- Email: [email protected]
Multilinear forms associated with the Yajima-Oikawa system
29 July 1996
PHYSICS
ELSEVIER
LETTERS
A
Physics Letters A 218 (1996) 16-24
Multilinear forms associated with the Yajima-Oikawa system Walter Strampp Fachbereich 17 - MathematiMnformatik, Universitiit-GH Kassel, Holliindische Strasse 36, 34109 Kassel, Germany
Received 18May 1995; accepted for publication 29 April 1996 Communicated by A.R. Bishop
Abstract It is well known that the Kadomtsev-Petviashvili hierarchy (KP hierarchy) is compatible with generalized k-constraints of the type ( LL) _ = qX’r. Those constraints are leading to integrable systems in 1 + 1 dimensions such as the AKNS system (k = 1) or the Yajima-Oikawa system (k = 2). The effect of the l-constraint on the r-function of the KP hierarchy has been studied: it adds trilinear equations to the set of bilinear ones. In this paper we shall derive a quattro and a quintolinear equation for the r-function of the KP hierarchy under the 2-constraint.
1. The KP hierarchy The members
L,” = [L,
of the infinite KP hierarchy
CL”)-I,
(1.1)
where L is a pseudodifferential L =
are governed by the Lax equations
operator,
ax+ uza;’+ u&y2 + . . . .
We introduce
a pseudodifferential
W = 1 + w,a;’
+ w2a,2
(1.2)
operator,
+ .. .
,
being related to L through L = W&W-‘. form,
(1.3) In terms of the operator W the Lax equations
( 1.1) take the following
w,, = -( wa;w-1)-w, known as Sate’s equations. r-function 1
Wj
=
;P,T
(1.4) The solutions
of Sato’s equations
( 1.4) can be expressed
(7).
0375~9601/%/$12.00 Copyright 8 1996 Elsevier Science B.V. All rights reserved. PII SO375-9601(96)00387-S
with the help of the
(1.5)
W. Strampp /Physics
Letters A 218 (1996) 16-24
17
Here we use the Schur polynomials pi(t), t = (tl , t2, t3,. . .), defined through 00
efw)
=
Gf(t,A)
pi(f)
c
=~tiA’v
i=O
t1
=
x,
(1.6)
i=l
and differential operators P;’ =J%(+J),
P,l- =p1(-a”),
d = (a,, ;a[*, &,
. . .)
(1.7)
(we prepare p[ for the later convenience). Note that pa(t) = 1, pt(t) =x, pz(t) =x2/2+ t2 and p3(t) = x3/6 + xt2 + f3. The r-function depending upon the infinite set of variables t is governed by Hirota’s equations which form an infinite set of bilinear equations. (See Refs. [ l-41 as a general reference for Sato’s theory.)
2. The KP hierarchy and k-constraints It is well-known that the KP hierarchy ( 1.1) is compatible with generalized k-constraints of the form ( Lk)_ = qCY;lr,
(2.1)
which are leading to ( 1 + 1)-dimensional integrable hierarchies,
(L9,”= W,L"_l,
(2.2)
qt, = (L”)+%
(2.3)
rt. = -(L*“)+r
(2.4)
(where L* is the adjoint of L) [5-g]. The first three of those hierarchies are formed by the AKNS (k = 1)) Yajima-Oikawa (k = 2) and a coupled Boussinesq-type (k = 3) hierarchy. Let us write down the first two members of the AKNS and the Yajima-Oikawa hierarchy, respectively, qtz = qxx + 2q2r,
(2.5)
rt2 = -rxx - 2qr2,
(2.6)
qtj =
qxxx+ 6qw,
(2.7)
rfj = rxxx+ 6qrr,
(2.8)
utz = (qr),,
(2.9)
and
2uq,
(2.10)
r 12= -rxx - 2ur,
(2.11)
Ut, = $UXXX+ 3uu, + a(qd- - qrx)x,
(2.12)
412= 4xX +
+ $fr,
(2.13)
rt3 = rxxx + 3ur, + ijug - &r2.
(2.14)
4ts = qxxx + 3%
+ $,q
18
W. Strampp /Physics
Letters A 2 I8 (I 996) 16-24
3. The trilinear form of the Jr&up-Broer system The Kaup-Broer
[ 9, lo]
system
k,, = (k, + 2X*k),,
(3.1)
x:, = c-x:
(3.2)
+ x** + 2h),,
plays an important role in the theory of nonlinear water waves and is closely Namely, the AKNS system transforms to the Kaup-Broer system through
system.
x* = -5.
h = qr, The following
related to he AKNS
I ansatz for obtaining
solutions
has been made in Ref. [ 111,
h = (logs),,,
(3.3)
hx* = ;[(hvL,~ - Uw~)n,l.
(3.4)
This ansatz becomes evident by the fact that Eq. (3.1) is nothing but the simplest equation of the KP hierarchy (l.l), namely U2,rz = (u2.x
So (3.1)
+ &lx.
is trivially
satisfied by (3.3))
(3.4)
and in addition
we obtain the following
trilinear
equation
for r
ill17
(3.5)
4. Multilinear forms and k-constraints From a more general point of view the k-constraint was studied by using Sato’s equations ( 1.4) in Ref. [ 121. The impact of the constraint has been formulated there as equations constraining the coefficients of the operator W. In the l-constraint case the simplest constraining equation for the coefficients Wj reads -Wj,x $
+P2+(Wj) =O,
while in the 2-constraint P:(Wj)Wl,x
- Wj,t*T
(4.1)
case we have the following + T(Wj)
=09
simplest constraining
equation, (4.2)
where the operator T is given by T = -p3+ + V,,
+ dt, = p;
+ c+~,
(4.3)
see Ref. [ 121. The constraining Eqs. (4.1) and (4.2) are difficult to handle because according to ( 1.5) the r-function appears in the denominators. Therefore we aim at expressing the constraining equations in terms of differential operators acting on the r-function.
W. Sirampp/Physics
Letters A218 (1996) 16-24
19
Considering Rq. (4.1) for j = 1 and j = 2 as a linear homogeneous system with a nontrivial solution vector gives d,(W)
P;(W)
Mw2)
P2+(W2>
=o
(4.4)
.
Considering Rq. (4.2) in the same manner for j = 1, j = 2 and j = 3 as a linear homogeneous system with a nontrivial solution vector gives AAWl)
4*(w)
T(w)
4(w2)
&(W2)
T(W2)
&(w3)
&z(W3)
T(W3)
(4.5)
=a
However, we have more possibilities for drawing consequences from (4.2). For instance, take (4.2) for j = 1 and j = 2 and eliminate qx/q, yielding &WI
J,Wl
~,w
T(w)
4w2
T(w2)
(4.6)
=
WI ,x
4w2
ayw2
Finally, take two samples of Eq. (4.2) and consider them as a linear inhomogeneous system for the unknowns ~1,~ and qx/q. This gives the following conditions which will not be discussed further in this paper, T(wj)
-wj,tz
T(w)
-Wi,r*
P;‘(Wj)
-wj,tz
P’(Wi)
-Wi.tz
Pl+(Wj)
T(wj)
= w~,~= independent of indices j, i
(4.7)
and
Pl+(Wi)T(wi) Pl+(Wj)
-wj,tz
p+ (WI
-Wi,Q
qx
=
independent of indices j, i.
(4.8)
= S
By choosing two pairs of indices on the left-hand side of (4.7) and (4.8) we obtain further conditions to be satisfied by the r-function if we set the corresponding expressions equal to each other. In Ref. [ 121 it has been shown that (4.4) becomes equivalent with the trilinear form (3.5). The purpose of the present paper is to cast the condition (4.5) into the quintolinear form P,-(7)
ax(Pip))
42 (Ply(d)
(&x(7)P2’ +
PF (7)
ax (Pi- (7))
at* (PF (7))
(4(7)
TT)
P2’ + 7n
(Pi
Cd)
(p; (7))
= 0, ~37)
a, (P;(T))
at2 (P;(T))
(a,(4
p; + 7~)
(P; (7))
P; (7)
4 (P; (7))
4, (~3 (7))
mm
p:
(~3 cd)
and (4.6) into the quattrolinear form
+ 7~)
(4.9)
20
W. Strampp/Physics
Letters A 218 (19%) 16-24
PO(T)
dtz (P&d)
(-P2+(7)a,+a,(7)P:+7T)
Pr
(7)
&, (P;(T))
(-P;(7)&
+ J,(7)
P:
+ 777
(P;
(7))
P;
(7)
62 (P;
(-P2+W&
+ ax(T)
P2f + TT)
(P;
(7))
(7))
(P&7))
(4.10)
= 0.
Note the remarkable similarities between Eqs. (4.9) and (4.10). In Section 6 we shall outline the derivation of the quintolinear form (4.9) in detail. Since the quattrolinear form (4.10) can be obtained by quite analogous means we shall not present the details. We shall rather discuss an obvious difference of the forms (4.9) and (4.10) in Section 7.
5. Derivation of the trilinear form for the KaupBroer
system
We shall briefly demonstrate the steps leading from (4.4) to (3.5) in a way differing from the derivation given in Ref. [ 121. The advantage of our present method is that it can be carried to the 2-constraint case. Let us first take care of the denominators by calculating
(5.1) (5.2) For the sake of simplicity
“2 =
( )
Furthermore
PC (7) P;(T)
we introduce
the notation (5.3)
.
we shall need the formulae
(5.5)
P,(f(f)g(f))=CPn_j(f(f))P,~(g(f)),
H which are obtained
from Taylor expansion.
a,(w)
P;(w)
Jx(w2)
P;(w)
= /& (3 = 4
-;
We shall now transform
(4.4)
I=~&(;v2)9P:(;v2)~
u2+po+
(gP:(v2,,p:
~Iv2,P:(v2)1+
(3
[ 81 ($q)
v2+P: -bP:
($P:(V2) (;)I
+d
Iv2,plf(v2)1
0 = $[7lP:(u2).P:(u2)l
with the help of (5.1)-(
-4(7)l~2~P;GJ2)1
Introducing T = pi (7) in the coefficients
(:)P:(v2)1
+
-lP;t(v2)~P2f(v2)I :
+P,+(7)I~;?~Pit(v2N.
of the determinants
we obtain the trilinear equation
(3.5).
5.3)
W. Smmpp/Physics
L&en
21
A 2I8 (19%) 1624
6. The quintolinear form
We now turn to the case k = 2. First of all we prepare a few formulae taking care of the denominators,
(6.1)
With the notation
(6.4)
we obtain immediately (6.5) (6.6) (6.7)
(6.8) (6.9)
It is a little more complicated T
1v3 = -&-3T(7) ( 7 1 + $[-Th(T)2
-
to show that +iMd3
$T2d,,(T)
+ $[-;72Md,,(s)
=P3
= (P;
(f)n+P;
- T&(T>&T)]u~
+ 37q(T)]ax(u3) + 73T(v3)]
+ ;T~&(T>&Y~)
holds. We proof (6.10) T ($3)
+nMr)a,,(~)
(6.10)
by first using (4.3) and (5.5),
+ hs) ($3)
($)P;(ui)+P;
Taking (6.1) -( 6.3) and the definition
(9
P1 (v3)
+ ;P;
(v3)
(4.3) of T into account we calculate
+ $w,(s)
- a,,(T)v31.
22
W. SrramPp/Physics Letters A 218 (1996) 16-24
T
1~3 = +W ( 7 >
+ 4m3
- $bm2+
;%(d
+ ;[-g(u,)
+ $w,,(~3)
= $+%c+%~~,3
+%w
+ ;$wd:(U3)
+ p&7)
+ -$MT~;,a:(u,)
- f&,(V3)1
+ $v3)
- &,(v,)]
- r2 [-#CT> .
+ $w,,w v T(7)
+&(d3
+ ; [4&u3) .
+n%(7)~%~(7)
- ;T2a1,(T)
+ &M&73) Y T(w)
&,(w>
&(w2)
4,(~2)
T(w2)
h(w3)
4,(~3)
7’(w3)
+73T(z73)l.
= 1% ($3)
+Md3+&W,,(d
+ +Md3+
$%w2a,,(~)
- $%wM7)
px(Y3)*Y3&v3>1
I~3,M~3M,(~3)1 I~37Mvs),T(~3)I
+ ;yI-&d
- $‘,,(d
6.10))
+3)~
I4(~3),&,(~3))7~3I b3,4*(~3)74(~3)1
;$m - if
Ia,(y3).d,,(U3),a,2(Ug)I (Mm*
IV39&,(s)&v3)I
l4(~3),~3,7’(~3)(
+7la,(u3),a,,(~3),T(~3)1 --&CT)
Iu~,&~(u~)~T(u~)II
~U3,&(s),at,(s>~+7~a,~u3~,a,,~U3~,~,2~U3~~ ~u3,&(u3),~,2~u3)~
--a,(7) h,(T)
=?i
($u1)
- $%w2d:(7)l
lAA~3),u3,4~(~3)1+
+ $lax(~3),ar*(v3>,T(u3)1
94
-~&W,2(dl
+ $%m,,w
1
use of (6.5)-(
T(w)
= $I--SW
+&CT)
/
+ ;T2a,2(79]ax(u3)
+ ;T~&(T)&u~)
J,(w)
+ ~&&,)I
- T&(T)~;(T)]Y~
Now we are able to show that (4.5) can be carried over into (4.9) by making
+&CT)
+ 5~&-)1}u3 /
- ~~&7)1&(~3)
+ -$~T~&(T)&,(v,)
= $J-UT)
- ;72dx&*(7)]Y3
- -$&)u,
- mda,2w
- ;f4w&3)
+ -&T&(T)2
- nfWa,2(7)
- ~~d,2(Tmx(s)
+&(7)&,(T)
- fw2+
= f[-~~T(d
+m7)&,(7)
(u3.&*(u3L&u3)(1 T(T)
p;;d
&;%,
a (P-(T))
UP-(~))
P;(T) P;(T)
a,&)) &(P;(~))
&p:(d) &(pf(~))
T&) UP;(T))
- $-WI
)~39&,(~3)9T(~3)I
W. Strampp/ Physics Letters A 218 (1996) 16-24
7 at209 am T(7) I p; (7) a,(pl-(T)1 a&- (4 > Updd 1 = 7 p;(7) a,(p;(7)) a,,(p;w VP;(~) p;(7) ax,(p;(T)) a,,(p;(m UP;(~)
+a, p,-(~) a, (P;(T)) af2(p;(d) =-
(axWp; + TV (PO(7))
P; tT) a, (p; (7)) a, (p; cd) (aAdp$ + 7T)(P;(T))
1
75
p; cT) a, (p; (7)) a, (p; (7)) (ah)p,+ + 7T)(P;W) p;(~) a, (p;(7)) a,, (p;(d)
7. Concluding
(4wp2+
+ 7T)
(p;(d)
remarks
We shall point out the difference between the quattrolinear form (4.9) and the quintolinear form (4.10) by considering r-functions of the unconstrained KP hierarchy given by [ I-41
(7.1) where fl and f2 satisfy the linear equations fl,fz=f,
(2)
9
fl,fj=fi
(3)
9
f2.h
= f2
(2)
,
(3) f2,rs
= f2
9
(7.2)
with f’k’
= akf
ax .
By the obvious fact that p;(Wronskian(fl,
f2))
=0
under (7.2), the quintolinear form (4.9) vanishes for r = Wronskian( f 1, f2). However, by straightforward calculations (which should be sustained by a computer algebra system) it turns out that, if fr, f2 satisfy (7.2) then (4.10) becomes
24
W. Strampp/Physics
Letters A 218 (19%) 16-24
(-&da,
+ axwp:
+ 7n
(P;(T))
(-&da,
+ axwp,+
+ 7T)
(PF(~)
(7.3)
This means that we have to require
=o
(7.4)
in order that the r-function (7.1) satisfies the quattrolinear form (4.10). To study the effect of condition (7.4) we define the entries of the r-function fi = e7)’+ e6,
r]i = a$ +
a32
+ v&l,
5i =
bix + bfr2 + &o,
(7.1)
through setting (7.5)
giving KP two-solitons. Obviously, the fi given by (7.5) satisfy (7.2). We now ask for necessary conditions under which they may give rise to solitons of the Yajima-Oikawa system. Inserting (7.5) into (7.4) yields 1 at 1 a* 1 b
u; u; a; a; b2 b3 (a; - 6;) (a; - b;)e~‘+f1+~2+~2 = 0.
1 b:
b;
(7.6)
bi
Eq. (7.6) shows necessary restrictions to be imposed on the KP two-soliton of the Yajima-Oikawa system (2.9) -( 2.11) .
in order to become a two-soliton
References [ I] Y. Ohta, J. Satsuma, D. Takahashi and T. Tokihiro, Progr. Theor. Phys. Suppl. 94 (1988) 219. [2] E. Date, M. Jimbo, M. Kashiwara and T. Miwa, in: Nonlinear integrable systems - classical and quantum theory, eds. M. Jimbo and T. Miwa (World Scientific, Singapore, 1983) p. 39. M. Jimbo and T. Miwa, Publ. RIMS Kyoto Univ. 19 (1983) 943. [3] M. Sato and Y. Sato, in: Nonlinear partial differential equations in applied sciences, eds. H. Fuji& PD. Lax and G. Strang (Kinokuniya/North-Holland, Tokyo/Amsterdam, 1983) p. 259. [4] L.A. Dickey, Advanced series in mathematical physics, Vol. 12. Soliton equations and Hamiltonian systems (World Scientific, Singapore, 1991). [S] B.G. Konopelchenko, J. Sidorenko and W. Strampp, Phys. Lett. A 157 (1991) 17. [6] Y. Cheng and Y.S. Li, Phys. Lett. A 157 (1991) 22. [7] B. Xu, Inverse Problems 8 (1992) L13; 9 (1993) 355. [ 81 W. Gevel and W. Strampp, Commun. Math. Phys. 157 (1993) 51. [9] D.J. Kaup, Prog. Theor. Phys. 54 ( 1975) 396. [lo] L.J.F. Broer, Appl. Sci. Res. 31 (1975) 377. [ 111 J. Matsukidaira, J. Satsuma and W. Strampp, Phys. Lett. A 147 (1990) 467. [ 121 Y. Cheng, W. Strampp and B. Zhang, Commun. Math. Phys. 168 (1995) 117.
PHYSICS
ELSEVIER
LETTERS
A
Physics Letters A 218 (1996) 16-24
Multilinear forms associated with the Yajima-Oikawa system Walter Strampp Fachbereich 17 - MathematiMnformatik, Universitiit-GH Kassel, Holliindische Strasse 36, 34109 Kassel, Germany
Received 18May 1995; accepted for publication 29 April 1996 Communicated by A.R. Bishop
Abstract It is well known that the Kadomtsev-Petviashvili hierarchy (KP hierarchy) is compatible with generalized k-constraints of the type ( LL) _ = qX’r. Those constraints are leading to integrable systems in 1 + 1 dimensions such as the AKNS system (k = 1) or the Yajima-Oikawa system (k = 2). The effect of the l-constraint on the r-function of the KP hierarchy has been studied: it adds trilinear equations to the set of bilinear ones. In this paper we shall derive a quattro and a quintolinear equation for the r-function of the KP hierarchy under the 2-constraint.
1. The KP hierarchy The members
L,” = [L,
of the infinite KP hierarchy
CL”)-I,
(1.1)
where L is a pseudodifferential L =
are governed by the Lax equations
operator,
ax+ uza;’+ u&y2 + . . . .
We introduce
a pseudodifferential
W = 1 + w,a;’
+ w2a,2
(1.2)
operator,
+ .. .
,
being related to L through L = W&W-‘. form,
(1.3) In terms of the operator W the Lax equations
( 1.1) take the following
w,, = -( wa;w-1)-w, known as Sate’s equations. r-function 1
Wj
=
;P,T
(1.4) The solutions
of Sato’s equations
( 1.4) can be expressed
(7).
0375~9601/%/$12.00 Copyright 8 1996 Elsevier Science B.V. All rights reserved. PII SO375-9601(96)00387-S
with the help of the
(1.5)
W. Strampp /Physics
Letters A 218 (1996) 16-24
17
Here we use the Schur polynomials pi(t), t = (tl , t2, t3,. . .), defined through 00
efw)
=
Gf(t,A)
pi(f)
c
=~tiA’v
i=O
t1
=
x,
(1.6)
i=l
and differential operators P;’ =J%(+J),
P,l- =p1(-a”),
d = (a,, ;a[*, &,
. . .)
(1.7)
(we prepare p[ for the later convenience). Note that pa(t) = 1, pt(t) =x, pz(t) =x2/2+ t2 and p3(t) = x3/6 + xt2 + f3. The r-function depending upon the infinite set of variables t is governed by Hirota’s equations which form an infinite set of bilinear equations. (See Refs. [ l-41 as a general reference for Sato’s theory.)
2. The KP hierarchy and k-constraints It is well-known that the KP hierarchy ( 1.1) is compatible with generalized k-constraints of the form ( Lk)_ = qCY;lr,
(2.1)
which are leading to ( 1 + 1)-dimensional integrable hierarchies,
(L9,”= W,L"_l,
(2.2)
qt, = (L”)+%
(2.3)
rt. = -(L*“)+r
(2.4)
(where L* is the adjoint of L) [5-g]. The first three of those hierarchies are formed by the AKNS (k = 1)) Yajima-Oikawa (k = 2) and a coupled Boussinesq-type (k = 3) hierarchy. Let us write down the first two members of the AKNS and the Yajima-Oikawa hierarchy, respectively, qtz = qxx + 2q2r,
(2.5)
rt2 = -rxx - 2qr2,
(2.6)
qtj =
qxxx+ 6qw,
(2.7)
rfj = rxxx+ 6qrr,
(2.8)
utz = (qr),,
(2.9)
and
2uq,
(2.10)
r 12= -rxx - 2ur,
(2.11)
Ut, = $UXXX+ 3uu, + a(qd- - qrx)x,
(2.12)
412= 4xX +
+ $fr,
(2.13)
rt3 = rxxx + 3ur, + ijug - &r2.
(2.14)
4ts = qxxx + 3%
+ $,q
18
W. Strampp /Physics
Letters A 2 I8 (I 996) 16-24
3. The trilinear form of the Jr&up-Broer system The Kaup-Broer
[ 9, lo]
system
k,, = (k, + 2X*k),,
(3.1)
x:, = c-x:
(3.2)
+ x** + 2h),,
plays an important role in the theory of nonlinear water waves and is closely Namely, the AKNS system transforms to the Kaup-Broer system through
system.
x* = -5.
h = qr, The following
related to he AKNS
I ansatz for obtaining
solutions
has been made in Ref. [ 111,
h = (logs),,,
(3.3)
hx* = ;[(hvL,~ - Uw~)n,l.
(3.4)
This ansatz becomes evident by the fact that Eq. (3.1) is nothing but the simplest equation of the KP hierarchy (l.l), namely U2,rz = (u2.x
So (3.1)
+ &lx.
is trivially
satisfied by (3.3))
(3.4)
and in addition
we obtain the following
trilinear
equation
for r
ill17
(3.5)
4. Multilinear forms and k-constraints From a more general point of view the k-constraint was studied by using Sato’s equations ( 1.4) in Ref. [ 121. The impact of the constraint has been formulated there as equations constraining the coefficients of the operator W. In the l-constraint case the simplest constraining equation for the coefficients Wj reads -Wj,x $
+P2+(Wj) =O,
while in the 2-constraint P:(Wj)Wl,x
- Wj,t*T
(4.1)
case we have the following + T(Wj)
=09
simplest constraining
equation, (4.2)
where the operator T is given by T = -p3+ + V,,
+ dt, = p;
+ c+~,
(4.3)
see Ref. [ 121. The constraining Eqs. (4.1) and (4.2) are difficult to handle because according to ( 1.5) the r-function appears in the denominators. Therefore we aim at expressing the constraining equations in terms of differential operators acting on the r-function.
W. Sirampp/Physics
Letters A218 (1996) 16-24
19
Considering Rq. (4.1) for j = 1 and j = 2 as a linear homogeneous system with a nontrivial solution vector gives d,(W)
P;(W)
Mw2)
P2+(W2>
=o
(4.4)
.
Considering Rq. (4.2) in the same manner for j = 1, j = 2 and j = 3 as a linear homogeneous system with a nontrivial solution vector gives AAWl)
4*(w)
T(w)
4(w2)
&(W2)
T(W2)
&(w3)
&z(W3)
T(W3)
(4.5)
=a
However, we have more possibilities for drawing consequences from (4.2). For instance, take (4.2) for j = 1 and j = 2 and eliminate qx/q, yielding &WI
J,Wl
~,w
T(w)
4w2
T(w2)
(4.6)
=
WI ,x
4w2
ayw2
Finally, take two samples of Eq. (4.2) and consider them as a linear inhomogeneous system for the unknowns ~1,~ and qx/q. This gives the following conditions which will not be discussed further in this paper, T(wj)
-wj,tz
T(w)
-Wi,r*
P;‘(Wj)
-wj,tz
P’(Wi)
-Wi.tz
Pl+(Wj)
T(wj)
= w~,~= independent of indices j, i
(4.7)
and
Pl+(Wi)T(wi) Pl+(Wj)
-wj,tz
p+ (WI
-Wi,Q
qx
=
independent of indices j, i.
(4.8)
= S
By choosing two pairs of indices on the left-hand side of (4.7) and (4.8) we obtain further conditions to be satisfied by the r-function if we set the corresponding expressions equal to each other. In Ref. [ 121 it has been shown that (4.4) becomes equivalent with the trilinear form (3.5). The purpose of the present paper is to cast the condition (4.5) into the quintolinear form P,-(7)
ax(Pip))
42 (Ply(d)
(&x(7)P2’ +
PF (7)
ax (Pi- (7))
at* (PF (7))
(4(7)
TT)
P2’ + 7n
(Pi
Cd)
(p; (7))
= 0, ~37)
a, (P;(T))
at2 (P;(T))
(a,(4
p; + 7~)
(P; (7))
P; (7)
4 (P; (7))
4, (~3 (7))
mm
p:
(~3 cd)
and (4.6) into the quattrolinear form
+ 7~)
(4.9)
20
W. Strampp/Physics
Letters A 218 (19%) 16-24
PO(T)
dtz (P&d)
(-P2+(7)a,+a,(7)P:+7T)
Pr
(7)
&, (P;(T))
(-P;(7)&
+ J,(7)
P:
+ 777
(P;
(7))
P;
(7)
62 (P;
(-P2+W&
+ ax(T)
P2f + TT)
(P;
(7))
(7))
(P&7))
(4.10)
= 0.
Note the remarkable similarities between Eqs. (4.9) and (4.10). In Section 6 we shall outline the derivation of the quintolinear form (4.9) in detail. Since the quattrolinear form (4.10) can be obtained by quite analogous means we shall not present the details. We shall rather discuss an obvious difference of the forms (4.9) and (4.10) in Section 7.
5. Derivation of the trilinear form for the KaupBroer
system
We shall briefly demonstrate the steps leading from (4.4) to (3.5) in a way differing from the derivation given in Ref. [ 121. The advantage of our present method is that it can be carried to the 2-constraint case. Let us first take care of the denominators by calculating
(5.1) (5.2) For the sake of simplicity
“2 =
( )
Furthermore
PC (7) P;(T)
we introduce
the notation (5.3)
.
we shall need the formulae
(5.5)
P,(f(f)g(f))=CPn_j(f(f))P,~(g(f)),
H which are obtained
from Taylor expansion.
a,(w)
P;(w)
Jx(w2)
P;(w)
= /& (3 = 4
-;
We shall now transform
(4.4)
I=~&(;v2)9P:(;v2)~
u2+po+
(gP:(v2,,p:
~Iv2,P:(v2)1+
(3
[ 81 ($q)
v2+P: -bP:
($P:(V2) (;)I
+d
Iv2,plf(v2)1
0 = $[7lP:(u2).P:(u2)l
with the help of (5.1)-(
-4(7)l~2~P;GJ2)1
Introducing T = pi (7) in the coefficients
(:)P:(v2)1
+
-lP;t(v2)~P2f(v2)I :
+P,+(7)I~;?~Pit(v2N.
of the determinants
we obtain the trilinear equation
(3.5).
5.3)
W. Smmpp/Physics
L&en
21
A 2I8 (19%) 1624
6. The quintolinear form
We now turn to the case k = 2. First of all we prepare a few formulae taking care of the denominators,
(6.1)
With the notation
(6.4)
we obtain immediately (6.5) (6.6) (6.7)
(6.8) (6.9)
It is a little more complicated T
1v3 = -&-3T(7) ( 7 1 + $[-Th(T)2
-
to show that +iMd3
$T2d,,(T)
+ $[-;72Md,,(s)
=P3
= (P;
(f)n+P;
- T&(T>&T)]u~
+ 37q(T)]ax(u3) + 73T(v3)]
+ ;T~&(T>&Y~)
holds. We proof (6.10) T ($3)
+nMr)a,,(~)
(6.10)
by first using (4.3) and (5.5),
+ hs) ($3)
($)P;(ui)+P;
Taking (6.1) -( 6.3) and the definition
(9
P1 (v3)
+ ;P;
(v3)
(4.3) of T into account we calculate
+ $w,(s)
- a,,(T)v31.
22
W. SrramPp/Physics Letters A 218 (1996) 16-24
T
1~3 = +W ( 7 >
+ 4m3
- $bm2+
;%(d
+ ;[-g(u,)
+ $w,,(~3)
= $+%c+%~~,3
+%w
+ ;$wd:(U3)
+ p&7)
+ -$MT~;,a:(u,)
- f&,(V3)1
+ $v3)
- &,(v,)]
- r2 [-#CT> .
+ $w,,w v T(7)
+&(d3
+ ; [4&u3) .
+n%(7)~%~(7)
- ;T2a1,(T)
+ &M&73) Y T(w)
&,(w>
&(w2)
4,(~2)
T(w2)
h(w3)
4,(~3)
7’(w3)
+73T(z73)l.
= 1% ($3)
+Md3+&W,,(d
+ +Md3+
$%w2a,,(~)
- $%wM7)
px(Y3)*Y3&v3>1
I~3,M~3M,(~3)1 I~37Mvs),T(~3)I
+ ;yI-&d
- $‘,,(d
6.10))
+3)~
I4(~3),&,(~3))7~3I b3,4*(~3)74(~3)1
;$m - if
Ia,(y3).d,,(U3),a,2(Ug)I (Mm*
IV39&,(s)&v3)I
l4(~3),~3,7’(~3)(
+7la,(u3),a,,(~3),T(~3)1 --&CT)
Iu~,&~(u~)~T(u~)II
~U3,&(s),at,(s>~+7~a,~u3~,a,,~U3~,~,2~U3~~ ~u3,&(u3),~,2~u3)~
--a,(7) h,(T)
=?i
($u1)
- $%w2d:(7)l
lAA~3),u3,4~(~3)1+
+ $lax(~3),ar*(v3>,T(u3)1
94
-~&W,2(dl
+ $%m,,w
1
use of (6.5)-(
T(w)
= $I--SW
+&CT)
/
+ ;T2a,2(79]ax(u3)
+ ;T~&(T)&u~)
J,(w)
+ ~&&,)I
- T&(T)~;(T)]Y~
Now we are able to show that (4.5) can be carried over into (4.9) by making
+&CT)
+ 5~&-)1}u3 /
- ~~&7)1&(~3)
+ -$~T~&(T)&,(v,)
= $J-UT)
- ;72dx&*(7)]Y3
- -$&)u,
- mda,2w
- ;f4w&3)
+ -&T&(T)2
- nfWa,2(7)
- ~~d,2(Tmx(s)
+&(7)&,(T)
- fw2+
= f[-~~T(d
+m7)&,(7)
(u3.&*(u3L&u3)(1 T(T)
p;;d
&;%,
a (P-(T))
UP-(~))
P;(T) P;(T)
a,&)) &(P;(~))
&p:(d) &(pf(~))
T&) UP;(T))
- $-WI
)~39&,(~3)9T(~3)I
W. Strampp/ Physics Letters A 218 (1996) 16-24
7 at209 am T(7) I p; (7) a,(pl-(T)1 a&- (4 > Updd 1 = 7 p;(7) a,(p;(7)) a,,(p;w VP;(~) p;(7) ax,(p;(T)) a,,(p;(m UP;(~)
+a, p,-(~) a, (P;(T)) af2(p;(d) =-
(axWp; + TV (PO(7))
P; tT) a, (p; (7)) a, (p; cd) (aAdp$ + 7T)(P;(T))
1
75
p; cT) a, (p; (7)) a, (p; (7)) (ah)p,+ + 7T)(P;W) p;(~) a, (p;(7)) a,, (p;(d)
7. Concluding
(4wp2+
+ 7T)
(p;(d)
remarks
We shall point out the difference between the quattrolinear form (4.9) and the quintolinear form (4.10) by considering r-functions of the unconstrained KP hierarchy given by [ I-41
(7.1) where fl and f2 satisfy the linear equations fl,fz=f,
(2)
9
fl,fj=fi
(3)
9
f2.h
= f2
(2)
,
(3) f2,rs
= f2
9
(7.2)
with f’k’
= akf
ax .
By the obvious fact that p;(Wronskian(fl,
f2))
=0
under (7.2), the quintolinear form (4.9) vanishes for r = Wronskian( f 1, f2). However, by straightforward calculations (which should be sustained by a computer algebra system) it turns out that, if fr, f2 satisfy (7.2) then (4.10) becomes
24
W. Strampp/Physics
Letters A 218 (19%) 16-24
(-&da,
+ axwp:
+ 7n
(P;(T))
(-&da,
+ axwp,+
+ 7T)
(PF(~)
(7.3)
This means that we have to require
=o
(7.4)
in order that the r-function (7.1) satisfies the quattrolinear form (4.10). To study the effect of condition (7.4) we define the entries of the r-function fi = e7)’+ e6,
r]i = a$ +
a32
+ v&l,
5i =
bix + bfr2 + &o,
(7.1)
through setting (7.5)
giving KP two-solitons. Obviously, the fi given by (7.5) satisfy (7.2). We now ask for necessary conditions under which they may give rise to solitons of the Yajima-Oikawa system. Inserting (7.5) into (7.4) yields 1 at 1 a* 1 b
u; u; a; a; b2 b3 (a; - 6;) (a; - b;)e~‘+f1+~2+~2 = 0.
1 b:
b;
(7.6)
bi
Eq. (7.6) shows necessary restrictions to be imposed on the KP two-soliton of the Yajima-Oikawa system (2.9) -( 2.11) .
in order to become a two-soliton
References [ I] Y. Ohta, J. Satsuma, D. Takahashi and T. Tokihiro, Progr. Theor. Phys. Suppl. 94 (1988) 219. [2] E. Date, M. Jimbo, M. Kashiwara and T. Miwa, in: Nonlinear integrable systems - classical and quantum theory, eds. M. Jimbo and T. Miwa (World Scientific, Singapore, 1983) p. 39. M. Jimbo and T. Miwa, Publ. RIMS Kyoto Univ. 19 (1983) 943. [3] M. Sato and Y. Sato, in: Nonlinear partial differential equations in applied sciences, eds. H. Fuji& PD. Lax and G. Strang (Kinokuniya/North-Holland, Tokyo/Amsterdam, 1983) p. 259. [4] L.A. Dickey, Advanced series in mathematical physics, Vol. 12. Soliton equations and Hamiltonian systems (World Scientific, Singapore, 1991). [S] B.G. Konopelchenko, J. Sidorenko and W. Strampp, Phys. Lett. A 157 (1991) 17. [6] Y. Cheng and Y.S. Li, Phys. Lett. A 157 (1991) 22. [7] B. Xu, Inverse Problems 8 (1992) L13; 9 (1993) 355. [ 81 W. Gevel and W. Strampp, Commun. Math. Phys. 157 (1993) 51. [9] D.J. Kaup, Prog. Theor. Phys. 54 ( 1975) 396. [lo] L.J.F. Broer, Appl. Sci. Res. 31 (1975) 377. [ 111 J. Matsukidaira, J. Satsuma and W. Strampp, Phys. Lett. A 147 (1990) 467. [ 121 Y. Cheng, W. Strampp and B. Zhang, Commun. Math. Phys. 168 (1995) 117.