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Partial Differential Equations that Lead to Solitons

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Encyclopedia of Complexity and Systems Science

Definition of the Subject

In this part, we introduce the reader to a certain class of nonlinear partial differential equations which are characterized by solitarywave solutions of the classical nonlinear equations that lead tosolitons . The classical nonlinear equations of interest show the existence of special types of travelingwave solutions which are either solitary waves or solitons. In this study, we will review a few solutions arising from the analytic work of theKorteweg–de Vries (KdV) equations, the generalized regularized long-wave RLW equation, Kadomtsev–Petviashvili (KP) equation, theKlein–Gordon (KG) equation, the Sine-Gordon (SG) equation, the Boussinesq equation, Pochhammer–Chree (PC) equation and the nonlinearSchrödinger (NLS) equation, the Fisher equation, Burgers equation, the Korteweg–de Vries Burgers' equation (KdVB), the two‐dimensionalKorteweg-deVries Burgers' (tdKdVB), the potential Kadomtsev–Petviashvili equation, the...

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Books and Reviews

  1. The following, referenced by the end of the paper, is intended to give some useful for further reading.

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  2. For another obtaining of the KdV equation for water waves, see Kevorkian and Cole (1981); one can see the work of the Johnson (1972) for a different water-wave application with variable depth, for waves on arbitrary shears in the work of Freeman and Johnson (1970) and Johnson (1980) for a review of one and two‐dimensional KdV equations. In addition to these; one can see the book of Drazin and Johnson (1989) for some numerical solutions of nonlinear evolution equations. In the work of the Zabusky, Kruskal and Deam (F1965) and Eilbeck (F1981), one can see the motion pictures of soliton interactions. See a comparison of the KdV equation with water wave experiments in Hammack and Segur (1974)

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  3. For further reading of the classical exact solutions of the nonlinear equations can be seen in the works: the Lax approach is described in Lax (1968); Calogero and Degasperis (1982, A.20), the Hirota's bilinear approach is developed in Matsuno (1984), the Bäckland transformations are described in Rogers and Shadwick (1982); Lamb (1980, Chap. 8), the Painleve properties is discussed by Ablowitz and Segur (1981, Sect. 3.8), In the book of Dodd, Eilbeck, Gibbon and Morris (1982, Chap. 10) can found review of the many numerical methods to solve nonlinear evolution equations and shown many of their solutions.

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  1. Department of Mathematics, Firat University, Elazig, Turkey

    Doğan Kaya

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  1. Doğan Kaya
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  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Kaya, D. (2009). Partial Differential Equations that Lead to Solitons. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_380

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