Asymptotic solutions of soliton type of the Korteweg–de Vries equation with variable coefficients and singular perturbation

2019;
: pp. 374–385
https://doi.org/10.23939/mmc2019.02.374
Received: May 15, 2019
Revised: November 07, 2019
Accepted: November 08, 2019
1
Taras Shevchenko National University of Kyiv
2
Taras Shevchenko National University of Kyiv
3
Taras Shevchenko National University of Kyiv
4
Taras Shevchenko National University of Kyiv
5
Borys Grinchenko Kyiv University

The paper deals with the singularly perturbed Korteweg–de Vries equation with variable coefficients.  The equation describes wave processes in various inhomogeneous media with variable characteristics and small dispersion.  We consider the general algorithm of construction of asymptotic solutions of soliton type to the equation and present its approximate solutions of this type.  We analyze properties of the constructed asymptotic solution depending on a small parameter.  The results are demonstrated by the examples of the studied equation.  We show that for an adequate description of qualitative properties of soliton type solutions to the singularly perturbed KdV equation with variable coefficients it is necessary to construct at least the first asymptotic approximation, that is, expansion containing both the main and the first term.

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Mathematical Modeling and Computing, Vol. 6, No. 2, pp. 374–385 (2019)