Solutions to the non-autonomous ABS lattice equations: Casoratians and bilinearization

0.22 MB·English

by  Ying Shi

Solutions to the non-autonomous ABS lattice equations:
Casoratians and bilinearization
2 Ying Shi∗, Da-jun Zhang†, Song-lin Zhao
1
0 Department of Mathematics, Shanghai University, Shanghai 200444, P.R. China
2
July 8, 2012
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a
J

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3
Abstract

]
I In the paper non-autonomous H1, H2, H3δ and Q1δ equations in the ABS list are bilin-
S
earized. Their solutions are derived in Casoratian form. We also list out some Casoratian
.
n shift formulae which are used to verify Casoratiansolutions.
i
l
n
Key words: non-autonomous ABS list, Casoratian, bilinear, soliton solutions
[
PACS: 02.30.Ik, 05.45.Yv
1
v
8
1 Introduction
7
4
6 The “discrete integrable systems” has been a popular topic and is still drawing more and more
.
1 attention. Particularly in the recent ten years it received much progress. The property of mul-
0
tidimensional consistency [1–3] provides an approach to investigate integrability for the discrete
2
1 systems defined on an elementary quadrilateral:
:
v
Q(u ,u ,u ,u ;p,q) = 0, (1.1)
i n,m n+1,m n,m+1 n+1,m+1
X
r where p,q are spacing parameters. The ABS list [3] contains all quadrilateral lattice equations
a
of the above form which are consistent-around-the-cube(CAC). In the ABS list the spacing pa-
rameters p,q can either constants or functions p and q , which corresponds to autonomous
n m
case or non-autonomous case, respectively. In general an autonomous system means a differen-
tial/difference model with constant coefficients while a non-autonomous one means the model
has coefficients varying with independent variables but the model can not be transformed back
to an autonomous one. Obviously, the ABS lattice equations themselves are automatically non-
autonomous in the sense of taking p = p ,q = q , and this non-autonomous case still keeps the
n m
CAC property.
Integrable non-autonomous systems have its own importance. It is known that most of
discrete Painlev´e equations are non-autonomous ordinary difference equations. In addition, for
integrable non-autonomous forms of partial difference equations, their reductions usually lead
to integrable non-autonomous mappings, which quite often are discrete Painlev´e equations. To
get a non-autonomous version of a partial difference equation, taking (1.1) as an example, one
∗Corresponding author. E-mail: [email protected]
†E-mail: djzhang@staff.shu.edu.cn
1
can replace constant lattice parameters (p,q) by (p ,q ), but the integrability should be
n,m n,m
kept. Many criterions, such as singularity confinement, conservation laws and algebraic entropy,
have been used to check integrability for the non-autonomous systems, for both ordinary and
partial difference cases [4–7]. Besides, it is also possible to deautonomise a discrete bilinear
system if it contains spacing parameters. With suitable deautonomisation the obtained non-
autonomous bilinear systems admit N-soliton solutions expressed through deformed discrete
exponential functions, (see [8,9] as examples).
Recently, many solving approaches have been developed to find solutions for autonomous
lattice equations in the ABS list [10–18]. In [13] the H1, H2, H3 and Q1 equations in the
δ δ
ABS list were bilinearized and their solutions were derived in Casoratian form. In the present
paperwe will repeat the treatment of [13] to get bilinear forms as well as solutions in Casoratian
form for some non-autonomous ABS lattice equations. As we have mentioned before, the ABS
lattice equations with spacing parameters (p ,q ) are automatically non-autonomous and still
n m
CAC. Their integrable aspects are also double checked by singularity confinement and algebraic
entropy approaches [7].
The paper is organized as follows. Section 2 contains some basic notations for discrete
systems and Casoratians and a list of non-autonomous ABS lattice equations. In Section 3
the non-autonomous H1, H2, H3 and Q1 equations are bilinearized and their solutions are
δ δ
derived in Casoratian form. The Appendix contains a collection of Casoratian formulae of
non-autonomous case.
2 Preliminaries
Conventionally, we use tilde/hat notations to express the shifts in n/m directions, for example,
u= u , u= u , u= u , u= u , u= u , u= u .
n,m n+1,m n−1,m n,m+1 n,m−1 n+1,m+1
˜ ˆ
By these notations the lattice equation (1.1) is rewritten as b
e b e
Q(u,u,u,u;p,q) = 0. (2.1)
The non-autonomous ABS list is as followsb[3,7]:
e b e
H1 : (u−u)(u−u)+q −p = 0, (2.2a)
m n
H2 : (u−u)(u−u)+(q −p )(u+u+u+u)+q2 −p2 = 0, (2.2b)
b m n m n
e e b
H3 : p (uu+uu)−q (uu+uu)+δ(p2 −q2 ) = 0, (2.2c)
δ n b m n mb
e e b e b e
Q1 : p (u−u)(u−u)−q (u−u)(u−u)+δ2p q (p −q ) = 0, (2.2d)
δ n b m b n m n m
e be b ee
Q2 : p (u−u)(u−u)−q (u−u)(u−u)+p q (p −q )(u+u+u+u)
n m n m n m
b b
b e e e b e
−p q (p −q )(p2 −p q +q2 ) = 0, (2.2e)
n m n m n n m m
b b b
b e e e b e e b e
Q3 : sin(p +q )(uu+uu)−sinp (uu+uu)−sinq (uu+uu)
δ n m n m
+δ2sinp sinq sin(p +q )= 0, (2.2f)
n m n m
b b b
e eb e be b ee
Q4 : sn(p +q )(uu+uu)−snp (uu+uu)−snq (uu+uu)
n m n m
+snp snq sn(p +q )(1+k2uuuu)= 0, (2.2g)
n m n m
b b b
e eb e be b ee
b
ebe
2
where δ is a constant, p = p(n) and q = q(m) are arbitrary non-zero functions of discrete
n m
variables n and m, respectively.
Here the forms of Q3 and Q4 are in accordance with the autonomous version via the
δ
parametrization introduced by Hietarinta [19]. We omit A1 and A2 from the above list because
δ
of the equivalence between A1 and Q1 by u → (−1)n+mu, as well as A2 and Q3 by
δ δ δ=0
u→ u(−1)n+m.
The discrete version of Wronskian is Casoratian, which is a determinant of the Casorati
matrix:
f = |ψ(n,m,l ),ψ(n,m,l ),··· ,ψ(n,m,l )| = |l ,l ,··· ,l |, (2.3a)
1 2 N 1 2 N
where the basic column vector is
ψ(n,m,l) = (ψ (n,m,l),ψ (n,m,l),··· ,ψ (n,m,l))T, (2.3b)
1 2 N
and the shifts are in l direction. Using the standard short-hand notations [20], we list the
following often-used Nth-order Casoratians
\ \ ^
|N −1| = |0,1,··· ,N−1|, |N −2,N| = |0,1,··· ,N−2,N|, |−1,N −1| = |−1,1,2,··· ,N−1|.
As in [13], since in (2.3) there are three direction variables, say n,m and l, one can introduce
the operators Eν (ν = 1,2,3) by
E1ψ = ψ = ψ(n+1,m,l), E2ψ = ψ = ψ(n,m+1,l), E3ψ = ψ¯= ψ(n,m,l+1), (2.4)
then define a Casoratian w.r.t Eν-shift,
e b
|N\−1| =|ψ,Eνψ,(Eν)2ψ,··· ,(Eν)N−1ψ|, (ν = 1,2,3). (2.5)
[ν]
For these Casoratians we have
Proposition 1. The Casoratians
\ \ \
|N −1| = |N −1| = |N −1| , (2.6)
[1] [2] [3]
if their column vector ψ(n,m,l) satisfies the relations
α ψ =ψ−ψ, β ψ = ψ−ψ, (2.7)
n m
where α and β are arbitrary functions of discrete variables n and m, respectively, i.e. α =
n m e b n
α(n), β = β(m).
m
Proof. By the definition of Eν in (2.4)the relations (2.7) can be rewritten as
E3ψ = (E1+α )ψ, E3ψ = (E2 +β )ψ, (2.8)
n m
from which one has
k k−j j
(E3)k = (E1+α )k = (E1)k + α (E1)k−j, k = 1,2,··· ,N −1, (2.9a)
n n+li
Xj=1 lXj=0 Yi=1
lj≤lj+1
k k−j j
(E3)k = (E2 +β )k = (E2)k + β (E2)k−j, k = 1,2,··· ,N −1. (2.9b)
m m+li
Xj=1 lXj=0 Yi=1
lj≤lj+1
Then, substituting them into (2.5) one can easily obtain (2.6).
3
This Proposition will bring more flexibility for Casoratian verifications. Besides, for later
convenience, we give the following Laplace expansion property [20]:
Proposition 2. Suppose that B is a N ×(N −2) matrix, and a,b,c,d are Nth-order column
vectors, then
|B,a,b||B,c,d|−|B,a,c||B,b,d|+|B,a,d||B,b,c| = 0. (2.10)
3 Bilinearization and Casoratian solutions
In the following we derive bilinear forms and Casoratian solutions for the non-autonomous H1,
H2, H3 and Q1 models in the non-autonomous ABS list (2.2). Singularity confinement might
δ δ
provide a possible transformation to connect discrete integrable systems with their bilinear
forms, (see [21,22] as examples), but here we will roughly use the same transformations as for
the autonomous lattice equations [13]. It then turns out that these non-autonomous lattice
equations can share the same bilinear forms with those autonomous ones except changing the
spacing parameters accordingly.
To get Casoratian solutions one needs to use deformed discrete exponential functions and
develop corresponding Caosratian shift formulae, which we have listed in Appendix.
3.1 Non-autonomous H1 equation
We note that the non-autonomous H1 has been solved in [8] through bilinear approach and the
Casoratian solutions were given, but here we give a more generalized result.
With the parametrization
p = c−a2, q = c−b2 , (c is an arbitrary constant), (3.1)
n n m m
and through the transformation
n−1 m−1
g
n,m
u = − a − b −γ, (γ is an arbitrary constant), (3.2)
n,m i j
f
n,m
i=n j=m
X0 X0
the non-autonomous H1 (2.2a) is bilinearized by
H ≡ (gf −gf)+(a −b )(ff −ff)= 0, (3.3a)
1 n m
H2 ≡ (gbfe−gefb)+(an+bm)(fbfe−ffbe)= 0, (3.3b)
where in the transformation (3.2) bne0,mbe0 are arbitrary inbteegerbse. The connection between (2.2a)
and (3.3) is
− H +(a −b )ff H +(a +b )ff /(ffff)+(a2 −b2 )≡ H1,
1 n m 2 n m n m
which is the sa(cid:2)me relation as thebea(cid:3)(cid:2)utonomous one [1b3e](cid:3). bebe
Solutions to the bilinear equations (3.3) can be given by
Proposition 3. The Casoratians
\ \
f(ψ)= |N −1| , g(ψ) = |N −2,N| , (3.4)
[3] [3]
4
solve the non-autonomous bilinear equations (3.3), if the column vector ψ(n,m,l) symmetri-
cally∗, in terms of the pairs (n,a ) and (m,b ), satisfies the shift relations
n m
a ψ = ψ−ψ, (3.5a)
n−1
˜ ˜
ψ = A φ, b φ= φ+φ, (3.5b)
[m] m
where φ(n,m,l) is an auxiliary vector, the N ×N transform matrix A is invertible, and the
b b [m]
subscript [m] specially means A only depends on m but is independent of (n,l).
[m]
Proof. We prove H in its down-tilde-hat version H :
1
ˆ˜1
H ≡ (gf −gf)+(a −b )(ff −ff). (3.6)
n−1 m−1
ˆ˜1 ˜ˆ ˆ˜ ˜ˆ ˆ˜
\
Using the formulae given in appendix A with c = 0. In (3.6) f = |N −1| , f, f, g+a f, f
n−1
[3]
and g+b f are (A.8c), (A.6e), (A.9a), (A.6b) and (A.9e), respectively, weˆ˜hˆave˜ ˜ ˜
m−1
ˆ ˆ
H ≡ −(a −b )ff +f(g+a f)−f(g+b f)
ˆ˜1 = −a−nN−+12 b−mN−+12ˆ˜[|N\−ˆ3,˜ψ(Nn−−1˜2), ψ˜(Nˆ −1m)|−1ˆ·|N\−3, ψ(N −2), ψ(N −2)|
n−1 m−1 [3] [3]
\ \ ˆ ˜
−|N −3, ψ(N −2), ψ(N −2)| ·|N −3, ψ(N −1), ψ(N −2)|
[3] [3]
\ ˆ \ ˜
+|N −3, ψ(N −2), ψ(N −2)| ·|N −3, ψ(N −1), ψ(N −2)| ]
[3] [3]
˜ ˆ
= 0,
\
wherewe have made use of Proposition 2 in which B = (N −3), (a,b,c,d) = (ψ(N−2),ψ(N−
1),ψ(N −2),ψ(N −2)).
ˆNext, we p˜rove the down-tilde version of H , which is
2
H ≡ (fg−gf)+(a +b )(ff −ff). (3.7)
2 n−1 m
˜ ˜ ˜ ˜ ˜
\
In (3.7) we take f = |N −1|[3], andb forbf, f, g−bmf, f anbd g+ban−1f we use (A.8b), (A.6b),
(A.9b), (A.6g) and (A.9a), with c = 0, r˜esp˜ectively. Then we ˜have ˜
b b b b
H ≡ −(a +b )ff −f(g−b f)+f(g+a f)
2 n−1 m m n−1
˜ = −a−N+2 b−N+2˜|A ˜ A−1|·[|N\−˜3 ψ(N −˜2) ψ(N −1)|
n−1 m [m+1] [m] [3]
×|N\−3, ψ(Nb −2),bE◦2ψ(Nb −2b)|
[3]
\ ˜ \ ◦
−|N −3, ψ(N −2), ψ(N −2)| ·|N −3, ψ(N −1), E2ψ(N −2)|
[3] [3]
\ ˜◦ \
+|N −3, ψ(N −2), E2ψ(N −2)| ·|N −3, ψ(N −1), ψ(N −2)| ]
[3] [3]
˜
= 0,
\
by using Proposition 2 in which B = (N −3), (a,b,c,d) = (ψ(N − 2), ψ(N − 1), ψ(N −
◦ ˜
2), E2ψ(N −2)).
For the explicit forms of ψ together with the transformation matrices we can take either
(A.3) with c= 0 or (A.4) with c = 0.
∗ Here the symmetric property between pairs (n,an) and (m,bm) means, for example, once we have (3.5), at
thesame time we have
bm−1ψ=ψ−ψ,
ˆ ˆ
and
ψ=A[n]ω, anωe =ω+ωe.
5
3.2 Non-autonomous H2 equation
By the parametrization (3.1) with c = 0, i.e., p = −a2, q = −b2 , we first rewrite the
n n m m
non-autonomous H2 (2.2b) into
H2 ≡ (u−u)(u−u)+(a2 −b2 )(u+u+u+u−(a2 +b2 )) = 0. (3.8)
n m n m
Then, taking the transformation
b b
e e b e b e
g h+s
u = U2 −2U + , (3.9a)
n,m n,m n,mf f
with
n−1 m−1
U = a + b +γ, (γ is an arbitrary constant), (3.9b)
n,m i j
i=n j=m
X0 X0
and
s−h= αf, (α is some constant), (3.9c)
one can bilinearize (3.8) by
H ≡ (gf −gf)+(a −b )(ff −ff)= 0, (3.10a)
1 n m
H2 ≡ (gbfe−gefb)+(an+bm)(fbfe−ffbe)= 0, (3.10b)
H3 ≡ −(bean+bebm)fg+anfg+bbemfgb+e fh−fh= 0, (3.10c)
H ≡ −(a −b )fg+a fg−b fg+fh−fh= 0, (3.10d)
4 n m be nbe m be be eb
H ≡ b (fg−fg)+fh+fs−gg = 0, (3.10e)
5 m
be eb be eb be
where the connection is
b b 5b b b
H2 = H P /(ffff),
i i
i=1
X
b
and ebe
P = −4(a +b )[(UU −a2 +b2 )ff −Ufg−(a −b )fg],
1 n m n m n m
P2 = −4[(an −bm)(UeUbe −a2n+b2m)fefb+(beUbUe−a2n+b2m)fgbe−UUfg−(an −bm)Ufg],
P = 4[(a −b )Uff +Ufg−Ufg−fh+fh],
3 n m ebe eb ebe eb ebe be e be
P = 4[(a +b )(Uff −fg)+U(fg−fg)],
4 n m eb beb ebe eb be
P5 = 4(a2n −b2m)ffb, be be e be be
where U = Un,m is deefibened in (3.9b). This is as same as the autonomous case [13].
For solutions we have
Proposition 4. The Casoratians
\ \ \ \
f = |N −1| , g = |N −2, N| , h= |N −3, N −1, N| , s = |N −2, N +1| , (3.12)
[3] [3] [3] [3]
solve the non-autonomous bilinear equations (3.10), where the basic column vector ψ satisfies
the same conditions in Proposition 3.
6
Proof. We skip the proof for H and H as they are just the bilinear H1. We prove H and
1 2 5
shifted H and H in the following forms
3 4
H ≡ −(a +b )fg+a fg+b fg+fh−fh, (3.13a)
3 n−1 m n−1 m
˜ ˜ ˜ ˜ ˜ ˜
H ≡ −(a −b )fg+f(a g+h)−f(b g+h). (3.13b)
4 n−1 m−1 n−1 m−1
ˆ˜ bˆ˜ ˆ b ˜ ˜b ˜b bˆ ˆ
We still use the formulae given in appendix A with c = 0.
\
For (3.13a), g = |N −2, N| , andf, f,b g−h, f anda g+hare(A.8b), (A.6b), (A.9d),
m n−1
(A.6g) and (A.9c) respectively.[3T] hen w˜e˜have ˜ ˜
b b b b
H ≡ −(a +b )gf +f(b g−h)+f(a g+h)
3 n−1 m m n−1
˜ = −a−N+2 b−N+2˜|A ˜ A−1|·[|N\−3,ψ(˜N −˜2),ψ(N)|
n−1 m [m+1] [m] [3]
×|N\−3,ψ(Nb−2),E◦2bψ(Nb−2)b|
[3]
\ ˜ \ ◦
−|N −3, ψ(N −2), ψ(N −2)| ·|N −3, ψ(N), E2ψ(N −2)|
[3] [3]
\ ˜◦ \
+|N −3, ψ(N −2), E2ψ(N −2)| ·|N −3, ψ(N), ψ(N −2)| ]
[3] [3]
˜
= 0,
\
where we have made use of Proposition 2 in which B = (N −3), (a, b, c, d) = (ψ(N −
◦
2), ψ(N), ψ(N −2), E2ψ(N −2)).
˜ \
For (3.13b), g = |N −2, N| , and f, f, a g +h, f and b g +h are (A.6a), (A.6e),
n−1 m−1
(A.9c), (A.6b) and (A.9f) respect[3i]vely. Tˆ˜henˆ we ha˜ve ˜ ˜ ˆ ˆ
H ≡ −(a −b )fg+f(a g+h)−f(b g+h)
4 n−1 m−1 n−1 m−1
˜ = −a−N+2 b−N+2[ˆ˜|N\−ˆ3, ψ(N˜−2˜), ψ˜(N)| ·ˆ|N\ˆ−3, ψ(N −2), ψ(N −2)|
n−1 m−1 [3] [3]
\ \ ˆ ˜
−|N −3, ψ(N −2), ψ(N −2)| ·|N −3, ψ(N), ψ(N −2)|
[3] [3]
\ ˆ \ ˜
+|N −3, ψ(N −2), ψ(N −2)| ·|N −3, ψ(N), ψ(N −2)| ]
[3] [3]
˜ ˆ
= 0,
\
where we have made use of Proposition 2 in which B = (N −3), (a, b, c, d) = (ψ(N −
2), ψ(N), ψ(N −2), ψ(N −2)).
ˆ ˜ \ \ \
For (3.10e), we have f = |N −1| , g = |N −2,N| , s = |N −3,N −1,N| , and h−b g,
m
[3] [3] [3]
g−b f and f are provided by (A.9d), (A.9b) and (A.6g). Now we obtain
m
b b
b H5 ≡b f(hb−bmg)−g(g−bmf)+fs
= b−N+2|A A−1|·[|N\−3,ψ(N −2),ψ(N −1)| ·|N\−3,ψ(N),E◦2ψ(N −2)|
m [m+1] [m] [3] [3]
−|bN\−3,bψ(N −b2), ψ(bN)|b·|N\−3, ψ(N −1), E◦2ψ(N −2)|
[3] [3]
\ ◦ \
+|N −3, ψ(N −2), E2ψ(N −2)| ·|N −3, ψ(N −1), ψ(N)| ]
[3] [3]
= 0,
\
where we have made use of Proposition 2 in which B = (N −3), (a, b, c, d) = (ψ(N −
◦
2), ψ(N −1), ψ(N), E2ψ(N −2)).
For the explicit forms of ψ together with the transformation matrices we can take either
(A.3) with c= 0 or (A.4) with c = 0.
7
3.3 Non-autonomous H3 equation
With the parametrization
1+α2 1+β2 a −c b −c
p = n, q = m, α2 = − n , β2 = − m ,
n 2α m 2β n a +c m b +c
n m n m
the non-autonomous H3 equation (2.2c) admits two different sets of bilinear forms. One is
B ≡ 2cff +(a −c)ff −(a +c)ff = 0, (3.14a)
1 n n
¯ ¯
B ≡ 2cff +(b −c)ff −(b +c)ff = 0, (3.14b)
2 e m e m e
¯ ¯
and the other is b b b
B′ ≡ (b +c)ff +(a −c)ff −(a +b )ff = 0, (3.15a)
1 m n n m
B2′ ≡ (an+c)ffbe+(bm−c)fbef −(an +bm)fefb = 0, (3.15b)
¯ ¯ ¯
B3′ ≡ (an−c)(bbem +c)ff −(bebm−c)(an +c)efbf −2c(an −bm)ff = 0. (3.15c)
¯ ¯
Both of them share same transfobremation eb be
un,m = A Vn,mffn,m +B Vn−,m1¯ffn,m, AB = −41δ, (3.16a)
n,m n,m
where
n−1 m−1
V = α β . (3.16b)
n,m i j
iY=n0 j=Ym0
The connections are respectively
−δ2B−2V2 (a −c)(b −c)P +4δP +16B2V−2(a +c)(b +c)P
n,m n m 1 2 n,m n m 3
H3 = ,
32(a2 −c2)(b2 −c2)ffff
n m
with b
ebe
P = f[(b −c)fB −(a −c)fB ]−f[(b +c)fB −(a +c)fB ],
1 m 1 n 2 m 1 n 2
P2 = 2bec[(bm +c)b(bm −c)(ffB1e+ffB1)−(an+ecc)(an −c)(ffBb2f+ffB2)],
P3 = f[(bm+c)fB1−(anb+bec)fB2]−ecf[(bm−c)fB1−(an −ecbe)fB2], bf
¯ ¯ ¯ ¯ ¯ ¯
and be b e ec bf
c ffB′ −ffB′ ffB′ −ffB′
H3 ≡ [A2V2 1 2 +B2V−2 ¯ 1 ¯ 2
n,m(a +c)(b +c) n,m(a −c)(b −c)
ffff n m n m
eb be be eb
ffB′ +ffB′ ffB′ +ffB′ 2(a +b )ffB′
+eAbbBe( ¯ 2 2 − ¯ 1 1 + n m 3 )],
(a +c)(b −c) (a −c)(b +c) (a2 −c2)(b2 −c2)
n m n m n m
be eb eb be eb
which are similar to the one in the autonomous case [13]. For solutions we have
8
Proposition 5. The Casoratians
\
f = |N −1| , ν = 1, 2, 3, (3.18)
[ν]
solve non-autonomous bilinear equations (3.14) and (3.15), if the basic column vector ψ(n,m,l)
is symmetric in terms of pairs (n,a ) and (m,b ), and together with auxiliary vectors ω(n,m,l)
n m
and ζ(n,m,l), satisfies the following shift relations
(c−a )ψ = ψ−ψ, (3.19a)
n
¯ ¯
ψ = A ω, (a +c)ω = ω+ω, (3.19b)
[n] n
e
ψ = B ζ, (c+b )ζ = ζ +ζ, (3.19c)
[l] m e e
where A and B are N×N transform matrices, and the mbatrix product B B−1 is indepen-
[n] [l] [l] [l+1]
dent of l.
Proof. We prove (3.14a) in its down-tilde version
B ≡ 2cff +(a −c)ff −(a +c)ff. (3.20)
1 n−1 n−1
˜ ˜ ¯˜ ˜¯
In (3.20), for f, f, f, f, f and f, we make use of (A.7b), (A.8i), (A.7h), (A.8h), (A.8l) and
˜ ¯˜ ˜ ¯
(A.7c) respectively, and get
N−3 N−3 N−3
B ≡ (a −b )−1 (c−b )−1 (c+b )−1|B B−1|
˜1 n−1 m+j m+j m+j [l+1] [l]
j=0 j=0 j=0
Y Y Y
×[−|N\−3, ψ(N −2), ψ(N −2)| ·|N\−3, ψ(N −2), E◦3ψ(N −2)|
˜ [2] ¯ [2]
+|N\−3, ψ(N −2), ψ(N −2)| ·|N\−3, ψ(N −2), E◦3ψ(N −2)|
¯ [2] ˜ [2]
−|N\−3, ψ(N −2), E◦3ψ(N −2)| ·|N\−3, ψ(N −2), ψ(N −2)| ]
[2] ˜ ¯ [2]
=0,
\
with the help of Proposition 2 in which B = (N −3), (a,b,c,d) = (ψ(N −2),ψ(N −2),ψ(N −
2),E◦3ψ(N −2)). Here to get the coefficient |B B−1| we request B B−1 to˜be indep¯endent
[l+1] [l] [l] [l+1]
of l, i.e. B B−1 = B B−1.
[l] [l−1] [l+1] [l]
B holds thanks to B and the n-m symmetric property of ψ(n,m,l).
2 1
We prove B′ in its down-tilde shifted version, i.e.
1
B′ ≡ (b +c)ff +(a −c)ff −(a +b )ff. (3.21)
1 m n n−1 m
˜ ˜ ˜ ˜
In (3.21), we take f = |N\−1| , and bf, f, f, f abnd f as (A.8e), (bA.6c), (A.6h), (A.6a) and
[3]
(A.8f), respectively. Then we have ˜ ˜ ˜
b b b
B′ ≡(a −c)−N+2(b +c)−N+2|A A−1|
˜1 n−1 m [m+1] [m]
×[−|ψ(0), N^−2, ψ(N −1)| ·|N^−2, ψ(N −1), E◦2ψ(N −1)|
[3] [3]
˜
+|ψ(0), N^−2, ψ(N −1)| ·|N^−2, ψ(N −1), E◦2ψ(N −1)|
[3] [3]
˜
9
−|ψ(0), N^−2, E◦2ψ(N −1)| ·|N^−2, ψ(N −1), ψ(N −1)| ]
[3] [3]
˜
=0,
^
where we have made use of Proposition 2, in which B = (N −2), (a,b,c,d) = (ψ(0),ψ(N −
◦
1),ψ(N −1),E2ψ(N −1)).
˜B′ holds thanks to B′ and the n-m symmetric property of ψ(n,m,l).
2 1
By a down-hat shift, B′ reads
3
B′ ≡(a −c)(b +c)ff −(a +c)(b −c)ff −2c(a −b )ff. (3.23)
3 n m−1 n m−1 n m−1
ˆ ¯ˆ ¯ˆ ˆ
e e e
In (3.23), for f, f, f, f, f and f we use (A.7d), (A.8n), (A.7e), (A.8k), (A.7j) and (A.8j)
¯ ˆ ¯ˆ ˆ
respectively. Then we have
e e e
N−2 N−2 N−2
B′ ≡ (b −a )−1 (c−a )−1 (c+a )−1|B B−1|
ˆ3 m−1 n+i n+i n+i [l+1] [l]
i=1 i=1 i=1
Y Y Y
×[|N^−2, ψ(0), ψ(N −1)| ·|N^−2, ψ(N −2), E◦3ψ(N −1)|
¯ [1] ˆ [1]
−|N^−2, ψ(0), ψ(N −1)| ·|N^−2, ψ(N −1), E◦3ψ(N −1)|
ˆ [1] ¯ [1]
+|N^−2, ψ(0), E◦3ψ(N −1)| ·|N^−2, ψ(N −1), ψ(N −1)| ]
[1] ¯ ˆ [1]
=0,
^
where we have made use of Proposition 2 in which B = (N −2), (a,b,c,d) = (ψ(0),ψ(N −
◦ ¯
1),ψ(N −1),E3ψ(N −1)).
ˆ
For the explicit forms of ψ together with the transformation matrices we can take either
(A.3) or (A.4).
3.4 Non-autonomous Q1 equation
The non-autonomous Q1 equation can have two different bilinearizations which are also similar
to their autonomous cases [13]. First, using the parametrization
rc2 rc2
p = , q = , (3.25a)
n a2 −c2 m b2 −c2
n m
where c and r are constants, and through the transformation
f fn,m r2δ2
un,m = AVn,mfn,m +BVn−,m1¯¯f , AB = 16 , (3.25b)
n,m n,m
where V is (3.16b) and α = an−c, β = bm−c, the non-autonomous Q1 equation (2.2d) can
n,m n an+c m bm+c
be transformed to (3.14), i.e., one of bilinear form for the non-autonomous H3 equation. So its
solutions consequently follow the Proposition 5. In this case, the connection is
r2
Q1≡ (α β U2 A2P + α−1β−1(a +c)−2(b +c)−2δ2P +α−1β−1U−2 B2P )/ffff,
n m n,m 1 16 n m n m 2 n m n,m 1
b
ebe
10