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

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.

the Korteweg–de Vries equation
singular perturbation
asymptotic solution
solution of soliton type
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