Sech-polynomial travelling solitary-wave solutions of odd-order generalized KdV equations

Parkes, E.J.; Zhu, Zuo-Nong; Duffy, Brian; Huang, Hongci

doi:10.1016/s0375-9601(98)00662-8

Cited by 35 publications

(21 citation statements)

References 10 publications

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“…The selection of α = 0 presents rational solutions. If v is a polynomial and r = 0, then u gives an exact solution in the form of a finite series in tanh ξ or tan ξ, which is obtainable by the tanh-function type method [9,12,13]. …”

Section: A Transformed Rational Function Methodsmentioning

confidence: 99%

A transformed rational function method and exact solutions to the 3+1 dimensional Jimbo–Miwa equation

Lee

2009

Chaos, Solitons & Fractals

541

224

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A direct approach to exact solutions of nonlinear partial differential equations is proposed, by using rational function transformations. The new method provides a more systematical and convenient handling of the solution process of nonlinear equations, unifying the tanh-function type methods, the homogeneous balance method, the expfunction method, the mapping method, and the F -expansion type methods. Its key point is to search for rational solutions to variable-coefficient ordinary differential equations transformed from given partial differential equations. As an application, the construction problem of exact solutions to the 3 + 1 dimensional Jimbo-Miwa equation is treated, together with a Bäcklund transformation. MSC: 35Q58, 37K10, 35Q53

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Section: A Transformed Rational Function Methodsmentioning

confidence: 99%

A transformed rational function method and exact solutions to the 3+1 dimensional Jimbo–Miwa equation

Lee

2009

Chaos, Solitons & Fractals

541

224

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“…For example the fifth-order KdV equations can be derived in fluid dynamics and in magneto-acoustic waves in plasma and its exact solutions was given in [47][48][49][50][51]. The higher-order KdV equation can be derived for magnetized plasmas by using the reductive perturbation technique.…”

Section: The Modified Fifth Order Kdv Equationmentioning

confidence: 99%

“…Traveling wave solutions of Kawahara equation and modified Kawahara equation have been studied [9,48,49]. Moreover, the solitary wave solutions of nonlinear equations with arbitrary odd-order derivatives were studied by many authors [47,51]. Consider the modified fifth order KdV equation where β, c 3 and c 5 are constants.…”

Section: The Modified Fifth Order Kdv Equationmentioning

confidence: 99%

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Exact Solutions Expressible in Hyperbolic and Jacobi Elliptic Functions of Some Important Equations of Ion-Acoustic Waves

Khater¹,

Hassan²

2011

Acoustic Waves - From Microdevices to Helioseismology

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“…Some methods have been proposed in order to solve nonlinear differential equations. For instance, the homogeneous balancement method [16], the method of expansion in hyperbolic functions [17][18][19], the method of test function [20,21], the method of nonlinear transformation [22,23], and sine-cosine method [24]. However, these methods yield only the solutions of solitary waves and shock waves.…”

mentioning

confidence: 99%

Heisenberg spin textures on a cylinder with topological defects

Paula¹

2009

Braz. J. Phys.

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The present work aims to study equilibrium configurations of spins on a cylinder with topological defects such as screw dislocation and deficit angle. By making use of elliptic-f expansion method, which in turn utilizes the Jacobi elliptic functions, we obtain exact solutions of the nonlinear sigma model in this geometry. We have significant changes in the qualitative behavior of the solutions due to the presence of the parameter k of screw dislocation. In particular, the behavior of soliton-like solutions, characteristic of a cylinder without dislocation, was not found in the model here proposed. Keywords: Heisenberg's spins -nanomagnetism The nonlinear sigma model is the continuous limit of Heisenberg's hamiltonian for spins and has gained interest recently. It allows the study of equilibrium configurations of spins on non-trivial geometries such as cylinders, tori, ellipsoids, surfaces with negative curvature and so on [1][2][3][4]. The study of the stability of spin configurations and geometric frustration on different kinds of geometry are the two effects most explored with this model [5][6][7][8]. Defects such as punctures and impurities in the structure have also played an important role related to spin topological textures. In particular, we have introduced topological defects in an ideal cylinder such as screw dislocation. This defect is obtained by producing a longitudinal displacement in the structure, distorting the lattice helically. In this way we can study how the spin textures are modified in relation to the simple cylinder without these defects.Connections with fundamental areas and important technological applications can also be made: the "similarity" between the nonlinear sigma model in 2D and Yang-Mills in (3+1)D [9]; out-of-plane vortex (its core is similar to the central region of a soliton of spins). These out-of-plane vortex are important in several mechanisms of magnetic logic (MRAM), ultra-precise magnetic sensors, magnetic recording, etc. [10,11] The Heisenberg model takes into account only the interaction of each spin of the lattice with its first neighbors. By applying an external magnetic field, the Hamiltonian of the system is thenHere h ab is the 3 × 3 dimensionless interaction law matrix (metric of the internal space of spins), J is the exchange energy, and S a i denotes the a-th component of the spin of the i-th cell of the lattice, while < i, j > includes in the sum only the first neighbor cells j of each cell i. For isotropic interaction we have h ab = diag (1, 1, 1). The parameter g is the gyromagnetic factor of the spins in the material medium, µ is their magnetic momentum and B a is the a-th component of the external magnetic field. Eq. (1) is the hamiltonian for J > 0, describing the ferromagnetic case. In the antiferromagnetic case (J < 0), the spin vector S must be replaced by the Néel vector η = 1 2 ( S 1 − S 2 ), where the index distinguish two distinct sub-lattices in the more simple case.For sufficiently small distances between neighbor cells (and large spins) we...

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Sech-polynomial travelling solitary-wave solutions of odd-order generalized KdV equations

Cited by 35 publications

References 10 publications

A transformed rational function method and exact solutions to the 3+1 dimensional Jimbo–Miwa equation

A transformed rational function method and exact solutions to the 3+1 dimensional Jimbo–Miwa equation

Exact Solutions Expressible in Hyperbolic and Jacobi Elliptic Functions of Some Important Equations of Ion-Acoustic Waves

Heisenberg spin textures on a cylinder with topological defects

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