Mirror transformations of Hamiltonian systems

Physica D 152–153 (2001) 110–123

Mirror transformations of Hamiltonian systems Jishan Hu∗ , Min Yan, Tat-Leung Yee Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, PR China

Abstract We demonstrate, through various examples of Hamiltonian systems, that symplectic structures have been encoded into the Painlevé test. Each principal balance in the Painlevé test induces a mirror transformation that regularizes movable singularities. Moreover, for finite-dimensional Hamiltonian systems, the mirror transformations are canonical. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Hamiltonian systems; Painlevé test; Symptectic structures

1. Introduction The dynamics of a system of differential equations may be studied by considering the geometry of the corresponding flow in the phase space. However, it may not be easy to get the global information because the flow stops at singularities. On the other hand, if the system is integrable, it is often possible to complete its phase space by “adding singularities” and extend the flow over the singularities. As a result, we get a global flow (i.e., defined for all time) on the completion of the phase space, and the study of the geometry of this global flow provides a complete and global picture on the dynamics of the system of differential equations. The key technical step in the above approach is the completion process. Moreover, if the given system is Hamiltonian, we would further like to preserve the symplectic structure in the completion process. In the late 1980s, the idea was carried out for autonomous finite-dimensional Hamiltonian systems by Adler and van Moerbeke [1], and Ercolani and Siggia [2,3]. It was assumed that only movable singularities are poles and the system passes certain hierarchical type Painlevé test (i.e., principal and lower balances forming a “coherent” tree structure). The completion of the phase space was constructed by introducing changes of variables at the pole singularities so that the Laurent series solutions are regularized. For principal balances, Adler and van Moerbeke used the Laurent series solutions directly as the change of variables. The change of variables constructed by Ercolani and Siggia is more refined and is often canonical (so the symplectic structure on the complete phase space is easily understood). However, their constructions are ad hoc and involve some guess works. In [4,5], we demonstrated a systematic way of regularizing the principal balances for general ODE systems passing the Painlevé test. In [6], we further showed that the method is equivalent to the Painlevé test. The resulting change of variable, which we call the mirror transformation, is very similar to Ercolani and Siggia and can be used to complete the phase space. In this paper, we demonstrate, through several examples, that he mirror transformation for ∗

Corresponding author.

0167-2789/01/$ – see front matter © 2001 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 2 7 8 9 ( 0 1 ) 0 0 1 6 4 - 6

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Hamiltonian systems can always be made compatible to symplectic structure. Our examples include autonomous, non-autonomous, as well as infinite-dimensional Hamiltonian systems. In fact, we have rigorously proved the canonicity of mirror transformations for almost weighted homogeneous (finite-dimensional) Hamiltonian systems in [6]. We note that Adler and van Moerbeke considered weighted homogeneous systems in [1], and all the examples in Ercolani and Siggia are in fact almost weighted homogeneous. Thus the completion process for the Hamiltonian systems is now as easy and systematic as the Painlevé test, at least for principal balances and almost weighted homogeneous systems. Technically, underlying our construction of the mirror transformation is the LU decomposition of the resonance matrix, up to a certain rearrangement of the dependent variables. We will demonstrate that for Hamiltonian systems, the resonance matrix can always be arranged to display symplectic structure, and there is a pairing property among resonances. For almost weighted homogeneous Hamiltonian systems, we have rigorously established such properties in [6] and used the properties to proved that the mirror transformation can be made canonical. In [7], Lochak suggested that symplectic structure and the resonance pairing property are already built into the Painlevé test of Hamiltonian systems. However, Lochak did not explain the relation between his argument and the Painlevé test, so that his argument is not directly applicable as a theoretical explanation. This paper serves as a collection of examples for the theory established in [6]. For finite-dimensional (almost weighted homogeneous) Hamiltonian systems, the observations made in this paper have been rigorously proved in [6]. We believe that the almost weighted homogeneous condition is unnecessary for finding canonical mirror transformations. Unfortunately, we do not know any example of integrable Hamiltonian systems that are not almost weighted homogeneous to test our theory. On the other hand, more work needs to be done in order to understand the change of variables needed around the lower balances. The infinite-dimensional examples are more exploratory. We believe many properties can be rigorously proved. However, the canonicity of the mirror transformation for infinite-dimensional Hamiltonian systems is still mysterious to us (because the meaning of canonicity of singular transformations is yet to be understood). This is why we have not tried to set up a rigorous theory regarding infinite-dimensional Hamiltonian systems in [6]. 2. The underlying theory 2.1. Painlevé test Consider a system of first order ODEs: u = f (t, u),

u = (u1 , . . . , un ), f = (f1 , . . . , fn ).

(1)

A balance is a Laurent series solution ui = ci (t − t0 )−gi + ui,1 (t − t0 )1−gi + ui,2 (t − t0 )2−gi + · · · ,

i = 1, . . . , n,

(2)

such that the vector (g1 c1 , . . . , gn cn ) = 0,

(3)

and some free variables (called resonance parameters) are involved. We remark that condition (3) allows some leading coefficients ci to vanish. Such relaxation is necessary for some of our examples. For any specific resonance parameter r, the smallest j such that r appears in some ui,j is the resonance. The corresponding resonance vector is

  ∂u1,j ∂un,j g1 −j +1 ∂u1 gn −j +1 ∂un v = (t − t0 ) = , . . . , (t − t0 ) ,..., . ∂r ∂r t=t0 ∂r ∂r

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We may also consider t0 as a resonance parameter. We assign j = −1 as the corresponding resonance, and 
∂u1 ∂un , . . . , (t − t0 )gn +1 = (g1 c1 , . . . , gn cn ) v = (t − t0 )g1 +1 ∂t0 ∂t0 t=t0 as the corresponding resonance vector. With all the resonance vectors as columns, in the order of increasing resonances, we get a resonance matrix. The balance is principal if the number of resonances (including j = −1) is n. The resonances and resonance vectors are often computed as the eigenvalues and eigenvectors of a certain Kowalevskian matrix. However, there are some cases that the choice of the Kowalevskian matrix becomes a subtle issue (see, for example, the discussion near the end of Ref. [3]). Therefore in [6], we established a rigorous theory on the relation between the resonance vectors and the Kowalevskian matrix. Specifically, we assume fi to be polynomials in u and analytic in t. We call the leading exponents g1 , . . . , gn Fuchsian for system (3) if the g∗ -weighted degree of fi is ≤ gi + 1. The condition means that at least formally, the dominant balance involves both the derivatives on the left and the functions fi on the right (instead of internal cancellation on the right). Under this condition, the Kowalevskian matrix is   g1 D ∂f   .. (t0 , c1 , . . . , cn ) +  K= , . ∂u gn where fiD =



{terms in fi with g∗ -weighted degree = gi + 1}.

In general, we expect no relation between resonances, nor between resonance vectors. For principal balances of Hamiltonian systems with u = (q1 , . . . , qm , p1 , . . . , pm ),

fi =

∂H , ∂pi

fi+m = −

∂H , ∂qi

1 ≤ i ≤ m,

the papers of Ercolani and Siggia [2,3], and Lochak [7] suggest that the following is true: the resonances −1 = j1 < j2 ≤ · · · ≤ j2m should satisfy the following resonance pairing property: j1 + j2m j2 + j2m−1 jm + jm+1

= = .. .

g1 + gm+1 , g2 + gm+2 ,

=

gm + g2m .

Moreover, it is possible satisfy  if 1 viT Jvj = −1 if  0 if where J =




O −Im

Im O

to choose resonance parameters elegantly, so that after rescaling, the resonance vectors i + j = 2m + 1, 1 ≤ i ≤ m, i + j = 2m + 1, m + 1 ≤ i ≤ 2m, i + j = 2m + 1,

 .

In other words, the matrix S = [v1 , v2 , . . . , vm , v2m , v2m−1 , . . . , vm+1 ] (obtained after reversing the order of the last half of the resonance vectors) is a symplectic matrix: S T JS = J . Again, we have established a rigorous theory

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on the above observations in [6] under the assumption that the Hamiltonian system satisfies the following almost weighted homogeneous condition: gi + gm+i + 1 = g∗ -weighted degree of H. The condition implies that the Kowalevskian matrix is   g1   .. K = J [Hessian of H D ] +  , . gn where H D is defined similar to fiD . As a consequence of the fact that the Hessian is symmetric, we may derive the symplectic structure of the resonance matrix described above. More details can be found in Section 2.4 of [6]. 2.2. Mirror transformation The mirror transformations are associated to principal balances. The first example of such transformations appeared in Painlevé’s proof of the Painlevé property of the first Painlevé equation [8]. In [4,5], we demonstrated a simple and systematic method for deriving the mirror transformations. Consider a principal balance (2) of system (1). From condition (3), we may assume g1 c1 = 0. Then we introduce the indicial normalization u1 = θ −g1 . Moreover, we postulate the Laurent series θ  = a0 + a1 θ + a2 θ 2 + · · · ,

ui = c¯i θ −gi + u¯ i,1 θ 1−gi + u¯ i,2 θ 2−gi + · · · ,

i = 2, . . . , n

(4)

θ

for and the other dependent variables. Substituting these into system (1), we get some recursive relations for the coefficients. The solutions of the recursive relations involves some arbitrary choices r¯2 , . . . , r¯n , which correspond to the resonance parameters. Moreover, the lowest powers where these arbitrary choices appear are the same as the resonance parameters in the principal balance. The second part is the derivation of the mirror transformation from the θ -series. Let −1 = j1 < j2 ≤ · · · ≤ jn be the resonances for the principal balance and assume the “resonance” for r¯i is ji . Then for some 2 ≤ i ≤ n, r¯2 appears in the coefficient of u¯ i,j2 θ j2 −gi in ui . Without loss of generality, we assume i = 2 and introduce a new variable ξ2 by truncating u2 at θ j2 −g2 u2 = c¯2 θ −g2 + u¯ 2,1 θ 1−g2 + · · · + u¯ 2,j2 −1 θ j2 −1−g2 + ξ2 θ j2 −g2 .

(5)

From ξ2 = u¯ 2,j2 + u¯ 2,j2 +1 θ + u¯ 2,j2 +2 θ 2 + · · · ,

(6)

we may express r¯2 as a θ -power series. Then we update the θ -series of u3 , . . . , un by replacing r¯2 with this power series ui = c¯¯i θ −gi + u¯¯ i,1 θ 1−gi + u¯¯ i,2 θ 2−gi + · · · ,

i = 3, . . . , n,

(7)

where the coefficients involve ξ2 and the arbitrary choices r¯3 , . . . , r¯n . Again, we may assume r¯3 appears in the coefficient of θ j3 −g3 in the updated θ -series for u3 and introduce a new variable ξ3 by truncating the updated θ -series at θ j3 −g3 . The process continues until we introduce the last new variable ξn . The theoretical study of the process above has been carried out in [6]. It is apparent that the first part is very similar (and in fact equivalent) to the Painlevé test. What is new is the second part and the key is to make sure that,

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up to some rearrangement of the dependent variables, r¯i appears in coefficient of θ ji −gi in the updated θ-series of ui . The condition for this is exactly (∂ u¯ 2,j2 /∂ r¯2 ) = 0, (∂ u¯¯ 3,j3 /∂ r¯3 ) = 0, and so on. As explained in [6], these partial derivatives are the diagonal terms in the U part of the LU-decomposition of the resonance matrix. In our strict definition, the columns of the resonance matrix of a principal balance is a basis consisting of eigenvectors of the Kowalevskian matrix. Since up to a permutation of rows, any invertible matrix has LU-decomposition, we see that, by permuting the dependent variables in the same way, we can always have the conditions (∂ u¯ 2,j2 /∂ r¯2 ) = 0, (∂ u¯¯ 3,j3 /∂ r¯3 ) = 0, . . . , satisfied. The symplectic structure in the resonance matrix is crucial for the mirror transformation of Hamiltonian systems to be canonical. The key point here is that symplectic invertible matrices have LU-decomposition up to special types of row permutations. Accordingly, the permutation needed for dependent variables in the construction of the mirror transformation is also of special type. This makes it possible to fine tune the mirror transformation and make it canonical.

3. Finite-dimensional Hamiltonian systems Now, let us first use two autonomous and one non-autonomous finite-dimensional Hamiltonian systems to demonstrate how to have the canonical mirror transformations. 3.1. A Hénon–Heiles Hamiltonian system The following Hénon–Heiles system: q˙1 = p1 ,

q˙2 = p2 ,

p˙ 1 = −q1 − 2q1 q2 ,

p˙ 2 = −q2 − q12 − 6q22 ,

(8)

is given by the Hamiltonian H = 21 (p12 + p22 + q12 + q22 ) + q12 q2 + 2q23 , and was studied by Ercolani and Siggia [3]. The system passes the Painlevé test and has one principal balance

  3 3 5 r 11r r 5r r 61r 2 3 2 2 q1 ∼ −r2 (t − t0 )−1 + − − 2 (t − t0 ) + (t − t0 )2 + + + 2 (t − t0 )3 12 12 2 720 144 72

  r 2 r3 13r23 r5 r27 r3 r2 r 2 r4 + − − 2 (t − t0 )4 + − − + 2 + + (t − t0 )5 + · · · , 24 24 243 25920 2592 15552 36

 r22 r22 r24 1 1 r 2 r3 r4 −2 q2 ∼ −(t − t0 ) − + + − + + (t − t0 )2 − (t − t0 )3 + (t − t0 )4 + · · · , 12 12 240 12 48 6 4  

3 3 5 2r r 11r r r 5r 61r r 3 2 2 3 2 p1 ∼ r2 (t − t0 )−2 − − 2 + r3 (t − t0 ) + + + 2 (t − t0 )2 + − − 2 (t − t0 )3 12 12 240 48 24 6 6

 13r23 5r25 5r27 5r2 5r2 r4 + − − + + + (t − t0 )4 + · · · , 243 5184 2592 15552 36

 r22 r24 1 r 2 r3 −3 p2 ∼ 2(t − t0 ) + − + + (t − t0 ) − (t − t0 )2 + r4 (t − t0 )3 + · · · . 120 6 24 2

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If we take the natural leading exponents 1, 2, 2, 3, then the resonances would be −1, 0, 3, 6 for the resonance parameters t0 , r2 , r3 , r4 , respectively. Although, we have the resonance pairing property −1 + 6 = 2 + 3, 0 + 3 = 1 + 2, the resonance matrix fails to become symplectic despite column permutations. The failure of the “naive” symplectic property has been noticed by Ercolani and Siggia [3]. They pointed out that one has to be very careful in applying the usual notions of Painlevé analysis to certain Hamiltonian systems. We may save the situation by adding 0(t − t0 )−2 and 0(t − t0 )−3 to q1 and p1 , making their leading exponents 2 and 3. This changes the resonances to −1, 1, 4, 6, which still satisfies the resonance pairing property −1 + 6 = 2 + 3, 1 + 4 = 2 + 3. Moreover, the resonance matrix becomes   0 −1 21 0    −2 0 0 1   4   .  0 1 1 0   6 0 0 1 After rescaling and reversing the last two columns, we get a symplectic matrix   0 −1 0 − 13    −2 0 − 1 0  14    . 2   0 1 0 −3   6

0

− 27

0

Note that the upper left part

 0 −1 −2

0

of the symplectic matrix has LU-decomposition after reversing the two rows. This implies that the original resonance matrix has LU decomposition after reversing the first two rows. Thus according to the theory in [6], we expect to obtain a canonical mirror transformation by introducing indicial normalization at q2 and then successively truncating q1 , p1 , and p2 at the resonance parameters. Specifically, we introduce q2 = θ −2 and find the Laurent θ-series:

  2 2 4 i¯ r 13i¯ r 7i¯ r i i i¯r2 r¯3 5 r¯4 6 2 2 θ ∼ i + − 2 θ2 + − + + θ4 + θ − θ + ··· , 8 8 128 64 128 2 2

  i¯r23 3i¯r23 i¯r25 3i¯r2 9i¯r2 i¯r3 2 −1 q1 ∼ i¯r2 θ + − − θ + − θ− − − θ3 8 2 8 128 64 128

5i¯r23 7i¯r25 11i¯r2 r¯2 r¯4 + + + − 2304 384 2304 18

 θ5 + · · · ,

 

5¯r23 r¯25 3¯r22 r¯3 r¯2 r¯3 2 p1 ∼ r¯2 θ + + θ + + θ3 4 16 16 8 8 

r¯23 37¯r25 r¯27 2i r¯2 + + + + r¯2 r¯4 θ 4 + · · · , + − 2304 768 2304 256 9 −2

r¯2 + r¯3 θ + + 2

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p2 ∼ −2iθ

−3

i¯r 2 i + − + 2 4 4

 θ

−1

+

13i¯r22 7i¯r24 i − − 64 32 64

 θ − i¯r2 r¯3 θ 2 + r¯4 θ 3 + · · · .

Then we truncate the θ-series of q1 at r¯2 by introducing ξ2 : q2 = ξ2 θ −1 .

(9)

From the θ -series of ξ2 , we find r¯2 = −iξ2 + (− 38 iξ2 + 18 iξ23 )θ 2 + 21 r¯3 θ 3 + · · · , where the coefficients are functions of ξ2 , r¯3 , and r¯4 . Substitute this into the θ -series of p1 and p2 and truncating the θ-series of p2 at r¯3 , we introduce ξ3 : p1 = −iξ2 θ −2 − 78 iξ2 + 18 iξ23 + ξ3 θ.

(10)

From the θ-series of ξ3 , we may derive the θ -series of r¯3 , with functions of ξ2 , ξ3 , and r¯4 as coefficients. Substituting this into the (already updated) θ -series of p2 and truncating at r¯4 , we introduce ξ4 :     1 7 2 3 4 i + 32 p2 = −2iθ −3 + − 41 i − 41 iξ22 θ −1 + 64 iξ2 − 64 iξ2 θ − 21 (ξ2 ξ3 )θ 2 + ξ4 θ 3 . (11) As pointed out in [6], the transformation (q1 , q2 , p1 , p2 ) ↔ (θ, ξ2 , ξ3 , ξ4 ) given by q1 = θ −2 , (9)–(11) converts system (8) into a regular system and converts the Laurent series solution into a power series solution. Since, dq1 ∧ dp1 + dq2 ∧ dp2 = −2dθ ∧ dξ4 + dξ2 ∧ dξ3 , we may further introduce Q1 = θ, Q2 = ξ2 , P1 = −2ξ4 , P2 = ξ3 . Then the following mirror transformation: q1 = Q2 Q−1 1 ,

q2 = Q−2 1 ,

3 7 1 p1 = −iQ2 Q−2 1 − 8 iQ2 + 8 iQ2 + P2 Q1 ,

−1 2 1 1 1 p2 = −2iQ−3 1 + (− 4 i − 4 iQ2 )Q1 + ( 64 i +

2 7 32 iQ2

−

4 3 64 iQ2 )Q1

− 21 (P2 Q2 )Q21 − 21 (P1 )Q31 ,

is canonical. In particular, it converts the Hamiltonian system (8) into a Hamiltonian system with the polynomial Hamiltonian

 39Q42 83Q22 5Q62 iP2 Q32 Q 1 3iP 2 2 + iP1 − + − + − + Q1 H¯ (Q1 , Q2 , P1 , P2 ) = 256 256 256 256 4 4

 P22 7Q22 iP1 Q22 95Q42 21Q62 9Q82 1 iP1 + − + + − + − + − Q21 8192 2 8 2048 8 4096 2048 8192

 7iP2 Q32 3iP2 Q52 iP2 Q2 + − − + Q31 128 64 128

 P22 Q22 7iP1 Q22 3iP1 Q42 iP1 P1 P2 Q2 5 P12 6 + − + − + Q41 + Q1 + Q , 128 8 64 128 4 8 1 which is obtained by substituting the mirror transformation into the original Hamiltonian.

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3.2. A 3-freedom spherical symmetric Hamiltonian system Consider a 3-freedom spherical symmetric Hamiltonian H = 21 (p12 + p22 + p32 ) + (q12 + q22 + q32 )2 .

(12)

The Hamiltonian system passes the Painlevé test and has two principal balances. One balance is characterized by the leading behavior q1 ∼ ρ1 (t − t0 )−1 , p1 ∼ −ρ1 (t − t0 )−2 ,

q2 ∼ ρ2 (t − t0 )−1 ,

q3 ∼ ρ3 (t − t0 )−1 ,

p2 ∼ −ρ2 (t − t0 )−2 ,

p3 ∼ −ρ3 (t − t0 )−2

with the leading coefficients satisfying 2ρ12 + 2ρ22 + 2ρ32 + 1 = 0. The resonances are −1, 0, 0, 3, 3, 4, satisfying −1 + 4 = 1 + 2, 0 + 3 = 1 + 2. The special feature of this example is that the resonances appear in the leading coefficients. We will see that this makes the discussion of the resonance matrix and the construction of the mirror transformation somewhat more complicated. Still we see the symplectic structure in Painlevé analysis and that the mirror transformation is canonical. For the double resonance 3, the resonance parameters form a vector subspace of dimension 2. In case ρ1 = 0, the vectors (−ρ2 , ρ1 , 0, −2ρ2 , 2ρ1 , 0), (−ρ3 , 0, ρ1 , −2ρ3 , 0, 2ρ1 ) form a basis of this subspace, so that any vector can be written as a linear combination of the two vectors, with the resonance parameters r4 and r5 as coefficients. Under such an arrangement, the Laurent series solutions are of the form q1 ∼ ρ1 (t − t0 )−1 − (ρ2 r4 + ρ3 r5 )(t − t0 )2 + ρ1 r6 (t − t0 )3 + · · · , q2 ∼ ρ2 (t − t0 )−1 + ρ1 r4 (t − t0 )2 + ρ2 r6 (t − t0 )3 + · · · , q3 ∼ ρ3 (t − t0 )−1 + ρ1 r5 (t − t0 )2 + ρ3 r6 (t − t0 )3 + · · · , p1 ∼ −ρ1 (t − t0 )−2 − 2(ρ2 r4 + ρ3 r5 )(t − t0 ) + 3ρ1 r6 (t − t0 )2 + · · · , p2 ∼ −ρ2 (t − t0 )−2 + 2ρ1 r4 (t − t0 ) + 3ρ2 r6 (t − t0 )2 + · · · , p3 ∼ −ρ3 (t − t0 )−2 + 2ρ1 r5 (t − t0 ) + 3ρ3 r6 (t − t0 )2 + · · · , where 2ρ12 + 2ρ22 + 2ρ32 + 1 = 0, ρ1 = 0, and r4 , r5 , and r6 are arbitrary. For the double resonance 0, the resonance vectors are of the form (a, b, c, −a, −b, −c),

ρ1 a + ρ2 b + ρ3 c = 0,

where the condition is the tangent space of the surface 2ρ12 + 2ρ22 + 2ρ32 + 1 = 0. Under the condition ρ1 = 0, we may choose (ρ1 ρ3 , ρ2 ρ3 , ρ32 + 21 ) and (ρ1 ρ2 , ρ22 + 21 , ρ2 ρ3 ) as a basis of the tangent space. With this arrangement, the resonance matrix becomes   ρ1 ρ1 ρ2 ρ1 ρ3 −ρ3 −ρ2 ρ1    ρ ρ22 + 21 ρ2 ρ3 0 ρ1 ρ2   2      1 2 ρ2 ρ3 ρ3 + 2 ρ1 0 ρ3   ρ3  .    −2ρ1 −ρ1 ρ2 −ρ1 ρ3 −2ρ3 −2ρ2 3ρ1       −2ρ2 −ρ22 − 21 −ρ2 ρ3 0 2ρ1 3ρ2    1 2 −2ρ3 −ρ2 ρ3 −ρ3 − 2 2ρ1 0 3ρ3

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After rescaling and reversing the last three columns, we get a symplectic matrix   ρ1 ρ1 ρ2 ρ1 ρ3 − 25 ρ1 − 23 ρ1−1 ρ2 − 23 ρ1−1 ρ3   2   ρ ρ22 + 21 ρ2 ρ3 − 25 ρ2 0   2 3     1 2 2 2 ρ2 ρ3 ρ3 + 2 − 5 ρ3 0   ρ3 3 .     −2ρ1 −ρ1 ρ2 −ρ1 ρ3 − 65 ρ1 − 43 ρ1−1 ρ2 − 43 ρ1−1 ρ3      6 4   −2ρ2 −ρ22 − 1 −ρ ρ − ρ 0 2 3 2 5 2 3   4 −ρ2 ρ3 −ρ32 − 21 − 65 ρ3 0 −2ρ3 3 Assume ρ1 =  0. Then the upper left part   ρ1 ρ1 ρ2 ρ1 ρ3    ρ2 ρ 2 + 1 ρ2 ρ3  2 2   ρ3

ρ2 ρ3

ρ32 +

1 2

of the symplectic matrix has LU-decomposition. This suggests that we can obtain a canonical mirror transformation by introducing indicial normalization at q1 and then successively truncating q2 , q3 , p3 , p2 and p1 at the resonance parameters. Thus, we define θ by q1 = θ −1 , expand θ  , q2 , q3 , p1 , p2 and p3 in powers of θ , and then introduce new variables at the resonances of these expansions. The result is the following mirror transformation: q1 = Q−1 q2 = Q2 Q−1 q3 = Q3 Q−1 1 , 1 , 1 ,  √ 2 p1 = − 2i 1 + Q22 + Q23 Q−2 1 + (−P2 Q2 − P3 Q3 )Q1 − P1 Q1 , √  √  + P Q , p = − 2i 1 + Q22 + Q23 Q3 Q−2 p2 = − 2i 1 + Q22 + Q23 Q2 Q−2 2 1 3 1 1 + P 3 Q1 . The mirror transformation is easily verified to be canonical. Moreover, the Hamiltonian for (Q1 , Q2 , Q3 , P1 , P2 , P3 ) is given by substituting the mirror transformation into the original Hamiltonian (12): H¯ =

√  2i 1 + Q22 + Q23 P1 + 21 (P22 + P32 + P22 Q22 + P32 Q23 + 2P2 P3 Q2 Q3 )Q21 +(Q2 P2 + Q3 P3 )P1 Q31 + 21 P12 Q41 ,

which is regular near the singularity of the original equation (where 1 + Q22 + Q23 is close to ρ1−2 (ρ12 + ρ22 + ρ32 ) = − 21 ρ1−2 = 0). 3.3. A non-autonomous Hamiltonian system Everything we have done for autonomous Hamiltonian systems is applicable to non-autonomous ones. The only difference is the construction of the new Hamiltonian function. The non-autonomous Hamiltonian system given by the function (α is some constant) H (q1 , q2 , p1 , p2 ) = p1 q2 + p2 (2q13 + tq1 + α) has two principal balances. For the balance with q1 ∼ (t − t0 )−1 , the resonance matrix demonstrates again the

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symplectic structure observed before and we can obtain the mirror transformation q1 = Q−1 1 ,

2 1 1 q2 = −Q−2 1 − 2 t − 2 (1 + 2α)Q1 + Q2 Q1 ,

2 1 p1 = 2P2 Q−3 1 − 2 (1 + 2α)P2 + 2P2 Q2 Q1 − P1 Q1 ,

p2 = P2 Q−2 1

by successively truncating at p2 , q2 , and p1 (which means we have permuted q2 and p2 ). The detailed calculation is omitted here. It is easy to verify the canonicity of the mirror transformation and show that the mirror system is a Hamiltonian system given by a polynomial H¯ = P1 + 41 (1 + 2α)tP2 + 41 [(1 + 2α)2 − 4tQ2 ]P2 Q1 + 21 [tP1 − 3(1 + 2α)P2 Q2 ]Q21 + 21 [(1 + 2α)P1 + 4P2 Q22 ]Q31 − P1 Q2 Q41 . Compared with the function 2 1 1 H = p1 q2 + p2 (2q13 + tq1 + α) = − 21 P2 Q−2 1 + P1 + 4 (1 + 2α)tP2 + 4 ((1 + 2α) − 4tQ2 )P2 Q1

+ 21 [tP1 − 3(1 + 2α)P2 Q2 ]Q21 + 21 [(1 + 2α)P1 + 4P2 Q22 ]Q31 − P1 Q2 Q41 , obtained by applying the mirror transformation to the original Hamiltonian, we see that the new Hamiltonian is the “regular part” of the original Hamiltonian under the mirror transformation. In Section 4.4 of [6], we have proved that this is always the case.

4. Infinite-dimensional Hamiltonian systems 4.1. Burgers’ equation Burgers’ equation ut + uux + uxx = 0 is equivalent to the first two equations in the following space-evolution Hamiltonian system: δH δH = q2 , q2x = = −q1t − q1 q2 , δp1 δp2 δH δH p1x = − = p2 q2 − p2t , p2x = − = −p1 + p2 q1 , δq1 δq2 q1x =

given by the Hamiltonian functional  ∞ [p1 q2 − p2 q1 q2 − p2 (q1 )t ] dt. H = −∞

(13)

(14)

Here we recall that for any functional  +∞ h(x, u, ut , . . . ) dt H [u] = −∞

of functions u = (u1 , . . . , un ) in (x, t) and of their derivatives in t, the functional derivatives are computed by
 ∞ δH ∂h ∂k . = (−1)k k δui ∂t ∂(ui )t k k=0

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System (13) passes the Painlevé test and has a principal balance given by q1 ∼ 2φ −1 + ψ  + r3 φ − 41 ψ  φ 2 + · · · ,

q2 ∼ −2φ −2 + r3 − 21 ψ  φ + · · · ,

p1 ∼ 2r2 φ −1 + r2 ψ  + r2 r3 φ + (−r4 − 21 r2 ψ  − 41 r2 ψ  )φ 2 + · · · ,

p2 ∼ r2 − 21 r2 φ 2 + r4 φ 3 + · · · ,

where φ(x, t) = x − ψ(t), and ψ, r2 , r3 , and r4 are arbitrary functions of t. The resonances −1, 0, 2, 3 satisfy the resonance pairing property −1 + 3 = 1 + 1, 0 + 2 = 2 + 0. Moreover, the resonance matrix is   2 0 1 0  −4 0 1 0     .  2r2 (t) 2 r2 (t) −1  0

1

0

1

After rescaling and reversing the last two columns, we get a symplectic matrix   2 0 0 1  −4 0 0 1    .  1   2r2 (t) − 23 r (t) 2 6 − 13

0

− 16

0

By permuting the second and the fourth rows, the upper left corner of the symplectic matrix has LU-decomposition. The experience in finite-dimensional systems suggests us to introduce indicial normalization at q1 and successively truncate p2 , q2 , and q1 . Letting q1 = θ −1 , we find the following expansions: θx ∼

1 2

+ 2θt θ − r¯3 θ 2 + (4¯r3 θt − 8θtt )θ 3 + (2¯r3 − 48¯r3 θt2 + 96θt θtt )θ 4 + · · · ,

q2 ∼ − 21 θ −2 − 2θt θ −1 + r¯3 + (−4¯r3 θt + 8θtt )θ + (−2¯r3 + 48¯r3 θt2 − 96θt θtt )θ 2 + · · · , p1 ∼ r¯2 θ −1 + (− 21 r¯4 + 8¯r2 θt )θ 2 + (2¯r2 − 48¯r2 θt2 )θ 3 + · · · , p2 ∼ r¯2 − 2¯r2 θ 2 + r¯4 θ 3 + (−4¯r3 r¯2 − 2¯r2 − 6¯r4 θt + 48¯r2 θt2 )θ 4 + · · · , where r¯2 , r¯3 and r¯4 are arbitrary functions of t. By introducing the new variables successively at the resonances, we get the mirror transformation q1 = Q−1 1 ,

−1 q2 = − 21 Q−2 1 − 2Q1t Q1 + Q2 ,

p2 = P2 ,

2 p1 = P2 Q−1 1 + 2P2t Q1 − P1 Q1 .

The transformation converts system (13) into Q1x =

1 2

+ 2Q1t Q1 − Q2 Q21 ,

Q2x = −2Q2 Q1t + 4Q1tt − 2Q2t Q1 ,

P1x = −2P2t Q2 − 4P2tt + (2P1 Q2 + 2P1t )Q1 ,

P2x = −2P2t Q1 + P1 Q21 .

(15)

This is a Hamiltonian system. In fact, its Hamiltonian functional can be obtained as follows. Applying the mirror transformation to the original Hamiltonian functional (14), we have  ∞ H= [p1 q2 − p2 (q1 )t − p2 q1 q2 ] dt −∞  
 ∞ P1 −1 2 P2 Q1t Q−2 + Q − 4P − P Q + Q Q + 2P Q − P Q Q = (2P ) 2t 1 2t 1t 2t 2 1 1t 1 1 2 1 dt. 1 2 −∞

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121

Then it is easy to verify that the “regular part” of H :    ∞ 
P1 H¯ = − 4P2t Q1t + (2P2t Q2 + 2P1 Q1t )Q1 − P1 Q2 Q21 dt 2 −∞ is a Hamiltonian functional for (15). So far it appears that everything we have observed for finite-dimensional Hamiltonian systems remains true for infinite-dimensional ones. In fact, we have also computed the mirror transformation of the KdV equation, given by the Hamiltonian functional  +∞ H = [p1 q2 + p2 q3 − p3 q1 q2 − p3 (q1 )t ] dt, −∞

and observed the similar properties. However, if we compute the Poisson brackets by the usual formula   +∞ n
δA δB δA δB {A, B} = dt, − δqi δpi δpi δqi −∞ i=1

we found the mirror transform is not canonical. We suspect the problem lies in the fact that the mirror transformation is singular, and the usual transformation rule for the differential form is no longer valid. 4.2. The nonlinear Schrödinger equation The (defocusing) nonlinear Schrödinger equation iut + uxx − 2u2 v = 0,

−ivt + vxx − 2uv 2 = 0,

(16)

is equivalent to the following space-evolution Hamiltonian system q1x = q2 ,

q2x = 2q12 p2 − iq1t ,

p1x = −2q1 p22 − ip2t ,

p2x = −p1 ,

given by the Hamiltonian functional   +∞  i i 2 2 H = p1 q2 + p2 q1 − p2 q1t + p2t q1 dt. 2 2 −∞

(17)

(18)

The system has a principal balance 1 q1 ∼ r2 φ −1 + 21 ir2 ψ  + (− 12 r2 ψ 2 + 61 ir2 )φ + (−r22 r3 − 41 r2 ψ  )φ 2

+(r22 r4 −

1 2 −1 12 r 2 r2

− 16 ir2 ψ  ψ  +

1  3 12 r2 )φ

+ ··· ,

1 q2 ∼−r2 φ −2 + (− 12 r2 ψ 2 + 16 ir2 ) + (−2r22 r3 − 21 r2 ψ  )φ+(3r22 r4 − 41 r 22 r2−1 − 21 ir2 ψ  ψ  + 41 r2 )φ 2 + · · · , 1 2 −1 p1 ∼ r2−1 φ −2 + ( 12 ψ r2 − 16 ir2 r2−2 ) − 2r3 φ − 3r4 φ 2 + · · · , 1 2 −1 p2 ∼ r2−1 φ −1 − 21 iψ  r2−1 + (− 12 ψ r2 + 16 ir2 r2−2 )φ + r3 φ 2 + r4 φ 3 + · · · ,

where φ(x, t) = x − ψ(t), and ψ, r2 , r3 , and r4 are arbitrary functions of t with r2 (t) = 0. The resonances −1, 0, 3, 4 satisfy the resonance pairing property −1 + 4 = 1 + 2, 0 + 3 = 2 + 1. The resonance matrix is   r2 (t) 1 −r22 (t) r22 (t)    −2r2 (t) −1 −2r22 (t) 3r22 (t)  .   2r −1 (t) −r −2 (t) −2 −3    2 2 −1 −2 1 1 r2 (t) −r2 (t)

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After rescaling and reversing the last two columns, we get a symplectic matrix 

r2 (t)

1

  −2r2 (t) −1   −1  2r (t) −r −2 (t)  2 2 r2−1 (t)

1 − 10 r2 (t) 3 − 10 r2 (t) 3 −1 10 r2 (t)0

1 −1 −r2−2 (t) − 10 r2 (t)

 1 2 6 r2 (t)  1 2  3 r2 (t)  1 3

.  

− 16

By the usual process, we find the mirror transformation q1 = Q−1 1 ,

q2 = Q2 Q−2 1 ,

−1 2 p1 = −Q32 Q−2 1 + 2iQ2 Q1t Q1 − iQ2t − 2P2 Q2 Q1 − P1 Q1 ,

2 p2 = Q22 Q−1 1 − iQ1t + P2 Q1 .

(19)

The new variables (Q1 , Q2 , P1 , P2 ) satisfy a regular differential system Q1x = −Q2 ,

Q2x = −iQ1t + 2P2 Q21 ,

P1x = −iP2t − 2P22 Q1 ,

P2x = P1 .

(20)

We can easily see that system (20) is again a Hamiltonian system (in fact not much different from (17)), with the Hamiltonian functional given by   +∞  i i 2 2 ¯ H = −P1 Q2 + P2 Q1 − P2 Q1t + Q1 P2t dt. 2 2 −∞ This functional is again the “regular part” of the original Hamiltonian (18) under the mirror transformation (19)   +∞  Q21t −2 Q1tt −1 i i 2 2 H= − Q + Q1 − P1 Q2 + P2 Q1 + Q1 P2t − P2 Q1t dt 2 1 2 2 2 −∞   +∞  2 Q Q1tt −1 = − 1t Q−2 dt + H¯ . 1 + 2 Q1 2 −∞ The mirror transformation for the nonlinear Schrödinger equation is canonical, in contrast to the last example. We feel this is only an accident. By eliminating two “intermediate” variables Q2 and P1 in (20), we have iQ1t − Q1xx − 2Q21 P2 = 0,

−iP2t − P2xx − 2Q1 P22 = 0.

This is a focusing nonlinear Schrödinger equation. The transformation from the defocusing case to the focusing case is given by u = Q−1 1 ,

2 v = Q21x Q−1 1 − iQ1t + P2 Q1 ,

which has not been seen in the literature.

Acknowledgements The authors would like to thank Prof. G.W. Meng for many discussions. JH is supported by grant DAG99/00.SC21 of Hong Kong RGC.

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