Analytical and numerical studies of the variable-coefficient Gardner equation

摘要

The Gardner or extended Korteweg–de Vries equation with the variable-sign cubic non-linear term is studied. This equation occurred in the theory of interfacial waves in the stratified shear flow at some certain conditions on the medium stratification. An asymptotic method is applied to study the solitary wave transformation when the coefficient of the cubic non-linear term is varied slowly. It is found the one branch of the solitary waves destroys in the vicinity of the bifurcation point, where the exponential soliton transforms into the breather through algebraic soliton. Other solitary waves change their amplitude accordingly to the cubic non-linear term variation. The structure of the tail formed behind the solitary wave is determined. Direct numerical simulation of the Gardner equation is performed to analyze the transformation and breaking of the solitary wave. It confirms most of the theoretical results. The calculated dynamics of the “table-top” soliton due to the large values of the mass and momentum integrals differs from the asymptotic prediction because the characteristic time of the variation of the cubic non-linear term has the same order as the duration of “table-top” soliton.