Samoilenko Y. Asymptotic soliton type solutions to singularly perturbed Korteweg-de Vries equation with variable coefficients

There is proposed an algorithm of constructing asymptotic soliton type solutions to singularly perturbed Korteweg-de Vries equation with variable coefficients. The theorems on the accuracy with which constructed solutions satisfy the equation are proved. The structure of asymptotic soliton type solutions to singularly perturbed Korteweg-de Vries equation with variable coefficients in depend on the degree of small parameter at the highest derivative of the given equation is studied. The notion of Huhgoniut type condition for existence of discontinuous solution to unperturbed for singularly perturbed Korteweg-de Vries equation with variable coefficients is proposed. The relationship the condition and the equation of discontinuity curve for singular part of asymptotic solutions is demonstrated. The problem of constructing asymptotic soliton type solutions to Cauchy problem for singularly perturbed Korteweg-de Vries equation with variable coefficients had been firstly studied. The algorithm of constructing the asymptotic solution is proposed. The theorems on the accuracy with which constructed solutions satisfy the Cauchy problem are obtained.