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.

scholarly journals Integrability via Functional Expansion for the KMN Model

Symmetry ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 1819
Author(s):  
Radu Constantinescu ◽  
Aurelia Florian
Keyword(s):  
Differential Equations ◽  
Traveling Wave ◽  
Nonlinear Differential Equations ◽  
Traveling Wave Solutions ◽  
Auxiliary Equation ◽  
First Order ◽  
Second Order Differential Equations ◽  
Functional Expansion ◽  
Wave Solutions ◽  
First Time

This paper considers issues such as integrability and how to get specific classes of solutions for nonlinear differential equations. The nonlinear Kundu–Mukherjee–Naskar (KMN) equation is chosen as a model, and its traveling wave solutions are investigated by using a direct solving method. It is a quite recent proposed approach called the functional expansion and it is based on the use of auxiliary equations. The main objectives are to provide arguments that the functional expansion offers more general solutions, and to point out how these solutions depend on the choice of the auxiliary equation. To see that, two different equations are considered, one first order and one second order differential equations. A large variety of KMN solutions are generated, part of them listed for the first time. Comments and remarks on the dependence of these solutions on the solving method and on form of the auxiliary equation, are included.

scholarly journals Solitons for the modified $(2 + 1)$-dimensional Konopelchenko–Dubrovsky equations

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Xiumei Lyu ◽  
Wei Gu
Keyword(s):  
Elliptic Function ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
High Order ◽  
Auxiliary Equation ◽  
Jacobi Elliptic Function ◽  
Equation Approach ◽  
Exact Traveling Wave Solutions ◽  
Wave Solutions ◽  
Triangular Function

Abstract In the paper, we consider the modified $(2 + 1)$ ( 2 + 1 ) -dimensional Konopelchenko–Dubrovsky equations which possess high order nonlinear terms. Under the aid of Maple, we derive the exact traveling wave solutions of the mKDs by the auxiliary equation approach. Under some special conditions, Jacobi elliptic function solutions, degenerated triangular function solutions, and solitons for the mKD equations are constructed.

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.

Abundant traveling wave solutions to the resonant nonlinear Schrödinger’s equation with variable coefficients

2020 ◽  
Vol 34 (12) ◽  
pp. 2050118 ◽  
Author(s):  
Hadi Rezazadeh ◽  
Kalim U. Tariq ◽  
Jamilu Sabi’u ◽  
Ahmet Bekir
Keyword(s):  
Traveling Wave ◽  
Dynamical Behavior ◽  
Traveling Wave Solutions ◽  
Auxiliary Equation ◽  
Mathematical Tool ◽  
Schrödinger’S Equation ◽  
Wave Solutions ◽  
Schrödinger's Equation ◽  
Nonlinear Schrodinger’S Equation ◽  
Nonlinear Schrödinger’S Equation

In this paper, some new traveling wave solutions to the resonant nonlinear Schrödinger’s equation (R-NLSE) with time-dependent coefficients are constructed. The well-known auxiliary equation method is applied to develop numerous interesting classes of nonlinearities, namely the Kerr law and parabolic law. Such approach provides an extensive mathematical tool to develop a family of traveling wave solutions such as bright, dark, singular and optical solutions to the nonlinear evolution model. Moreover, with the aid of symbolic computation the three-dimensional plot and contour plot have been carried out to demonstrate the dynamical behavior of the nonlinear complex model.

Traveling Wave Exact Solutions for Nonlinear Coupled Scalar Field Equations

2011 ◽  
Vol 284-286 ◽  
pp. 2053-2056
Author(s):  
Dong Bo Cao ◽  
Jia Ren Yan
Keyword(s):  
Scalar Field ◽  
Traveling Wave ◽  
Soliton Solutions ◽  
Traveling Wave Solutions ◽  
Trigonometric Function ◽  
Field Equations ◽  
Transform Method ◽  
Wave Solutions ◽  
Trigonometric Function Solutions ◽  
Scalar Field Equations

In the present paper, with the aid of symbolic computation, the nonlinear coupled scalar field equations relevant to materials physics are investigated by using the trigonometric function transform method. More exact traveling wave solutions are obtained for nonlinear coupled scalar field equations. The solutions obtained in this paper include four kinds of soliton solutions and four kinds of trigonometric function solutions.

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.

scholarly journals Symbolic Computation and Construction of New Exact Traveling Wave Solutions to Fitzhugh-Nagumo and Klein-Gordon Equations

2009 ◽  
Vol 64 (1-2) ◽  
pp. 15-20 ◽  
Author(s):  
Turgut Öziş ◽  
İsmail Aslan
Keyword(s):  
Symbolic Computation ◽  
Traveling Wave ◽  
Gordon Equation ◽  
Traveling Wave Solutions ◽  
Expansion Method ◽  
Auxiliary Equation ◽  
Auxiliary Equation Method ◽  
Wave Solutions ◽  
Klein Gordon ◽  
Nagumo Equation

With the aid of the symbolic computation system Mathematica, many exact solutions for the Fitzhugh-Nagumo equation and the Klein-Gordon equation with a quadratic nonlinearity are constructed by an auxiliary equation method, the so-called (G'/G)-expansion method, where the new and more general forms of solutions are also obtained. Periodic and solitary traveling wave solutions capable of moving in both directions are observed.

scholarly journals New Traveling Wave Solutions by the Extended Generalized Riccati Equation Mapping Method of the(2+1)-Dimensional Evolution Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
Hasibun Naher ◽  
Farah Aini Abdullah
Keyword(s):  
Riccati Equation ◽  
Traveling Wave ◽  
Nonlinear Partial Differential Equations ◽  
Traveling Wave Solutions ◽  
Rational Functions ◽  
Mapping Method ◽  
Auxiliary Equation ◽  
Mathematical Tool ◽  
Wave Solutions ◽  
Generalized Riccati Equation

The generalized Riccati equation mapping is extended with the basic(G′/G)-expansion method which is powerful and straightforward mathematical tool for solving nonlinear partial differential equations. In this paper, we construct twenty-seven traveling wave solutions for the (2+1)-dimensional modified Zakharov-Kuznetsov equation by applying this method. Further, the auxiliary equationG′(η)=w+uG(η)+vG2(η)is executed with arbitrary constant coefficients and called the generalized Riccati equation. The obtained solutions including solitons and periodic solutions are illustrated through the hyperbolic functions, the trigonometric functions, and the rational functions. In addition, it is worth declaring that one of our solutions is identical for special case with already established result which verifies our other solutions. Moreover, some of obtained solutions are depicted in the figures with the aid of Maple.

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