# Encyclopedia of Complexity and Systems Science

Living Edition
| Editors: Robert A. Meyers

# Semi-analytical Methods for Solving the KdV and mKdV Equations

• Doğan Kaya
Living reference work entry

DOI: https://doi.org/10.1007/978-3-642-27737-5_305-3

## Glosarry

Adomian decomposition method, homotopy analysis method, homotopy perturbation method, and variational perturbation method

These are some of the semi-analytical/numerical methods for solving Ordinary Differential Equation (in short ODE) or Partial Differential Equation (in short PDE) in literature. – 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. – 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.

## Definition of the Subject

In this study, some semi-analytical/numerical methods are applied to solve the...

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## Primary Literature

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58. Sajid M, Hayat T, Asghar S (2007) Comparison between HAM and HPM solutions of thin film flows of non-Newtonian fluids on a moving belt. Nonlinear Dyn 50:27–35
59. Shawagfeh N, Kaya D (2004) Series solution to the Pochhammer-Chree equation and comparison with exact solutions. Comp Math Appl 47:1915–1920
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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 readingGoogle Scholar
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)Google Scholar
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 solutionsGoogle Scholar