  
pp. 1059-1065 ◽  
Author(s):  
Huizhang Yang
Keyword(s):  
Exact Solutions ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Explicit Solutions ◽  
Bifurcation Method ◽  
Characteristic Equations ◽  
Symmetry Reductions ◽  
Wave Solutions ◽  
Nonclassical Symmetries

AbstractIn this article, based on the compatibility method, some nonclassical symmetries of Kudryashov–Sinelshchikov equation are derived. By solving the corresponding characteristic equations associated with symmetry equations, some new exact explicit solutions of Kudryashov–Sinelshchikov equation are obtained. For the exact explicit traveling wave solutions, the exact parametric representations are investigated by the integral bifurcation method.

Research on sensitivity analysis and traveling wave solutions of the (4+1)-dimensional nonlinear Fokas equation via three different techniques

2021 ◽  
Author(s):  
Melike Kaplan Yalçın ◽  
Arzu Akbulut ◽  
Nauman Raza
Keyword(s):  
Sensitivity Analysis ◽  
Nonlinear Equation ◽  
Traveling Wave ◽  
Function Method ◽  
Graphical Representation ◽  
Traveling Wave Solutions ◽  
Analytical Techniques ◽  
Wave Solutions ◽  
Generalized Kudryashov Method ◽  
Kudryashov Method

Abstract In the current manuscript, (4+1) dimensional Fokas nonlinear equation is considered to obtain traveling wave solutions. Three renowned analytical techniques, namely the generalized Kudryashov method (GKM), the modified extended tanh technique, exponential rational function method (ERFM) are applied to analyze the considered model. Distinct structures of solutions are successfully obtained. The graphical representation of the acquired results is displayed to demonstrate the behavior of dynamics of nonlinear Fokas equation. Finally, the proposed equation is subjected to a sensitive analysis.

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.

Abundant mixed lump-kink solutions to a (2+1)-dimensional bSK equation

2018 ◽  
Vol 32 (26) ◽  
pp. 1850313 ◽  
Author(s):  
Zeguang Liu
Keyword(s):  
Bilinear Form ◽  
Logarithmic Transformation ◽  
Symbolic Computations ◽  
Hirota Bilinear Form ◽  
Hirota Bilinear Equation ◽  
Simulation Results ◽  
Hirota Bilinear ◽  
Bilinear Equation

In this paper, we study lump-kink solutions of a (2+1)-dimensional bidirectional Sawada–Kotera equation and discuss their dynamics. A Hirota bilinear form of a (2+1)-dimensional bidirectional Sawada–Kotera equation is deduced via a dependent logarithmic transformation. Based on this Hirota bilinear equation, we obtain eight classes of lump-kink solutions which combine stripe soliton and lump soliton by using symbolic computations. Our simulation results with the appropriate choice of the arbitrary parameters that show the motion of lump soliton and the process of interaction between lump soliton and a stripe soliton.

scholarly journals Travelling Wave Solutions for Two Generalized Hirota-Satsuma Coupled KdV Systems

2001 ◽  
Vol 56 (3-4) ◽  
pp. 312-318 ◽  
Author(s):  
Engui Fan
Keyword(s):  
Full Advantage ◽  
Riccati Equation ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Travelling Wave ◽  
Travelling Wave Solutions ◽  
Symbolic Computations ◽  
Extended Tanh Method ◽  
Wave Solutions ◽  
Tanh Method

Abstract In this paper we present an extended tanh method that utilizes symbolic computations to obtain more travelling wave solutions for two generalized Hirota-Satsuma coupled KdV systems in a unified way. The key idea of this method is to take full advantage of a Riccati equation involving a parameter and use its solutions to replace the tanh-function by the tanh method. It is quite interesting that the numbers and types of the traveling wave solutions can be judged from the sign of the parameter. In this paper we investigate the two generalized Hirota-Satsuma coupled KdV systems

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