Application of Inverse Scattering Method to Problems of Differential Geometry

This article presents, in a brief form, the recent results on application of the Inverse scattering method to some problems of Differential geometry. A connection between the theory of Integrable systems and Differential geometry is not a new concept. The Sine–Gordon equation was introduced in the theory of surfaces of constant negative curvature around 1860. Actually, it should be called the “Bonnet equation”. The Backlund transformations appeared in Differential geometry also in nineteenth century. At present, it is established that some important integrable equations, found in the last three decades in the theory of solitons (Bullough–Dodd equation, Dawey– Stewartson stationary equation, etc.), have a geometrical interpretation. All of the equations mentioned above are in 1 + 1 dimensions. In this article we explore some geometrical applications of integrable systems in 2 + 1 dimensions. The closest relative of the famous “three–wave equation”,