1978
.
—
. XXXITI,
MAT
E M AT
1
. .
1978
5 (203)
X
.
1.
.
.
.
.
«
».
[1] — [3].
*
[1] D. J.
n n e , Some properties of long nonlinear waves, Studies in Appl. Math.
52 : 1 (1973), 45—50.
[2] . .
,
. .
,
. I.
,
.
11 : 3 (1977), 31—42.
[3] . .
,
. .
,
. II.
,
.
12 : 1 (1978), 25—37.
15
1.
.
1978 .
.
,
.
—
, «
(
I.
-
)».
(
)
1970 .
-
.
,
(
[2], [3]
. [1]).
.
(1)
3(
+ uxx) = 4wx;
3
— uux — uxxx
wy — ut = wxx
:
»
L-e-4- i-"]- 0 -
2
3
L = -jj-r+u(x,
+u(x
, t), =- - '+-2
+ —v(t),
, tV w^w
.
^
+ y ^ ; --JJ-==\I
, t).
-*•
.
- — \ udx = 0.
(t) +
\i(t)
,
,
|
t
| -*
\ udxdy,
,
.
.
, XXXIII,
(1),
\ udx — 0.
.
14
+ ">(*>
. 5
\
(1)
2
dx dy
210
.
,
;
.
1.
,
2.
[2],
,
,
-
.
,
/, t
(
.
[4]),
g
D
g (
-
g).
3.
2
2
+
, /, t
«
( 2 +
—>-
,
->- .
( — at,
»
2 1
)' ,
[5].
(
(3)
( ,
Xj(y, t)
»
,
— bt),
, *)=-2 2
)
)- 2 ,
(*-*;(?.
z/, t
«
—
*
| + 2 2 (* -*;)-2
( , )=± 2
( . [4], [6]).
III.
[7]
-
-
,
. .
,
Z]1, . . ., t\i,
.
.
.
60-
lg
g,
.
,
g,
: )
;
...,aZg,
(4)
2
1
. . ., yig 6 ,,
)
aj = aj
^ =
(g = l, . , . , Z), .
('' = 1' ••" **>•
°
9=1
,
a j = 1.
)
(
\ ( ,
);
= (
. . .,
)
),
£
0 6
Vi> • • • » yig( )
)
, xj —
:
,
|>
?
XJ
-
, = (if) , . . ., |)*).
.
|)9
—
1
-;
^
q
a(5)
|
)
->
-
0
=
? ^).
"
:
* (*, / ) = ( + 2 « (*) * " * ) % (*, *),
=1
£0 = (1, 0, . . . , 0), gj (#) —
<>.
6
(>
,
(1X1)
.
At(x,
1
-1
~ =/
W0 ( ,
( )—
)
^i = - ^ - 4 f o 1
,
)
dAt
dA
1 ~1
j
r
=1
,
^
|1
"
-
2
1.
), ),
^,
}( ,
-
tf>(x, P)
^
)\
{ ,
)}
'
0,
» ) = 1.
[8]
= (xi<7-)>
2,
3
=
1=
=2'
*2.
*="2>
° )+**
Z = 3.
, t
2.
(—
,
L\ if —0,
(—
,
w=
'— I-
A3 = K3 +
0,
(Pij),
. . .,
^-, ^
, t),
, t, ),
0(
^ ,
, t, P)
) - = 0.
( ,
h"T-2) ii-
I X I
*=
.
' * = * + * ( ! ! „ 1)+ >
/0
...0,
0
...0 ,
A2 = x* + (qij),
1== +
W
U 0z, zMi, . . . 0/
.
(6)
.
,
1=2
.
if.
—
- ( ,
, q tj
,
»
.)
xfJ- = 0.
, Kj,j+i = 1,
*=(°
/?
L
, i) =—2
, w
^i (#> V-, t),
.
[7].
Ai =
• ( , ).
=-(#!, . . .,
*(* = 1, . . ., Kg + 1) — 1 = N).
4?0{ , ),
,
=
xN).
3.
(l(g + 1) — 1)-
lg
-
1g.
.
/2#-
,
/( ),
.
.
1=1
I > 1
.
GL(l,
.
Kg + 1 ) — 1
[1]
.
GL(l,
.
.
,
.
.
,
[2]
.
.
.
8:3
.
),
,
.
.
)
,
192 : 4 (1970), 291—294.
,
. I,
[3]
(1974), 43—53.
,
,
[4]
.
-
,
.
—
19 : 12 (1973), 219-225.
.
,
32 : 6 (1977), 183—208.
[5] L. .
d a g, A. R. I t s , V. . M a t v e e v , S. V. M a n a c o v , V. . Z k h a r o v , Two-dimension solitons of Kadomtzev—Petviashvely equations, Phys..
Lett. (1978).
[6] . .
,
—
N
,
.
12 : 1 (1978).
{7] . .
,
.
,
—
,
.
|12 : 4 (1978).
,
14*
212
[8]
.
.
,
,
12 : 3 (1978), 20—31.
22
1.
-
.
.
1978
.
.
,
.
.
«
».
X
Y —
), ( , t){t > tQ, £ X) —
h.
dU(t
d[
(1)
X)
(
,
, [\7 (g) u(x)]h = u'(x)h —
(
,
)
+* ( (*, *)) = 0,
( ,
)=
( ),
-
*;< .
1
lv ( )~ — (A0A*V, V)u-\-(av, V) u-\-bv ( * , u)-\-cvu —
(
v — v(t, x))
Av = A(t, , ) 6
£ ©2( )» av = a(t, , v) £ X , bD = b(t, x, v) £ L2(Y X X, Y), cv = c(t, x, v) £ L(Y),
(t0 ^ t ^ , x 6 X, f 6 )»
L(X), 8 2
» ^ ( ^ X X, ) —
,
*
,
.
(1),
t
t
(2)
l(t) = x+^
a(s, l(s), u(s, l(s)))ds+
\ A{s, l(s), u(s, l(s))) dw (s) ,
e
e
(3)
T| (t, 6) y^y
t
+ j c* (s, £ (s), u (*, I (s)))
T) (*,
)
tfs +
+ ( b*(«, U ^ u(s, 6(*)))( (*.
) , A*(*)).
0
(4)
( ,
( *( ,
), z)y = ( /, b(z,
(*, *)> =
)> , z,
( ( , 0 ,
6
,
0(£(
))> ,
6 X.
u{t, x)
]( , 9)
£(£)
(1),
,
(4).
,
av, Av
|| v \\
v
, bv —
{cvh, h )y < \\h ||2[p0 + Pi || v \\P],
|| $
, a cv
|| v \\ (
> 0)
0
<0.
.
.
—
^-+
(wfc+i)=0
-
,
(
)
.
.
.
X
.
[1],
1.
[2].
-
= A(t,
,
, V®
)
(1)
+
^ * ^ , V)w + b(t,
,
, V®
) = 0,