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Korteweg–de Vries Equation (KdV) and Modified Korteweg–de Vries Equations (mKdV), Semi-analytical Methods for Solving the

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Encyclopedia of Complexity and Systems Science

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In this study, some semi‐analytical/numerical methods are applied to solve the Korteweg–de Vries (KdV) equation and the modifiedKorteweg–de Vries (mKdV) equation, which are characterized by the solitary wave solutions of the classical nonlinear equations that lead tosolitons. Here, the classical nonlinearequations of interest usually admit for the existence of a special type of the traveling wave solutions which are either solitary waves orsolitons. These approaches are based on the choice of a suitable differential operator which may be ordinary or partial, linear or nonlinear,deterministic or stochastic. It does not require discretization, and consequently massive computation.

In this scheme the solution is performed in the form of a convergent power series with easily computable components. This section isparticularly concerned with the Adomian decomposition method (ADM ) and the results obtained are compared to those obtained by the...

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Abbreviations

Korteweg–de Vries equation :

The classical nonlinear equations of interest usually admit for the existence of a special type of the traveling wave solutions , which are either solitary waves or solitons.

Modified Korteweg–de Vries:

This equation is a modified form of the classical KdV equation in the nonlinear term.

Soliton :

This concept can be regarded as solutions of nonlinear partial differential equations.

Exact solution :

A solution to a problem that contains the entire physics and mathematics of a problem, as opposed to one that is approximate, perturbative, closed, etc.

Adomian decomposition method , Homotopy analysis method, Homotopy perturbation method and Variational perturbation method :

These are some of the semi‐analytic/numerical methods for solving ODE or PDE in literature.

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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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Author information

Authors and Affiliations

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

    Doğan Kaya

  2. Dean of Engineering Faculty, Ardahan University, Ardahan, Turkey

    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). Korteweg–de Vries Equation (KdV) and Modified Korteweg–de Vries Equations (mKdV), Semi-analytical Methods for Solving the. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_305

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