Nonexistence results for the Korteweg-deVries and Kadomtsev-Petviashvili equations

Abstract

We study characteristic Cauchy problems for the Korteweg-deVries (KdV) equation $u_t=u{u}_x+u_{xxx}$, and the Kadomtsev-Petviashvili (KP) equation $u_{yy}=(u_{xxx}+u{u}_x+u_t{)}_x$ with holomorphic initial data possessing nonnegative Taylor coefficients around the origin. For the KdV equation with initial value $u(0,x)=u_0(x)$, we show that there is no solution holomorphic in any neighbourhood of $(t,x)=(0,0)$ in ${\mathbb{C}}^2$ unless $u_0(x)=a_0+a_{1}x$. This also furnishes a nonexistence result for a class of $y$-independent solutions of the KP equation. We extend this to $y$-dependent cases by considering 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 $u_0$ and $u_1$ around $t=0$, $x=0$ are assumed nonnegative. We prove that there is no holomorphic solution around the origin in ${\mathbb{C}}^3$ unless $u_0$ and $u_1$ are polynomials of degree 2 or lower.

17 pages in LaTeX2e, to appear in Stud. Appl. Math