soliton

Asymptotic stepwise solutions of the Korteweg-de Vries equation with a singular perturbation and their accuracy

The paper deals with the Korteweg-de Vries equation with variable coefficients and a small parameter at the highest derivative.  The asymptotic step-like solution to the equation is obtained by the non-linear WKB technique.  An algorithm of constructing the higher terms of the asymptotic step-like solutions is presented.  The theorem on the accuracy of the higher asymptotic approximations is proven.  The proposed technique is demonstrated by example of the equation with given variable coefficients.  The main term and the first asymptotic approximation of the given example are found, their a

Asymptotic analysis of the Korteweg-de Vries equation by the nonlinear WKB technique

The paper deals with the Korteweg-de Vries equation with variable coefficients and a small parameter at the highest derivative.  The non-linear WKB technique has been used to construct the asymptotic step-like solution to the equation.  Such a solution contains regular and singular parts of the asymptotics.  The regular part of the solution describes the background of the wave process, while its singular part reflects specific features associated with soliton properties.  The singular part of the searched asymptotic solution has the main term that, like the soliton solu