Nonexistence results for the Korteweg-de Vries and Kadomtsev-Petviashvili equations

We study characteristic Cauchy problems for the Korteweg-de Vries (KdV) equ ation u(t) = uu(x) + u(xxx), and the Kadomtsev-Petviashvili (KP) equation u (yy) = (u(xxx) + uu(x) + u(t))(x) with holomorphic initial data possessing non-negative Taylor coefficients around the origin. For the KdV equation wi th initial value u(0, x) = u(0)(X), we show that there is no solution holom orphic in any neighborhood of (t, x) = (0,0) in C-2 unless u(0)(x) = a(0) a(1)x. This also furnishes a nonexistence result for a class of y-independ ent solutions of the KP equation. We extend this to y-dependent cases by co nsidering initial values given at y = 0, u(t, x, 0) = U-0(x, t), U-y(t, x, 0) = u(1)(x, t), where the Taylor coefficients of u, and u, around t = O, x = O are assumed non-negative. We prove that there is no holomorphic soluti on around the origin in C-3, unless u, and u, are polynomials of degree 2 o r lower. MSC 2000: 35Q53, 35B30, 35C10.