Differential Geometry and Hydrodynamics of Soliton Lattices

I am going to discuss here some results in the soliton theory of a Moscow group during the last years. The group of people who worked here include B.A.Dubrovin, I.M. Krichever, S.P. Tsarev (and the present author). More details may be found in the survey article [1]. Modern needs in the large new classes of hydrodynamic type systems appear in connection with very interesting asymptotic method - so called “nonlinear analog of WKB-method”, method of the slow modulations of parameters or “Whitham method” in the theory of solitons (see for example the book [2], chapter 4). This method is based on the large family of exact solutions φ0(x,t,u)periodic or quasiperiodic in x and t, of the form which we call “soliton lattice ”: $${{\varphi }_{0}}(x,t;u) = F({{\eta }_{0}} + Ux + Vt;{{u}^{1}}, \ldots ,{{u}^{N}}).$$ (0.1) Here \(F({{\eta }_{1}}, \ldots ,{{\eta }_{m}};{{u}^{1}}, \ldots ,{{u}^{N}})\) is the function, which is 2π-periodic in each variable η j, and depends on the N parameters up. All quantities (aj, bk,up,) are constants and φ0(x,t,u) satisfies some nonlinear P.D. equation $${{\varphi }_{t}}(x,t) = K(\varphi ,{{\varphi }_{x}}, \ldots ,{{\varphi }^{{(s)}}}),$$ (0.2) describing the propagation of solitons or some other nonlinear waves.