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Binary Darboux transformations for Manakov triad
12 December 1994 PHYSICS LETTERS A Physics Letters A 195 (1994) 339-345
ELSEVIER
Binary Darboux transformations for Manakov triad Zixiang Zhou Institute of Mathematics, Fudan University,Shanghai,China Received 8 February 1994; revised manuscript received 13 July 1994;accepted for publication 4 October 1994 Communicated by A.P. Fordy
Abstract
The binary Darboux transformations for Manakov triads of second order are constructed, which generalize the work for the KP equation and Novikov-Veselov equation. An equation given by Konopelchenko is discussed. 1. Introduction
A Manakov triad is a generalization o f a Lax pair. It can represent more integrable systems than the Lax pair. In the present Letter, we discuss the Manakov triad (L, N, W) with
Zt= W L - L N ,
(I.1)
L~U=0,
(1.2)
~=N~,
where 2
2--i
L= E Li Oi,
L~= E LijO~,
i=O
N=N, Ox+No,
j=O
Ni= ~ NoO~ ,
W= WI Ox + Wo ,
W, = ~ WaO~ ,
j=O
(1.3)
j=O
and the coefficients Lij, Nij, W,j are r X r matrix functions o f (x, y, t). Moreover, suppose L2o = L ( 1.1 ) is the integrability condition of ( 1.2). We can determine W explicitly as
Wt =N1,
Wo=No+2N~.x+ [L~, N~] ,
(1.4)
and N satisfies
LLt + [Lo, N~] - [L~, N~ ]L~ + ILl, No] -2N1,xL~ -N~ L~,x +L~ N~,x + 2No,x +Nl,xx=O, Lo.t + [Lo, No ] - [L,, N~ ]Lo - 2 N,,xLo + L~ No,x-N, Lo,~ + No,~ = O .
( 1.5 )
This system contains several interesting equations. Supported by National Research Project "Nonlinear Science", FEYUT-SEDC-CHINAand the Fok Ying-TungEducation Foundation of China. Elsevier Science B.V. SSD10375-9601 (94)00799-3
340
Z. Zhou / Physics LettersA 195 (1994) 339-345
Example 1. L=O2+OlOy+U
(ol=l ori),
N=4aOxOy-2uOx+u~+3aw,
W=N-4ux.
Eq. ( 1.5 ) gives U t + 6 U U x + U x x x = -- 3 0 / 2 W y ,
Wx=Uy,
which is the well-known KP equation. This Manakov triad is equivalent to the usual Lax pair.
Example 2. L = O x2 + O r2 + u ,
N=-8OxO~+(3vR-2u)O~+3VxOy-2Ux,
W=N+6VR.~-4u~.
We have the equations
U,=2Uxxx--6Uxyy+ 3VRUx+ 3VxUy+ 2VR,xU--6UUx ,
"
VR,x--Vl,y=2Ux ,
PR,y-l-Ul.x=--2Uy ,
which is called the Novikov-Veselov equation, and is discussed in Ref. [4]. In Refs. [ 3,4 ], the binary Darboux transformation for the DS equation, whose L is of first order, and the KP equation, Novikov-Veselov equation, which are special cases of ( 1.2 ), are discussed. In the present Letter, we use the binary Darboux transformation to discuss the more general system ( 1.2).
Remark. Any system in ( 1.2) can be transformed to a linear system of first order whose coefficients are 3 r × 3r matrices. However, since there are many constraints among these coefficients (e.g., some coefficients should be given constants), it is more difficult to solve these systems than to solve ( 1.2 ). Therefore we would rather study (1.2) than study a general system of first order with many reductions.
2. Binary Darboux transformation In order to discuss the binary Darboux transformation, we need the following lemma about a closed one-form. Let
~[
~] =span{ O~O~L T/, O~xO~(~ - NT/) l i, j>_.0},
(2.1)
and ~ 2 [ ~] = ~
(2.2)
[ 7/] ® span{dx ^ dy, dx ^ dt, dy A dt}.
Lemma 1. For given L, N in ( 1.3 ), there exists a one-form 09[ 7/] =a7/ dx + flT/ dy+ ~7/ dt ,
(2.3)
where the differential operators
ot=OtolOy+Otoo, fl=fllOx+flo,
fl,=fl,o,
flo=/3o, Oy+Ooo, ~=~lax+7O,
7; = ~ 7ua~, j=O
(2.4)
(Oto;, flu, 7u and r × r matrix functions of (x, y, t) ) such that do9 [ 7/] ~ ~.~2 [ 7/], provided that the system L*2*=0,
(2.5a)
27 = - W*2*,
(2.5b)
Z. Zhou / PhysicsLettersA 195 (1994)339-345
341
has a nondegenerate solution 2. Here L*, W* are the formal conjugations of L and W.
Proof Denote do)[ T] =Dxydx ^ dy+Dxt d x ^ dt+Dyt dy^ dt. The highest order of 0x in Dxy does not exceed 2, and the coefficients of a~o k (k>_- 1 ) are all zero. Hence let Dxy= 2L, where 2 is an r × r matrix function to be determined later. Comparing the coefficients in both sides, we have o~ol = -2Lo2 , fl,o = 2 ,
aoo = -2Lot + (~.LI 1 )x "]- (-~Lo2)y,
floo =2Llo - 2 x ,
flOl = 2 L , , ,
(2.6)
and 2 satisfies
Z (-1 )j + / jaxOy(2Lj,)=O,
j+l<~2
(2.7)
which is (2.5a). Next we consider Dyt. Since
flN~= (fl, Ox + flo ) ( Nl O~ + No) ~ = ( - f l l N i L l +fl,, Nl,x+floN~ +fllNo)Ox+ (-fllNiLo+fllNo.~+floNo)~ (rood L¢~ [ 7j] ) , and
Drt = 7~y + Ty T - f l N T - fl, T , the requirement Dyt =-0 (rood ~ 2 [ T] ) implies --fll,t + 710y + yl,y = - - f l t N t L l + flmNl,x + floNl + fllNo ,
(2.8a)
-flo,t + yoOy + 7O.y= -fllN~Lo + fl~No,x + floNo .
(2.8b)
71 can be solved recursively by comparing the coefficients of0~ + 1, ..., Oy in (2.8a). With (1.4), it can be checked that the right-hand side of (2.8a) equals to 2Wo-(2W1)x. Hence the terms without Oy in (2.8a) are equal identically by (2.5b). Similarly, 7o can be solved from (2.8b) and (2.8b) holds identically because of ( 1.5 ) and (2.5). Therefore, we have proved Dy t ~ 0 ( m o d ~ 2 [ ~] ). The definition of Dxy, Oxt and By t leads to OyDxt =-0 (mod ~ [ ~] ). Suppose l
ar
Dx~-- ~, ~ d~O~O~(mod~.~2[Tt]), i=0j=0
then 1
Z
J
Z (dij + Oydi,j-i ) 0 j*l ~J+ doo ~ - 0 (mod ~ 2 [ gJ] ) ,
i=0j=0
which means do=O for all i,j. Lemma 1 is proved. Now we proceed to the discussion of binary Darboux transformations for ( 1.2).
Theorem 1. Let ~r and 2 be n ondegenerate r × r matrix solutions of ( 1.2 ) and (2.5) respectively, S(x, y, t) be an arbitrary nondegenerate matrix function and C be a constant r × r matrix. Suppose ~is a solution of ( 1.2). Let g2[ ~] =fo)[ ~] + C, I2o=~2[~z], R = S£2o7r-1, and ~ = R T t - S I 2 [ 7t] .
(2.9)
342
z. Zhou /Physics Letters A 195 (1994) 339-345
Then, at the points where det g20# 0 and det ( 1 - R - ~S2L t t + (R - t$2 ) 2Lo2 ) # 0, ~ satisfies
E~=o,
~, =R~.
(2.10)
Here 2
2--i
£= E £iaix'
£i= E ~-,iJaJy,
i=0
j=0
]V=Nlax~'~fo,
(2.11)
j=O
and the coefficients are given by (/-~2o /~,,
(L,o
Eo,)(
/~o~)
R -Sfl~o
R
-SO~ol
-SfllO
R-Sflo~
L~
Lo~),
--S°(°l ) R-Sflo~ f
=R(L~o
=R(L2o
Lo,)-(L2o
L,,
- 2Sxao~ - Sao~,x
2Rx-Saoo
Eo~)| R~-Sxfl,o \ -- 2Syfllo -- SfllO,y
'~
R ~ - Sxflo ~ - Sy ao , - Saoo - S a o ~,y) .
2Ry - 2Syflo~ - S floo - S flo~.~ R~:, - 2Sx Otoo - Saoo,~,
( R~ - S o~oo Loo R = R Loo - ( L ~o Lo ~) \ R ~ - S floo ] - ( L ~o
co )
(2.12) \
Ryr -- 2Syfloo-- Sfloo,y
/
can be determined by the recursion formulae N,~R=RN,.-ST,u-
Z C{N,~(O~ - ~ R ) + E c?c~_,R.(Oxa~-*s)(o~-'-.#,) k~+l j~k~+l
j~k~+l NouR-N, uS~=RNo~-STou+Rt~o~ +
Z
j~k~+l
~., C ~ N o k ( O ~ - U R ) + ~. C ~ N L k [ O ~ - u ( S ~ ) ] k~+l k~+l
C~C~-,N,/OxO~-~S)(O~-I-%)
+
Z
j~k~+l
~ J-~ S ) ( O y~ - ' -~flo). C~C#_,Noj(Oy
(2.13)
R e m a r k . For given n and 2, we can always have det g2o# 0 and det ( 1 - R - ~S2L ~~+ ( R - ~$2) 2Lo2 ) # 0 locally by a suitable choice of the arbitrary matrices S and C. Proof. Since det( 1 - R - IS2L~I + ( R - |S,,q.)2L02)5 ~ 0, the coefficient matrices on the left-hand side of all the equations in (2.12 ) are nondegenerate. Hence (2.12) determines/~j uniquely. Direct calculations show that
/S~= (/~S)D[ ~ ] ,
(2.14)
holds for any solution ~ of ( 1.2). Taking ~ = n, we have/SS= 0. Hence/S~_= 0 for any solution of ~ of ( 1.2 ). 57 is uniquely determined by (2.13). Using (2.13), we can check that (0, - N ) q~= - (S, - I ~ S ) D [ ~ ] .
(2.15)
Let ~ = n , then S , = N S . Hence ~, =]V~. Theorem 1 is proved. R e m a r k . Since L 7 = ( - N*) L * - L* ( - W*), if (n, 2) creates a binary Darboux transformation as in Theorem
Z. Zhou / Physics Letters A 195 (1994) 339-345
343
1, then (2*, n*) creates a binary Darboux transformation for (2.5). Therefore, not only the solution of (1.2) with respect to (L, N), but also the solution of (2.5) with respect to (L, W) is constructed. This means that the binary Darboux transformation can be constructed successively by using differentiation and integration, without solving any differential equations.
3. Examples
For the KP equation and the Novikov-Veselov equation, the binary Darboux transformations given by Theorem 1 are just those given by Ref. [4 ]. Here we consider the following example. Let
L=OxOy+POy+Q,
N=O2+F,
(3.1)
then (1.1) holds if and only if
Fy=ZQx,
Pt-Pxx+ZPxP+Fx+[P,F]=O,
Qt+Qxx+2(PQ)~+[Q,F]=O.
(3.2)
(3.1) is not of form (1.2). However, a simple change of variables can lead to L2o=L For r = 1, this is a special case of the equation discussed in Refs. [ 1,2], which will be discussed here. Under the reduction Q = F = 0, the system (3.2) becomes the matrix Burgers equation in 1 + 1 dimensions. Let ~=x+y, q=x-y, then (3.1) becomes
L=O~-O~+P(O¢-O.)+Q,
N=(O~+O.)2+F.
(3.3)
For r = 1,
W= (O¢+O.)Z+F-2P.=O2x+f - P x +Py, a=20,-2.+2P,
(3.4)
fl=AO¢-2¢+2P,
y=220¢0, +2202 - 2 ( 2 ¢ + 2 . ) (0¢ + 0.) +2/'0¢ + 32P0.+22¢. +22~. +2 ( P c - P.) -2¢P-32,P-22Q.
(3.5)
To construct the binary Darboux transformation for (3.3), let 7rbe a solution of
rct=N~z,
Ln=0,
(3.6)
and 2 be a solution of L*2*=0,
2* = - W*2*.
(3.7)
Suppose t'2o~ 0 and g2o+ 27r¢ 0, and let 7~
s=
~o-,l---~
(3.8)
As in Theorem 1, R=
12o
g2o -2~r'
(3.9)
(2.12) leads to £2 0 = 1 ,
£ , , = O,
£o~ = - 1 ,
344
Z. Zhou / PhysicsLettersA 195 (1994)339-345 I2o --Art 2n 2 £2o(0~+0~)(2n) --2n(O¢+O~)Oo ~ (O¢+O,) f2o-2--'---~= P 0 2-22n 2 '
/S,o=-/So, = P = P - 2
£oo=O_=Q-R-~P(S¢-S,)2 - R -~ [R~¢-Rnn - 2 ( S ¢ - S n ) 2 P - S [ ( 2 P ) ¢ - (~-P) n] + 2S~2n - 2Sn2¢],
(3.10)
which gives the new solution (P, Q) of (3.2). We can obtain many explicit solutions using (3.10). For example, take the seed solution as P = 2p > 0 (p is a real constant), Q = F = 0, then (3.6) has a solution n=exp [a(~+ q) +4a2t] + exp [ b ( ~ - q) - p ( ~ + q) + 4p2t] ,
(3.11 )
and (3.7) has a solution 2= - e x p [ c ( ~ + q ) - 4 c : t ] +exp [ d ( ~ - q) + p ( ~ + r/) -4pZt],
(3.12)
where the constants a, b, c and d satisfy
(p+a)(a+c)
b(d+b)
Let p+a #= - a+c
>10
'
v=-
~
b
>0,
and
A=a(~+q)+4a2t,
B=b(~-q)-p(~+q)+4p2t,
C=c(~+q)-4c2t,
D=d(~-rl)+p(~+~l)-4p2t,
then direct calculation gives
P=-2p+(a+c) ~ , tr2 where
a, = (1~+ 1 )2exp(2A+C+ D) +/t 2 exp(A+ B+ 2C) + #vexp(A+ B+ 2D) + (/z+ 1 ) (v+ 1 ) exp(2B+ C+D) + (3/t+ v+ 2) exp(A + B + C + D ) , a2 =/t(/~+ 1 ) exp(2A + 2C) + v(v+ 1 ) exp(2B+ 2D) + (#+ 1 ) exp(2A+C+D) +l~exp(A+B+2C)
+ vexp(A+B+2D) + (~+ 1) exp(2B+C+D) + ( 2 # v + # + v + 2 ) e x p ( A + B + C + D ) . Clearly, it is bounded for all (~, q, t). However, it does not decay at infinity. It is still interesting to look for solutions which decay at infinity.
Remark. In this Letter, we have only discussed the binary Darboux transformations for the Manakov triads of second order. It is naturally interesting to discuss the binary Darboux transformation for the Manakov triads of higher order. This is still open due to some technical problems.
Z. Zhou / Physics Letters A 195 (1994) 339-345
345
Acknowledgement The author would like to thank Professor Gu Chaohao, Professor Hu Hesheng and Professor D. Levi for helpful advice. The author would also like to thank the referees for comments and one of the referees also for valuable suggestions.
References [ I ] B.G. Konopelchenko, Mod. Phys. Lett. A 3 (1988) 1807. [2] B.G. Konopelchenko, Inverse Probl. 4 (1988) 151. [3] S.B. Leble, M.A. Salle and A.V. Yurov, Inverse Probl. 8 (1992) 207. [4 ] V.B. Matveev and M.A. Salle, Darboux transformations and solitons (Springer, Berlin, 1991 ).
ELSEVIER
Binary Darboux transformations for Manakov triad Zixiang Zhou Institute of Mathematics, Fudan University,Shanghai,China Received 8 February 1994; revised manuscript received 13 July 1994;accepted for publication 4 October 1994 Communicated by A.P. Fordy
Abstract
The binary Darboux transformations for Manakov triads of second order are constructed, which generalize the work for the KP equation and Novikov-Veselov equation. An equation given by Konopelchenko is discussed. 1. Introduction
A Manakov triad is a generalization o f a Lax pair. It can represent more integrable systems than the Lax pair. In the present Letter, we discuss the Manakov triad (L, N, W) with
Zt= W L - L N ,
(I.1)
L~U=0,
(1.2)
~=N~,
where 2
2--i
L= E Li Oi,
L~= E LijO~,
i=O
N=N, Ox+No,
j=O
Ni= ~ NoO~ ,
W= WI Ox + Wo ,
W, = ~ WaO~ ,
j=O
(1.3)
j=O
and the coefficients Lij, Nij, W,j are r X r matrix functions o f (x, y, t). Moreover, suppose L2o = L ( 1.1 ) is the integrability condition of ( 1.2). We can determine W explicitly as
Wt =N1,
Wo=No+2N~.x+ [L~, N~] ,
(1.4)
and N satisfies
LLt + [Lo, N~] - [L~, N~ ]L~ + ILl, No] -2N1,xL~ -N~ L~,x +L~ N~,x + 2No,x +Nl,xx=O, Lo.t + [Lo, No ] - [L,, N~ ]Lo - 2 N,,xLo + L~ No,x-N, Lo,~ + No,~ = O .
( 1.5 )
This system contains several interesting equations. Supported by National Research Project "Nonlinear Science", FEYUT-SEDC-CHINAand the Fok Ying-TungEducation Foundation of China. Elsevier Science B.V. SSD10375-9601 (94)00799-3
340
Z. Zhou / Physics LettersA 195 (1994) 339-345
Example 1. L=O2+OlOy+U
(ol=l ori),
N=4aOxOy-2uOx+u~+3aw,
W=N-4ux.
Eq. ( 1.5 ) gives U t + 6 U U x + U x x x = -- 3 0 / 2 W y ,
Wx=Uy,
which is the well-known KP equation. This Manakov triad is equivalent to the usual Lax pair.
Example 2. L = O x2 + O r2 + u ,
N=-8OxO~+(3vR-2u)O~+3VxOy-2Ux,
W=N+6VR.~-4u~.
We have the equations
U,=2Uxxx--6Uxyy+ 3VRUx+ 3VxUy+ 2VR,xU--6UUx ,
"
VR,x--Vl,y=2Ux ,
PR,y-l-Ul.x=--2Uy ,
which is called the Novikov-Veselov equation, and is discussed in Ref. [4]. In Refs. [ 3,4 ], the binary Darboux transformation for the DS equation, whose L is of first order, and the KP equation, Novikov-Veselov equation, which are special cases of ( 1.2 ), are discussed. In the present Letter, we use the binary Darboux transformation to discuss the more general system ( 1.2).
Remark. Any system in ( 1.2) can be transformed to a linear system of first order whose coefficients are 3 r × 3r matrices. However, since there are many constraints among these coefficients (e.g., some coefficients should be given constants), it is more difficult to solve these systems than to solve ( 1.2 ). Therefore we would rather study (1.2) than study a general system of first order with many reductions.
2. Binary Darboux transformation In order to discuss the binary Darboux transformation, we need the following lemma about a closed one-form. Let
~[
~] =span{ O~O~L T/, O~xO~(~ - NT/) l i, j>_.0},
(2.1)
and ~ 2 [ ~] = ~
(2.2)
[ 7/] ® span{dx ^ dy, dx ^ dt, dy A dt}.
Lemma 1. For given L, N in ( 1.3 ), there exists a one-form 09[ 7/] =a7/ dx + flT/ dy+ ~7/ dt ,
(2.3)
where the differential operators
ot=OtolOy+Otoo, fl=fllOx+flo,
fl,=fl,o,
flo=/3o, Oy+Ooo, ~=~lax+7O,
7; = ~ 7ua~, j=O
(2.4)
(Oto;, flu, 7u and r × r matrix functions of (x, y, t) ) such that do9 [ 7/] ~ ~.~2 [ 7/], provided that the system L*2*=0,
(2.5a)
27 = - W*2*,
(2.5b)
Z. Zhou / PhysicsLettersA 195 (1994)339-345
341
has a nondegenerate solution 2. Here L*, W* are the formal conjugations of L and W.
Proof Denote do)[ T] =Dxydx ^ dy+Dxt d x ^ dt+Dyt dy^ dt. The highest order of 0x in Dxy does not exceed 2, and the coefficients of a~o k (k>_- 1 ) are all zero. Hence let Dxy= 2L, where 2 is an r × r matrix function to be determined later. Comparing the coefficients in both sides, we have o~ol = -2Lo2 , fl,o = 2 ,
aoo = -2Lot + (~.LI 1 )x "]- (-~Lo2)y,
floo =2Llo - 2 x ,
flOl = 2 L , , ,
(2.6)
and 2 satisfies
Z (-1 )j + / jaxOy(2Lj,)=O,
j+l<~2
(2.7)
which is (2.5a). Next we consider Dyt. Since
flN~= (fl, Ox + flo ) ( Nl O~ + No) ~ = ( - f l l N i L l +fl,, Nl,x+floN~ +fllNo)Ox+ (-fllNiLo+fllNo.~+floNo)~ (rood L¢~ [ 7j] ) , and
Drt = 7~y + Ty T - f l N T - fl, T , the requirement Dyt =-0 (rood ~ 2 [ T] ) implies --fll,t + 710y + yl,y = - - f l t N t L l + flmNl,x + floNl + fllNo ,
(2.8a)
-flo,t + yoOy + 7O.y= -fllN~Lo + fl~No,x + floNo .
(2.8b)
71 can be solved recursively by comparing the coefficients of0~ + 1, ..., Oy in (2.8a). With (1.4), it can be checked that the right-hand side of (2.8a) equals to 2Wo-(2W1)x. Hence the terms without Oy in (2.8a) are equal identically by (2.5b). Similarly, 7o can be solved from (2.8b) and (2.8b) holds identically because of ( 1.5 ) and (2.5). Therefore, we have proved Dy t ~ 0 ( m o d ~ 2 [ ~] ). The definition of Dxy, Oxt and By t leads to OyDxt =-0 (mod ~ [ ~] ). Suppose l
ar
Dx~-- ~, ~ d~O~O~(mod~.~2[Tt]), i=0j=0
then 1
Z
J
Z (dij + Oydi,j-i ) 0 j*l ~J+ doo ~ - 0 (mod ~ 2 [ gJ] ) ,
i=0j=0
which means do=O for all i,j. Lemma 1 is proved. Now we proceed to the discussion of binary Darboux transformations for ( 1.2).
Theorem 1. Let ~r and 2 be n ondegenerate r × r matrix solutions of ( 1.2 ) and (2.5) respectively, S(x, y, t) be an arbitrary nondegenerate matrix function and C be a constant r × r matrix. Suppose ~is a solution of ( 1.2). Let g2[ ~] =fo)[ ~] + C, I2o=~2[~z], R = S£2o7r-1, and ~ = R T t - S I 2 [ 7t] .
(2.9)
342
z. Zhou /Physics Letters A 195 (1994) 339-345
Then, at the points where det g20# 0 and det ( 1 - R - ~S2L t t + (R - t$2 ) 2Lo2 ) # 0, ~ satisfies
E~=o,
~, =R~.
(2.10)
Here 2
2--i
£= E £iaix'
£i= E ~-,iJaJy,
i=0
j=0
]V=Nlax~'~fo,
(2.11)
j=O
and the coefficients are given by (/-~2o /~,,
(L,o
Eo,)(
/~o~)
R -Sfl~o
R
-SO~ol
-SfllO
R-Sflo~
L~
Lo~),
--S°(°l ) R-Sflo~ f
=R(L~o
=R(L2o
Lo,)-(L2o
L,,
- 2Sxao~ - Sao~,x
2Rx-Saoo
Eo~)| R~-Sxfl,o \ -- 2Syfllo -- SfllO,y
'~
R ~ - Sxflo ~ - Sy ao , - Saoo - S a o ~,y) .
2Ry - 2Syflo~ - S floo - S flo~.~ R~:, - 2Sx Otoo - Saoo,~,
( R~ - S o~oo Loo R = R Loo - ( L ~o Lo ~) \ R ~ - S floo ] - ( L ~o
co )
(2.12) \
Ryr -- 2Syfloo-- Sfloo,y
/
can be determined by the recursion formulae N,~R=RN,.-ST,u-
Z C{N,~(O~ - ~ R ) + E c?c~_,R.(Oxa~-*s)(o~-'-.#,) k~+l j~k~+l
j~k~+l NouR-N, uS~=RNo~-STou+Rt~o~ +
Z
j~k~+l
~., C ~ N o k ( O ~ - U R ) + ~. C ~ N L k [ O ~ - u ( S ~ ) ] k~+l k~+l
C~C~-,N,/OxO~-~S)(O~-I-%)
+
Z
j~k~+l
~ J-~ S ) ( O y~ - ' -~flo). C~C#_,Noj(Oy
(2.13)
R e m a r k . For given n and 2, we can always have det g2o# 0 and det ( 1 - R - ~S2L ~~+ ( R - ~$2) 2Lo2 ) # 0 locally by a suitable choice of the arbitrary matrices S and C. Proof. Since det( 1 - R - IS2L~I + ( R - |S,,q.)2L02)5 ~ 0, the coefficient matrices on the left-hand side of all the equations in (2.12 ) are nondegenerate. Hence (2.12) determines/~j uniquely. Direct calculations show that
/S~= (/~S)D[ ~ ] ,
(2.14)
holds for any solution ~ of ( 1.2). Taking ~ = n, we have/SS= 0. Hence/S~_= 0 for any solution of ~ of ( 1.2 ). 57 is uniquely determined by (2.13). Using (2.13), we can check that (0, - N ) q~= - (S, - I ~ S ) D [ ~ ] .
(2.15)
Let ~ = n , then S , = N S . Hence ~, =]V~. Theorem 1 is proved. R e m a r k . Since L 7 = ( - N*) L * - L* ( - W*), if (n, 2) creates a binary Darboux transformation as in Theorem
Z. Zhou / Physics Letters A 195 (1994) 339-345
343
1, then (2*, n*) creates a binary Darboux transformation for (2.5). Therefore, not only the solution of (1.2) with respect to (L, N), but also the solution of (2.5) with respect to (L, W) is constructed. This means that the binary Darboux transformation can be constructed successively by using differentiation and integration, without solving any differential equations.
3. Examples
For the KP equation and the Novikov-Veselov equation, the binary Darboux transformations given by Theorem 1 are just those given by Ref. [4 ]. Here we consider the following example. Let
L=OxOy+POy+Q,
N=O2+F,
(3.1)
then (1.1) holds if and only if
Fy=ZQx,
Pt-Pxx+ZPxP+Fx+[P,F]=O,
Qt+Qxx+2(PQ)~+[Q,F]=O.
(3.2)
(3.1) is not of form (1.2). However, a simple change of variables can lead to L2o=L For r = 1, this is a special case of the equation discussed in Refs. [ 1,2], which will be discussed here. Under the reduction Q = F = 0, the system (3.2) becomes the matrix Burgers equation in 1 + 1 dimensions. Let ~=x+y, q=x-y, then (3.1) becomes
L=O~-O~+P(O¢-O.)+Q,
N=(O~+O.)2+F.
(3.3)
For r = 1,
W= (O¢+O.)Z+F-2P.=O2x+f - P x +Py, a=20,-2.+2P,
(3.4)
fl=AO¢-2¢+2P,
y=220¢0, +2202 - 2 ( 2 ¢ + 2 . ) (0¢ + 0.) +2/'0¢ + 32P0.+22¢. +22~. +2 ( P c - P.) -2¢P-32,P-22Q.
(3.5)
To construct the binary Darboux transformation for (3.3), let 7rbe a solution of
rct=N~z,
Ln=0,
(3.6)
and 2 be a solution of L*2*=0,
2* = - W*2*.
(3.7)
Suppose t'2o~ 0 and g2o+ 27r¢ 0, and let 7~
s=
~o-,l---~
(3.8)
As in Theorem 1, R=
12o
g2o -2~r'
(3.9)
(2.12) leads to £2 0 = 1 ,
£ , , = O,
£o~ = - 1 ,
344
Z. Zhou / PhysicsLettersA 195 (1994)339-345 I2o --Art 2n 2 £2o(0~+0~)(2n) --2n(O¢+O~)Oo ~ (O¢+O,) f2o-2--'---~= P 0 2-22n 2 '
/S,o=-/So, = P = P - 2
£oo=O_=Q-R-~P(S¢-S,)2 - R -~ [R~¢-Rnn - 2 ( S ¢ - S n ) 2 P - S [ ( 2 P ) ¢ - (~-P) n] + 2S~2n - 2Sn2¢],
(3.10)
which gives the new solution (P, Q) of (3.2). We can obtain many explicit solutions using (3.10). For example, take the seed solution as P = 2p > 0 (p is a real constant), Q = F = 0, then (3.6) has a solution n=exp [a(~+ q) +4a2t] + exp [ b ( ~ - q) - p ( ~ + q) + 4p2t] ,
(3.11 )
and (3.7) has a solution 2= - e x p [ c ( ~ + q ) - 4 c : t ] +exp [ d ( ~ - q) + p ( ~ + r/) -4pZt],
(3.12)
where the constants a, b, c and d satisfy
(p+a)(a+c)
b(d+b)
Let p+a #= - a+c
>10
'
v=-
~
b
>0,
and
A=a(~+q)+4a2t,
B=b(~-q)-p(~+q)+4p2t,
C=c(~+q)-4c2t,
D=d(~-rl)+p(~+~l)-4p2t,
then direct calculation gives
P=-2p+(a+c) ~ , tr2 where
a, = (1~+ 1 )2exp(2A+C+ D) +/t 2 exp(A+ B+ 2C) + #vexp(A+ B+ 2D) + (/z+ 1 ) (v+ 1 ) exp(2B+ C+D) + (3/t+ v+ 2) exp(A + B + C + D ) , a2 =/t(/~+ 1 ) exp(2A + 2C) + v(v+ 1 ) exp(2B+ 2D) + (#+ 1 ) exp(2A+C+D) +l~exp(A+B+2C)
+ vexp(A+B+2D) + (~+ 1) exp(2B+C+D) + ( 2 # v + # + v + 2 ) e x p ( A + B + C + D ) . Clearly, it is bounded for all (~, q, t). However, it does not decay at infinity. It is still interesting to look for solutions which decay at infinity.
Remark. In this Letter, we have only discussed the binary Darboux transformations for the Manakov triads of second order. It is naturally interesting to discuss the binary Darboux transformation for the Manakov triads of higher order. This is still open due to some technical problems.
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Acknowledgement The author would like to thank Professor Gu Chaohao, Professor Hu Hesheng and Professor D. Levi for helpful advice. The author would also like to thank the referees for comments and one of the referees also for valuable suggestions.
References [ I ] B.G. Konopelchenko, Mod. Phys. Lett. A 3 (1988) 1807. [2] B.G. Konopelchenko, Inverse Probl. 4 (1988) 151. [3] S.B. Leble, M.A. Salle and A.V. Yurov, Inverse Probl. 8 (1992) 207. [4 ] V.B. Matveev and M.A. Salle, Darboux transformations and solitons (Springer, Berlin, 1991 ).