Solitary wave solutions of nonlinear evolution and wave equations using a direct method and MACSYMA

Hereman, Willy; Takaoka, Masanori

doi:10.1088/0305-4470/23/21/021

Cited by 154 publications

(67 citation statements)

References 28 publications

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“…With the development of computer science and technology, symbolic computation as a new branch of artificial intelligence drastically increases the ability of a computer to deal with the complicated and tedious calculations of the coefficient functions in the NLEEs under investigation. In soliton theory, the computerized symbolic computation system like MATHEMATICA or MAPLE [17 -22] has been playing a more and more important role in analytically investigating NLEEs, such as, transforming a wide class of variable-coefficient models to the standard ones [17,18], constructing exact analytical solutions [17,19,22] (including the solitonic solutions, periodic solutions, rational solutions and so on), and testing the integrability of NLEEs [17,20,21].…”

Section: Motivations For a Generalizedmentioning

confidence: 99%

“…If soliton solutions exist this expansion will always truncate and then finite series will lead to an exact solution. Many perfect software packages available for the bilinear method have already been proposed see [21] for details. (10), integrating twice with respect to x, and taking the integration constants as zero, we get the variablecoefficient bilinear form of (10) as…”

Section: Symbolic Computation Study Of (10) With the Extended Bilineamentioning

confidence: 99%

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Symbolic Computation Study of a Generalized Variable-Coefficient Two-Dimensional Korteweg-de Vries Model with Various External-Force Terms from Shallow Water Waves, Plasma Physics, and Fluid Dynamics

Yao

et al. 2009

Zeitschrift Für Naturforschung A

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The variable-coefficient two-dimensional Korteweg-de Vries (KdV) model is of considerable significance in describing many physical situations such as in canonical and cylindrical cases, and in the propagation of surface waves in large channels of varying width and depth with nonvanishing vorticity. Under investigation hereby is a generalized variable-coefficient two-dimensional KdV model with various external-force terms. With the extended bilinear method, this model is transformed into a variable-coefficient bilinear form, and then a Bäcklund transformation is constructed in bilinear form. Via symbolic computation, the associated inverse scattering scheme is simultaneously derived on the basis of the aforementioned bilinear Bäcklund transformation. Certain constraints on coefficient functions are also analyzed and finally some possible cases of the external-force terms are discussed. Among the most important models of nonlinear evolution equations (NLEEs), constant-coefficient Korteweg-de Vries (KdV) and KdV-type equations are encountered in many apparently unrelated phenomena in different physical areas such as shallow water waves, plasmas, fluids and lattice vibrations of a crystal at low temperatures [1]. In all of these applications, the physical situations described via the KdV (or KdVtype) models tend to be highly idealized, owing to the assumption of constant coefficients. Considering the inhomogeneities of media, nonuniformities of boundaries and external forces, the variable-coefficient models are much more powerful and realistic than their constant-coefficient counterparts in describing various 0932-0784 / 09 / 0300-0222 $ 06.00 c 2009 Verlag der Zeitschrift für Naturforschung, Tübingen · http://znaturforsch.com situations, e. g., in the coastal waters of oceans, space and laboratory plasmas, superconductors and opticalfiber communications [2 -6].In the past decades, it has been shown that various physical phenomena in nature, actual physics and engineering can be described by the variable-coefficient KdV model with perturbed, dissipative and externalforce terms as [7] where v(x,t) is a function of the variables x and t, µ 1 (t) = 0, µ 2 (t) = 0, µ 3 (t), µ 4 (t) and µ 5 (t) represent the coefficients of the nonlinear, dispersive, dissipative, perturbed and external-force terms, respectively, all of which are real functions. In recent studies, of physical and mechanical interests, many important examples of (1), among others, can be listed [8]:Unauthenticated Download Date | 5/8/18 5:12 AM

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Section: Motivations For a Generalizedmentioning

confidence: 99%

Section: Symbolic Computation Study Of (10) With the Extended Bilineamentioning

confidence: 99%

Symbolic Computation Study of a Generalized Variable-Coefficient Two-Dimensional Korteweg-de Vries Model with Various External-Force Terms from Shallow Water Waves, Plasma Physics, and Fluid Dynamics

Yao

et al. 2009

Zeitschrift Für Naturforschung A

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“…If a = −1, one gets the real Newell-Whitehead equation describing the dynamical behaviour near the bifurcation point for the Rayleigh-Bénard convection of binary fluid mixtures. The travelling frame form of (59) has been discussed in detail by Hereman and Takaoka [5] …”

Section: Fitzhugh-nagumo Equationmentioning

confidence: 99%

Supersymmetric pairing of kinks for polynomial nonlinearities

Rosu

Cornejo-Pérez

2005

Phys. Rev. E

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We show how one can obtain kink solutions of ordinary differential equations with polynomial nonlinearities by an efficient factorization procedure directly related to the factorization of their nonlinear polynomial part. We focus on reaction-diffusion equations in the travelling frame and damped-anharmonic-oscillator equations. We also report an interesting pairing of the kink solutions, a result obtained by reversing the factorization brackets in the supersymmetric quantum mechanical style. In this way, one gets ordinary differential equations with a different polynomial nonlinearity possessing kink solutions of different width but propagating at the same velocity as the kinks of the original equation. This pairing of kinks could have many applications. We illustrate the mathematical procedure with several important cases, among which the generalized Fisher equation, the FitzHugh-Nagumo equation, and the polymerization fronts of microtubules.

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“…In this case, the solution (2.3) reduces to the solitary wave solution [7] u(x, t) = 225v 2 64 sech 4 1 4…”

Section: Generalized Korteweg-de Vries Equationsmentioning

confidence: 99%

“…In Section 2, we consider a generalized Korteweg-de Vries equation of third order [7] which has the maximum nonlinearity term u 2p u x . This equation is known to have stable soliton solutions for p < 4.…”

Section: Introductionmentioning

confidence: 99%