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Bäcklund transformations and homoclinic structures for the integrable discretization of the NLS equation
PhysicsLettersA North-Holland
163 (1992) 181-187
PHYSICS
LETTERS
A
Bkklund transformations and homoclinic structures for the integrable discretization of the NLS equation Yanguang Li Department of Mathematics, Princeton University, Princeton, NJ 08544, USA
Received 3 January 1992; accepted for publication 13 January 1992 Communicated by D.D. Holm
Bticklund transformations for the integrable discretization ofthe NLS equation are found, and used to generate representations of homoclinic orbits and whiskers of whiskered tori for that discrete system of arbitrary dimension.
1. Introduction
It is well-known that the nonlinear Schriidinger (NLS) equation iq,=q,+21q12q-2wq
(1.1)
(where q is a complex-valued function of two real variables t and x and COis a real constant) is a completely integrable Hamiltonian system. It is also known that the following specific discretization (called the integrable discretization) of the NLS equation, i& = 4n+l -%l+q,-, h*
-2wq?I+qdL,(q*+,
+qn-l)
7
(1.2)
is a completely integrable Hamiltonian system [ 1,2]. In order to understand homoclinic and chaotic behavior in partial differential equations, it is natural to begin with a study of perturbations of the integrable NLS equation ( 1.1). Furthermore, one natural way to study perturbations of NLS p.d.e.‘s, is to study perturbations of the finite dimensional system ( 1.2). In these studies, Bgicklund transformations play fundamental roles because they provide analytic representations of the homoclinic structures of the unperturbed systems. The Bgcklund transformations for p.d.e. ( 1.1) are well-known [ 31, but the BIcklund transformations for discretization ( 1.2) have been elusive. Schober has given a nice formula for the transformation of potentials [ 41 (qn+Qn). In this note, we will derive the Darboux form of the Bticklund transformations for both potentials and eigenfunctions, which has the advantage that it is easy to iterate. Iteration is necessary for representations of whiskers of dimension > 2.
2. Derivation of the Bffcklund transformations Eq. ( 1.2) can be rewritten in the form
(2.1)
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Its Lax pair (or discrete Zakharov-Shabat system) takes the form [2 ] ~',,+ ~=L~qG,
(2.2a) (2.2b1
where L. :(ih~_ ~
B.=
ihq.~ 1/z.] "
zsi (1-z2+2i2h-haq.4._ , +o)h 2 -zihq.+(1/z)ihq~_, ) -zihgl._~ +(1/z)ih4. l/z2-1+2iRh+h2(l,,q,,_~-~h2
in which z=exp(i2h). Compatibility of overdetermined system (2.2) gives the "Lax representation"
L~ =B~+IL. - L . B .
( 2.3
of the discrete NLS (2.1).
Theorem (B~icklund transformation). Fix a solution q% of the linear system (2.2) at define a 2 × 2 matrix F. by F~ = ( z +
(q., z+ ), and use 0,, to
b~
(1/z)a. cn
/z+zd~) - 1 "
where a . = ~ ( 1z+0 . 2 b~
12+ l z + [ 210.1 12) ,
Iz+l~-I -
z~+~. 0.,<2,
d n = - - - ( 110 n 2
Lz+14-1
Z+~n
-
c~- z+e+~O"'O"2'
12+lz+l'lq)n, . 12) ,
&=-(l/e+)(10.~12+iz+1210.212).
Note that t~. = - d . ,
6.
~-~-C n
.
Further, define
~.=F.~.,
i
Q.=~b.+,-a.+,q..
Then, if q/. solves linear system (2.2) at (z, q.), ~P. solves linear system (2.2) at (z, Q.). Moreover, by compatibility of linear system (2.2), q. and Q. both solve the discrete NLS.
Proof The proof will be finished if we can show that ~P. solves eq. (2.2) at (z, Q. ) provided that ~u. solves eq. (2.2) at (z, q.). From eq. (2.2) we get ~.+~ - L . ( Q . ) ~e. =
[F.+,L.(q)-L.(Q)F.I~.,
~n -B.(Qn)~. ={/~n - [B.(Q)Fn-F.B.(q)}q/..
Hence, if
F~+,L.(q)-L.(Q)F.=O,
(2.4)
F~- [B~(Q)F~-F~B.(q) ]=O ,
(2.5)
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the theorem is established. In the component form, eqs. (2.4), (2.5), together with the definitions of Qn and On, are equivalent to the following ten equations: i Q.=-h bn+,-an+,qn,
i
O n = - ~¢n+l +dn+lqn,
(2.6)
and an+ 1 an h =iQ,,c,, -i#nb,,+ l,
(2.7a)
dn+, - d , =iQnbn --iqnCn+l h
(2.7b)
bn+l -bn = i Q , ( d , - 1 ) -iqn(an+, + 1 ) h
(2.7c)
c.+, - c . = i 0 n ( a . + 1 ) -it]n(dn+ ~- 1 ) h
(2.7d)
-
-
i
fin = ~5 [anh2(qn(ln-~-QnO,_l)+C, ihQn_~-bnih#n] ,
(2.8a)
i (ln= -~ [dnh2(QnQn_l-Ctnqn-,)+cnihq,-b, ihQ,_,] ,
(2.8b)
i bn = ~-5 [2bn( 1 + o)h 2) - bnh2(QnOn_~ +tinqn--~ ) +ihQ, -ihqn_~ +anihqn +dnihQn_~ ],
(2.8c)
i ?, = ~ [ -2cn( 1 +o~h 2) +cnh2(O,Q,_, +qn#n-, ) + i h Q , - i h # , _ , - a , ihQn_l - d , ih#n].
(2.8d)
Now the problem is reduced to checking eqs. ( 2.7a)- ( 2.7d ), ( 2.8a)- ( 2.8d ). From the representations of an, b,, c,, dn given in the theorem, we know that (2.7a)=~ (2.7b),
(2.7c)=~ (2.7d),
(2.8a)=~ (2.8b),
(2.8c)=~ (2.8d).
So we only need to check (2.7a), (2.7c), (2.8a), (2.8c). And this is done explicitly, establishing the theorem.
Remark. In the process of checking (2.8a), (2.8c), the relation Z~n+ 1 ~--- - -
~------z/~ a~;~+ ( 1 +h2[ q~ [ 2 ) Z+
is crucial.
3. Reality and periodicity
Here, "reality" means that Qn is the complex conjugate of Qn; that is, i
[ --qnan+l -F (i/h)bn+, 1- =#ndn+, - -h cn+,. This is obvious from the representations of an, bn, cn, dn given in the theorem. 183
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Now we consider periodicity: Qn + N= Qn, Vn, provided that q, + N= q,, Vn. The analogue of this periodicity for the NLS p.d.e, is guaranteed by choosing the analogue of the 0, in the above theorem to be periodic (or antiperiodic) eigenfunctions [5,6 ]. When we try to define periodic (or antiperiodic) eigenfunctions of eq. (2.2a), we have the trouble: the Wronskian relation for eq. (2.2a) W(~,, 0 ) ( n + 1 ) = p , W(gt, O ) ( n ) ,
(3.1)
where
W(ql, O)(n)--ql~,C),,2-ql,2Om,
p , = l + h Z l q , I: ,
is not standard (p,,~ 1 ). However, we can transform nonstandard linear problem (2.2) into a standard one [4].
Proposition. Through the transformations [/J(n+ 1 ) 1 q& I
--
~/'/(n + 1 ) I q& I
~
@(n+l)2 ~" ~/.2
~L/(n4- 1 )2 -
-
001 = ~(t)
I)/JOl ,
~/~t02 = ~ ( [ ) ~()2 ,
q/..'
the Lax pair (2.2) is transformed into an equivalent Lax pair
P,,+,=I,,P.,
~,,=~.P.,,
(3.2)
where 7(t) is some real-valued function of t (to be defined in the proof of the proposition),
B~= ~5
-zihq._, + ( l/z)ihq.
1 / z 2 - 1 + 2i2h+h2(q.q,, , +(t.q. 1 ) / 2 - e ) h 2 "
Proof of the proposition. From the recursion relations given in the proposition, we see that the relation g/,,, ~u,,l -- 0,2 ~,,2
(3.3a)
holds for any n. With relation (3.3a), (3.2a) can be easily derived. We also have, from the recursion relations in the proposition, that rz--I
q),=y(t) I-I /=0
~,=r(t)
fi /----1
p[-,/2~,,
(n>0),
(3.3b)
p)/2gt,,
(n<0).
(3.3ct
From these two relations, by choosing the factor y(t) to satisfy the following equation,
2+ i ?' ~
(qoq-1 - q o q - 1 ) = 0
(3.3d)
and by using eq. (2.1), we can easily derive the matrix/~,. Then we complete the proof of the proposition. The Wronskian relation for the linear problem (3.2)
W(CJ, 6 ) ( n + 1 ) = W(t~, 6 ) ( n ) , 184
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is standard (cf. (3.1) ). However, we will not work with linear problem (3.2) directly because its Gauge matrix in the Darboux transformations (cf. the theorem) is too complicated as the following corollary indicates. Corollary. In the coordinate {~n }, the representations of fin, /~n,0n, an, Qn have the same forms with those given in the theorem, but -Pn now takes the form
n-i \/ n-l \-l --1/2 -1/2 /~n: 7t2(t) t=fIoP',¢2) )~o(,),__n° p,(q) ) Fn
(n>0),
/ ~ = Ye(t)
(n<0);
Pl~¢~) Ya(t) l=--1
p}[2)
Fn
I=--1
where Fn takes the same form with that given in the theorem with an, b~, cn, dn replaced by ~n, 6n, 6n, tin. The proof of the corollary follows from relations (3.3b)-(3.3d) in the proof of the proposition; and the theorem. Now we know that the advantage of linear problem (3.2) is that it has a standard Wronskian relation, so that there is standard definition of the periodic (or antiperiodic) eigenfunction for it; the disadvantage (fatal to B~icklund transformations) of linear problem (3.2) is that the corresponding B~icklund transformations are impossible to iterate; in contrast, the advantage of linear problem (2.2) is that the corresponding B~icklund transformations are easy to iterate; the disadvantage of linear problem (2.2) is that it has a nonstandard Wronskian relation, and the natural replacements for the periodic (or antiperiodic) eigenfunctions must be identified. To do so, we will work with linear problem (2.2), and transfer the periodic (or antiperiodic) eigenfunction of linear problem (3.2) to some corresponding objects for linear problem (2.2). All content of the rest of this section is based on this idea. Assume that the period interval of eq. (1.1) is [0, 1 ], and we discretize it into N subintervals, then (3.4)
h=l/N.
Let M, be the fundamental matrix solution of (2.2), such that Mo=I ,
where I is the 2 × 2 identity matrix. The Floquet discriminant ,5 is defined to be ,5= trace{Mu }.
( 3.5 )
From the Wronskian relation (3.1), we have N--I
det{Mu}= ~ Pn=D2.
(3.6)
n=0
The eigenfunction ~, of (2.2) is said to be subbounded if its Floquet multiplier a satisfies lal = D .
(3.7)
The spectrum of Ln is defined to be S ( L n ) = {zl,5(z) is real, and -2D~<,5(z) ~<2D}.
(3.8)
Denote fro = e / D ,
then 185
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A=D(oo+ 1/aD) •
(3.9)
The simple point z s of A is defined by AI . . . . .
_+2D,
dz~ dz
z=z s
(3.10)
¢0,
and the double point z d of A is defined by ~1:=_~. =
_+2D,
(3.11)
Uzz dA z=2 d =o.
The subperiodic (anti-subperiodic) eigenfunctions of L, are defined to be the subbounded eigenfunctions associated with simple or double points with A= 2D (A= - 2 D ) . Notice that at simple or double points, the two Floquet multipliers are equal and real ( a = _+D), moreover, the Q, in the theorem is in the form of a quotient of two quadratic forms in G (cf. the theorem), so if we choose the 0n in the theorem to be a subperiodic (or anti-subperiodic) eigenfunction, then the periodicity is guaranteed.
Remark. D2 = II N-I . = o P n < ~ ( l + h 2C) l/h , w h e r e C = m a x n { I q n l 2 } . m o r e o v e r , (l+h2C)~/h-.1, ash-~O;onthe other hand, D2>~ 1; so as h ~ 0 , D2-~ 1.
4. Homoclinic structures
In this section, following the pattern set forth in refs. [ 5,8 ], we shall discuss briefly an example ofhomoclinic structures in the system (2.1). A more thorough discussion will be presented later. We will use the B~icklund transformations in section 2 to generate the whiskers [ 7 ] of the one parameter family of whiskered tori
q=cexp{-2i[ (c2-og)t+ 7]} ,
(4.1)
where c>~ 0, y is an arbitrary real number. We set the z+ to be a real double point not on the unit circle, choose the 0, to be an arbitrary combination of the subperiodic (or anti-subperiodic) eigenfunctions located at z+, then we obtain the Qn,
Q n = ( E / F - I )q,
(4.2)
where E = 2 ( I +z2+)ix/p sinp [ [rl2(z+ ' -~fpS+)+z+ - , ~ 6 +
] ,
_2 ~l ~s2.+l_fq52.+l ) . F = (1 + I rt2)h21ql2(1 + z 2) +ihx/p (1-,:+1~zq,.+
p=l+h2lql 2,
z+=xfpcosp+~fl-1,
mg
/t=~
(the integer m is chosen so that z+ is a double point of the kind required above), #
6+=exp(i/t),
6 =exp(-i/l),
r=roexp{i[2(og-lql2)]t}exp(vt),
v=2~z sin~(z+-l/z +
Zo is a complex parameter. Examination of the formulas shows that, as t ~ + co, Qn (t) approaches a torus translate of q at an exponential rate e x p ( - vt). Explicitly, as t - , - c o , 186
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(z+ Q.--, - q
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6_) 2 h21ql 2
,
as t--. + 09,
(z+ Q,,~-q
2
h2lq[2
'
which are torus translates o f q (because Iz+ - x / P 6_ I = I z+ - x / ~ 6+ I = h lql, moreover, they are i n d e p e n d e n t o f t i m e ) . In the representation (4.2) o f Q,, there are three free p a r a m e t e r s (to, c, to), for each specified pair (Zo, c), eq. (4.2) represents a homoclinic orbit to q; while if we only specify c a n d keep (to, to) free, it represents the (three real d i m e n s i o n a l ) whisker to a fixed circle; if we keep all these three parameters free, it represents a family o f whiskers. If there are m o r e than one positive real double point not on the unit circle, one can iterate the B~icklund transformations in o r d e r to get a representation o f the whole whisker. In that case, eq. (4.2) only represents one part o f the b o u n d a r y o f that whole whisker. The iterated Bgcklund transformation formulae are also computable by hand, but look more complicated, and will not be presented here.
5. Conclusion In this note, we give the B~icklund transformations for the integrable discretization o f the NLS equation. We also use the B~icklund transformations to generate a specific example o f homoclinic structures. More complete results using these representations for integrable and near-integrable discretizations will be presented later.
Acknowledgement The a u t h o r is greatly i n d e b t e d to Professor D a v i d McLaughlin for leading him to the p r o b l e m and for m a n y extremely useful discussions with him.
References [ 1] P.P. Kulish, Lett. Math. Phys. 5 ( 1981 ) 191. [2] M.J. Ablowitz, and J.F. Ladik, Stud. Appl. Math. 55 (1976) 213. [ 3 ] D.H. Sattinger and V.D. Zurkowski, Physica D 26 ( 1987 ) 225. [4] C.A. Schober, Ph.D. thesis, University of Arizona ( 1991 ). [5] N. Ercolani, M.G. Forest and D.W. McLaughlin, Physica D 43 (1990) 349. [ 6 ] Y. Li and D.W. McLaughlin, Instabilities, homoclinic orbits, Morse functions and Melnikov functions of a soliton equation, Princeton University, preprint ( 1991 ). [ 7 ] D.W. McLaughlin and E.A. Overman, Whiskered tori for integrable p.d.e.'s: chaotic behavior in near integrable p.d.e.'s, Princeton University, preprint ( 1991 ). [ 8 ] D.W. McLaughlin, Notes on the NLS equation, unpublished (1988).
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LETTERS
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Bkklund transformations and homoclinic structures for the integrable discretization of the NLS equation Yanguang Li Department of Mathematics, Princeton University, Princeton, NJ 08544, USA
Received 3 January 1992; accepted for publication 13 January 1992 Communicated by D.D. Holm
Bticklund transformations for the integrable discretization ofthe NLS equation are found, and used to generate representations of homoclinic orbits and whiskers of whiskered tori for that discrete system of arbitrary dimension.
1. Introduction
It is well-known that the nonlinear Schriidinger (NLS) equation iq,=q,+21q12q-2wq
(1.1)
(where q is a complex-valued function of two real variables t and x and COis a real constant) is a completely integrable Hamiltonian system. It is also known that the following specific discretization (called the integrable discretization) of the NLS equation, i& = 4n+l -%l+q,-, h*
-2wq?I+qdL,(q*+,
+qn-l)
7
(1.2)
is a completely integrable Hamiltonian system [ 1,2]. In order to understand homoclinic and chaotic behavior in partial differential equations, it is natural to begin with a study of perturbations of the integrable NLS equation ( 1.1). Furthermore, one natural way to study perturbations of NLS p.d.e.‘s, is to study perturbations of the finite dimensional system ( 1.2). In these studies, Bgicklund transformations play fundamental roles because they provide analytic representations of the homoclinic structures of the unperturbed systems. The Bgcklund transformations for p.d.e. ( 1.1) are well-known [ 31, but the BIcklund transformations for discretization ( 1.2) have been elusive. Schober has given a nice formula for the transformation of potentials [ 41 (qn+Qn). In this note, we will derive the Darboux form of the Bticklund transformations for both potentials and eigenfunctions, which has the advantage that it is easy to iterate. Iteration is necessary for representations of whiskers of dimension > 2.
2. Derivation of the Bffcklund transformations Eq. ( 1.2) can be rewritten in the form
(2.1)
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Its Lax pair (or discrete Zakharov-Shabat system) takes the form [2 ] ~',,+ ~=L~qG,
(2.2a) (2.2b1
where L. :(ih~_ ~
B.=
ihq.~ 1/z.] "
zsi (1-z2+2i2h-haq.4._ , +o)h 2 -zihq.+(1/z)ihq~_, ) -zihgl._~ +(1/z)ih4. l/z2-1+2iRh+h2(l,,q,,_~-~h2
in which z=exp(i2h). Compatibility of overdetermined system (2.2) gives the "Lax representation"
L~ =B~+IL. - L . B .
( 2.3
of the discrete NLS (2.1).
Theorem (B~icklund transformation). Fix a solution q% of the linear system (2.2) at define a 2 × 2 matrix F. by F~ = ( z +
(q., z+ ), and use 0,, to
b~
(1/z)a. cn
/z+zd~) - 1 "
where a . = ~ ( 1z+0 . 2 b~
12+ l z + [ 210.1 12) ,
Iz+l~-I -
z~+~. 0.,<2,
d n = - - - ( 110 n 2
Lz+14-1
Z+~n
-
c~- z+e+~O"'O"2'
12+lz+l'lq)n, . 12) ,
&=-(l/e+)(10.~12+iz+1210.212).
Note that t~. = - d . ,
6.
~-~-C n
.
Further, define
~.=F.~.,
i
Q.=~b.+,-a.+,q..
Then, if q/. solves linear system (2.2) at (z, q.), ~P. solves linear system (2.2) at (z, Q.). Moreover, by compatibility of linear system (2.2), q. and Q. both solve the discrete NLS.
Proof The proof will be finished if we can show that ~P. solves eq. (2.2) at (z, Q. ) provided that ~u. solves eq. (2.2) at (z, q.). From eq. (2.2) we get ~.+~ - L . ( Q . ) ~e. =
[F.+,L.(q)-L.(Q)F.I~.,
~n -B.(Qn)~. ={/~n - [B.(Q)Fn-F.B.(q)}q/..
Hence, if
F~+,L.(q)-L.(Q)F.=O,
(2.4)
F~- [B~(Q)F~-F~B.(q) ]=O ,
(2.5)
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the theorem is established. In the component form, eqs. (2.4), (2.5), together with the definitions of Qn and On, are equivalent to the following ten equations: i Q.=-h bn+,-an+,qn,
i
O n = - ~¢n+l +dn+lqn,
(2.6)
and an+ 1 an h =iQ,,c,, -i#nb,,+ l,
(2.7a)
dn+, - d , =iQnbn --iqnCn+l h
(2.7b)
bn+l -bn = i Q , ( d , - 1 ) -iqn(an+, + 1 ) h
(2.7c)
c.+, - c . = i 0 n ( a . + 1 ) -it]n(dn+ ~- 1 ) h
(2.7d)
-
-
i
fin = ~5 [anh2(qn(ln-~-QnO,_l)+C, ihQn_~-bnih#n] ,
(2.8a)
i (ln= -~ [dnh2(QnQn_l-Ctnqn-,)+cnihq,-b, ihQ,_,] ,
(2.8b)
i bn = ~-5 [2bn( 1 + o)h 2) - bnh2(QnOn_~ +tinqn--~ ) +ihQ, -ihqn_~ +anihqn +dnihQn_~ ],
(2.8c)
i ?, = ~ [ -2cn( 1 +o~h 2) +cnh2(O,Q,_, +qn#n-, ) + i h Q , - i h # , _ , - a , ihQn_l - d , ih#n].
(2.8d)
Now the problem is reduced to checking eqs. ( 2.7a)- ( 2.7d ), ( 2.8a)- ( 2.8d ). From the representations of an, b,, c,, dn given in the theorem, we know that (2.7a)=~ (2.7b),
(2.7c)=~ (2.7d),
(2.8a)=~ (2.8b),
(2.8c)=~ (2.8d).
So we only need to check (2.7a), (2.7c), (2.8a), (2.8c). And this is done explicitly, establishing the theorem.
Remark. In the process of checking (2.8a), (2.8c), the relation Z~n+ 1 ~--- - -
~------z/~ a~;~+ ( 1 +h2[ q~ [ 2 ) Z+
is crucial.
3. Reality and periodicity
Here, "reality" means that Qn is the complex conjugate of Qn; that is, i
[ --qnan+l -F (i/h)bn+, 1- =#ndn+, - -h cn+,. This is obvious from the representations of an, bn, cn, dn given in the theorem. 183
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Now we consider periodicity: Qn + N= Qn, Vn, provided that q, + N= q,, Vn. The analogue of this periodicity for the NLS p.d.e, is guaranteed by choosing the analogue of the 0, in the above theorem to be periodic (or antiperiodic) eigenfunctions [5,6 ]. When we try to define periodic (or antiperiodic) eigenfunctions of eq. (2.2a), we have the trouble: the Wronskian relation for eq. (2.2a) W(~,, 0 ) ( n + 1 ) = p , W(gt, O ) ( n ) ,
(3.1)
where
W(ql, O)(n)--ql~,C),,2-ql,2Om,
p , = l + h Z l q , I: ,
is not standard (p,,~ 1 ). However, we can transform nonstandard linear problem (2.2) into a standard one [4].
Proposition. Through the transformations [/J(n+ 1 ) 1 q& I
--
~/'/(n + 1 ) I q& I
~
@(n+l)2 ~" ~/.2
~L/(n4- 1 )2 -
-
001 = ~(t)
I)/JOl ,
~/~t02 = ~ ( [ ) ~()2 ,
q/..'
the Lax pair (2.2) is transformed into an equivalent Lax pair
P,,+,=I,,P.,
~,,=~.P.,,
(3.2)
where 7(t) is some real-valued function of t (to be defined in the proof of the proposition),
B~= ~5
-zihq._, + ( l/z)ihq.
1 / z 2 - 1 + 2i2h+h2(q.q,, , +(t.q. 1 ) / 2 - e ) h 2 "
Proof of the proposition. From the recursion relations given in the proposition, we see that the relation g/,,, ~u,,l -- 0,2 ~,,2
(3.3a)
holds for any n. With relation (3.3a), (3.2a) can be easily derived. We also have, from the recursion relations in the proposition, that rz--I
q),=y(t) I-I /=0
~,=r(t)
fi /----1
p[-,/2~,,
(n>0),
(3.3b)
p)/2gt,,
(n<0).
(3.3ct
From these two relations, by choosing the factor y(t) to satisfy the following equation,
2+ i ?' ~
(qoq-1 - q o q - 1 ) = 0
(3.3d)
and by using eq. (2.1), we can easily derive the matrix/~,. Then we complete the proof of the proposition. The Wronskian relation for the linear problem (3.2)
W(CJ, 6 ) ( n + 1 ) = W(t~, 6 ) ( n ) , 184
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is standard (cf. (3.1) ). However, we will not work with linear problem (3.2) directly because its Gauge matrix in the Darboux transformations (cf. the theorem) is too complicated as the following corollary indicates. Corollary. In the coordinate {~n }, the representations of fin, /~n,0n, an, Qn have the same forms with those given in the theorem, but -Pn now takes the form
n-i \/ n-l \-l --1/2 -1/2 /~n: 7t2(t) t=fIoP',¢2) )~o(,),__n° p,(q) ) Fn
(n>0),
/ ~ = Ye(t)
(n<0);
Pl~¢~) Ya(t) l=--1
p}[2)
Fn
I=--1
where Fn takes the same form with that given in the theorem with an, b~, cn, dn replaced by ~n, 6n, 6n, tin. The proof of the corollary follows from relations (3.3b)-(3.3d) in the proof of the proposition; and the theorem. Now we know that the advantage of linear problem (3.2) is that it has a standard Wronskian relation, so that there is standard definition of the periodic (or antiperiodic) eigenfunction for it; the disadvantage (fatal to B~icklund transformations) of linear problem (3.2) is that the corresponding B~icklund transformations are impossible to iterate; in contrast, the advantage of linear problem (2.2) is that the corresponding B~icklund transformations are easy to iterate; the disadvantage of linear problem (2.2) is that it has a nonstandard Wronskian relation, and the natural replacements for the periodic (or antiperiodic) eigenfunctions must be identified. To do so, we will work with linear problem (2.2), and transfer the periodic (or antiperiodic) eigenfunction of linear problem (3.2) to some corresponding objects for linear problem (2.2). All content of the rest of this section is based on this idea. Assume that the period interval of eq. (1.1) is [0, 1 ], and we discretize it into N subintervals, then (3.4)
h=l/N.
Let M, be the fundamental matrix solution of (2.2), such that Mo=I ,
where I is the 2 × 2 identity matrix. The Floquet discriminant ,5 is defined to be ,5= trace{Mu }.
( 3.5 )
From the Wronskian relation (3.1), we have N--I
det{Mu}= ~ Pn=D2.
(3.6)
n=0
The eigenfunction ~, of (2.2) is said to be subbounded if its Floquet multiplier a satisfies lal = D .
(3.7)
The spectrum of Ln is defined to be S ( L n ) = {zl,5(z) is real, and -2D~<,5(z) ~<2D}.
(3.8)
Denote fro = e / D ,
then 185
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16 March 1992
A=D(oo+ 1/aD) •
(3.9)
The simple point z s of A is defined by AI . . . . .
_+2D,
dz~ dz
z=z s
(3.10)
¢0,
and the double point z d of A is defined by ~1:=_~. =
_+2D,
(3.11)
Uzz dA z=2 d =o.
The subperiodic (anti-subperiodic) eigenfunctions of L, are defined to be the subbounded eigenfunctions associated with simple or double points with A= 2D (A= - 2 D ) . Notice that at simple or double points, the two Floquet multipliers are equal and real ( a = _+D), moreover, the Q, in the theorem is in the form of a quotient of two quadratic forms in G (cf. the theorem), so if we choose the 0n in the theorem to be a subperiodic (or anti-subperiodic) eigenfunction, then the periodicity is guaranteed.
Remark. D2 = II N-I . = o P n < ~ ( l + h 2C) l/h , w h e r e C = m a x n { I q n l 2 } . m o r e o v e r , (l+h2C)~/h-.1, ash-~O;onthe other hand, D2>~ 1; so as h ~ 0 , D2-~ 1.
4. Homoclinic structures
In this section, following the pattern set forth in refs. [ 5,8 ], we shall discuss briefly an example ofhomoclinic structures in the system (2.1). A more thorough discussion will be presented later. We will use the B~icklund transformations in section 2 to generate the whiskers [ 7 ] of the one parameter family of whiskered tori
q=cexp{-2i[ (c2-og)t+ 7]} ,
(4.1)
where c>~ 0, y is an arbitrary real number. We set the z+ to be a real double point not on the unit circle, choose the 0, to be an arbitrary combination of the subperiodic (or anti-subperiodic) eigenfunctions located at z+, then we obtain the Qn,
Q n = ( E / F - I )q,
(4.2)
where E = 2 ( I +z2+)ix/p sinp [ [rl2(z+ ' -~fpS+)+z+ - , ~ 6 +
] ,
_2 ~l ~s2.+l_fq52.+l ) . F = (1 + I rt2)h21ql2(1 + z 2) +ihx/p (1-,:+1~zq,.+
p=l+h2lql 2,
z+=xfpcosp+~fl-1,
mg
/t=~
(the integer m is chosen so that z+ is a double point of the kind required above), #
6+=exp(i/t),
6 =exp(-i/l),
r=roexp{i[2(og-lql2)]t}exp(vt),
v=2~z sin~(z+-l/z +
Zo is a complex parameter. Examination of the formulas shows that, as t ~ + co, Qn (t) approaches a torus translate of q at an exponential rate e x p ( - vt). Explicitly, as t - , - c o , 186
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(z+ Q.--, - q
PHYSICS LETTERS A
16 March 1992
6_) 2 h21ql 2
,
as t--. + 09,
(z+ Q,,~-q
2
h2lq[2
'
which are torus translates o f q (because Iz+ - x / P 6_ I = I z+ - x / ~ 6+ I = h lql, moreover, they are i n d e p e n d e n t o f t i m e ) . In the representation (4.2) o f Q,, there are three free p a r a m e t e r s (to, c, to), for each specified pair (Zo, c), eq. (4.2) represents a homoclinic orbit to q; while if we only specify c a n d keep (to, to) free, it represents the (three real d i m e n s i o n a l ) whisker to a fixed circle; if we keep all these three parameters free, it represents a family o f whiskers. If there are m o r e than one positive real double point not on the unit circle, one can iterate the B~icklund transformations in o r d e r to get a representation o f the whole whisker. In that case, eq. (4.2) only represents one part o f the b o u n d a r y o f that whole whisker. The iterated Bgcklund transformation formulae are also computable by hand, but look more complicated, and will not be presented here.
5. Conclusion In this note, we give the B~icklund transformations for the integrable discretization o f the NLS equation. We also use the B~icklund transformations to generate a specific example o f homoclinic structures. More complete results using these representations for integrable and near-integrable discretizations will be presented later.
Acknowledgement The a u t h o r is greatly i n d e b t e d to Professor D a v i d McLaughlin for leading him to the p r o b l e m and for m a n y extremely useful discussions with him.
References [ 1] P.P. Kulish, Lett. Math. Phys. 5 ( 1981 ) 191. [2] M.J. Ablowitz, and J.F. Ladik, Stud. Appl. Math. 55 (1976) 213. [ 3 ] D.H. Sattinger and V.D. Zurkowski, Physica D 26 ( 1987 ) 225. [4] C.A. Schober, Ph.D. thesis, University of Arizona ( 1991 ). [5] N. Ercolani, M.G. Forest and D.W. McLaughlin, Physica D 43 (1990) 349. [ 6 ] Y. Li and D.W. McLaughlin, Instabilities, homoclinic orbits, Morse functions and Melnikov functions of a soliton equation, Princeton University, preprint ( 1991 ). [ 7 ] D.W. McLaughlin and E.A. Overman, Whiskered tori for integrable p.d.e.'s: chaotic behavior in near integrable p.d.e.'s, Princeton University, preprint ( 1991 ). [ 8 ] D.W. McLaughlin, Notes on the NLS equation, unpublished (1988).
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