An Improved Modified Extended tanh-Function Method

Yang, Zonghang; Hon, Y.C.

doi:10.1515/zna-2006-3-401

Cited by 71 publications

(9 citation statements)

References 3 publications

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“…where c, k and m are arbitrary constants and cn(x, m) is the Jacobi elliptic function of elliptic modulus m that reduces to cos(x) when m = 0. In this equation, this family of traveling waves and many others have been found in [39]. For this case…”

Section: Introduction and Main Resultssupporting

confidence: 52%

“…We remark that explicit TWS have also been searched for by using several direct methods, such as the exp-function method and the tanh-function method and its variants, see for instance [14,24,25,26,39]. These methods are essentially based on the following idea: fix a class of functions with several free parameters and then impose conditions on the parameters to find some particular cases satisfying the corresponding equations.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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Explicit travelling waves and invariant algebraic curves

Gasull

Giacomini

2015

Nonlinearity

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In this paper we introduce a precise definition of algebraic traveling wave solution for general n-th order partial differential equations. All examples of explicit traveling waves known by the authors fall in this category. Our main result proves that algebraic traveling waves exist if and only if an associated ndimensional first order ordinary differential system has some invariant algebraic curve. As a paradigmatic application we prove that, for the celebrated Fisher-Kolmogorov equation, the only algebraic traveling waves solutions are the ones found in 1979 by Ablowitz and Zeppetella. To the best of our knowledge, this is the first time that this type of results have been obtained.

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Section: Introduction and Main Resultssupporting

confidence: 52%

Section: Introduction and Main Resultsmentioning

confidence: 99%

Explicit travelling waves and invariant algebraic curves

Gasull

Giacomini

2015

Nonlinearity

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“…In this section, the improved modified extended tanh-function method is described as follows [1,28]: Assuming the non-linear evolution equation (NLEE) with two independent variables t and x:…”

Section: Improved Modified Extended Tanh-function Methodsmentioning

confidence: 99%

Optical solutions and other solutions for Radhakrishnan-Kundu-Laksmannan equation by using improved modified extended tanh-function method

Elsherbeny

Elbarkouky

Ahmed

et al. 2021

Preprint

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This paper studies Radhakrishnan-Kundu-Laksmannan equation which is used to describe the pulse propagation in optical fiber communications. By using improved modified extended tanh-function method various types of solutions are extracted such as bright solitons, singular solitons, singular periodic wave solutions, Jacobi elliptic solutions, periodic wave solutions and Weierstrass elliptic doubly periodic solutions. Moreover, some of the obtained solutions are represented graphically.

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“…The key footsteps of the extended Fan sub-equation method are given as [44,45]: For a given nonlinear partial differential equation (PDE) where L is a polynomial of v(x, y, t) and its partial derivatives in which the highest order derivatives and nonlinear terms are contained.…”

Section: Outlines Of the Extended Fan Sub-equation Approachmentioning

confidence: 99%

“…The first order Eq. (5) provides various types of fundamental solutions [45]. From these solutions, more new type of exact solutions of Eq.…”

Section: Stepmentioning

confidence: 99%

Traveling, periodic and localized solitary waves solutions of the (4+1)-dimensional nonlinear Fokas equation

Khatri¹,

Malik

Gautam

2020

SN Appl. Sci.

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By utilizing the three distinctive approaches specifically, the extended Fan sub-equation method, the exp[−G()]-function expansion method, and the fractional transformation method, the traveling waves and soliton solutions of the (4+1)-dimensional nonlinear Fokas equation are extracted. Meanwhile, some parametric constraint conditions are described. The acquired solutions are singular and nonsingular soliton solutions, periodic solutions, breather solution, rational solutions, trigonometric periodic wave solutions, hyperbolic solutions, Weierstrass, and Jacobi elliptic doubly periodic wave solutions. The dynamics of some of the obtained solutions are investigated and described in 2-dimensional figures by choosing appropriate parameter values. The comparison of our obtained results with the other solutions in literature shows that the obtained solutions of this paper are new and have not been formulated before by other techniques. We believe that all these results are useful to enrich the knowledge of the important physical phenomenon characterized by the Fokas equation. The reported solutions illustrate the straightforwardness, reliability, and effectiveness of the used techniques that can be further employed to higher-dimensional nonlinear evolution equations.

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An Improved Modified Extended tanh-Function Method

Cited by 71 publications

References 3 publications

Explicit travelling waves and invariant algebraic curves

Explicit travelling waves and invariant algebraic curves

Optical solutions and other solutions for Radhakrishnan-Kundu-Laksmannan equation by using improved modified extended tanh-function method

Traveling, periodic and localized solitary waves solutions of the (4+1)-dimensional nonlinear Fokas equation

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