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A note on the quasi-periodic solutions of the modified Boussinesq hierarchy
Accepted Manuscript A note on the quasi-periodic solutions of the modified Boussinesq hierarchy Lihua Wu, Guoliang He, Xianguo Geng PII: DOI: Reference:
S0393-0440(15)00142-4 http://dx.doi.org/10.1016/j.geomphys.2015.06.014 GEOPHY 2572
To appear in:
Journal of Geometry and Physics
Received date: 6 November 2014 Accepted date: 16 June 2015 Please cite this article as: L. Wu, G. He, X. Geng, A note on the quasi-periodic solutions of the modified Boussinesq hierarchy, Journal of Geometry and Physics (2015), http://dx.doi.org/10.1016/j.geomphys.2015.06.014 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
A Note on the Quasi-Periodic Solutions of the Modified Boussinesq Hierarchy Lihua Wu1∗, Guoliang He2 , Xianguo Geng3 1
Department of Mathematics, Huaqiao University, Quanzhou 362021, China
2
Department of Mathematics and Information Science, Zhengzhou University of Light Industry, Zhengzhou 450002, China
3
Department of Mathematics, Zhengzhou University, Zhengzhou 450001, China E-mail address: [email protected], [email protected], [email protected]
Abstract Based on the theory of trigonal curve and the properties of three kinds of the Abel differentials on it, we deduce the explicit theta function representations of the Baker-Akhiezer function and the meromorphic function associated with the modified Boussinesq hierarchy. The modified Boussinesq flows are straightened using the Abel map and the Lagrange interpolation formula. The explicit theta function representations of solutions for the entire modified Boussinesq hierarchy are constructed with the aid of the asymptotic properties and the algebrogeometric characters of the meromorphic function. Keywords: the modified Boussinesq hierarchy, explicit quasi-periodic solutions MR(2000) Subject Classification: 35Q51, 37K10, 14H70, 35C99
1
Introduction
In the previous paper [1], based on a trigonal curve related to the characteristic polynomial of Lax matrix associated with a 3 × 3 matrix spectral problem, we discussed algebro-geometric constructions of the modified Boussinesq hierarchy proposed by Fordy and Gibbons [2] and its quasi-periodic solutions. However, the theta function representations of solutions of every equation in the modified Boussinesq hierarchy are related to a Riccati equation, which is nonlinear. The principal aim of the present paper is to improve the main result in [1], by which explicit quasi-periodic solutions of the entire modified Boussinesq hierarchy are deduced through introducing another meromorphic function and using its asymptotic properties and the algebrogeometric characters. During the last four decades, finding quasi-periodic solutions of soliton equations aroused interest of many mathematicians and physicists. Quasi-periodic solutions for a lot of soliton ∗
Corresponding author
1
equations associated with 2 × 2 matrix spectral problems have been obtained such as the KdV, nonlinear Schr¨odinger, sine-Gordon, and Toda lattice, Camassa-Holm equations, and others [3-14]. As compared with the 2 × 2 case, the study of quasi-periodic solutions of soliton equations associated with 3 × 3 matrix spectral problems is very few. The primary cause of this discrepancy is enormous complexity for the theory of non-hyperelliptic curves or trigonal ones [15-19] associated with 3 × 3 matrix spectral problems. In Refs. [4,15,16,20-26], certain algebrogeometric solutions of the Boussinesq equation related to a third-order differential operator were found as special solutions of the Kadomtsev-Peiviashvili equation or by the reduction theory of Riemann theta functions. In Refs. [27,28], a unified framework was proposed which yields all algebro-geometric solutions of the entire Boussinesq hierarchy. The present paper is organized as follows. In section 2, we summarize the main results about the algebro-geometric constructions of the modified Boussinesq hierarchy in Ref. [1]. In section 3, we introduce the Baker-Akhiezer function and another associated meromorphic function from which the modified Boussinesq equations are decomposed into the system of Dubrovin-type ordinary differential equations. In terms of the properties of three kinds of the Abel differentials on Km−1 , the explicit theta function representations of the Baker-Akhiezer function and the meromorphic functions are presented. Using the Abel map and the Lagrange interpolation formula, the corresponding flows are straightened. The explicit quasi-periodic solutions for the entire modified Boussinesq hierarchy are constructed according to the asymptotic properties and the algebro-geometric characters of the meromorphic functions.
2
The previous main results
In this section, we simply recall the primary results in Ref. [1]. We first reconstruct the modified Boussinesq hierarchy associated with the 3 × 3 matrix spectral problem ψ1 u+v λ 0 ψx = U ψ, ψ = ψ2 , U = 0 (2.1) −2v 1 , 1
ψ3
0
by introducing two sets of Lenard recursion equations
with two starting points
Kgj−1 = Jgj ,
gj |(u,v)=0 = 0,
j ≥ 0,
K gˆj−1 = J gˆj ,
gˆj |(u,v)=0 = 0,
j≥0
g−1 = (2v, 1, 1, 1)T , and
gˆ−1 = (1, 0, 0, 0)T ,
+ ∂u 0 0 0 + ∂v K22 0 0 , −3∂ K32 K33 0 0 K42 ∂ − 2u K44
1 2 ∂ 21 2 2∂
K=
v−u
2
1 1 −1 0 2 2 1 1 0 0 − 2 2 , J = K33 0 −3∂ K32 0 K42 ∂ − 2u K44
(2.2) (2.3)
(2.4)
1 K22 = − (∂ 2 + 3∂v − ∂u)(∂ + u + 3v), K32 = 2(∂ − u)(∂ + u + 3v), 3 K33 = (∂ − u − 3v)(∂ − 2u), K42 = ∂ + u + 3v, K44 = ∂ + u − 3v. Let ψ satisfy the spectral problem (2.1) and an auxiliary problem ψtn = V (n) ψ, (n)
where each entry Vij
(n)
(2.5)
V (n) = (Vij )3×3 ,
= Vij (a(n) , b(n) , c(n) , d(n) ) is a Laurent expansion in λ:
(n)
V11 = (∂ + u + v)a(n) − 13 (∂ + 3v − u)(∂ + u + 3v)b(n) , (n)
V12 = a(n) − (∂ + u + 3v)b(n) − (∂ − 2u)c(n) , (n)
V21 = b(n) , (n)
(n)
(n)
V13 = c(n) ,
V22 = −(∂ + 2v)a(n) + 23 (∂ + 3v − u)(∂ + u + 3v)b(n) , (n)
V23 = a(n) − (∂ + u + 3v)b(n) ,
V31 = a(n) ,
(n)
(n)
V32 = d(n) ,
V33 = (v − u)a(n) − 13 (∂ + 3v − u)(∂ + u + 3v)b(n) , a(n) =
n P
aj−1 λn−j ,
b(n) =
c(n) =
bj−1 λn−j ,
j=0
j=0
with Gj = (aj , bj , cj , dj
n P
)T
n P
cj−1 λn−j ,
d(n) =
j=0
n P
dj−1 λn−j
j=0
(2.6)
determined by
Gj = α0 gj + β0 gˆj + · · · + αj g0 + βj gˆ0 + αj+1 g−1 + βj+1 gˆ−1 ,
j ≥ −1,
(2.7)
where αj and βj are arbitrary constants. Then the compatibility condition of (2.1) and (2.5) (n) yields the zero-curvature equation, Utn −Vx +[U, V (n) ] = 0, which is equivalent to the hierarchy of nonlinear evolution equations (utn , vtn )T = Xn ,
n ≥ 0,
(2.8)
where the vector fields Xj = L(aj−1 , bj−1 )T with à ! 1 2 ∂ + ∂u 0 2 L= 1 2 . 2 ∂ + ∂v K22 The first nontrivial member in the hierarchy (2.8) is ut0 = α0 (vxx + 2(uv)x ) + β0 ux , vt0 = α0 (− 13 uxx + 23 uux − 2vvx ) + β0 vx .
(2.9)
For α0 = 1, β0 = 0, (t0 = t), equation (2.9) is reduced to the modified Boussinesq equation ut = vxx + 2(uv)x , vt = − 13 uxx + 32 uux − 2vvx .
(2.10)
Then we defined a trigonal curve associated with the modified Boussinesq hierarchy or its compactification engendering the three-sheeted Riemann surface of arithmetic genus m − 1 by Km−1 :
Fm (λ, y) = y 3 + ySm (λ) − Tm (λ) = 0, 3
(2.11)
where m = 3n + 2 or 3n + 1 as α0 = 1, β0 an arbitrary constant or α0 = 0, β0 = 1. Moreover, We introduced the Baker-Akhiezer function ψ2 (P, x, x0 , tr , t0,r ) and the corresponding meromorphic function φ2 (P, x, tr ) on Km−1 by ψx (P, x, x0 , tr , t0,r ) = U (u(x, tr ), v(x, tr ); λ(P ))ψ(P, x, x0 , tr , t0,r ), ψtr (P, x, x0 , tr , t0,r ) = Ve (r) (u(x, tr ), v(x, tr ); λ(P ))ψ(P, x, x0 , tr , t0,r ), V (n) (u(x, tr ), v(x, tr ); λ(P ))ψ(P, x, x0 , tr , t0,r ) = y(P )ψ(P, x, x0 , tr , t0,r ), ψ2 (P, x0 , x0 , t0,r , t0,r ) = 1, P = (λ, y) ∈ Km−1 \ {P∞ }, x, x0 , tr , t0,r ∈ C,
(2.12)
and φ2 (P, x, tr ) =
∂ψ2 (P, x, x0 , tr , t0,r ) + 2v(x, tr ), ψ2 (P, x, x0 , tr , t0,r )
P ∈ Km−1 , x, x0 , tr , t0,r ∈ C.
(2.13)
Through a direct calculation, we had φ2 = where
(n)
(n)
yV31 + Cm y 2 V21 − yAm + Bm ε(m)λFm−1 , = 2 (n) = (n) −ε(m)Em−1 yV21 + Am y V31 − yCm + Dm (n)
(n)
(n)
(n)
Am = V23 V31 − V33 V21 , (n)
(n)
(n)
(2.14)
(n)
(n)
(n)
(n)
(n)
(n)
(n)
Bm = V22 (V11 V21 + V23 V31 ) − λV21 (V12 V21 + V23 V32 ), (n)
(n)
(n)
(n)
Cm = λV21 V32 − V22 V31 , (n)
(n)
(n)
(n)
(n)
(n)
(n)
(n)
(n)
(n)
Dm = V31 (V11 V33 − λV13 V31 ) + λV32 (V21 V33 − V23 V31 ), (n)
(n)
(n)
(n)
(n)
(n)
(n)
(n)
(n)
Em−1 = −ε(m)[V23 (V21 V33 − V11 V21 − V23 V31 ) + λV13 (V21 )2 )], (n)
(n)
(n)
(n)
(n)
(n)
(n)
(n)
(n)
Fm−1 = ε(m)[V31 (V22 V32 − V11 V32 + V12 V31 ) − λV21 (V32 )2 ].
(2.15)
By observation, we knew Em−1 and Fm−1 are monic polynomials with respect to λ of degree m − 1. Hence we may write them in the following form Em−1 (λ, x, tr ) =
m−1 Y j=1
(λ − µj (x, tr )),
Fm−1 (λ, x, tr ) =
m−1 Y j=1
(λ − νj (x, tr )).
(2.16)
After discussing the singularies of the meromorphic function φ2 (P, x, tr ) as well as the BakerAkhiezer function ψ2 (P, x, x0 , tr , t0,r ) and with the help of the theory of trigonal curve, we obtained the Riemann theta function representations of them as ³ θ(λ(P, νˆ(x, tr )))θ(λ(P∞ , µ ˆ (x, tr ))) φ2 (P, x, tr ) = exp e(3) (Q0 ) − θ(λ(P∞ , νˆ(x, tr )))θ(λ(P, µ ˆ (x, tr )))
Z
P
Q0
θ(λ(P, µ ˆ (x, tr )))θ(λ(P∞ , µ ˆ (x0 , t0,r ))) ψ2 (P, x, x0 , tr , t0,r ) = θ(λ(P∞ , µ ˆ (x, tr )))θ(λ(P, µ ˆ (x0 , t0,r ))) Z P Z ³ (2) (2) (2) × exp (x − x0 )(e2 (Q0 ) − ωP∞ ,2 ) + (tr − t0,r )(es+1 (Q0 ) − Q0
P
Q0
´ (3) ωP∞ ,P0 , ´ ˜ (2) Ω ) P∞ ,s+1 .
(2.17)
(2.18)
Combining the asymptotic properties and Riemann theta function representations of the meromorphic function φ2 (P, x, tr ) and the Baker-Akhiezer function ψ2 (P, x, x0 , tr , t0,r ), we got the 4
following quasi-periodic solutions of modified Boussinesq hierarchy ux (x, tr ) − u2 (x, tr ) = 32 ∂x2 ln θ(λ(P∞ , νˆ(x, tr )))θ(λ(P∞ , µ ˆ(x, tr ))) + 43 [∂x ln v(x, tr ) = − 12 ∂x ln
θ(λ(P∞ , νˆ(x, tr ))) 1 + w . θ(λ(P∞ , µ ˆ(x, tr ))) 2 0
θ(λ(P∞ , νˆ(x, tr ))) − w0 ]2 − 3q1 , θ(λ(P∞ , µ ˆ(x, tr )))
(2.19) Unsatisfactorily, the expression of u(x, tr ) in (2.19) is given by a Riccati equation and it is implicit. Our purpose of this note is to arrive at explicit Riemann theta representations of u(x, tr ), v(x, tr ) by introducing another meromorphic function on Km−1 together with the above results.
3
Explicit quasi-periodic solutions of the modified Boussinesq hierarchy
In this section, we shall introduce another meromorphic function closely related to the BakerAkhiezer function. Then we obtain Riemann theta function representations for the BakerAkhiezer function, the meromorphic function and explicit quasi-periodic solutions of the modified Boussinesq hierarchy. Throughout this paper we use the same notations as [1], that is, the same notations express the same quantities in the two papers. Now, we introduce another Baker-Akhiezer function ψ3 (P, x, x0 , tr , t0,r ) on Km−1 by (2.12) and ψ3 (P, x0 , x0 , t0,r , t0,r ) = 1. Then, another meromorphic function φ3 (P, x, tr ) closely related to ψ3 (P, x, x0 , tr , t0,r ) is defined by φ3 (P, x, tr ) =
∂x ψ3 (P, x, x0 , tr , t0,r ) + u(x, tr ) − v(x, tr ), ψ3 (P, x, x0 , tr , t0,r )
P ∈ Km−1 , x, x0 , tr , t0,r ∈ C,
(3.1)
which implies from (2.12) that φ3 (P, x, tr ) = = = where
(n)
ε(m)λHm−1 (λ, x, tr ) − yCm (λ, x, tr ) + Em (λ, x, tr ) − yAm (λ, x, tr ) + Bm (λ, x, tr ) −ε(m)Fm−1 (λ, x, tr ) (n) yV12 (λ, x, tr ) + Cm (λ, x, tr ) , (n) yV32 (λ, x, tr ) + Am (λ, x, tr ) (n) y V12 (λ, x, tr ) (n) y 2 V32 (λ, x, tr ) 2
(n)
(n)
(n)
Am = V12 V31 − V11 V32 , (n)
(n)
(n)
(n)
(n)
(n)
(n)
(n)
(n)
(n)
Bm = V33 (V22 V32 + V12 V31 ) − λV32 (V23 V32 + V13 V31 ), (n)
(n)
(n)
(n)
Cm = λV13 V32 − V12 V33 , (n)
(n)
(n)
(3.2)
(n)
(n)
(n)
(n)
(n)
(n)
(n)
Em = V12 (V11 V22 − λV12 V21 ) + λV13 (V11 V32 − V12 V31 ),
5
(3.3)
(n)
(n)
(n)
(n)
(n)
(n)
(n)
(n)
Fm−1 = −ε(m)[λV21 (V32 )2 + V31 V32 (V11 − V22 ) − V12 (V31 )2 ], (n)
(n)
(n)
(n)
(n)
(n)
(n)
(n)
Hm−1 =(ε(m)[(V12 )2 V23 + V12 V13 (V33 − V22 ) − λ(V13 )2 V32 ], −1, if m = 3n + 2, ε(m) = 1, if m = 3n + 1.
(3.4)
We can easily show that there exist various interrelationships among the polynomials Am , Bm , Cm , Em , Fm−1 , Hm−1 , Sm , Tm , some of which are listed below: (n)
(n)
(n)
2, ε(m)λV32 Hm−1 = V12 Em − (V12 )2 Sm − Cm (n)
ε(m)λAm Hm−1 = (V12 )2 Tm + Cm Em , (n)
(n)
(n)
ε(m)V12 Fm−1 = (V32 )2 Sm − V32 Bm + A2m , (n)
−ε(m)Cm Fm−1 = (V32 )2 Tm + Am Bm ,
(n)
(n)
(n)
(n)
(n)
(n)
V12 V32 Tm + V12 Am Sm + V32 Cm Sm − Bm Cm − Am Em = 0, (n) V12 Am Tm
+
(n) V32 Cm Tm
(3.6)
(n)
V12 Bm + V32 Em − V12 V32 Sm + Am Cm = 0, (n)
(3.5)
(3.7)
− λFm−1 Hm−1 − Bm Em = 0,
(n)
ε(m)Fm−1,x = 2V32 Sm − 3Bm − 3ε(m)(u − v)Fm−1 , (n)
(n)
(n)
(n)
(n)
V12 Hm−1,x = 3[(u + v)V12 − V11 ]Hm−1 + ε(m)V13 (2V12 Sm − 3Em ).
(3.8)
Further properties of φ3 (P, x, tr ) are summarized in:
φ3,xx (P, x, tr ) + 3φ3 (P, x, tr )φ3,x (P, x, tr ) + 3[v(x, tr ) − u(x, tr )][φ3,x (P, x, tr ) + φ23 (P, x, tr )] +φ33 (P, x, tr ) − [2ux (x, tr ) + 6u(x, tr )v(x, tr ) − 2u2 (x, tr )]φ3 (P, x, tr ) = λ, (3.9) (φ3 (P, x, tr ) − u(x, tr ) + v(x, tr ))tr ¡ (r) (3.10) = ∂x Ve32 (λ, x, tr )[φ3,x (P, x, tr ) + φ23 (P, x, tr ) − 2u(x, tr )φ3 (P, x, tr )] ¢ (r) (r) +Ve (λ, x, tr )φ3 (P, x, tr ) + Ve (λ, x, tr ) , 31
33
λHm−1 (λ, x, tr ) , Fm−1 (λ, x, tr )
(3.11)
Fm−1,x (λ, x, tr ) + 3u(x, tr ) − 3v(x, tr ), Fm−1 (λ, x, tr )
(3.12)
φ3 (P, x, tr )φ3 (P ∗ , x, tr )φ3 (P ∗∗ , x, tr ) =
φ3 (P, x, tr ) + φ3 (P ∗ , x, tr ) + φ3 (P ∗∗ , x, tr ) =
1 1 1 + + φ3 (P, x, tr ) φ3 (P ∗ , x, tr ) φ3 (P ∗∗ , x, tr ) (n) (n) (n) V12 (λ, x, tr ) Hm−1,x (λ, x, tr ) [u(x, tr ) + v(x, tr )]V12 (λ, x, tr ) − V11 (λ, x, tr ) , = − (n) +3 (n) λV13 (λ, x, tr ) Hm−1 (λ, x, tr ) λV13 (λ, x, tr ) (3.13) ∗ ∗∗ 2 2 ∗ 2 ∗∗ φ3,x (P, x, tr ) + φ3,x (P , x, tr ) + φ3,x (P , x, tr ) + φ3 (P, x, tr ) + φ3 (P , x, tr ) + φ3 (P , x, tr ) (n)
= −3
V33 (λ, x, tr ) (n) V32 (λ, x, tr )
(n)
−[
V31 (λ, x, tr ) (n) V32 (λ, x, tr )
− 2u(x, tr )][
6
Fm−1,x (λ, x, tr ) + 3u(x, tr ) − 3v(x, tr )]. Fm−1 (λ, x, tr ) (3.14)
Lemma 3.1. Assume (2.12) and let (λ, x, tr ) ∈ C3 . Then (r) Ve32 (n) (n) V31 ) V32 (r) e (r) Ve (n) (r) e (r) − V32 V (n) )], +3Fm−1 (λ, x, tr )[Ve33 − 32 V + (u − v)( V 31 (n) 33 (n) 31 V32 V32 (r) Ve (n) (r) Hm−1,tr (λ, x, tr ) = Hm−1,x (λ, x, tr )(Ve12 − 13 (n) V12 ) V13 (r) e (r) Ve (r) (n) e (r) − V13 V (n) )]. +3Hm−1 (λ, x, tr )[Ve11 − 13 V − (u + v)( V 12 (n) 11 (n) 12 V13 V13 (3.15) Then ψ3 (P, x, x0 , tr , t0,r ) satisfy the following properties ³ (r) ψ3,tr (P, x, x0 , tr , t0,r ) = Ve32 (λ, x, tr )[φ3,x (P, x, tr ) + φ23 (P, x, tr ) − 2u(x, tr )φ3 (P, x, tr )] ´ (r) (r) +Ve31 (λ, x, tr )φ3 (P, x, tr ) + Ve33 (λ, x, tr ) ψ3 (P, x, x0 , tr , t0,r ), (3.16) ³Z x 0 0 0 0 ψ3 (P, x, x0 , tr , t0,r ) = exp [φ3 (P, x , tr ) − u(x , tr ) + v(x , tr )]dx x 0 Z tr ¡ (r) Ve32 (λ, x0 , t0 )[φ3,x (P, x0 , t0 ) + φ23 (P, x0 , t0 ) − 2u(x0 , t0 )φ3 (P, x0 , t0 )] + t0,r ¢ ´ (r) (r) +Ve31 (λ, x0 , t0 )φ3 (P, x0 , t0 ) + Ve33 (λ, x0 , t0 ) dt0 , (3.17) F (λ, x, t ) m−1 r ψ3 (P, x, x0 , tr , t0,r )ψ3 (P ∗ , x, x0 , tr , t0,r )ψ3 (P ∗∗ , x, x0 , tr , t0,r ) = . (3.18) Fm−1 (λ, x0 , t0,r ) (r)
Fm−1,tr (λ, x, tr ) = Fm−1,x (λ, x, tr )(Ve31 −
Due to the observation of (3.4), one infers that Fm−1 and Hm−1 are monic polynomials with respect to λ of degree m − 1. Hence we may write them in the following forms Fm−1 (λ, x, tr ) =
m−1 Y j=1
Hm−1 (λ, x, tr ) = Defining
(λ − νj (x, tr )),
m−1 Y j=1
(3.19)
(λ − ξj (x, tr )).
(3.20)
³ ´ ³ ´ Am (νj (x, tr ), x, tr ) νˆj (x, tr ) = νj (x, tr ), y(ˆ νj (x, tr )) = νj (x, tr ), − (n) ∈ Km−1 , V32 (νj (x, tr ), x, tr ) ³ ´ ³ ´ Cm (ξj (x, tr ), x, tr ) ξˆj (x, tr ) = ξj (x, tr ), y(ξˆj (x, tr )) = ξj (x, tr ), − (n) ∈ Km−1 , V12 (ξj (x, tr ), x, tr ) 1 ≤ j ≤ m − 1, (x, tr ) ∈ C2 , (n)
(3.21)
(3.22)
(n)
and P0 = (0, V33 |λ=0 ), where V33 |λ=0 = (v − u)an−1 − 13 (∂ − u + 3v)(∂ + u + 3v)bn−1 . In fact, the definition of νˆj (x, tr ) in (3.21) is equivalent to (3.20) in [1] by noting that (n)
(n)
(n)
(n)
(n)
(n)
(n)
(n)
Fm−1 (νj (x, tr ), x, tr ) = −ε(m)[λV21 (V32 )2 + V31 V32 (V11 − V22 ) − V12 (V31 )2 ]|λ=νj (x,tr ) (n)
(n)
= ε(m)[V31 Am − V32 Cm ]|λ=νj (x,tr ) = 0, 7
namely,
Am (νj (x, tr ), x, tr )
(n) V32 (νj (x, tr ), x, tr )
=
Cm (νj (x, tr ), x, tr ) (n)
V31 (νj (x, tr ), x, tr )
.
m−1 The dynamics of the zeros {νj (x, tr )}m−1 j=1 and {ξj (x, tr )}j=1 of Fm−1 (λ, x, tr ) and Hm−1 (λ, x, tr ) with respect to x and tr are then described in terms of Dubrovin-type equations. m−1 Lemma 3.2. Suppose the zeros {νj (x, tr )}m−1 j=1 and {ξj (x, tr )}j=1 of Fm−1 (λ, x, tr ) and m−1 Hm−1 (λ, x, tr ) remain distinct, respectively. Then {νj (x, tr )}m−1 j=1 and {ξj (x, tr )}j=1 satisfy the system of differential equations
νj,x (x, tr ) =
(n)
ε(m)V32 (νj (x, tr ), x, tr )[3y 2 (ˆ νj (x, tr )) + Sm (νj (x, tr ))] , m−1 Y (νj (x, tr ) − νk (x, tr )) k=1 k6=j
νj,tr (x, tr ) =
(n) (r) [V32 (νj (x, tr ), x, tr )Ve31 (νj (x, tr ), x, tr ) ε(m)[3y 2 (ˆ νj (x, tr )) + Sm (νj (x, tr ))] × , m−1 Y k=1 k6=j
ξj,x (x, tr ) =
(νj (x, tr ) − νk (x, tr ))
(n) ε(m)V13 (ξj (x, tr ), x, tr )[3y 2 (ξˆj (x, tr )) m−1 Y k=1 k6=j
1 ≤ j ≤ m − 1,
(3.23) (r) (n) e − V32 (νj (x, tr ), x, tr )V31 (νj (x, tr ), x, tr )] 1 ≤ j ≤ m − 1;
(3.24)
+ Sm (ξj (x, tr ))]
,
(ξj (x, tr ) − ξk (x, tr ))
1 ≤ j ≤ m − 1,
(3.25)
(n) (r) (r) (n) ξj,tr (x, tr ) = [V13 (ξj (x, tr ), x, tr )Ve12 (ξj (x, tr ), x, tr ) − Ve13 (ξj (x, tr ), x, tr )V12 (ξj (x, tr ), x, tr )] ε(m)[3y 2 (ξˆj (x, tr )) + Sm (ξj (x, tr ))] × , 1 ≤ j ≤ m − 1. m−1 Y (ξj (x, tr ) − ξk (x, tr )) k=1 k6=j
(3.26) For investigating the asymptotic expansion of φ3 (P, x, tr ) near P∞ ∈ Km−1 , we choose the local 1 coordinate ζ = λ− 3 . Substituting a power series ansatz into (3.9) and comparing the same powers of ζ, we arrive at ∞ 1 P χj (x, tr )ζ j , ζ ζ→0 j=0
φ3 (P, x, tr ) =
8
as
P → P∞ ,
(3.27)
where χ0 = 1, χ1 = u − v, χ2 = 13 (u2 − ux ) + v 2 + vx , χ3 = 13 (−2vxx − 2ux v − 6vvx + 2u2 v − 2v 3 ), χ4 = 91 (uxx + 3vxx − 2uux + 6ux v + 12vvx − 6u2 v + 6v 3 )x , j−1 j−1 j−1 P P P χj = − 13 [χj−2,xx + 3(v − u)χj−2,x + 3 χj−1−i χi,x + 3(v − u) χj−1−i χi + χi χj−i +
j−1 P j−i P
i=1 l=0
i=0
i=0
i=1
χi χl χj−i−l − (2ux + 6uv − 2u2 )χj−2 ], (j ≥ 3).
(3.28)
One infers from (3.2) and (3.27) that the divisor (φ3 (P, x, tr )) of φ3 (P, x, tr ) is given by (φ3 (P, x, tr )) = DP0 ,ξˆ1 (x,tr ),...,ξˆm−1 (x,tr ) (P ) − DP∞ ,ˆν1 (x,tr ),...,ˆνm−1 (x,tr ) (P ).
(3.29)
That is, P0 , ξˆ1 (x, tr ), . . . , ξˆm−1 (x, tr ) are the m zeros of φ3 (P, x, tr ) and P∞ , νˆ1 (x, tr ), . . . , νˆm−1 (x, tr ) its m poles. To obtain Riemann theta function representation for the Baker-Akhiezer function ψ3 (P, x, x0 , tr , t0,r ), we first consider the second integration in (3.17) and define the function Js (P, x, tr ), meromorphic on Km−1 × C2 , by (r) (r) Js (P, x, tr ) = Ve31 (λ, x, tr )φ3 (P, x, tr ) + Ve32 (λ, x, tr )[φ3,x (P, x, tr ) + φ23 (P, x, tr ) (r) −2u(x, tr )φ3 (P, x, tr )] + Ve (λ, x, tr ).
(3.30)
33
(r) (r) (r) Denote by J¯s (P, x, tr ) the associated homogeneous quantity replacing Ve31 , Ve32 , Ve33 by the ¯ (r,s) ¯ (r,s) ¯ (r,s) corresponding homogeneous polynomials Ve 31 , Ve 32 , Ve 33 , where ( (r) (r,s) Ve3j |α˜ 0 =1,α˜ 1 =···=α˜ r =β˜0 =···=β˜r =0 s = 3r + 2, ¯ Ve 3j = j = 1, 2, 3, (r) Ve3j |α˜ 0 =0,β˜0 =1,α˜ 1 =···=α˜ r =β˜1 =···=β˜r =0 s = 3r + 1,
and
(˜ α0 , β˜0 ) =
(
(1, β˜0 ), s = 3r + 2, ˜ β0 ∈ R. (0, 1), s = 3r + 1,
Lemma 3.3. Let s = 3r + 2 or 3r + 1, r ∈ N0 , (x, tr ) ∈ C2 , and λ = ζ −3 be the local coordinate near P∞ . Then J¯s (P, x, tr ) = ζ −s + O(ζ), as P → P∞ .
(3.31)
ζ→0
Proof. In order to make the proof more clear, we introduce some notations ( a ˜(r) |α˜ 0 =1,α˜ 1 =···=α˜ r =β˜0 =···=β˜r =0 s = 3r + 2, (r,s) ¯ a ˜ = ; (r) a ˜ |α˜ 0 =0,β˜0 =1,α˜ 1 =···=α˜ r =β˜1 =···=β˜r =0 s = 3r + 1, ¯˜(r,s) a = r
(
a ˜r |α˜ 0 =1,α˜ 1 =···=α˜ r =β˜0 =···=β˜r =0
a ˜r |α˜ 0 =0,β˜0 =1,α˜ 1 =···=α˜ r =β˜1 =···=β˜r =0 9
s = 3r + 2, s = 3r + 1,
.
From (2.12) and (3.30), one easily gets ¯ (r,s) ¯ (r,s) J¯s (P, x, tr ) = Ve 31 (λ, x, tr )φ3 (P, x, tr ) + Ve 32 (λ, x, tr )[φ3,x (P, x, tr ) + φ23 (P, x, tr ) ¯ (r,s) −2uφ3 (P, x, tr )] + Ve 33 (λ, x, tr ) ¯˜(r,s) (λ, x, tr )φ3 (P, x, tr ) + d¯˜(r,s) (λ, x, tr )[φ3,x (P, x, tr ) + φ2 (P, x, tr ) = a 3 ¯˜(r,s) (λ, x, tr ) − 1 (∂ + 3v − u)(∂ + u + 3v)˜¯b(r,s) (λ, x, tr ). −2uφ3 (P, x, tr )] + (v − u)a 3 (3.32) Using (3.27), we have J¯1 = φ3 − u + v = ζ −1 + O(ζ), as P → P∞ , J¯2 = φ3,x + φ23 + 2(v − u)φ3 − 31 (ux + 3vx − u2 + 3v 2 + 6uv) = ζ −2 + O(ζ), as P → P∞ . Therefore (3.31) holds for s = 1 and s = 2. Suppose one may rewrite (3.31) as J¯s (P, x, tr ) = ζ −s + ζ→0
∞ X j=1
γj (x, tr )ζ j , as P → P∞ ,
(3.33)
for some coefficients {γj (x, tr )}j∈N . From (3.10) and (3.30), it is easy to see that ¯ (r,s) ¯ (r,s) ∂x J¯s (P, x, tr ) = ∂x [Ve 31 (λ, x, tr )φ3 (P, x, tr ) + Ve 32 (λ, x, tr )(φ3,x (P, x, tr ) + φ23 (P, x, tr ) ¯ (r,s) −2uφ3 (P, x, tr )) + Ve 33 (λ, x, tr )] = ∂tr [φ3 (P, x, tr ) − u(x, tr ) + v(x, tr )], namely, ∂x (ζ −s +
∞ X
γj (x, tr )ζ j ) = ∂tr (ζ −1 +
j=1
∞ X j=2
∞ X χj (x, tr )ζ j−1 ) = ∂tr ( χj+1 (x, tr )ζ j ).
(3.34)
j=1
Using (3.27) and comparing coefficients of the same powers of ζ in (3.34) give rise to γj,x (x, tr ) = χj+1,tr (x, tr ), j = 1, 2, . . . ,
¯(r,s) γ1,x (x, tr ) = χ2,tr (x, tr ) = − 31 (ux − u2 − 3vx − 3v 2 )tr = −d˜r,x (x, tr ), γ2,x (x, tr ) = χ3,tr (x, tr ) = 31 (−2vxx − 6vvx − 2v 3 + 2u2 v − 2ux v)tr ¯˜(r,s) ¯˜(r,s) = −a (x, tr ))x , r,x (x, tr ) + 2(v(x, tr )dr
γ3,x (x, tr ) = χ4,tr (x, tr ) = 19 (uxx + 3vxx − 2uux + 12vvx + 6ux v − 6u2 v + 6v 3 )x,tr ¯(r,s) ¯(r,s) = 1 [(∂ − u + 3v)(∂ + u + 3v)˜br (x, tr ) − (ux + 3vx − u2 + 9v 2 )d˜r (x, tr )]x , 3
(3.35)
from which we deduce ¯(r,s) γ1 (x, tr ) = ²1 (tr ) − d˜r (x, tr ), ¯(r,s) ¯˜(r,s) γ2 (x, tr ) = ²2 (tr ) − a (x, tr ) + 2v(x, tr )d˜r (x, tr ), r ¯(r,s) ¯(r,s) γ3 (x, tr ) = ²3 (tr ) + 1 [(∂ − u + 3v)(∂ + u + 3v)˜br (x, tr ) − (ux + 3vx − u2 + 9v 2 )d˜r (x, tr )], 3
(3.36)
10
where ²1 (tr ), ²2 (tr ), ²3 (tr ) are integration constants. Next we note that the coefficients of the power series for φ3 (P, x, tr ) in the coordinate ζ near P∞ , and the coefficients of the homoge¯˜(r,s) (ζ, x, tr ), ¯˜b(r,s) (ζ, x, tr ) , c¯˜(r,s) (ζ, x, tr ) and d¯˜(r,s) (ζ, x, tr ), are differential neous polynomials a polynomials in u and v, with no arbitrary integration constants in their construction. From the definition of J¯s it follows that it also have no arbitrary integration constants, and must consist purely of differential polynomials of u and v. From these considerations, we know that ²1 (tr ) = ²2 (tr ) = ²3 (tr ) = 0. Hence one concludes ¯(r,s) ¯(r,s) ¯˜(r,s) J¯s (P, x, tr ) = ζ −s − d˜r ζ + [−a (x, tr ) + 2v(x, tr )˜br (x, tr )]ζ 2 r ¯(r,s) ¯(r,s) + 1 [(∂ − u + 3v)(∂ + u + 3v)˜br (x, tr ) − (ux + 3vx − u2 + 9v 2 )d˜r (x, tr )]ζ 3 3
+O(ζ 4 ), as P → P∞ .
(3.37)
On the other hand we have ¯ J¯s+3 (P, x, tr ) = d˜(r+1,s+3) (λ, x, tr )[φ3,x (P, x, tr ) + φ2 (P, x, tr ) − 2uφ3 (P, x, tr )] 3
¯˜(r+1,s+3) (λ, x, tr )φ3 (P, x, tr ) + (v − u)a ¯˜(r+1,s+3) (λ, x, tr ) +a ¯ − 31 (∂ − u + 3v)(∂ + u + 3v)˜b(r+1,s+3) (λ, x, tr ) ¯(r+1,s+3) = ζ −3 J¯s + d˜r (λ, x, tr )[φ3,x (P, x, tr ) + φ23 (P, x, tr ) − 2uφ3 (P, x, tr )] ¯˜r(r+1,s+3) (λ, x, tr )φ3 (P, x, tr ) + (v − u)a ¯˜(r+1,s+3) +a (λ, x, tr ) r ¯˜(r+1,s+3) 1 (λ, x, tr ) − (∂ − u + 3v)(∂ + u + 3v)br 3
= ζ −s−3 + O(ζ),
(3.38)
which implies (3.31). From (2.12), we know that Js (P, x, tr ) =
r X
α ˜ r−l J¯3l+2 (P, x, tr ) +
r X
Therefore Z tr r X Js (P, x, τ )dτ = (tr − t0,r ) (˜ αr−l ζ→0
t0,r
β˜r−l J¯3l+1 (P, x, tr ), s = 3r + 2 or 3r + 1.
(3.39)
1 1 + β˜r−l 3l+1 ) + O(ζ), as P → P∞ . ζ 3l+2 ζ
(3.40)
l=0
l=0
l=0
(2)
Let ωP∞ ,j , j = 3l + 2 or 3l + 1, l ∈ N0 , be the normalized differential of the second kind holomorphic on Km−1 \ {P∞ } with a pole of order j at P∞ , (2)
ωP∞ ,j (P ) = (ζ −j + O(1))dζ, ζ→0
as
P → P∞ .
(3.41)
Furthermore, define the normalized differential of the second kind by e (2) Ω P∞ ,s+1 =
r X l=0
α ˜ r−l (3l +
(2) 2)ωP∞ ,3l+3
+
r X l=0
(2) β˜r−l (3l + 1)ωP∞ ,3l+2 ,
(3.42)
where s = 3r +2 or 3r +1, r ∈ N0 . In addition, we define the vector of b-periods of the differential e (2) of the second kind Ω P∞ ,s+1 , Z 1 (2) (2) (2) e (2) e e e e (2) U = ( U , . . . , U ), U = Ω , j = 1, . . . , m − 1, (3.43) s+1 s+1,1 s+1,m−1 s+1,j 2πi bj P∞ ,s+1 11
with s = 3r + 2 or 3r + 1, r ∈ N0 . Integrating (3.42) yields Z
P
Q0
e (2) Ω P∞ ,s+1
r X
=
α ˜ r−l (3l + 2)
=
−
ζ→0
l=0
ζ
ζ0
l=0
r X
Z
α ˜ r−l
1 ζ 3l+2
(2) ωP∞ ,3l+3
r X
−
β˜r−l
l=0
+ 1
r X
Z
β˜r−l (3l + 1)
ζ0
l=0
ζ 3l+1
ζ
(2)
ωP∞ ,3l+2
(2)
+ es+1 (Q0 ) + O(ζ), as P → P∞ , (3.44)
(2)
where es+1 (Q0 ) is a constant. Combining (3.40) and (3.44) yields Z
tr
t0,r
(2)
Js (P, x, τ )dτ = (tr − t0,r )(es+1 (Q0 ) − ζ→0
Z
P
Q0
e (2) Ω P∞ ,s+1 ) + O(ζ), as P → P∞ .
(3.45)
(3)
Next, we introduce the normalized Abel differential ωP∞ ,P0 (P ) of the third kind which is the unique differential holomorphic on Km−1 \{P∞ , P0 } with simple poles at P∞ and P0 with residues ±1, respectively, that is, (3)
ωP∞ ,P0 (P ) = (ζ −1 − w1 + O(ζ))dζ, ζ→0
(3)
ωP∞ ,P0 (P ) = (−ζ −1 + O(ζ))dζ, ζ→0
then
Z
P
ZQP0
Q0
as as
(3)
ωP∞ ,P0 (P ) = ln ζ + e(3) (Q0 ) − w1 ζ + O(ζ 2 ), (3) ωP∞ ,P0 (P )
= − ln ζ + e
(3)
(Q0 ) + O(ζ),
as
P → P∞ ,
(3.46)
P → P0 ,
as
P → P∞ ,
(3.47)
P → P0
with w1 and e(3) (Q0 ) two constants. Given these preparations, the theta function representations of φ3 (P, x, tr ) and ψ3 (P, x, x0 , tr , t0,r ) read as follows. Theorem 3.4. Assume that the curve Km−1 is nonsingular. Let P = (λ, y) ∈ Km−1 \ {P∞ } and let (x, tr ), (x0 , t0,r ) ∈ Ων , where Ων ⊆ C2 is open and connected. Suppose that Dνˆ(x,tr ) , or equivalently, Dˆξ(x,tr ) is nonspecial for (x, tr ) ∈ Ων . Then ³ θ(λ(P, ˆξ(x, tr )))θ(λ(P∞ , νˆ(x, tr ))) φ3 (P, x, tr ) = exp e(3) (Q0 ) − θ(λ(P∞ , ˆξ(x, tr )))θ(λ(P, νˆ(x, tr )))
Z
P
Q0
θ(λ(P, νˆ(x, tr )))θ(λ(P∞ , νˆ(x0 , t0,r ))) ψ3 (P, x, x0 , tr , t0,r ) = θ(λ(P∞ , νˆ(x, tr ))θ(λ(P, νˆ(x0 , t0,r ))) Z Z P ³ (2) (2) (2) × exp (x − x0 )(e2 (Q0 ) − ωP∞ ,2 ) + (tr − t0,r )(es+1 (Q0 ) −
P
Q0
Q0
´ (3) ωP∞ ,P0 . ´ ˜ (2) Ω ) P∞ ,s+1 .
Proof. It follows from (3.47) that Z P ´ ³ (3) (3) ωP∞ ,P0 = ζ −1 + O(1), as P → P∞ , exp e (Q0 ) − ζ→0 ZQP0 ´ ³ (3) ωP∞ ,P0 = ζ + O(1), as P → P0 . exp e(3) (Q0 ) − Q0
ζ→0
12
(3.48)
(3.49)
(3.50)
Using (3.29) we immediately know that φ3 has simple poles at νˆ(x, tr ) and P∞ , and simple zeros ˆ tr ). Let Φ3 be defined by the right hand side of (3.48) with the aim to prove at P0 and ξ(x, that φ3 = Φ3 . By (3.50) and a special case of Riemann’s vanishing theorem [29-31], we see that Φ3 has the same properties. Using the Riemann-Roch theorem [29-31], we conclude that the holomorphic function Φ3 = ς, where ς is a constant with respect to P . Using (3.27) and (3.50), φ3 we have −1 Φ3 = (1 + O(ζ))(ζ + O(1)) = 1 + O(ζ), as P → P , (3.51) ∞ −1 φ3 ζ→0 ζ→0 (ζ + O(1)) from which we conclude ς = 1. Let ψ3 (P, x, x0 , tr , t0,r ) be defined as in (3.17) and denote the right-hand side of (3.49) by Ψ3 (P, x, x0 , tr , t0,r ). In order to prove that ψ3 = Ψ3 , one uses (3.15) and (n) (n) (n) V31 φ3 + V32 [φ3,x + φ23 − 2uφ3 ] + V33 = y, to compute φ3 (P, x, tr ) − u(x, tr ) + v(x, tr )
(n)
= = = =
λ→νj (x)
=
λ→νj (x)
Js (P, x, tr )
= = = =
λ→νj (x,tr )
y 2 V32 − yAm + Bm −u+v −ε(m)Fm−1 (n) (n) y 2 V32 − yAm + 23 V32 Sm − 13 ε(m)Fm−1,x −ε(m)Fm−1 (n) m V32 y(y + A(n) ) 1 Fm−1,x 2 (n) 3y 2 + Sm V32 V32 + + 3 −ε(m)Fm−1 3 Fm−1 ε(m)Fm−1 ν − j,x + O(1) λ − νj ∂x ln(λ − νj (x, tr )) + O(1),
(3.52) (r) (r) (r) Ve31 φ3 + Ve32 [φ3,x + φ23 − 2uφ3 ] + Ve33 (r) (r) (r) ³ ´ Ve32 (n) Ve32 Ve (n) (r) (r) e V φ + V − V + y Ve31 − 32 3 33 (n) 31 (n) 33 (n) V32 V32 V32 (n) m yV32 (y + A(n) ) (r) (n) (r) 1 Fm−1,tr ³ e (r) Ve32 (n) ´ 2 V32 (3y 2 + Sm ) Ve32 V32 + V31 − (n) V31 [ + ] + y (n) 3 Fm−1 3 −ε(m)Fm−1 ε(m)Fm−1 V32 V32 νj,tr + O(1). − λ − νj (3.53)
More concisely, φ3 (P, x0 , tr ) − u(x0 , tr ) + v(x0 , tr ) = ∂x0 ln(λ − νj (x0 , tr )) + O(1), as P → νˆj (x0 , tr ), Js (P, x0 , t0 ) = ∂t0 ln(λ − νj (x0 , t0 )) + O(1), as P → νˆj (x0 , t0 ). Hence
³Z
´ (∂x0 ln(λ − νj (x0 , tr )) + O(1))dx0 x0 ´ ³Z tr (∂t0 ln(λ − νj (x0 , t0 )) + O(1))dt0 × exp
ψ3 (P, x, x0 , tr , t0,r ) = exp
x
t0,r
13
(3.54)
λ − νj (x, tr ) λ − νj (x0 , tr ) × O(1) λ − νj (x0 , tr ) λ − νj (x0 , t0,r ) for P → νˆj (x0 , tr ) 6= νˆj (x0 , t0,r ), (λ − νj (x0 , tr ))O(1) = O(1) for P → νˆj (x0 , tr ) = νˆj (x0 , t0,r ), −1 (λ − νj (x0 , t0,r )) O(1) for P → νˆj (x0 , t0,r ) 6= νˆj (x0 , tr ),
=
(3.55)
where O(1) 6= 0 in (3.55). Consequently, all zeros and poles of ψ3 and Ψ3 on Km−1 \ {P∞ } are simple and coincident. It remains to identify the essential singularity of ψ3 and Ψ3 at P∞ . By (3.45) we see that the singularities in the exponential terms of ψ3 and Ψ3 coincide. The uniqueness result for Baker-Akhiezer functions completes the proof that ψ3 = Ψ3 . ∼ DP∞ νˆ(x,tr ) . Using Abel Theorem, we have From (3.29), we know that DP0 ξ(x,t ˆ r) ). AQ0 (P∞ ) + αQ0 (Dνˆ(x,tr ) ) = AQ0 (P0 ) + αQ0 (Dξ(x,t ˆ r)
(3.56)
As we have shown in [1] that (2) e (2) (tr − t0,r ). αQ0 (Dνˆ(x,tr ) ) = αQ0 (Dνˆ(x0 ,t0,r ) ) + U 2 (x − x0 ) + U s+1
(3.57)
(2) e (2) αQ0 (Dξ(x,t ) = αQ0 (Dξ(x ˆ ˆ 0 ,t0,r ) ) + U 2 (x − x0 ) + U s+1 (tr − t0,r ). r)
(3.58)
Combining (3.56) and (3.57), we arrive at
Then the modified Boussinesq flows are straightened by the Abel map.
Theorem 3.5. Assume that the curve Km−1 is nonsingular and let (x, tr ), (x0 , t0,r ) ∈ Ων , where Ων ⊆ C2 is open and connected. Then (2) e (2) αQ0 (Dνˆ(x,tr ) ) = αQ0 (Dνˆ(x0 ,t0,r ) ) + U 2 (x − x0 ) + U s+1 (tr − t0,r ), (2) (2) e s+1 (tr − t0,r ). αQ (D ˆ ) = αQ (D ˆ ) + U (x − x0 ) + U 0
ξ(x,tr )
0
ξ(x0 ,t0,r )
(3.59)
2
Immediately, we shall show the explicit theta function representations of solutions of the modified Boussinesq hierarchy. Theorem 3.6. Assume that the curve Km−1 is nonsingular and let (x, tr ) ∈ Ων , where Ων ⊆ C2 is open and connected. Suppose also that Dνˆ(x,tr ) , or equivalently, Dξ(x,t is nonspecial ˆ r) for (x, tr ) ∈ Ων . Then θ(λ(P∞ , νˆ(x, tr ))) 1 + w , θ(λ(P∞ , µ ˆ(x, tr ))) 2 0 ˆ tr ))) + 1 ∂x ln θ(λ(P∞ , µ ˆ(x, tr )))θ(λ(P∞ , νˆ(x, tr ))) + w1 + 12 w0 u(x, tr ) = −∂x ln θ(λ(P∞ , ξ(x, 2 (3.60) with w0 and w1 are constants. v(x, tr ) = − 21 ∂x ln
Proof. The theta function representation of potential v(x, tr ) has been obtained in [1]. Now we only need to derive explicit theta functions representation of u(x, tr ). Using Theorem 3.4
14
and Theorem 3.5, one can derive θ(ΞQ0 − AQ0 (P ) + αQ0 (Dξ(x,t )) ˆ tr ))) ˆ θ(λ(P, ξ(x, r) = ˆ tr ))) θ(ΞQ0 − AQ0 (P∞ ) + αQ0 (Dξ(x,t )) ˆ θ(λ(P∞ , ξ(x, r) Z P∞ θ(ΞQ0 − AQ0 (P∞ ) + αQ0 (Dξ(x,t )+ ω) ˆ r) P = θ(ΞQ0 − AQ0 (P∞ ) + αQ0 (Dξ(x,t )) ˆ r)
1 ) − ρ0,j ζ − ρ1,j ζ 2 + O(ζ 4 ), . . . ) θ(. . . , ΞQ0 ,j − AQ0 ,j (P∞ ) + αQ0 ,j (Dξ(x,t ˆ r) 2 = θ(ΞQ0 − AQ0 (P∞ ) + αQ0 (Dξ(x,t )) ζ→0 ˆ ) r =
ζ→0
(3.61)
m−1 X ∂θ2 ¤ 1£ ρ0,j ζ + O(ζ 2 ) θ2 − θ2 ∂λj j=1
= 1 − ∂x ln θ2 ζ + O(ζ 2 ),
ζ→0
P → P∞ ,
where θ2 = θ(ΞQ0 − AQ0 (P∞ ) + αQ0 (Dξ(x,t )). Similarly, we have ˆ r) θ(λ(P, νˆ(x, tr ))) = 1 − ∂x ln θ1 ζ + O(ζ 2 ), θ(λ(P∞ , νˆ(x, tr ))) ζ→0 with θ1 = θ(ΞQ0 − AQ0 (P∞ ) + αQ0 (Dνˆ(x,tr ) )). Then we have φ3 (P, x, tr )
= =
ζ→0
=
ζ→0
(3.62)
P → P∞ ,
³ θ(λ(P, ˆξ(x, tr )))θ(λ(P∞ , νˆ(x, tr ))) exp e(3) (Q0 ) − θ(λ(P∞ , ˆξ(x, tr )))θ(λ(P, νˆ(x, tr ))) θ2 (1 − ∂x ln ζ + O(ζ 2 ))(ζ −1 + w1 + O(ζ)) θ1 θ2 −1 + w1 + O(ζ), P → P∞ . ζ − ∂x ln θ1
Z
P
Q0
(3)
ωP∞ ,P0
´
(3.63)
On the other hand, we have from (3.27) that φ3 (P, x, tr ) = ζ −1 + u − v + O(ζ), ζ→0
P → P∞ .
(3.64)
By comparing (3.63) and (3.64), we arrive at u − v = −∂x ln
θ2 + w1 , θ1
(3.65)
which together with the first equality of (3.60) indicates the second one.
Acknowledgments This work was supported by National Natural Science Foundation of China (project nos. 11171312, 11326163 and 11401230).
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ˆ [18] J.C. Eilbeck, V.Z. Enolski, S. Matsutani, Y. Onishi, E. Previato, Abelian functions for purely trigonal curves of genus three, Int. Math. Res. Not. (2008) Art. ID rnm 140, no. 1, 38 pp. [19] Y.V. Brezhnev: Finite-band potentials with trigonal curves, Theor. Math. Phys. 133 (3) (2002) 1657-1662. [20] V.B. Matveev, A.O. Smirnov, Symmetric reductions of the Riemann-function and some of their applications to the Schr¨odinger and Boussinesq equations, Amer. Math. Soc. Transl. 157 (1993) 227-237. [21] E. Previato, The Calogero-Moser-Krichever system and elliptic Boussinesq solitons, in: J. Harnad, J.E. Marsden (Eds.), Hamiltonian Systems, Transformation Groups and Spectral Transform Methods, Monreal, CRM 1990, pp. 57-67. [22] E. Previato, Monodromy of Boussinesq elliptic operators, Acta Appl. Math. 36 (1-2) (1994) 49-55. [23] E. Previato, J.L. Verdier, Boussinesq elliptic solitons: the cyclic case, in: S. Ramanan, A. Beauville (Eds.), Proceedings of the Indo-French Conference on Geometry, Hindustan Book Agency, Dehli, 1993, pp. 173-185. [24] A.O. Smirnov, A matrix analogue of Appell’s theorem and reductions of multidimensional Riemann theta-functions, Math. USSR-Sb. 61 (2) (1988) 379-388. [25] H. Airault, H.P. McKean, J. Moser, Rational and elliptic solutions of the Korteweg-de Vries equation a related many-body problem, Comm. Pure Appl. Math. 30 (1) (1977) 95-148. [26] H.P. McKean, Integrable systems and algebraic curves, in: M. Grmela, J.E. Marsden (Eds.), Global Analysis, in: Lecture Notes in Mathematics 755, Springer, Berlin 1979, pp. 83-200. [27]R. Dickson, F. Gesztesy, K. Unterkofler, A new approach to the Boussinesq hierarchy, Math. Nachr. 198 (1) (1999) 51-108. [28] R. Dickson, F. Gesztesy, K. Unterkofler, Algebro-geometric solutions of the Boussinesq hierarchy, Rev. Math. Phys. 11 (7) (1999) 823-879. [29] P. Griffiths, J. Harris, Principles of Algebraic Geometry, Wiley, New York, 1994. [30] D. Mumford, Tata lectures on theta II, Birkh¨auser, Boston, 1984. [31] H. M. Farkas, I. Kra, Riemann Surfaces, Second ed., Springer, New York, 1992.
17
S0393-0440(15)00142-4 http://dx.doi.org/10.1016/j.geomphys.2015.06.014 GEOPHY 2572
To appear in:
Journal of Geometry and Physics
Received date: 6 November 2014 Accepted date: 16 June 2015 Please cite this article as: L. Wu, G. He, X. Geng, A note on the quasi-periodic solutions of the modified Boussinesq hierarchy, Journal of Geometry and Physics (2015), http://dx.doi.org/10.1016/j.geomphys.2015.06.014 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
A Note on the Quasi-Periodic Solutions of the Modified Boussinesq Hierarchy Lihua Wu1∗, Guoliang He2 , Xianguo Geng3 1
Department of Mathematics, Huaqiao University, Quanzhou 362021, China
2
Department of Mathematics and Information Science, Zhengzhou University of Light Industry, Zhengzhou 450002, China
3
Department of Mathematics, Zhengzhou University, Zhengzhou 450001, China E-mail address: [email protected], [email protected], [email protected]
Abstract Based on the theory of trigonal curve and the properties of three kinds of the Abel differentials on it, we deduce the explicit theta function representations of the Baker-Akhiezer function and the meromorphic function associated with the modified Boussinesq hierarchy. The modified Boussinesq flows are straightened using the Abel map and the Lagrange interpolation formula. The explicit theta function representations of solutions for the entire modified Boussinesq hierarchy are constructed with the aid of the asymptotic properties and the algebrogeometric characters of the meromorphic function. Keywords: the modified Boussinesq hierarchy, explicit quasi-periodic solutions MR(2000) Subject Classification: 35Q51, 37K10, 14H70, 35C99
1
Introduction
In the previous paper [1], based on a trigonal curve related to the characteristic polynomial of Lax matrix associated with a 3 × 3 matrix spectral problem, we discussed algebro-geometric constructions of the modified Boussinesq hierarchy proposed by Fordy and Gibbons [2] and its quasi-periodic solutions. However, the theta function representations of solutions of every equation in the modified Boussinesq hierarchy are related to a Riccati equation, which is nonlinear. The principal aim of the present paper is to improve the main result in [1], by which explicit quasi-periodic solutions of the entire modified Boussinesq hierarchy are deduced through introducing another meromorphic function and using its asymptotic properties and the algebrogeometric characters. During the last four decades, finding quasi-periodic solutions of soliton equations aroused interest of many mathematicians and physicists. Quasi-periodic solutions for a lot of soliton ∗
Corresponding author
1
equations associated with 2 × 2 matrix spectral problems have been obtained such as the KdV, nonlinear Schr¨odinger, sine-Gordon, and Toda lattice, Camassa-Holm equations, and others [3-14]. As compared with the 2 × 2 case, the study of quasi-periodic solutions of soliton equations associated with 3 × 3 matrix spectral problems is very few. The primary cause of this discrepancy is enormous complexity for the theory of non-hyperelliptic curves or trigonal ones [15-19] associated with 3 × 3 matrix spectral problems. In Refs. [4,15,16,20-26], certain algebrogeometric solutions of the Boussinesq equation related to a third-order differential operator were found as special solutions of the Kadomtsev-Peiviashvili equation or by the reduction theory of Riemann theta functions. In Refs. [27,28], a unified framework was proposed which yields all algebro-geometric solutions of the entire Boussinesq hierarchy. The present paper is organized as follows. In section 2, we summarize the main results about the algebro-geometric constructions of the modified Boussinesq hierarchy in Ref. [1]. In section 3, we introduce the Baker-Akhiezer function and another associated meromorphic function from which the modified Boussinesq equations are decomposed into the system of Dubrovin-type ordinary differential equations. In terms of the properties of three kinds of the Abel differentials on Km−1 , the explicit theta function representations of the Baker-Akhiezer function and the meromorphic functions are presented. Using the Abel map and the Lagrange interpolation formula, the corresponding flows are straightened. The explicit quasi-periodic solutions for the entire modified Boussinesq hierarchy are constructed according to the asymptotic properties and the algebro-geometric characters of the meromorphic functions.
2
The previous main results
In this section, we simply recall the primary results in Ref. [1]. We first reconstruct the modified Boussinesq hierarchy associated with the 3 × 3 matrix spectral problem ψ1 u+v λ 0 ψx = U ψ, ψ = ψ2 , U = 0 (2.1) −2v 1 , 1
ψ3
0
by introducing two sets of Lenard recursion equations
with two starting points
Kgj−1 = Jgj ,
gj |(u,v)=0 = 0,
j ≥ 0,
K gˆj−1 = J gˆj ,
gˆj |(u,v)=0 = 0,
j≥0
g−1 = (2v, 1, 1, 1)T , and
gˆ−1 = (1, 0, 0, 0)T ,
+ ∂u 0 0 0 + ∂v K22 0 0 , −3∂ K32 K33 0 0 K42 ∂ − 2u K44
1 2 ∂ 21 2 2∂
K=
v−u
2
1 1 −1 0 2 2 1 1 0 0 − 2 2 , J = K33 0 −3∂ K32 0 K42 ∂ − 2u K44
(2.2) (2.3)
(2.4)
1 K22 = − (∂ 2 + 3∂v − ∂u)(∂ + u + 3v), K32 = 2(∂ − u)(∂ + u + 3v), 3 K33 = (∂ − u − 3v)(∂ − 2u), K42 = ∂ + u + 3v, K44 = ∂ + u − 3v. Let ψ satisfy the spectral problem (2.1) and an auxiliary problem ψtn = V (n) ψ, (n)
where each entry Vij
(n)
(2.5)
V (n) = (Vij )3×3 ,
= Vij (a(n) , b(n) , c(n) , d(n) ) is a Laurent expansion in λ:
(n)
V11 = (∂ + u + v)a(n) − 13 (∂ + 3v − u)(∂ + u + 3v)b(n) , (n)
V12 = a(n) − (∂ + u + 3v)b(n) − (∂ − 2u)c(n) , (n)
V21 = b(n) , (n)
(n)
(n)
V13 = c(n) ,
V22 = −(∂ + 2v)a(n) + 23 (∂ + 3v − u)(∂ + u + 3v)b(n) , (n)
V23 = a(n) − (∂ + u + 3v)b(n) ,
V31 = a(n) ,
(n)
(n)
V32 = d(n) ,
V33 = (v − u)a(n) − 13 (∂ + 3v − u)(∂ + u + 3v)b(n) , a(n) =
n P
aj−1 λn−j ,
b(n) =
c(n) =
bj−1 λn−j ,
j=0
j=0
with Gj = (aj , bj , cj , dj
n P
)T
n P
cj−1 λn−j ,
d(n) =
j=0
n P
dj−1 λn−j
j=0
(2.6)
determined by
Gj = α0 gj + β0 gˆj + · · · + αj g0 + βj gˆ0 + αj+1 g−1 + βj+1 gˆ−1 ,
j ≥ −1,
(2.7)
where αj and βj are arbitrary constants. Then the compatibility condition of (2.1) and (2.5) (n) yields the zero-curvature equation, Utn −Vx +[U, V (n) ] = 0, which is equivalent to the hierarchy of nonlinear evolution equations (utn , vtn )T = Xn ,
n ≥ 0,
(2.8)
where the vector fields Xj = L(aj−1 , bj−1 )T with à ! 1 2 ∂ + ∂u 0 2 L= 1 2 . 2 ∂ + ∂v K22 The first nontrivial member in the hierarchy (2.8) is ut0 = α0 (vxx + 2(uv)x ) + β0 ux , vt0 = α0 (− 13 uxx + 23 uux − 2vvx ) + β0 vx .
(2.9)
For α0 = 1, β0 = 0, (t0 = t), equation (2.9) is reduced to the modified Boussinesq equation ut = vxx + 2(uv)x , vt = − 13 uxx + 32 uux − 2vvx .
(2.10)
Then we defined a trigonal curve associated with the modified Boussinesq hierarchy or its compactification engendering the three-sheeted Riemann surface of arithmetic genus m − 1 by Km−1 :
Fm (λ, y) = y 3 + ySm (λ) − Tm (λ) = 0, 3
(2.11)
where m = 3n + 2 or 3n + 1 as α0 = 1, β0 an arbitrary constant or α0 = 0, β0 = 1. Moreover, We introduced the Baker-Akhiezer function ψ2 (P, x, x0 , tr , t0,r ) and the corresponding meromorphic function φ2 (P, x, tr ) on Km−1 by ψx (P, x, x0 , tr , t0,r ) = U (u(x, tr ), v(x, tr ); λ(P ))ψ(P, x, x0 , tr , t0,r ), ψtr (P, x, x0 , tr , t0,r ) = Ve (r) (u(x, tr ), v(x, tr ); λ(P ))ψ(P, x, x0 , tr , t0,r ), V (n) (u(x, tr ), v(x, tr ); λ(P ))ψ(P, x, x0 , tr , t0,r ) = y(P )ψ(P, x, x0 , tr , t0,r ), ψ2 (P, x0 , x0 , t0,r , t0,r ) = 1, P = (λ, y) ∈ Km−1 \ {P∞ }, x, x0 , tr , t0,r ∈ C,
(2.12)
and φ2 (P, x, tr ) =
∂ψ2 (P, x, x0 , tr , t0,r ) + 2v(x, tr ), ψ2 (P, x, x0 , tr , t0,r )
P ∈ Km−1 , x, x0 , tr , t0,r ∈ C.
(2.13)
Through a direct calculation, we had φ2 = where
(n)
(n)
yV31 + Cm y 2 V21 − yAm + Bm ε(m)λFm−1 , = 2 (n) = (n) −ε(m)Em−1 yV21 + Am y V31 − yCm + Dm (n)
(n)
(n)
(n)
Am = V23 V31 − V33 V21 , (n)
(n)
(n)
(2.14)
(n)
(n)
(n)
(n)
(n)
(n)
(n)
Bm = V22 (V11 V21 + V23 V31 ) − λV21 (V12 V21 + V23 V32 ), (n)
(n)
(n)
(n)
Cm = λV21 V32 − V22 V31 , (n)
(n)
(n)
(n)
(n)
(n)
(n)
(n)
(n)
(n)
Dm = V31 (V11 V33 − λV13 V31 ) + λV32 (V21 V33 − V23 V31 ), (n)
(n)
(n)
(n)
(n)
(n)
(n)
(n)
(n)
Em−1 = −ε(m)[V23 (V21 V33 − V11 V21 − V23 V31 ) + λV13 (V21 )2 )], (n)
(n)
(n)
(n)
(n)
(n)
(n)
(n)
(n)
Fm−1 = ε(m)[V31 (V22 V32 − V11 V32 + V12 V31 ) − λV21 (V32 )2 ].
(2.15)
By observation, we knew Em−1 and Fm−1 are monic polynomials with respect to λ of degree m − 1. Hence we may write them in the following form Em−1 (λ, x, tr ) =
m−1 Y j=1
(λ − µj (x, tr )),
Fm−1 (λ, x, tr ) =
m−1 Y j=1
(λ − νj (x, tr )).
(2.16)
After discussing the singularies of the meromorphic function φ2 (P, x, tr ) as well as the BakerAkhiezer function ψ2 (P, x, x0 , tr , t0,r ) and with the help of the theory of trigonal curve, we obtained the Riemann theta function representations of them as ³ θ(λ(P, νˆ(x, tr )))θ(λ(P∞ , µ ˆ (x, tr ))) φ2 (P, x, tr ) = exp e(3) (Q0 ) − θ(λ(P∞ , νˆ(x, tr )))θ(λ(P, µ ˆ (x, tr )))
Z
P
Q0
θ(λ(P, µ ˆ (x, tr )))θ(λ(P∞ , µ ˆ (x0 , t0,r ))) ψ2 (P, x, x0 , tr , t0,r ) = θ(λ(P∞ , µ ˆ (x, tr )))θ(λ(P, µ ˆ (x0 , t0,r ))) Z P Z ³ (2) (2) (2) × exp (x − x0 )(e2 (Q0 ) − ωP∞ ,2 ) + (tr − t0,r )(es+1 (Q0 ) − Q0
P
Q0
´ (3) ωP∞ ,P0 , ´ ˜ (2) Ω ) P∞ ,s+1 .
(2.17)
(2.18)
Combining the asymptotic properties and Riemann theta function representations of the meromorphic function φ2 (P, x, tr ) and the Baker-Akhiezer function ψ2 (P, x, x0 , tr , t0,r ), we got the 4
following quasi-periodic solutions of modified Boussinesq hierarchy ux (x, tr ) − u2 (x, tr ) = 32 ∂x2 ln θ(λ(P∞ , νˆ(x, tr )))θ(λ(P∞ , µ ˆ(x, tr ))) + 43 [∂x ln v(x, tr ) = − 12 ∂x ln
θ(λ(P∞ , νˆ(x, tr ))) 1 + w . θ(λ(P∞ , µ ˆ(x, tr ))) 2 0
θ(λ(P∞ , νˆ(x, tr ))) − w0 ]2 − 3q1 , θ(λ(P∞ , µ ˆ(x, tr )))
(2.19) Unsatisfactorily, the expression of u(x, tr ) in (2.19) is given by a Riccati equation and it is implicit. Our purpose of this note is to arrive at explicit Riemann theta representations of u(x, tr ), v(x, tr ) by introducing another meromorphic function on Km−1 together with the above results.
3
Explicit quasi-periodic solutions of the modified Boussinesq hierarchy
In this section, we shall introduce another meromorphic function closely related to the BakerAkhiezer function. Then we obtain Riemann theta function representations for the BakerAkhiezer function, the meromorphic function and explicit quasi-periodic solutions of the modified Boussinesq hierarchy. Throughout this paper we use the same notations as [1], that is, the same notations express the same quantities in the two papers. Now, we introduce another Baker-Akhiezer function ψ3 (P, x, x0 , tr , t0,r ) on Km−1 by (2.12) and ψ3 (P, x0 , x0 , t0,r , t0,r ) = 1. Then, another meromorphic function φ3 (P, x, tr ) closely related to ψ3 (P, x, x0 , tr , t0,r ) is defined by φ3 (P, x, tr ) =
∂x ψ3 (P, x, x0 , tr , t0,r ) + u(x, tr ) − v(x, tr ), ψ3 (P, x, x0 , tr , t0,r )
P ∈ Km−1 , x, x0 , tr , t0,r ∈ C,
(3.1)
which implies from (2.12) that φ3 (P, x, tr ) = = = where
(n)
ε(m)λHm−1 (λ, x, tr ) − yCm (λ, x, tr ) + Em (λ, x, tr ) − yAm (λ, x, tr ) + Bm (λ, x, tr ) −ε(m)Fm−1 (λ, x, tr ) (n) yV12 (λ, x, tr ) + Cm (λ, x, tr ) , (n) yV32 (λ, x, tr ) + Am (λ, x, tr ) (n) y V12 (λ, x, tr ) (n) y 2 V32 (λ, x, tr ) 2
(n)
(n)
(n)
Am = V12 V31 − V11 V32 , (n)
(n)
(n)
(n)
(n)
(n)
(n)
(n)
(n)
(n)
Bm = V33 (V22 V32 + V12 V31 ) − λV32 (V23 V32 + V13 V31 ), (n)
(n)
(n)
(n)
Cm = λV13 V32 − V12 V33 , (n)
(n)
(n)
(3.2)
(n)
(n)
(n)
(n)
(n)
(n)
(n)
Em = V12 (V11 V22 − λV12 V21 ) + λV13 (V11 V32 − V12 V31 ),
5
(3.3)
(n)
(n)
(n)
(n)
(n)
(n)
(n)
(n)
Fm−1 = −ε(m)[λV21 (V32 )2 + V31 V32 (V11 − V22 ) − V12 (V31 )2 ], (n)
(n)
(n)
(n)
(n)
(n)
(n)
(n)
Hm−1 =(ε(m)[(V12 )2 V23 + V12 V13 (V33 − V22 ) − λ(V13 )2 V32 ], −1, if m = 3n + 2, ε(m) = 1, if m = 3n + 1.
(3.4)
We can easily show that there exist various interrelationships among the polynomials Am , Bm , Cm , Em , Fm−1 , Hm−1 , Sm , Tm , some of which are listed below: (n)
(n)
(n)
2, ε(m)λV32 Hm−1 = V12 Em − (V12 )2 Sm − Cm (n)
ε(m)λAm Hm−1 = (V12 )2 Tm + Cm Em , (n)
(n)
(n)
ε(m)V12 Fm−1 = (V32 )2 Sm − V32 Bm + A2m , (n)
−ε(m)Cm Fm−1 = (V32 )2 Tm + Am Bm ,
(n)
(n)
(n)
(n)
(n)
(n)
V12 V32 Tm + V12 Am Sm + V32 Cm Sm − Bm Cm − Am Em = 0, (n) V12 Am Tm
+
(n) V32 Cm Tm
(3.6)
(n)
V12 Bm + V32 Em − V12 V32 Sm + Am Cm = 0, (n)
(3.5)
(3.7)
− λFm−1 Hm−1 − Bm Em = 0,
(n)
ε(m)Fm−1,x = 2V32 Sm − 3Bm − 3ε(m)(u − v)Fm−1 , (n)
(n)
(n)
(n)
(n)
V12 Hm−1,x = 3[(u + v)V12 − V11 ]Hm−1 + ε(m)V13 (2V12 Sm − 3Em ).
(3.8)
Further properties of φ3 (P, x, tr ) are summarized in:
φ3,xx (P, x, tr ) + 3φ3 (P, x, tr )φ3,x (P, x, tr ) + 3[v(x, tr ) − u(x, tr )][φ3,x (P, x, tr ) + φ23 (P, x, tr )] +φ33 (P, x, tr ) − [2ux (x, tr ) + 6u(x, tr )v(x, tr ) − 2u2 (x, tr )]φ3 (P, x, tr ) = λ, (3.9) (φ3 (P, x, tr ) − u(x, tr ) + v(x, tr ))tr ¡ (r) (3.10) = ∂x Ve32 (λ, x, tr )[φ3,x (P, x, tr ) + φ23 (P, x, tr ) − 2u(x, tr )φ3 (P, x, tr )] ¢ (r) (r) +Ve (λ, x, tr )φ3 (P, x, tr ) + Ve (λ, x, tr ) , 31
33
λHm−1 (λ, x, tr ) , Fm−1 (λ, x, tr )
(3.11)
Fm−1,x (λ, x, tr ) + 3u(x, tr ) − 3v(x, tr ), Fm−1 (λ, x, tr )
(3.12)
φ3 (P, x, tr )φ3 (P ∗ , x, tr )φ3 (P ∗∗ , x, tr ) =
φ3 (P, x, tr ) + φ3 (P ∗ , x, tr ) + φ3 (P ∗∗ , x, tr ) =
1 1 1 + + φ3 (P, x, tr ) φ3 (P ∗ , x, tr ) φ3 (P ∗∗ , x, tr ) (n) (n) (n) V12 (λ, x, tr ) Hm−1,x (λ, x, tr ) [u(x, tr ) + v(x, tr )]V12 (λ, x, tr ) − V11 (λ, x, tr ) , = − (n) +3 (n) λV13 (λ, x, tr ) Hm−1 (λ, x, tr ) λV13 (λ, x, tr ) (3.13) ∗ ∗∗ 2 2 ∗ 2 ∗∗ φ3,x (P, x, tr ) + φ3,x (P , x, tr ) + φ3,x (P , x, tr ) + φ3 (P, x, tr ) + φ3 (P , x, tr ) + φ3 (P , x, tr ) (n)
= −3
V33 (λ, x, tr ) (n) V32 (λ, x, tr )
(n)
−[
V31 (λ, x, tr ) (n) V32 (λ, x, tr )
− 2u(x, tr )][
6
Fm−1,x (λ, x, tr ) + 3u(x, tr ) − 3v(x, tr )]. Fm−1 (λ, x, tr ) (3.14)
Lemma 3.1. Assume (2.12) and let (λ, x, tr ) ∈ C3 . Then (r) Ve32 (n) (n) V31 ) V32 (r) e (r) Ve (n) (r) e (r) − V32 V (n) )], +3Fm−1 (λ, x, tr )[Ve33 − 32 V + (u − v)( V 31 (n) 33 (n) 31 V32 V32 (r) Ve (n) (r) Hm−1,tr (λ, x, tr ) = Hm−1,x (λ, x, tr )(Ve12 − 13 (n) V12 ) V13 (r) e (r) Ve (r) (n) e (r) − V13 V (n) )]. +3Hm−1 (λ, x, tr )[Ve11 − 13 V − (u + v)( V 12 (n) 11 (n) 12 V13 V13 (3.15) Then ψ3 (P, x, x0 , tr , t0,r ) satisfy the following properties ³ (r) ψ3,tr (P, x, x0 , tr , t0,r ) = Ve32 (λ, x, tr )[φ3,x (P, x, tr ) + φ23 (P, x, tr ) − 2u(x, tr )φ3 (P, x, tr )] ´ (r) (r) +Ve31 (λ, x, tr )φ3 (P, x, tr ) + Ve33 (λ, x, tr ) ψ3 (P, x, x0 , tr , t0,r ), (3.16) ³Z x 0 0 0 0 ψ3 (P, x, x0 , tr , t0,r ) = exp [φ3 (P, x , tr ) − u(x , tr ) + v(x , tr )]dx x 0 Z tr ¡ (r) Ve32 (λ, x0 , t0 )[φ3,x (P, x0 , t0 ) + φ23 (P, x0 , t0 ) − 2u(x0 , t0 )φ3 (P, x0 , t0 )] + t0,r ¢ ´ (r) (r) +Ve31 (λ, x0 , t0 )φ3 (P, x0 , t0 ) + Ve33 (λ, x0 , t0 ) dt0 , (3.17) F (λ, x, t ) m−1 r ψ3 (P, x, x0 , tr , t0,r )ψ3 (P ∗ , x, x0 , tr , t0,r )ψ3 (P ∗∗ , x, x0 , tr , t0,r ) = . (3.18) Fm−1 (λ, x0 , t0,r ) (r)
Fm−1,tr (λ, x, tr ) = Fm−1,x (λ, x, tr )(Ve31 −
Due to the observation of (3.4), one infers that Fm−1 and Hm−1 are monic polynomials with respect to λ of degree m − 1. Hence we may write them in the following forms Fm−1 (λ, x, tr ) =
m−1 Y j=1
Hm−1 (λ, x, tr ) = Defining
(λ − νj (x, tr )),
m−1 Y j=1
(3.19)
(λ − ξj (x, tr )).
(3.20)
³ ´ ³ ´ Am (νj (x, tr ), x, tr ) νˆj (x, tr ) = νj (x, tr ), y(ˆ νj (x, tr )) = νj (x, tr ), − (n) ∈ Km−1 , V32 (νj (x, tr ), x, tr ) ³ ´ ³ ´ Cm (ξj (x, tr ), x, tr ) ξˆj (x, tr ) = ξj (x, tr ), y(ξˆj (x, tr )) = ξj (x, tr ), − (n) ∈ Km−1 , V12 (ξj (x, tr ), x, tr ) 1 ≤ j ≤ m − 1, (x, tr ) ∈ C2 , (n)
(3.21)
(3.22)
(n)
and P0 = (0, V33 |λ=0 ), where V33 |λ=0 = (v − u)an−1 − 13 (∂ − u + 3v)(∂ + u + 3v)bn−1 . In fact, the definition of νˆj (x, tr ) in (3.21) is equivalent to (3.20) in [1] by noting that (n)
(n)
(n)
(n)
(n)
(n)
(n)
(n)
Fm−1 (νj (x, tr ), x, tr ) = −ε(m)[λV21 (V32 )2 + V31 V32 (V11 − V22 ) − V12 (V31 )2 ]|λ=νj (x,tr ) (n)
(n)
= ε(m)[V31 Am − V32 Cm ]|λ=νj (x,tr ) = 0, 7
namely,
Am (νj (x, tr ), x, tr )
(n) V32 (νj (x, tr ), x, tr )
=
Cm (νj (x, tr ), x, tr ) (n)
V31 (νj (x, tr ), x, tr )
.
m−1 The dynamics of the zeros {νj (x, tr )}m−1 j=1 and {ξj (x, tr )}j=1 of Fm−1 (λ, x, tr ) and Hm−1 (λ, x, tr ) with respect to x and tr are then described in terms of Dubrovin-type equations. m−1 Lemma 3.2. Suppose the zeros {νj (x, tr )}m−1 j=1 and {ξj (x, tr )}j=1 of Fm−1 (λ, x, tr ) and m−1 Hm−1 (λ, x, tr ) remain distinct, respectively. Then {νj (x, tr )}m−1 j=1 and {ξj (x, tr )}j=1 satisfy the system of differential equations
νj,x (x, tr ) =
(n)
ε(m)V32 (νj (x, tr ), x, tr )[3y 2 (ˆ νj (x, tr )) + Sm (νj (x, tr ))] , m−1 Y (νj (x, tr ) − νk (x, tr )) k=1 k6=j
νj,tr (x, tr ) =
(n) (r) [V32 (νj (x, tr ), x, tr )Ve31 (νj (x, tr ), x, tr ) ε(m)[3y 2 (ˆ νj (x, tr )) + Sm (νj (x, tr ))] × , m−1 Y k=1 k6=j
ξj,x (x, tr ) =
(νj (x, tr ) − νk (x, tr ))
(n) ε(m)V13 (ξj (x, tr ), x, tr )[3y 2 (ξˆj (x, tr )) m−1 Y k=1 k6=j
1 ≤ j ≤ m − 1,
(3.23) (r) (n) e − V32 (νj (x, tr ), x, tr )V31 (νj (x, tr ), x, tr )] 1 ≤ j ≤ m − 1;
(3.24)
+ Sm (ξj (x, tr ))]
,
(ξj (x, tr ) − ξk (x, tr ))
1 ≤ j ≤ m − 1,
(3.25)
(n) (r) (r) (n) ξj,tr (x, tr ) = [V13 (ξj (x, tr ), x, tr )Ve12 (ξj (x, tr ), x, tr ) − Ve13 (ξj (x, tr ), x, tr )V12 (ξj (x, tr ), x, tr )] ε(m)[3y 2 (ξˆj (x, tr )) + Sm (ξj (x, tr ))] × , 1 ≤ j ≤ m − 1. m−1 Y (ξj (x, tr ) − ξk (x, tr )) k=1 k6=j
(3.26) For investigating the asymptotic expansion of φ3 (P, x, tr ) near P∞ ∈ Km−1 , we choose the local 1 coordinate ζ = λ− 3 . Substituting a power series ansatz into (3.9) and comparing the same powers of ζ, we arrive at ∞ 1 P χj (x, tr )ζ j , ζ ζ→0 j=0
φ3 (P, x, tr ) =
8
as
P → P∞ ,
(3.27)
where χ0 = 1, χ1 = u − v, χ2 = 13 (u2 − ux ) + v 2 + vx , χ3 = 13 (−2vxx − 2ux v − 6vvx + 2u2 v − 2v 3 ), χ4 = 91 (uxx + 3vxx − 2uux + 6ux v + 12vvx − 6u2 v + 6v 3 )x , j−1 j−1 j−1 P P P χj = − 13 [χj−2,xx + 3(v − u)χj−2,x + 3 χj−1−i χi,x + 3(v − u) χj−1−i χi + χi χj−i +
j−1 P j−i P
i=1 l=0
i=0
i=0
i=1
χi χl χj−i−l − (2ux + 6uv − 2u2 )χj−2 ], (j ≥ 3).
(3.28)
One infers from (3.2) and (3.27) that the divisor (φ3 (P, x, tr )) of φ3 (P, x, tr ) is given by (φ3 (P, x, tr )) = DP0 ,ξˆ1 (x,tr ),...,ξˆm−1 (x,tr ) (P ) − DP∞ ,ˆν1 (x,tr ),...,ˆνm−1 (x,tr ) (P ).
(3.29)
That is, P0 , ξˆ1 (x, tr ), . . . , ξˆm−1 (x, tr ) are the m zeros of φ3 (P, x, tr ) and P∞ , νˆ1 (x, tr ), . . . , νˆm−1 (x, tr ) its m poles. To obtain Riemann theta function representation for the Baker-Akhiezer function ψ3 (P, x, x0 , tr , t0,r ), we first consider the second integration in (3.17) and define the function Js (P, x, tr ), meromorphic on Km−1 × C2 , by (r) (r) Js (P, x, tr ) = Ve31 (λ, x, tr )φ3 (P, x, tr ) + Ve32 (λ, x, tr )[φ3,x (P, x, tr ) + φ23 (P, x, tr ) (r) −2u(x, tr )φ3 (P, x, tr )] + Ve (λ, x, tr ).
(3.30)
33
(r) (r) (r) Denote by J¯s (P, x, tr ) the associated homogeneous quantity replacing Ve31 , Ve32 , Ve33 by the ¯ (r,s) ¯ (r,s) ¯ (r,s) corresponding homogeneous polynomials Ve 31 , Ve 32 , Ve 33 , where ( (r) (r,s) Ve3j |α˜ 0 =1,α˜ 1 =···=α˜ r =β˜0 =···=β˜r =0 s = 3r + 2, ¯ Ve 3j = j = 1, 2, 3, (r) Ve3j |α˜ 0 =0,β˜0 =1,α˜ 1 =···=α˜ r =β˜1 =···=β˜r =0 s = 3r + 1,
and
(˜ α0 , β˜0 ) =
(
(1, β˜0 ), s = 3r + 2, ˜ β0 ∈ R. (0, 1), s = 3r + 1,
Lemma 3.3. Let s = 3r + 2 or 3r + 1, r ∈ N0 , (x, tr ) ∈ C2 , and λ = ζ −3 be the local coordinate near P∞ . Then J¯s (P, x, tr ) = ζ −s + O(ζ), as P → P∞ .
(3.31)
ζ→0
Proof. In order to make the proof more clear, we introduce some notations ( a ˜(r) |α˜ 0 =1,α˜ 1 =···=α˜ r =β˜0 =···=β˜r =0 s = 3r + 2, (r,s) ¯ a ˜ = ; (r) a ˜ |α˜ 0 =0,β˜0 =1,α˜ 1 =···=α˜ r =β˜1 =···=β˜r =0 s = 3r + 1, ¯˜(r,s) a = r
(
a ˜r |α˜ 0 =1,α˜ 1 =···=α˜ r =β˜0 =···=β˜r =0
a ˜r |α˜ 0 =0,β˜0 =1,α˜ 1 =···=α˜ r =β˜1 =···=β˜r =0 9
s = 3r + 2, s = 3r + 1,
.
From (2.12) and (3.30), one easily gets ¯ (r,s) ¯ (r,s) J¯s (P, x, tr ) = Ve 31 (λ, x, tr )φ3 (P, x, tr ) + Ve 32 (λ, x, tr )[φ3,x (P, x, tr ) + φ23 (P, x, tr ) ¯ (r,s) −2uφ3 (P, x, tr )] + Ve 33 (λ, x, tr ) ¯˜(r,s) (λ, x, tr )φ3 (P, x, tr ) + d¯˜(r,s) (λ, x, tr )[φ3,x (P, x, tr ) + φ2 (P, x, tr ) = a 3 ¯˜(r,s) (λ, x, tr ) − 1 (∂ + 3v − u)(∂ + u + 3v)˜¯b(r,s) (λ, x, tr ). −2uφ3 (P, x, tr )] + (v − u)a 3 (3.32) Using (3.27), we have J¯1 = φ3 − u + v = ζ −1 + O(ζ), as P → P∞ , J¯2 = φ3,x + φ23 + 2(v − u)φ3 − 31 (ux + 3vx − u2 + 3v 2 + 6uv) = ζ −2 + O(ζ), as P → P∞ . Therefore (3.31) holds for s = 1 and s = 2. Suppose one may rewrite (3.31) as J¯s (P, x, tr ) = ζ −s + ζ→0
∞ X j=1
γj (x, tr )ζ j , as P → P∞ ,
(3.33)
for some coefficients {γj (x, tr )}j∈N . From (3.10) and (3.30), it is easy to see that ¯ (r,s) ¯ (r,s) ∂x J¯s (P, x, tr ) = ∂x [Ve 31 (λ, x, tr )φ3 (P, x, tr ) + Ve 32 (λ, x, tr )(φ3,x (P, x, tr ) + φ23 (P, x, tr ) ¯ (r,s) −2uφ3 (P, x, tr )) + Ve 33 (λ, x, tr )] = ∂tr [φ3 (P, x, tr ) − u(x, tr ) + v(x, tr )], namely, ∂x (ζ −s +
∞ X
γj (x, tr )ζ j ) = ∂tr (ζ −1 +
j=1
∞ X j=2
∞ X χj (x, tr )ζ j−1 ) = ∂tr ( χj+1 (x, tr )ζ j ).
(3.34)
j=1
Using (3.27) and comparing coefficients of the same powers of ζ in (3.34) give rise to γj,x (x, tr ) = χj+1,tr (x, tr ), j = 1, 2, . . . ,
¯(r,s) γ1,x (x, tr ) = χ2,tr (x, tr ) = − 31 (ux − u2 − 3vx − 3v 2 )tr = −d˜r,x (x, tr ), γ2,x (x, tr ) = χ3,tr (x, tr ) = 31 (−2vxx − 6vvx − 2v 3 + 2u2 v − 2ux v)tr ¯˜(r,s) ¯˜(r,s) = −a (x, tr ))x , r,x (x, tr ) + 2(v(x, tr )dr
γ3,x (x, tr ) = χ4,tr (x, tr ) = 19 (uxx + 3vxx − 2uux + 12vvx + 6ux v − 6u2 v + 6v 3 )x,tr ¯(r,s) ¯(r,s) = 1 [(∂ − u + 3v)(∂ + u + 3v)˜br (x, tr ) − (ux + 3vx − u2 + 9v 2 )d˜r (x, tr )]x , 3
(3.35)
from which we deduce ¯(r,s) γ1 (x, tr ) = ²1 (tr ) − d˜r (x, tr ), ¯(r,s) ¯˜(r,s) γ2 (x, tr ) = ²2 (tr ) − a (x, tr ) + 2v(x, tr )d˜r (x, tr ), r ¯(r,s) ¯(r,s) γ3 (x, tr ) = ²3 (tr ) + 1 [(∂ − u + 3v)(∂ + u + 3v)˜br (x, tr ) − (ux + 3vx − u2 + 9v 2 )d˜r (x, tr )], 3
(3.36)
10
where ²1 (tr ), ²2 (tr ), ²3 (tr ) are integration constants. Next we note that the coefficients of the power series for φ3 (P, x, tr ) in the coordinate ζ near P∞ , and the coefficients of the homoge¯˜(r,s) (ζ, x, tr ), ¯˜b(r,s) (ζ, x, tr ) , c¯˜(r,s) (ζ, x, tr ) and d¯˜(r,s) (ζ, x, tr ), are differential neous polynomials a polynomials in u and v, with no arbitrary integration constants in their construction. From the definition of J¯s it follows that it also have no arbitrary integration constants, and must consist purely of differential polynomials of u and v. From these considerations, we know that ²1 (tr ) = ²2 (tr ) = ²3 (tr ) = 0. Hence one concludes ¯(r,s) ¯(r,s) ¯˜(r,s) J¯s (P, x, tr ) = ζ −s − d˜r ζ + [−a (x, tr ) + 2v(x, tr )˜br (x, tr )]ζ 2 r ¯(r,s) ¯(r,s) + 1 [(∂ − u + 3v)(∂ + u + 3v)˜br (x, tr ) − (ux + 3vx − u2 + 9v 2 )d˜r (x, tr )]ζ 3 3
+O(ζ 4 ), as P → P∞ .
(3.37)
On the other hand we have ¯ J¯s+3 (P, x, tr ) = d˜(r+1,s+3) (λ, x, tr )[φ3,x (P, x, tr ) + φ2 (P, x, tr ) − 2uφ3 (P, x, tr )] 3
¯˜(r+1,s+3) (λ, x, tr )φ3 (P, x, tr ) + (v − u)a ¯˜(r+1,s+3) (λ, x, tr ) +a ¯ − 31 (∂ − u + 3v)(∂ + u + 3v)˜b(r+1,s+3) (λ, x, tr ) ¯(r+1,s+3) = ζ −3 J¯s + d˜r (λ, x, tr )[φ3,x (P, x, tr ) + φ23 (P, x, tr ) − 2uφ3 (P, x, tr )] ¯˜r(r+1,s+3) (λ, x, tr )φ3 (P, x, tr ) + (v − u)a ¯˜(r+1,s+3) +a (λ, x, tr ) r ¯˜(r+1,s+3) 1 (λ, x, tr ) − (∂ − u + 3v)(∂ + u + 3v)br 3
= ζ −s−3 + O(ζ),
(3.38)
which implies (3.31). From (2.12), we know that Js (P, x, tr ) =
r X
α ˜ r−l J¯3l+2 (P, x, tr ) +
r X
Therefore Z tr r X Js (P, x, τ )dτ = (tr − t0,r ) (˜ αr−l ζ→0
t0,r
β˜r−l J¯3l+1 (P, x, tr ), s = 3r + 2 or 3r + 1.
(3.39)
1 1 + β˜r−l 3l+1 ) + O(ζ), as P → P∞ . ζ 3l+2 ζ
(3.40)
l=0
l=0
l=0
(2)
Let ωP∞ ,j , j = 3l + 2 or 3l + 1, l ∈ N0 , be the normalized differential of the second kind holomorphic on Km−1 \ {P∞ } with a pole of order j at P∞ , (2)
ωP∞ ,j (P ) = (ζ −j + O(1))dζ, ζ→0
as
P → P∞ .
(3.41)
Furthermore, define the normalized differential of the second kind by e (2) Ω P∞ ,s+1 =
r X l=0
α ˜ r−l (3l +
(2) 2)ωP∞ ,3l+3
+
r X l=0
(2) β˜r−l (3l + 1)ωP∞ ,3l+2 ,
(3.42)
where s = 3r +2 or 3r +1, r ∈ N0 . In addition, we define the vector of b-periods of the differential e (2) of the second kind Ω P∞ ,s+1 , Z 1 (2) (2) (2) e (2) e e e e (2) U = ( U , . . . , U ), U = Ω , j = 1, . . . , m − 1, (3.43) s+1 s+1,1 s+1,m−1 s+1,j 2πi bj P∞ ,s+1 11
with s = 3r + 2 or 3r + 1, r ∈ N0 . Integrating (3.42) yields Z
P
Q0
e (2) Ω P∞ ,s+1
r X
=
α ˜ r−l (3l + 2)
=
−
ζ→0
l=0
ζ
ζ0
l=0
r X
Z
α ˜ r−l
1 ζ 3l+2
(2) ωP∞ ,3l+3
r X
−
β˜r−l
l=0
+ 1
r X
Z
β˜r−l (3l + 1)
ζ0
l=0
ζ 3l+1
ζ
(2)
ωP∞ ,3l+2
(2)
+ es+1 (Q0 ) + O(ζ), as P → P∞ , (3.44)
(2)
where es+1 (Q0 ) is a constant. Combining (3.40) and (3.44) yields Z
tr
t0,r
(2)
Js (P, x, τ )dτ = (tr − t0,r )(es+1 (Q0 ) − ζ→0
Z
P
Q0
e (2) Ω P∞ ,s+1 ) + O(ζ), as P → P∞ .
(3.45)
(3)
Next, we introduce the normalized Abel differential ωP∞ ,P0 (P ) of the third kind which is the unique differential holomorphic on Km−1 \{P∞ , P0 } with simple poles at P∞ and P0 with residues ±1, respectively, that is, (3)
ωP∞ ,P0 (P ) = (ζ −1 − w1 + O(ζ))dζ, ζ→0
(3)
ωP∞ ,P0 (P ) = (−ζ −1 + O(ζ))dζ, ζ→0
then
Z
P
ZQP0
Q0
as as
(3)
ωP∞ ,P0 (P ) = ln ζ + e(3) (Q0 ) − w1 ζ + O(ζ 2 ), (3) ωP∞ ,P0 (P )
= − ln ζ + e
(3)
(Q0 ) + O(ζ),
as
P → P∞ ,
(3.46)
P → P0 ,
as
P → P∞ ,
(3.47)
P → P0
with w1 and e(3) (Q0 ) two constants. Given these preparations, the theta function representations of φ3 (P, x, tr ) and ψ3 (P, x, x0 , tr , t0,r ) read as follows. Theorem 3.4. Assume that the curve Km−1 is nonsingular. Let P = (λ, y) ∈ Km−1 \ {P∞ } and let (x, tr ), (x0 , t0,r ) ∈ Ων , where Ων ⊆ C2 is open and connected. Suppose that Dνˆ(x,tr ) , or equivalently, Dˆξ(x,tr ) is nonspecial for (x, tr ) ∈ Ων . Then ³ θ(λ(P, ˆξ(x, tr )))θ(λ(P∞ , νˆ(x, tr ))) φ3 (P, x, tr ) = exp e(3) (Q0 ) − θ(λ(P∞ , ˆξ(x, tr )))θ(λ(P, νˆ(x, tr )))
Z
P
Q0
θ(λ(P, νˆ(x, tr )))θ(λ(P∞ , νˆ(x0 , t0,r ))) ψ3 (P, x, x0 , tr , t0,r ) = θ(λ(P∞ , νˆ(x, tr ))θ(λ(P, νˆ(x0 , t0,r ))) Z Z P ³ (2) (2) (2) × exp (x − x0 )(e2 (Q0 ) − ωP∞ ,2 ) + (tr − t0,r )(es+1 (Q0 ) −
P
Q0
Q0
´ (3) ωP∞ ,P0 . ´ ˜ (2) Ω ) P∞ ,s+1 .
Proof. It follows from (3.47) that Z P ´ ³ (3) (3) ωP∞ ,P0 = ζ −1 + O(1), as P → P∞ , exp e (Q0 ) − ζ→0 ZQP0 ´ ³ (3) ωP∞ ,P0 = ζ + O(1), as P → P0 . exp e(3) (Q0 ) − Q0
ζ→0
12
(3.48)
(3.49)
(3.50)
Using (3.29) we immediately know that φ3 has simple poles at νˆ(x, tr ) and P∞ , and simple zeros ˆ tr ). Let Φ3 be defined by the right hand side of (3.48) with the aim to prove at P0 and ξ(x, that φ3 = Φ3 . By (3.50) and a special case of Riemann’s vanishing theorem [29-31], we see that Φ3 has the same properties. Using the Riemann-Roch theorem [29-31], we conclude that the holomorphic function Φ3 = ς, where ς is a constant with respect to P . Using (3.27) and (3.50), φ3 we have −1 Φ3 = (1 + O(ζ))(ζ + O(1)) = 1 + O(ζ), as P → P , (3.51) ∞ −1 φ3 ζ→0 ζ→0 (ζ + O(1)) from which we conclude ς = 1. Let ψ3 (P, x, x0 , tr , t0,r ) be defined as in (3.17) and denote the right-hand side of (3.49) by Ψ3 (P, x, x0 , tr , t0,r ). In order to prove that ψ3 = Ψ3 , one uses (3.15) and (n) (n) (n) V31 φ3 + V32 [φ3,x + φ23 − 2uφ3 ] + V33 = y, to compute φ3 (P, x, tr ) − u(x, tr ) + v(x, tr )
(n)
= = = =
λ→νj (x)
=
λ→νj (x)
Js (P, x, tr )
= = = =
λ→νj (x,tr )
y 2 V32 − yAm + Bm −u+v −ε(m)Fm−1 (n) (n) y 2 V32 − yAm + 23 V32 Sm − 13 ε(m)Fm−1,x −ε(m)Fm−1 (n) m V32 y(y + A(n) ) 1 Fm−1,x 2 (n) 3y 2 + Sm V32 V32 + + 3 −ε(m)Fm−1 3 Fm−1 ε(m)Fm−1 ν − j,x + O(1) λ − νj ∂x ln(λ − νj (x, tr )) + O(1),
(3.52) (r) (r) (r) Ve31 φ3 + Ve32 [φ3,x + φ23 − 2uφ3 ] + Ve33 (r) (r) (r) ³ ´ Ve32 (n) Ve32 Ve (n) (r) (r) e V φ + V − V + y Ve31 − 32 3 33 (n) 31 (n) 33 (n) V32 V32 V32 (n) m yV32 (y + A(n) ) (r) (n) (r) 1 Fm−1,tr ³ e (r) Ve32 (n) ´ 2 V32 (3y 2 + Sm ) Ve32 V32 + V31 − (n) V31 [ + ] + y (n) 3 Fm−1 3 −ε(m)Fm−1 ε(m)Fm−1 V32 V32 νj,tr + O(1). − λ − νj (3.53)
More concisely, φ3 (P, x0 , tr ) − u(x0 , tr ) + v(x0 , tr ) = ∂x0 ln(λ − νj (x0 , tr )) + O(1), as P → νˆj (x0 , tr ), Js (P, x0 , t0 ) = ∂t0 ln(λ − νj (x0 , t0 )) + O(1), as P → νˆj (x0 , t0 ). Hence
³Z
´ (∂x0 ln(λ − νj (x0 , tr )) + O(1))dx0 x0 ´ ³Z tr (∂t0 ln(λ − νj (x0 , t0 )) + O(1))dt0 × exp
ψ3 (P, x, x0 , tr , t0,r ) = exp
x
t0,r
13
(3.54)
λ − νj (x, tr ) λ − νj (x0 , tr ) × O(1) λ − νj (x0 , tr ) λ − νj (x0 , t0,r ) for P → νˆj (x0 , tr ) 6= νˆj (x0 , t0,r ), (λ − νj (x0 , tr ))O(1) = O(1) for P → νˆj (x0 , tr ) = νˆj (x0 , t0,r ), −1 (λ − νj (x0 , t0,r )) O(1) for P → νˆj (x0 , t0,r ) 6= νˆj (x0 , tr ),
=
(3.55)
where O(1) 6= 0 in (3.55). Consequently, all zeros and poles of ψ3 and Ψ3 on Km−1 \ {P∞ } are simple and coincident. It remains to identify the essential singularity of ψ3 and Ψ3 at P∞ . By (3.45) we see that the singularities in the exponential terms of ψ3 and Ψ3 coincide. The uniqueness result for Baker-Akhiezer functions completes the proof that ψ3 = Ψ3 . ∼ DP∞ νˆ(x,tr ) . Using Abel Theorem, we have From (3.29), we know that DP0 ξ(x,t ˆ r) ). AQ0 (P∞ ) + αQ0 (Dνˆ(x,tr ) ) = AQ0 (P0 ) + αQ0 (Dξ(x,t ˆ r)
(3.56)
As we have shown in [1] that (2) e (2) (tr − t0,r ). αQ0 (Dνˆ(x,tr ) ) = αQ0 (Dνˆ(x0 ,t0,r ) ) + U 2 (x − x0 ) + U s+1
(3.57)
(2) e (2) αQ0 (Dξ(x,t ) = αQ0 (Dξ(x ˆ ˆ 0 ,t0,r ) ) + U 2 (x − x0 ) + U s+1 (tr − t0,r ). r)
(3.58)
Combining (3.56) and (3.57), we arrive at
Then the modified Boussinesq flows are straightened by the Abel map.
Theorem 3.5. Assume that the curve Km−1 is nonsingular and let (x, tr ), (x0 , t0,r ) ∈ Ων , where Ων ⊆ C2 is open and connected. Then (2) e (2) αQ0 (Dνˆ(x,tr ) ) = αQ0 (Dνˆ(x0 ,t0,r ) ) + U 2 (x − x0 ) + U s+1 (tr − t0,r ), (2) (2) e s+1 (tr − t0,r ). αQ (D ˆ ) = αQ (D ˆ ) + U (x − x0 ) + U 0
ξ(x,tr )
0
ξ(x0 ,t0,r )
(3.59)
2
Immediately, we shall show the explicit theta function representations of solutions of the modified Boussinesq hierarchy. Theorem 3.6. Assume that the curve Km−1 is nonsingular and let (x, tr ) ∈ Ων , where Ων ⊆ C2 is open and connected. Suppose also that Dνˆ(x,tr ) , or equivalently, Dξ(x,t is nonspecial ˆ r) for (x, tr ) ∈ Ων . Then θ(λ(P∞ , νˆ(x, tr ))) 1 + w , θ(λ(P∞ , µ ˆ(x, tr ))) 2 0 ˆ tr ))) + 1 ∂x ln θ(λ(P∞ , µ ˆ(x, tr )))θ(λ(P∞ , νˆ(x, tr ))) + w1 + 12 w0 u(x, tr ) = −∂x ln θ(λ(P∞ , ξ(x, 2 (3.60) with w0 and w1 are constants. v(x, tr ) = − 21 ∂x ln
Proof. The theta function representation of potential v(x, tr ) has been obtained in [1]. Now we only need to derive explicit theta functions representation of u(x, tr ). Using Theorem 3.4
14
and Theorem 3.5, one can derive θ(ΞQ0 − AQ0 (P ) + αQ0 (Dξ(x,t )) ˆ tr ))) ˆ θ(λ(P, ξ(x, r) = ˆ tr ))) θ(ΞQ0 − AQ0 (P∞ ) + αQ0 (Dξ(x,t )) ˆ θ(λ(P∞ , ξ(x, r) Z P∞ θ(ΞQ0 − AQ0 (P∞ ) + αQ0 (Dξ(x,t )+ ω) ˆ r) P = θ(ΞQ0 − AQ0 (P∞ ) + αQ0 (Dξ(x,t )) ˆ r)
1 ) − ρ0,j ζ − ρ1,j ζ 2 + O(ζ 4 ), . . . ) θ(. . . , ΞQ0 ,j − AQ0 ,j (P∞ ) + αQ0 ,j (Dξ(x,t ˆ r) 2 = θ(ΞQ0 − AQ0 (P∞ ) + αQ0 (Dξ(x,t )) ζ→0 ˆ ) r =
ζ→0
(3.61)
m−1 X ∂θ2 ¤ 1£ ρ0,j ζ + O(ζ 2 ) θ2 − θ2 ∂λj j=1
= 1 − ∂x ln θ2 ζ + O(ζ 2 ),
ζ→0
P → P∞ ,
where θ2 = θ(ΞQ0 − AQ0 (P∞ ) + αQ0 (Dξ(x,t )). Similarly, we have ˆ r) θ(λ(P, νˆ(x, tr ))) = 1 − ∂x ln θ1 ζ + O(ζ 2 ), θ(λ(P∞ , νˆ(x, tr ))) ζ→0 with θ1 = θ(ΞQ0 − AQ0 (P∞ ) + αQ0 (Dνˆ(x,tr ) )). Then we have φ3 (P, x, tr )
= =
ζ→0
=
ζ→0
(3.62)
P → P∞ ,
³ θ(λ(P, ˆξ(x, tr )))θ(λ(P∞ , νˆ(x, tr ))) exp e(3) (Q0 ) − θ(λ(P∞ , ˆξ(x, tr )))θ(λ(P, νˆ(x, tr ))) θ2 (1 − ∂x ln ζ + O(ζ 2 ))(ζ −1 + w1 + O(ζ)) θ1 θ2 −1 + w1 + O(ζ), P → P∞ . ζ − ∂x ln θ1
Z
P
Q0
(3)
ωP∞ ,P0
´
(3.63)
On the other hand, we have from (3.27) that φ3 (P, x, tr ) = ζ −1 + u − v + O(ζ), ζ→0
P → P∞ .
(3.64)
By comparing (3.63) and (3.64), we arrive at u − v = −∂x ln
θ2 + w1 , θ1
(3.65)
which together with the first equality of (3.60) indicates the second one.
Acknowledgments This work was supported by National Natural Science Foundation of China (project nos. 11171312, 11326163 and 11401230).
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