Canonical reduction of self-dual Yang–Mills equations to Fitzhugh–Nagumo equation and exact solutions

Sayed, S. M.; Gharib, G. M.

doi:10.1016/j.chaos.2007.01.076

Cited by 14 publications

(10 citation statements)

References 27 publications

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“…Now we describe the sine -cosine method [19], for a given nonlinear evolution equation, say, in two variables [25][26][27][28][29][30],…”

Section: The Sine -Cosine Methodsmentioning

confidence: 99%

Traveling wave solutions for a generalized Kawahara and Hunter-Saxton equations

Alali¹

2013

ijma

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Traveling wave solutions to Kawahara equation (KE), transmission line (TL), and Korteweg-de Vries (KdV) equation are found by using an elliptic function method which is more general than the tanh-method. The method works by assuming that a polynomial ansatz satisfies a Weierstrass equation, and has two advantages: first, it reduces the number of terms in the ansatz by an order of two, and second, it uses Weierstrass functions which satisfy an elliptic equation for the dependent variable instead of the hyperbolic tangent functions which only satisfy the Riccati equation with constant coefficients. When the polynomial ansatz in the traveling wave variable is of first order, the equation reduces to the KdV equation with only a cubic dispersion term, while for the KE which includes a fifth order dispersion term the polynomial ansatz must necessary be of quadratic type. By solving the elliptic equation with coefficients that depend on the boundary conditions, velocity of the traveling waves, nonlinear strength, and dispersion coefficients, in the case of KdV equation we find the well-known solitary waves (solitons) for zero boundary conditions, as well as wave-trains of cnoidal waves for nonzero boundary conditions. Both solutions are either compressive (bright) or rarefactive (dark), and either propagate to the left or right with arbitrary velocity. In the case of KE with nonzero boundary conditions and zero cubic dispersion, we obtain cnoidal wave-trains which represent solutions to the TL equation. For KE with zero boundary conditions and all the dispersion terms present, we obtain again solitary waves, while for KE with all coefficients present and nonzero boundary condition, the solutions are written in terms of Weierstrass elliptic functions. For all cases of the KE we only find bright waves that are propagating to the right with velocity that is a function of both dispersion coefficients.

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“…Now we describe the sine -cosine method [19], for a given nonlinear evolution equation, say, in two variables [25][26][27][28][29][30],…”

Section: The Sine -Cosine Methodsmentioning

confidence: 99%

Traveling wave solutions for a generalized Kawahara and Hunter-Saxton equations

Alali¹

2013

ijma

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“…In addition, this equation arises in heat and mass transfer. Sayed and Gharib have found traveling wave solution class for FN and NW equations by using an improved sine-cosine method and Wu's elimination method [30]. Also Abbasbandy has employed the Homotopy Analysis Method (HAM) to obtain the solitary solutions of FN equation [34].…”

Section: Example 1: Fitzhugh-nagumo (Fn) Equationmentioning

confidence: 99%

Exact Traveling Wave Solutions for Fitzhugh-Nagumo (FN) Equation and Modified Liouville Equation

Abdelrahman¹,

Khater²

2015

IJCA

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In this paper, we employ the exp(−ϕ(ξ))-expansion method to find the exact traveling wave solutions involving parameters of nonlinear evolution equations Fitzhugh-Nagumo (FN) equation and Modified Liouville equation. When these parameters are taken to be special values, the solitary wave solutions are derived from the exact traveling wave solutions. It is shown that the proposed method provides a more powerful mathematical tool for constructing exact traveling wave solutions for many other nonlinear evolution equations.

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“…In particular, it has been known for a while that there is a relationship between surfaces of a constant negative Gaussian curvature in Euclidean three-space, the Sine-Gordon Equation and Bäcklund transformations which are relevant to the given equation [2]. Moreover, the original Bäcklund transformation for the Sine-Gordon Equation is also a simple geometric construction for pseudospherical surfaces [3]- [5]. It is well known that nonlinear complex physical phenomena are related to nonlinear partial differential equations (NLPDEs) which are involved in many fields from physics to biology, chemistry, mechanics, etc.…”

Section: Introductionmentioning

confidence: 99%

Travelling Wave Solutions of Kaup-Kupershmidt Equation Which Describes Pseudo Spherical Surfaces

Gharib¹

2015

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Canonical reduction of self-dual Yang–Mills equations to Fitzhugh–Nagumo equation and exact solutions

Cited by 14 publications

References 27 publications

Traveling wave solutions for a generalized Kawahara and Hunter-Saxton equations

Traveling wave solutions for a generalized Kawahara and Hunter-Saxton equations

Exact Traveling Wave Solutions for Fitzhugh-Nagumo (FN) Equation and Modified Liouville Equation

Travelling Wave Solutions of Kaup-Kupershmidt Equation Which Describes Pseudo Spherical Surfaces

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