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
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
C
3
unless
u
0
and
u
1
are polynomials of degree 2 or lower.