Nonexistence Results for the Korteweg–de Vries and Kadomtsev–Petviashvili Equations

We study characteristic Cauchy problems for the Korteweg–de Vries (KdV) equation 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 with initial value u (0,  x )= u 0 ( x ), we show that there is no solution holomorphic 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 ‐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 non‐negative. We prove that there is no holomorphic solution around the origin in C 3 , unless u 0 and u 1 are polynomials of degree 2 or lower. MSC 2000: 35Q53, 35B30, 35C10.