ar X iv : m at h - ph / 9 90 90 28 v 2 1 6 O ct 1 99 9 Asymptotics of soliton solution for the perturbed

The dromion of the Davey-Stewartson-1 equation is studied under perturbation on the large time. In this work we construct the asymptotic solution of the Davey-Stewart-son-1 equations (DS-1): Here ε is small positive parameter, F is operator of the perturbation, the values of the parameter σ = ±1 correspond so called focusing or defocusing DS-1 equations. The equations (1) at ε = 0 describe the interaction of long and short waves on the liquid surface if the capillary effects and potential flow are taken into account [1, 2]. The theorems about the existence of the solutions for this equations in the different functional classes are known [3, 4]. The inverse scattering transform method for the DS equations was formulated in [5]–[8]. This method allows to construct the soliton solutions [9] and to study the global properties of the solutions, for instance, the asymptotic