The Cauchy problem for the generalized Korteweg-de-Vries equation

Abstract

This lecture is devoted to a brief (and partial) survey of the Cauchy problem for the generalized Korteweg-de Vries (GKdV) equation, and to the presentation of some recent results on that problem obtained in collaboration wlth Y. Tsutsumi and G. Velo. We refer to [12][16] for a general survey and to [8] [9] [10] for a detailed exposition of the latter results. The GKdV equation can be written as 8 t u. D 3 u = D V' (u) (I) where u is a real function defined in space time IR, ® I~, D -dldx, the prime denotes the derivative and V e C,I(~, IR) with V(O) -V'(O) = O. The ordinary KdV equation is the special case V'(u) -u z and the modified KdV equation is the special case V'(u) = u 3. The equation (I) is the Euler-Lagrange equation of a variational problem with Lagrangian density ~.,(v) = (I/2) Sty Dv (112) (D z v) z V(Dv) (2) where v is a function of space time. In fact, the equation associated with (2) is 8 tDV÷D 4V=DV'(DV) (3) which coincides with (I) for Dv =. u. The equation (3) satisfies in general three