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The application of trigonal curve theory to the second-order Benjamin-Ono hierarchy

Advances in Difference Equations20142014:195

https://doi.org/10.1186/1687-1847-2014-195

  • Received: 11 March 2014
  • Accepted: 3 July 2014
  • Published:

Abstract

By introducing two sets of Lenard recursion equations, the second-order Benjamin-Ono hierarchy is proposed. In view of the characteristic polynomial of Lax matrix, a trigonal curve of arithmetic genus m 1 is deduced. Then the trigonal curve theory is used to derive the explicit algebro-geometric solutions represented in theta functions to the second-order Benjamin-Ono hierarchy with the help of the properties of Baker-Akhiezer function, the meromorphic function and the three kinds of Abel differentials.

MSC:35Q51, 37K10, 14H70, 35C99.

Keywords

  • second-order Benjamin-Ono hierarchy
  • algebro-geometric solutions
  • trigonal curve

1 Introduction

The principal aim of the present paper concerns the algebro-geometric solutions of the second-order Benjamin-Ono hierarchy with the aid of the theory of trigonal curves [13]. To the best of the authors’ knowledge, there have been no results about the algebro-geometric solutions of the second-order Benjamin-Ono equation [4, 5]
u t t = α ( u 2 ) x x + β u x x x x ,
(1.1)

which is used in the analysis of long waves in shallow water and many other physical applications, where α is a constant controlling nonlinearity and the characteristic speed of the long waves, and β is the depth of the fluid, although there are some results about the exact solutions of (1.1), such as the pulse-type and kink-type solutions, periodic solitary wave and double periodic solutions, soliton solutions etc., by using the following methods: the Jacobi elliptic function expansion method, the bilinear method, the extended homoclinic test approach, the homogeneous balance method and the lattice Boltzmann method etc. [610].

Before turning to the contents of each section, it seems appropriate to review the existing literature on algebro-geometric solutions, which are of great importance for revealing inherent structure mechanism of solutions and describing the quasi-periodic behavior of nonlinear phenomena. During the last few years, there have been fairly mature techniques to construct algebro-geometric solutions of soliton equations associated with 2 × 2 matrix spectral problems, such as the KdV, nonlinear Schrödinger, sine-Gordon, Toda equations and so on [1115]. Unfortunately, the situation is not so good for soliton equations associated with 3 × 3 matrix spectral problems, which are more complicated and more difficult. In [16], a unified framework was proposed to yield all algebro-geometric solutions of the entire Boussinesq hierarchy. Recently, based on the characteristic polynomial of Lax matrix associated with the 3 × 3 matrix spectral problems, we have developed the method in [16] to deal with some important soliton equations by introducing the trigonal curves of arithmetic genus m 1 and deriving the explicit Riemann theta function representations of the entire hierarchies, such as the modified Boussinesq, the Kaup-Kupershmidt hierarchies and others [1719].

The present paper is organized as follows. In Section 2, based on two kinds of different Lenard recursion equations, we derive the second-order Benjamin-Ono hierarchy, which relates to a 3 × 3 matrix spectral problem. In Section 3, we introduce the Baker-Akhiezer function and the associated meromorphic function. Then the second-order Benjamin-Ono hierarchy is decomposed into the system of Dubrovin-type ordinary differential equations. In Section 4, the explicit Riemann theta function representations of the Baker-Akhiezer function and the meromorphic function, and especially of the solutions to the entire second-order Benjamin-Ono hierarchy are displayed by resorting to the Riemann theta functions, the holomorphic differentials, and the Abel map.

2 The zero-curvature representation to the second-order Benjamin-Ono hierarchy

In this section, we shall derive the second-order Benjamin-Ono hierarchy associated with the 3 × 3 matrix spectral problem
ψ x = U ψ , ψ = ( ψ 1 ψ 2 ψ 3 ) , U = ( 0 1 0 u 0 1 v + λ u 0 ) ,
(2.1)
where u and v are two potentials, and λ is a constant spectral parameter. To this end, we introduce two sets of Lenard recursion equations
K g j 1 = J g j , g j | ( u , v ) = 0 = 0 , j 0 ,
(2.2)
K g ˆ j 1 = J g ˆ j , g ˆ j | ( u , v ) = 0 = 0 , j 0
(2.3)
with two starting points
g 1 = ( 1 , 0 ) T , g ˆ 1 = ( 0 , 1 ) T ,
where the initial conditions mean to identify constants of integration as zero, and two operators are defined as follows:
K = ( u + u 3 v + 1 2 v 2 v + v 1 6 5 1 3 ( 3 u + u 3 ) 1 2 ( 2 u + u 2 ) + u 2 + u 2 + 2 3 u u ) , J = ( 0 3 2 3 0 ) .
Hence g j and g ˆ j are uniquely determined, for example, the first two members read as
g 0 = 1 3 ( v 2 u ) , g ˆ 0 = 1 9 ( u x x 4 u 2 6 v ) .
In order to generate a hierarchy of evolution equations associated with the spectral problem (2.1), we solve the stationary zero-curvature equation
V x [ U , V ] = 0 , V = ( V i j ) 3 × 3 ,
(2.4)
which is equivalent to
V 11 , x + u V 12 + ( v + λ ) V 13 V 21 = 0 , V 12 , x + u V 13 + V 11 V 22 = 0 , V 13 , x V 23 + V 12 = 0 , V 21 , x + u ( V 22 V 11 ) + ( v + λ ) V 23 V 31 = 0 , V 22 , x + u ( V 23 V 12 ) + V 21 V 32 = 0 , V 23 , x u V 13 + V 22 V 33 = 0 , V 31 , x + u ( V 32 V 21 ) + ( v + λ ) ( V 33 V 11 ) = 0 , V 32 , x + u ( V 33 V 22 ) ( v + λ ) V 12 + V 31 = 0 , V 33 , x u V 23 ( v + λ ) V 13 + V 32 = 0 ,
(2.5)
where each entry V i j = V i j ( a , b ) is a Laurent expansion in λ:
V 11 = 1 3 ( 1 2 2 u ) b a , V 12 = a 1 2 b , V 13 = b , V 21 = ( 1 6 3 1 3 u 1 2 u + v + λ ) b + ( u 2 ) a , V 22 = 1 3 ( 2 + 2 u ) b , V 23 = a + 1 2 b , V 31 = ( 1 6 4 1 3 2 u 1 2 u 1 2 u 2 + u 2 ) b + ( v + λ ) a ,
(2.6)
V 32 = ( 1 6 3 + 1 3 u + 1 2 u + v + λ ) b + ( u 2 ) a , V 33 = 1 3 ( 1 2 2 u ) b + a , a = j 0 a j 1 λ j , b = j 0 b j 1 λ j .
(2.7)
A direct calculation shows that (2.5) and (2.6) imply the Lenard equation
K G = λ J G , G = ( a , b ) T .
(2.8)
Substituting (2.7) into (2.8) and collecting terms with the same powers of λ, we arrive at the following recursion relation:
K G j 1 = J G j , J G 1 = 0 , j 0 ,
(2.9)
where G j = ( a j , b j ) T . Since the equation J G 1 = 0 has the general solution
G 1 = α 0 g 1 + β 0 g ˆ 1 ,
(2.10)
then G j can be expressed as
G j = α 0 g j + β 0 g ˆ j + + α j g 0 + β j g ˆ 0 + α j + 1 g 1 + β j + 1 g ˆ 1 , j 0 ,
(2.11)

where α j and β j are arbitrary constants.

Let ψ satisfy the spectral problem (2.1) and its auxiliary problem
ψ t r = V ˜ ( r ) ψ , V ˜ ( r ) = ( V ˜ i j ( r ) ) 3 × 3 ,
(2.12)
where each entry V ˜ i j ( r ) = V ˜ i j ( a ˜ ( r ) , b ˜ ( r ) ) ,
a ˜ ( r ) = j = 0 r a ˜ j 1 λ r j , b ˜ ( r ) = j = 0 r a ˜ j 1 λ r j
with
G ˜ j = ( a ˜ j , b ˜ j ) T = α ˜ 0 g j + β ˜ 0 g ˆ j + + α ˜ j g 0 + β ˜ j g ˆ 0 + α ˜ j + 1 g 1 + β ˜ j + 1 g ˆ 1 , j 1 .
Then the compatibility condition of (2.1) and (2.12) yields the zero-curvature equation, U t r V ˜ x ( r ) + [ U , V ˜ ( r ) ] = 0 , which is equivalent to the hierarchy of nonlinear evolution equations
( u t r , v t r ) T = X ˜ r , r 0 ,
(2.13)
where the vector fields X ˜ j = X ˜ j ( u , v ; α ̲ ˜ ( j ) , β ̲ ˜ ( j ) ) = K G ˜ j 1 = J G ˜ j , and α ̲ ˜ ( j ) = ( α ˜ 0 , , α ˜ j ) , β ̲ ˜ ( j ) = ( β ˜ 0 , , β ˜ j ) . The first nontrivial member in the hierarchy (2.13) is as follows:
u t 0 = α ˜ 0 u x + β ˜ 0 v x , v t 0 = α ˜ 0 v x 1 3 β ˜ 0 ( u x x x 8 u u x ) .
(2.14)
For α ˜ 0 = 0 , β ˜ 0 = 1 ( t 0 = t ), equation (2.14) is reduced to the second-order Benjamin-Ono equation by canceling the variable v
u t t = 4 3 ( u 2 ) x x 1 3 u x x x x .
(2.15)
The second one in the hierarchy (2.13) (as α ˜ 1 = 0 , β ˜ 1 = 0 ) can be written as
u t 1 = 1 3 α ˜ 0 ( v x x 4 u v ) x 1 54 β ˜ 0 ( 6 u x x x x 60 u u x x 45 u x 2 + 40 u 3 + 45 v 2 ) x , v t 1 = 1 27 α ˜ 0 ( 3 u x x x x 36 u u x x 18 u x 2 + 32 u 3 + 18 v 2 ) x v t 1 = 1 9 β ˜ 0 ( v x x x x 5 u x x v 10 u v x x 5 u x v x + 20 u 2 v ) x .
(2.16)
For α ˜ 0 = 0 , β ˜ 0 = 9 ( t 1 = t ), equation (2.16) is reduced to a 5-order coupled equation
u t = u x x x x x ( 10 u u x x + 9 u x 2 9 v 2 20 3 u 3 ) x , v t = v x x x x x ( 5 u x x v + 10 u v x x + 5 u x v x 20 u 2 v ) x .
(2.17)

3 The meromorphic function and Dubrovin-type equations

In this section, we shall consider the Baker-Akhiezer function and the associated meromorphic function. By introducing the elliptic kind coordinates, we decompose the second-order Benjamin-Ono equation into the system of Dubrovin-type differential equations.

We first introduce the Baker-Akhiezer function ψ ( P , x , x 0 , t r , t 0 , r ) by
ψ x ( P , x , x 0 , t r , t 0 , r ) = U ( u ( x , t r ) , v ( x , t r ) ; λ ( P ) ) ψ ( P , x , x 0 , t r , t 0 , r ) , ψ t r ( P , x , x 0 , t r , t 0 , r ) = V ˜ ( r ) ( u ( x , t r ) , v ( x , t r ) ; λ ( P ) ) ψ ( P , x , x 0 , t r , t 0 , r ) , V ( n ) ( u ( x , t r ) , v ( x , t r ) ; λ ( P ) ) ψ ( P , x , x 0 , t r , t 0 , r ) = y ( P ) ψ ( P , x , x 0 , t r , t 0 , r ) , ψ 1 ( P , x 0 , x 0 , t 0 , r , t 0 , r ) = 1 ,
(3.1)
where V ( n ) = ( λ n V ) + = ( V i j ( n ) ) 3 × 3 and V i j ( n ) = V i j ( a ( n ) , b ( n ) ) ,
a ( n ) = j = 0 n a j 1 λ n j , b ( n ) = j = 0 n b j 1 λ n j
with a j , b j determined by (2.11). The compatibility conditions of the first three expressions in (3.1) yield that
U t r V ˜ x ( r ) + [ U , V ˜ ( r ) ] = 0 ,
(3.2)
V x ( n ) + [ U , V ( n ) ] = 0 ,
(3.3)
V t r ( n ) + [ V ˜ ( r ) , V ( n ) ] = 0 .
(3.4)
Through a direct calculation we can show that y I V ( n ) satisfies equations (3.3) and (3.4). So is an independent constant of the variables x and t r , from which we can define a trigonal curve K m 1 : F m ( λ , y ) = 0 with the expansion
(3.5)
where
S m = 1 i < j 3 | V i i ( n ) V i j ( n ) V j i ( n ) V j j ( n ) | , T m = | V 11 ( n ) V 12 ( n ) V 13 ( n ) V 21 ( n ) V 22 ( n ) V 23 ( n ) V 31 ( n ) V 32 ( n ) V 33 ( n ) | .

Immediately, from (2.10) if we choose β 0 = 1 , α 0 an arbitrary constant or β 0 = 0 , α 0 = 1 , we shall know that the corresponding values of m in (3.5) are 3 n + 2 or 3 n + 1 , respectively. For the convenience, the compactification of the curve K m 1 is denoted by the same symbol K m 1 . Thus K m 1 becomes a three-sheeted Riemann surface of arithmetic genus m 1 when it is nonsingular or smooth.

Next we shall introduce the meromorphic function ϕ 1 ( P , x , t r ) , which is closely related to ψ ( P , x , x 0 , t r , t 0 , r ) , by
ϕ 1 ( P , x , t r ) = x ψ 1 ( P , x , x 0 , t r , t 0 , r ) ψ 1 ( P , x , x 0 , t r , t 0 , r ) , P K m 1 , x C ,
(3.6)
which implies from (3.1) that
ϕ 1 ( P , x , t r ) = ε ( m ) F m ( λ , x , t r ) y 2 V 23 ( n ) ( λ , x , t r ) y C m ( λ , x , t r ) + D m ( λ , x , t r ) = y 2 V 13 ( n ) ( λ , x , t r ) y A m ( λ , x , t r ) + B m ( λ , x , t r ) ε ( m ) E m 1 ( λ , x , t r ) = y V 23 ( n ) ( λ , x , t r ) + C m ( λ , x , t r ) y V 13 ( n ) ( λ , x , t r ) + A m ( λ , x , t r ) ,
(3.7)
where P = ( λ , y ) K m 1 , ( x , t r ) C 2 ,
A m = V 12 ( n ) V 23 ( n ) V 13 ( n ) V 22 ( n ) , B m = V 13 ( n ) ( V 11 ( n ) V 33 ( n ) V 13 ( n ) V 31 ( n ) ) + V 12 ( n ) ( V 11 ( n ) V 23 ( n ) V 13 ( n ) V 21 ( n ) ) , C m = V 13 ( n ) V 21 ( n ) V 11 ( n ) V 23 ( n ) , D m = V 23 ( n ) ( V 22 ( n ) V 33 ( n ) V 23 ( n ) V 32 ( n ) ) + V 21 ( n ) ( V 13 ( n ) V 22 ( n ) V 12 ( n ) V 23 ( n ) ) ,
(3.8)
E m 1 = ε ( m ) [ V 13 ( n ) ( V 13 ( n ) V 32 ( n ) V 12 ( n ) V 33 ( n ) ) + V 12 ( n ) ( V 13 ( n ) V 22 ( n ) V 12 ( n ) V 23 ( n ) ) ] , F m = ε ( m ) [ V 23 ( n ) ( V 23 ( n ) V 31 ( n ) V 21 ( n ) V 33 ( n ) ) + V 21 ( n ) ( V 11 ( n ) V 23 ( n ) V 13 ( n ) V 21 ( n ) ) ] ,
(3.9)
and
ε ( m ) = { 1 if  m = 3 n + 2 , 1 if  m = 3 n + 1 ,
which is introduced to ensure that E m 1 , F m are both monic polynomials. It is easy to see that there exist various interrelationships between polynomials A m , B m , C m , D m , E m 1 , F m and S m , T m , some of which are summarized as follows:
ε ( m ) V 13 ( n ) F m = V 23 ( n ) D m S m ( V 23 ( n ) ) 2 C m 2 , ε ( m ) A m F m = T m ( V 23 ( n ) ) 2 + C m D m , ε ( m ) V 23 ( n ) E m 1 = S m ( V 13 ( n ) ) 2 V 13 ( n ) B m + A m 2 , ε ( m ) C m E m 1 = T m ( V 13 ( n ) ) 2 + A m B m ,
(3.10)
V 23 ( n ) B m + V 13 ( n ) D m V 13 ( n ) V 23 ( n ) S m + A m C m = 0 , V 13 ( n ) V 23 ( n ) T m + V 23 ( n ) A m S m + V 13 ( n ) C m S m B m C m A m D m = 0 , V 23 ( n ) A m T m + V 13 ( n ) C m T m E m 1 F m B m D m = 0 ,
(3.11)
ε ( m ) E m 1 , x = 2 S m V 13 ( n ) 3 B m , V 23 ( n ) F m , x = 3 V 22 ( n ) F m + ε ( m ) ( V 21 ( n ) u V 23 ( n ) ) ( 2 V 23 ( n ) S m 3 D m ) .
(3.12)
For displaying the properties of ϕ 1 ( P , x , t r ) exactly, we introduce the holomorphic map , changing sheets, as
: { K m 1 K m 1 , P = ( λ , y i ( λ ) ) P = ( λ , y i + 1 ( mod 3 ) ( λ ) ) , i = 0 , 1 , 2 , P : = ( P ) , etc.,
where y i ( λ ) , i = 0 , 1 , 2 , denote the three branches of y ( P ) satisfying F m ( λ , y ) = 0 . Then it is easy to show the properties of ϕ 1 ( P , x , t r ) immediately:
ϕ 1 , x x ( P , x , t r ) + 3 ϕ 1 ( P , x , t r ) ϕ 1 , x ( P , x , t r ) + ϕ 1 3 ( P , x , t r ) 2 u ( x , t r ) ϕ 1 ( P , x , t r ) = u x ( x , t r ) + v ( x , t r ) + λ ,
(3.13)
ϕ 1 , t r ( P , x , t r ) = x [ V ˜ 11 ( r ) ( λ , x , t r ) + V ˜ 12 ( r ) ( λ , x , t r ) ϕ 1 ( P , x , t r ) ϕ 1 , t r ( P , x , t r ) = + V ˜ 13 ( r ) ( λ , x , t r ) ( ϕ 1 , x ( P , x , t r ) + ϕ 1 2 ( P , x , t r ) u ( x , t r ) ) ] ,
(3.14)
ϕ 1 ( P , x , t r ) ϕ 1 ( P , x , t r ) ϕ 1 ( P , x , t r ) = F m ( λ , x , t r ) E m 1 ( λ , x , t r ) ,
(3.15)
ϕ 1 ( P , x , t r ) + ϕ 1 ( P , x , t r ) + ϕ 1 ( P , x , t r ) = E m 1 , x ( λ , x , t r ) E m 1 ( λ , x , t r ) ,
(3.16)
y ( P ) ϕ 1 ( P , x , t r ) + y ( P ) ϕ 1 ( P , x , t r ) + y ( P ) ϕ 1 ( P , x , t r ) = 3 T m ( λ ) V 32 ( n ) ( λ , x , t r ) + 2 S m ( λ ) A m ( λ , x , t r ) ε ( m ) E m 1 ( λ , x , t r ) ,
(3.17)
1 ϕ 1 ( P , x , t r ) + 1 ϕ 1 ( P , x , t r ) + 1 ϕ 1 ( P , x , t r ) = 3 V 22 ( n ) ( λ , x , t r ) V 21 ( n ) ( λ , x , t r ) u ( x , t r ) V 23 ( n ) ( λ , x , t r ) V 23 ( n ) ( λ , x , t r ) V 21 ( n ) ( λ , x , t r ) u ( x , t r ) V 23 ( n ) ( λ , x , t r ) F m , x ( λ , x , t r ) F m ( λ , x , t r ) .
(3.18)

After tedious calculations, we have the following lemma.

Lemma 3.1 Assume (3.1), (3.2), and let ( λ , x , x 0 , t r , t 0 , r ) C 5 . Then
E m 1 , t r ( λ , x , t r ) = E m 1 , x ( V ˜ 12 ( r ) V ˜ 13 ( r ) V 13 ( n ) V 12 ( n ) ) + 3 E m 1 ( V ˜ 11 ( r ) V ˜ 13 ( r ) V 13 ( n ) V 11 ( n ) ) , F m , t r ( λ , x , t r ) = F m , x ( V ˜ 23 ( r ) V ˜ 21 ( r ) u V ˜ 23 ( r ) V 21 ( n ) u V 23 ( n ) V 23 ( n ) ) F m , t r ( λ , x , t r ) = + 3 F m ( V ˜ 22 ( r ) V ˜ 21 ( r ) u V ˜ 23 ( r ) V 21 ( n ) u V 23 ( n ) V 22 ( n ) ) .
(3.19)

Moreover, by institute of (3.2), (3.6), (3.16), and (3.19), we arrive at the properties of ψ 1 ( P , x , x 0 , t r , t 0 , r ) immediately.

Lemma 3.2 Assume (3.1), (3.6), P = ( λ , y ( P ) ) K m 1 { P } , and let ( λ , x , x 0 , t r , t 0 , r ) C 5 . Then
ψ 1 , t r ( P , x , x 0 , t r , t 0 , r ) ψ 1 ( P , x , x 0 , t r , t 0 , r ) = V ˜ 13 ( r ) ( λ , x , t r ) [ ϕ 1 , x ( P , x , t r ) + ϕ 1 2 ( P , x , t r ) u ( x , t r ) ] ψ 1 , t r ( P , x , x 0 , t r , t 0 , r ) ψ 1 ( P , x , x 0 , t r , t 0 , r ) = + V ˜ 12 ( r ) ( λ , x , t r ) ϕ 1 ( P , x , t r ) + V ˜ 11 ( r ) ( λ , x , t r ) ,
(3.20)
ψ 1 ( P , x , x 0 , t r , t 0 , r ) ψ 1 ( P , x , x 0 , t r , t 0 , r ) ψ 1 ( P , x , x 0 , t r , t 0 , r ) = E m 1 ( λ , x , t r ) E m 1 ( λ , x 0 , t 0 , r ) ,
(3.21)
ψ 1 , x ( P , x , x 0 , t r , t 0 , r ) ψ 1 , x ( P , x , x 0 , t r , t 0 , r ) ψ 1 , x ( P , x , x 0 , t r , t 0 , r ) = F m ( λ , x , t r ) E m 1 ( λ , x 0 , t 0 , r ) ,
(3.22)
ψ 1 ( P , x , x 0 , t r , t 0 , r ) = exp ( x 0 x ϕ 1 ( P , x , t r ) d x + t 0 , r t r [ V ˜ 13 ( r ) ( λ , x 0 , t ) ( y ( P ) V 11 ( n ) ( λ , x 0 , t ) V 13 ( n ) ( λ , x 0 , t ) V 12 ( n ) ( λ , x 0 , t ) V 13 ( n ) ( λ , x 0 , t ) ϕ 1 ( P , x 0 , t ) ) + V ˜ 12 ( r ) ( λ , x 0 , t ) ϕ 1 ( P , x 0 , t ) + V ˜ 11 ( r ) ( λ , x 0 , t ) ] d t ) ,
(3.23)
ψ 1 ( P , x , x 0 , t r , t 0 , r ) = [ E m 1 ( λ , x , t r ) E m 1 ( λ , x 0 , t 0 , r ) ] 1 / 3 × exp ( x 0 x y ( P ) 2 V 13 ( n ) ( λ , x , t r ) y ( P ) A m ( λ , x , t r ) + 2 3 S m ( λ ) V 13 ( n ) ( λ , x , t r ) ε ( m ) E m 1 ( λ , x , t r ) d x + t 0 , r t r [ y ( P ) 2 V 13 ( n ) ( λ , x 0 , t ) y ( P ) A m ( λ , x 0 , t ) + 2 3 S m ( λ ) V 13 ( n ) ( λ , x 0 , t ) ε ( m ) E m 1 ( λ , x 0 , t ) × ( V ˜ 12 ( r ) ( λ , x 0 , t ) V ˜ 13 ( r ) ( λ , x 0 , t ) V 13 ( n ) ( λ , x 0 , t ) V 12 ( n ) ( λ , x 0 , t ) ) + y ( P ) V ˜ 13 ( r ) ( λ , x 0 , t ) V 13 ( n ) ( λ , x 0 , t ) ] d t ) .
(3.24)
By inspection of (3.9), one shall know that E m 1 and F m are both monic polynomials with respect to λ of degree m 1 and m, respectively. Hence we may decompose them into
E m 1 ( λ , x , t r ) = j = 1 m 1 ( λ μ j ( x , t r ) ) ,
(3.25)
F m ( λ , x , t r ) = l = 0 m 1 ( λ ν l ( x , t r ) ) .
(3.26)
Define
μ ˆ j ( x , t r ) = ( μ j ( x , t r ) , y ( μ ˆ j ( x , t r ) ) ) = ( μ j ( x , t r ) , A m ( μ j ( x , t r ) , x , t r ) V 13 ( n ) ( μ j ( x , t r ) , x , t r ) ) K m 1 , 1 j m 1 , ( x , t r ) C 2 ,
(3.27)
ν ˆ l ( x , t r ) = ( ν l ( x , t r ) , y ( ν ˆ l ( x , t r ) ) ) = ( ν l ( x , t r ) , C m ( ν l ( x , t r ) , x , t r ) V 23 ( n ) ( ν l ( x , t r ) , x , t r ) ) K m 1 , 0 l m 1 , ( x , t r ) C 2 .
(3.28)

The dynamics of the zeros μ j ( x , t r ) and ν l ( x , t r ) of E m 1 ( λ , x , t r ) and F m ( λ , x , t r ) are then described in terms of Dubrovin-type equations as follows.

Lemma 3.3 (i) Suppose that the zeros μ j ( x , t r ) j = 1 , , m 1 of E m 1 ( P , x , t r ) remain distinct for ( x , t r ) Ω μ , where Ω μ C 2 is open and connected. Then μ j ( x , t r ) j = 1 , , m 1 satisfy the system of differential equations
μ j , x ( x , t r ) = ε ( m ) V 13 ( n ) ( μ j ( x , t r ) , x , t r ) [ 3 y 2 ( μ ˆ j ( x , t r ) ) + S m ( μ j ( x , t r ) ) ] k = 1 m 1 k j ( μ j ( x , t r ) μ k ( x , t r ) ) , 1 j m 1 ,
(3.29)
μ j , t r ( x , t r ) = [ V 13 ( n ) ( μ j ( x , t r ) , x , t r ) V ˜ 12 ( r ) ( μ j ( x , t r ) , x , t r ) μ j , t r ( x , t r ) = V ˜ 13 ( r ) ( μ j ( x , t r ) , x , t r ) V 12 ( n ) ( μ j ( x , t r ) , x , t r ) ] μ j , t r ( x , t r ) = × ε ( m ) [ 3 y 2 ( μ ˆ j ( x , t r ) ) + S m ( μ j ( x , t r ) ) ] k = 1 m 1 k j ( μ j ( x , t r ) μ k ( x , t r ) ) , 1 j m 1 .
(3.30)
  1. (ii)
    Suppose that the zeros ν l ( x , t r ) l = 0 , , m 1 of F m ( P , x , t r ) remain distinct for ( x , t r ) Ω ν , where Ω ν C 2 is open and connected. Then ν l ( x , t r ) l = 0 , , m 1 satisfy the system of differential equations
    ν l , x ( x , t r ) = ε ( m ) [ V 21 ( n ) ( ν l ( x , t r ) , x , t r ) u V 23 ( n ) ( ν l ( x , t r ) , x , t r ) ] [ 3 y 2 ( ν ˆ l ( x , t r ) ) + S m ( ν l ( x , t r ) ) ] k = 0 m 1 k l ( ν l ( x , t r ) ν k ( x , t r ) ) , 0 l m 1 ,
    (3.31)
    ν l , t r ( x , t r ) = [ ( V 21 ( n ) ( ν l ( x , t r ) , x , t r ) u V 23 ( n ) ( ν l ( x , t r ) , x , t r ) ) V ˜ 23 ( r ) ( ν l ( x , t r ) , x , t r ) ν l , t r ( x , t r ) = ( V ˜ 21 ( r ) ( ν l ( x , t r ) , x , t r ) u V ˜ 23 ( r ) ( ν l ( x , t r ) , x , t r ) ) V 23 ( n ) ( ν l ( x , t r ) , x , t r ) ] ν l , t r ( x , t r ) = × ε ( m ) [ 3 y 2 ( ν ˆ l ( x , t r ) ) + S m ( ν l ( x , t r ) ) ] k = 0 m 1 k l ( ν l ( x , t r ) ν k ( x , t r ) ) , 0 l m 1 .
    (3.32)
     
Proof Using (3.10), we have ( λ = μ j ( x , t r ) )
S m ( μ j ( x , t r ) ) ( V 13 ( n ) ( μ j ( x , t r ) , x , t r ) ) 2 B m ( μ j ( x , t r ) , x , t r ) V 13 ( n ) ( μ j ( x , t r ) , x , t r ) + A m 2 ( μ j ( x , t r ) , x , t r ) = 0 ,
(3.33)
that is,
B m ( μ j ( x , t r ) , x , t r ) = S m ( μ j ( x , t r ) ) V 13 ( n ) ( μ j ( x , t r ) , x , t r ) + A m 2 ( μ j ( x , t r ) , x , t r ) V 13 ( n ) ( μ j ( x , t r ) , x , t r ) = [ S m ( μ j ( x , t r ) ) + y 2 ( μ ˆ j ( x , t r ) ) ] V 13 ( n ) ( μ j ( x , t r ) , x , t r ) .
After substituting B m into (3.12), we get
ε ( m ) E m 1 , x ( μ j ( x , t r ) , x , t r ) = V 13 ( n ) ( μ j ( x , t r ) , x , t r ) [ 3 y 2 ( μ ˆ j ( x , t r ) ) + S m ( μ j ( x , t r ) ) ] .
(3.34)
On the other hand, derivatives of the expression in (3.25) with respect to x and t r respectively, are
E m 1 , x ( μ j ( x , t r ) , x , t r ) = μ j , x ( x , t r ) k = 1 k j m 1 ( μ j ( x , t r ) μ k ( x , t r ) ) ,
(3.35)
E m 1 , t r ( μ j ( x , t r ) , x , t r ) = μ j , t r ( x , t r ) k = 1 k j m 1 ( μ j ( x , t r ) μ k ( x , t r ) ) .
(3.36)
Comparing (3.34) and (3.35), we can obtain (3.29). From (3.19), one can know
E m 1 , t r ( μ j ( x , t r ) , x , t r ) = E m 1 , x ( μ j ( x , t r ) , x , t r ) V 13 ( n ) V ˜ 12 ( r ) V ˜ 13 ( r ) V 12 ( n ) V 13 ( n ) = μ j , x ( x , t r ) k = 1 k j m 1 ( μ j ( x , t r ) μ k ( x , t r ) ) V 13 ( n ) V ˜ 12 ( r ) V ˜ 13 ( r ) V 12 ( n ) V 13 ( n ) = ε ( m ) [ 3 y 2 ( μ ˆ j ( x , t r ) ) + S m ( μ j ( x , t r ) ) ] ( V 13 ( n ) V ˜ 12 ( r ) V ˜ 13 ( r ) V 12 ( n ) ) ,
(3.37)

then we have (3.30). Similarly, we can prove (3.31) and (3.32). □

4 Algebro-geometric solutions to the second-order Benjamin-Ono hierarchy

In our final and principal section, we obtain Riemann theta function representations for the Baker-Akhiezer function and the meromorphic function; especially, the theta function representations for general algebro-geometric solutions u, v of the second-order Benjamin-Ono hierarchy. For the convenience, we assume that the curve K m 1 is nonsingular.

For investigating the asymptotic expansion of ϕ 1 ( P , x , t r ) near P , we choose the local coordinate ζ = λ 1 3 , then we get the following lemma.

Lemma 4.1 Let ( x , t r ) C 2 , near P K m 1 , we have
ϕ 1 ( P , x , t r ) = ζ 0 1 ζ j = 0 κ j ( x , t r ) ζ j as  P P ,
(4.1)
where
κ 0 = 1 , κ 1 = 0 , κ 2 = 2 3 u , κ 3 = 1 3 ( v u x ) , κ 4 = 1 9 u x x 1 3 v x , κ 5 = 2 9 ( v x x u u x u v ) , κ j = 1 3 [ κ j 2 , x x + 3 i = 2 j 1 κ j 1 i κ i , x + i = 2 j 1 κ i κ j i + i = 2 j 1 l = 0 j i κ i κ l κ j i l 2 u κ j 2 ] ( j 4 ) .
(4.2)
Proof In terms of the local coordinate ζ = λ 1 3 , (3.13) reads
ϕ 1 , x x + 3 ϕ 1 ϕ 1 , x + ϕ 1 3 2 u ϕ 1 = u x + v + ζ 3 .
(4.3)
Then, by inserting the power series ansatz of ϕ 1 ( P , x , t r ) in ζ as follows:
ϕ 1 ( P , x , t r ) = ζ 0 1 ζ j = 0 κ j ( x , t r ) ζ j
(4.4)
into (4.3)
ζ 1 j = 0 κ j , x x ζ j + 3 ζ 2 j = 0 i = 0 κ j κ i , x ζ ( j + i ) + ζ 3 j = 0 i = 0 l = 0 κ j κ i κ l ζ ( j + i + l ) 2 u ζ 1 j = 0 κ j ζ j = u x + v + ζ 3 ,
(4.5)

and comparing the same powers of ζ in (4.5), we arrive at (4.2). □

One infers, from (3.7), (3.25), (3.26), and (4.1), that the divisor ( ϕ 1 ( P , x , t r ) ) of ϕ 1 ( P , x , t r ) is given by
( ϕ 1 ( P , x , t r ) ) = D ν ˆ 0 ( x , t r ) , , ν ˆ m 1 ( x , t r ) ( P ) D P , μ ˆ 1 ( x , t r ) , , μ ˆ m 1 ( x , t r ) ( P ) .
(4.6)

That is, ν ˆ 0 ( x , t r ) , , ν ˆ m 1 ( x , t r ) are the m zeros of ϕ 1 ( P , x , t r ) and P , μ ˆ 1 ( x , t r ) , , μ ˆ m 1 ( x , t r ) are its m poles.

A straightforward calculation reveals that the asymptotic behaviors of y ( P ) and S m ( λ ) near P are
y ( P ) = ζ 0 { ζ 3 n 2 [ 1 + α 0 ζ + β 1 ζ 3 + α 1 ζ 4 + O ( ζ 6 ) ] as  P P , m = 3 n + 2 , ζ 3 n 1 [ 1 + β 1 ζ 2 + α 1 ζ 3 + O ( ζ 5 ) ] as  P P , m = 3 n + 1 ,
(4.7)
S m ( λ ) = ζ 0 { 3 ζ 6 n 3 [ α 0 + ( α 1 + β 1 α 0 ) ζ 3 + O ( ζ 6 ) ] as  P P , m = 3 n + 2 , 3 ζ 6 n [ β 1 + O ( ζ 3 ) ] as  P P , m = 3 n + 1 .
(4.8)
Next we will introduce the three kinds of holomorphic differentials and show some properties of them. The holomorphic differentials η l ( P ) on K m 1 are defined by
η l ( P ) = 1 3 y ( P ) 2 + S m { λ l 1 d λ , 1 l m n 1 , y ( P ) λ l + n m d λ , m n l m 1 .
(4.9)
To construct the theta function and normalize the holomorphic differentials, we choose a homology basis on K m 1 so that they satisfy
Introducing an invertible matrix E = ( E j , k ) ( m 1 ) × ( m 1 ) and e ̲ ( k ) = ( e 1 ( k ) , , e m 1 ( k ) ) , where
and the normalized holomorphic differentials ω j for j = 1 , , m 1 ,
(4.10)
Let ω P , 2 ( 2 ) ( P ) denote the normalized second Abel differential defined by
ω P , 2 ( 2 ) ( P ) = j = 1 m 1 z j η j ( P ) 1 3 y ( P ) 2 + S m { λ 2 n d λ , m = 3 n + 1 , y ( P ) λ n d λ , m = 3 n + 2 ,
(4.11)
which is holomorphic on K m 1 { P } with a pole of order 2 at P , and the constants { z j } j = 1 , , m 1 are determined by the normalization condition
The -periods of the differential ω P , 2 ( 2 ) are denoted by
(4.12)
On the other hand, ω P , 3 ( 2 ) ( P ) denotes the normalized third Abel differential which is holomorphic on K m 1 { P } with a pole of order 3 at P
ω P , 3 ( 2 ) ( P ) = ζ 0 ( ζ 3 + O ( 1 ) ) d ζ as  P P ,
(4.13)
and the -periods of it are defined by
Furthermore, the normalized third Abel differential ω P , ν ˆ 0 ( x ) ( 3 ) ( P ) is holomorphic on K m 1 { P , ν ˆ 0 ( x ) } with simple poles at P and ν ˆ 0 ( x ) with residues ±1, respectively, that is,
ω P , ν ˆ 0 ( x ) ( 3 ) ( P ) = ζ 0 ( ζ 1 + O ( 1 ) ) d ζ as  P P , ω P , ν ˆ 0 ( x ) ( 3 ) ( P ) = ζ 0 ( ζ 1 + O ( 1 ) ) d ζ as  P ν ˆ 0 ( x ) .
(4.14)
Then
P 0 P ω P , ν ˆ 0 ( x ) ( 3 ) ( P ) = ln ζ + e ( 3 ) ( P 0 ) + O ( ζ ) as  P P , P 0 P ω P , ν ˆ 0 ( x ) ( 3 ) ( P ) = ln ζ + e ( 3 ) ( P 0 ) + O ( ζ ) as  P ν ˆ 0 ( x )
(4.15)

with e ( 3 ) ( P 0 ) being an integration constant.

A straightforward Laurent expansion of (4.9), (4.10), and (4.11) near P yields the following results.

Lemma 4.2 Near P in the local coordinate ζ = λ 1 3 , the differentials ω ̲ and ω P , 2 ( 2 ) have the Laurent series
ω ̲ = ( ω 1 , , ω m 1 ) = ζ 0 ( ρ ̲ 0 + ρ ̲ 1 ζ + ρ ̲ 2 ζ 3 + O ( ζ 4 ) ) d ζ ,
(4.16)
with
ρ ̲ 0 = { e ̲ ( m n 1 ) , m = 3 n + 2 , e ̲ ( m 1 ) , m = 3 n + 1 , ρ ̲ 1 = { e ̲ ( m 1 ) + α 0 e ̲ ( m n 1 ) , m = 3 n + 2 , e ̲ ( m n 1 ) , m = 3 n + 1 , ρ ̲ 2 = { ( 2 β 1 α 0 3 ) e ̲ ( m n 1 ) + α 0 2 e ̲ ( m 1 ) e ̲ ( m n 2 ) , m = 3 n + 2 , α 1 e ̲ ( m 1 ) + β 1 e ̲ ( m n 1 ) e ̲ ( m 2 ) , m = 3 n + 1 , ω P , 2 ( 2 ) ( P ) = ζ 0 { ( ζ 2 + z m n 1 α 0 2 + ( β 1 + α 0 3 α 0 z m n 1 + z m 1 ) ζ + O ( ζ 2 ) ) d ζ , m = 3 n + 2 , ( ζ 2 + z m 1 β 1 + ( z m n 1 2 α 1 ) ζ + O ( ζ 2 ) ) d ζ , m = 3 n + 1 .
(4.17)
From Lemma 4.2 we infer
P 0 P ω P , 2 ( 2 ) ( P ) = ζ 0 ζ 1 + e 2 ( 2 ) ( P 0 ) q 1 ζ + q 2 ζ 2 + O ( ζ 3 ) as  P P ,
(4.18)
where e 2 ( 2 ) ( P 0 ) is an appropriate constant, and
q 1 = { z m n 1 + α 0 2 , m = 3 n + 2 , z m 1 + β 1 , m = 3 n + 1 , q 2 = { 1 2 ( β 1 + α 0 3 α 0 z m n 1 + z m 1 ) , m = 3 n + 2 , 1 2 z m n 1 α 1 , m = 3 n + 1 .
(4.19)
Let θ ( λ ̲ ) denote the Riemann theta function [2022] associated with K m 1 and the appropriately fixed homology basis . Next we choose a convenient base point P 0 K m 1 { P } . For brevity, define the function λ ̲ : K m 1 × σ m 1 K m 1 C by
λ ̲ ( P , Q ̲ ) = Ξ ̲ P 0 A ̲ P 0 ( P ) + α ̲ P 0 ( D Q ̲ ) , P K m 1 , Q ̲ = ( Q 1 , , Q m 1 ) σ m 1 K m 1 ,
where Ξ ̲ P 0 is the vector of Riemann constants, and the Abel maps A ̲ P 0 ( P ) and α ̲ P 0 ( P ) are defined by (period lattice L m 1 = { z ̲ C m 1 | z ̲ = N ̲ + τ M ̲ , N ̲ , M ̲ Z m 1 } )
A ̲ P 0 : K m 1 J ( K m 1 ) = C m 1 / L m 1 , P A ̲ P 0 ( P ) = ( A P 0 , 1 ( P ) , , A P 0 , m 1 ( P ) ) = ( P 0 P ω 1 , , P 0 P ω m 1 ) ( mod L m 1 ) ,
and
α ̲ P 0 : Div ( K m 1 ) J ( K m 1 ) , D α ̲ P 0 ( D ) = P K m 1 D ( P ) A ̲ P 0 ( P ) .

In view of these preparations, we give the theta function representation of our fundamental object ϕ 1 ( P , x , t r ) .

Theorem 4.3 Let P = ( λ , y ) K m 1 { P } , and let ( x , t r ) , ( x 0 , t 0 , r ) Ω μ , where Ω μ C 2 is open and connected. Suppose also that D μ ̲ ˆ ( x , t r ) , or equivalently, D ν ̲ ˆ ( x , t r ) is nonspecial for ( x , t r ) Ω μ . Then
ϕ 1 ( P , x , t r ) = θ ( λ ̲ ( P , ν ̲ ˆ ( x , t r ) ) ) θ ( λ ̲ ( P , μ ̲ ˆ ( x , t r ) ) ) θ ( λ ̲ ( P , ν ̲ ˆ ( x , t r ) ) ) θ ( λ ̲ ( P , μ ̲ ˆ ( x , t r ) ) ) exp ( e ( 3 ) ( P 0 ) P 0 P ω P , ν ˆ 0 ( x , t r ) ( 3 ) ) .
(4.20)
Proof Let Φ denote the right-hand side of (4.20). From (4.15) it follows that
exp ( e ( 3 ) ( P 0 ) P 0 P ω P , ν ˆ 0 ( x , t r ) ( 3 ) ) = ζ 0 ζ 1 + O ( 1 ) .
(4.21)
Using (4.6) we immediately know that ϕ 1 has simple poles at μ ˆ ̲ ( x , t r ) and P , and simple zeros at ν ˆ 0 ( x , t r ) , ν ˆ ̲ ( x , t r ) . By (4.20) and the Riemann vanishing theorem, we see that Φ has the same properties. Using the Riemann-Roch theorem [21, 22], we conclude that the holomorphic function Φ ϕ 1 = γ , where γ is a constant. Using (4.21) and Lemma 4.1, we have
Φ ϕ 1 = ζ 0 ( 1 + O ( ζ ) ) ( ζ 1 + O ( 1 ) ) ζ 1 + O ( ζ ) = ζ 0 1 + O ( ζ ) as  P P ,
(4.22)

from which we conclude γ = 1 . □

Let ω P , s ( 2 ) , s = 3 r + 2 (or 3 r + 1 ), r N 0 , be the normalized differential of the second kind holomorphic on K m 1 { P } , with a pole of order s at P ,
ω P , s ( 2 ) ( P ) = ζ 0 ( ζ s + O ( 1 ) ) d ζ as  P P .
Then we define the normalized differentials as
Ω ˜ P , s + 1 ( 2 ) = l = 0 r β ˜ r l ( 3 l + 2 ) ω ˜ P , 3 l + 3 ( 2 ) + l = 0 r α ˜ r l ( 3 l + 1 ) ω ˜ P , 3 l + 2 ( 2 ) , s = 3 r + 2 ( or  3 r + 1 ) , r N 0 ,
(4.23)
where
( α ˜ 0 , β ˜ 0 ) = { ( α ˜ 0 , 1 ) , s = 3 r + 2 , ( 1 , 0 ) , s = 3 r + 1 , α ˜ 0 C .
In addition, we define the vector of -periods of them as
(4.24)
Motivated by the second integration in (3.23), one defines the function I s ( P , x , t r ) , meromorphic on K m 1 × C 2 , by
I s ( P , x , t r ) = V ˜ 11 ( r ) ( λ , x , t r ) + V ˜ 12 ( r ) ( λ , x , t r ) ϕ 1 ( P , x , t r ) + V ˜ 13 ( r ) ( λ , x , t r ) ( ϕ 1 , x ( P , x , t r ) + ϕ 1 2 ( P , x , t r ) u ( x , t r ) ) .
(4.25)
Denote by I ¯ s ( P , x , t r ) the associated homogeneous one replacing V ˜ 1 j ( r ) by V ˜ ¯ 1 j ( r ) , where
V ˜ ¯ 1 j ( r ) = { V ˜ 1 j ( r ) | α ˜ 0 = 1 , α ˜ 1 = = α ˜ r = β ˜ 0 = β ˜ 1 = = β ˜ r = 0 , s = 3 r + 1 , V ˜ 1 j ( r ) | β ˜ 0 = 1 , α ˜ 0 = α ˜ 1 = = α ˜ r = β ˜ 1 = = β ˜ r = 0 , s = 3 r + 2 , j = 1 , 2 , 3 .
Lemma 4.4 Let s = 3 r + 2 (or 3 r + 1 ), r N 0 , ( x , t r ) C 2 , and λ = ζ 3 be the local coordinate near P . Then
I ¯ s ( P , x , t r ) = ζ 0 ζ s + O ( ζ ) as  P P .
(4.26)
Proof For the sake of convenience, we introduce the notation V ˜ 1 j ( r , s ) = V ˜ 1 j ( r ) , j = 1 , 2 , 3 . From (2.12) and (4.25), one easily gets
I ¯ s ( P , x , t r ) = V ˜ ¯ 11 ( r , s ) ( λ , x , t r ) + V ˜ ¯ 12 ( r , s ) ( λ , x , t r ) ϕ 1 ( P , x , t r ) + V ˜ ¯ 13 ( r , s ) ( λ , x , t r ) ( ϕ 1 , x ( P , x , t r ) + ϕ 1 2 ( P , x , t r ) u ) = 1 6 b ˜ ¯ x x ( r , s ) ( λ , x , t r ) 1 3 u b ˜ ¯ ( r , s ) ( λ , x , t r ) a ˜ ¯ x ( r , s ) ( λ , x , t r ) [ a ˜ ¯ ( r , s ) ( λ , x , t r ) 1 2 b ˜ ¯ x ( r , s ) ( λ , x , t r ) ] ϕ 1 ( P , x , t r ) + b ˜ ¯ ( r , s ) [ ϕ 1 , x ( P , x , t r ) + ϕ 1 2 ( P , x , t r ) u ( x , t r ) ] .
From (4.1), we can see
I ¯ 1 = ϕ 3 ( P , x , t r ) = ζ 1 + O ( ζ ) , I ¯ 2 = 1 3 u ( x , t r ) + ϕ 1 , x ( P , x , t r ) ϕ 1 2 ( P , x , t r ) u ( x , t r ) = ζ 2 + O ( ζ ) .
So (4.26) is correct for s = 1 and s = 2 . Then one may rewrite (4.26) as
I ¯ s ( P , x , t r ) = ζ 0 ζ s + j = 1 δ j ( x , t r ) ζ j as  P P
(4.27)
for some coefficients { δ j ( x , t r ) } j N . From (3.20) and (4.25), we can see
x I ¯ s ( P , x , t r ) = x ( V ˜ ¯ 12 ( r , s ) ( λ , x , t r ) ϕ 1 ( P , x , t r ) + V ˜ ¯ 13 ( r , s ) ( λ , x , t r ) ( ϕ 1 , x ( P , x , t r ) + ϕ 1 2 ( P , x , t r ) u ) + V ˜ ¯ 11 ( r , s ) ( λ , x , t r ) ) = ϕ 1 , t r ( P , x , t r ) ,
that is,
x ( ζ s + j = 1 δ j ( x , t r ) ζ j ) = ( ζ 1 + j = 1 κ j ( x , t r ) ζ j 1 ) t r = ( j = 1 κ j + 1 ( x , t r ) ζ j ) t r .
(4.28)
Using (3.2), (4.2), and comparing coefficients of ζ in (4.28), we should obtain
δ j , x ( x , t r ) = κ j + 1 , t r ( x , t r ) , j = 1 , 2 , δ 1 , x ( x , t r ) = κ 2 , t r ( x , t r ) = 2 3 u t r ( x , t r ) = b ˜ ¯ r , x ( r , s ) ( x , t r ) , δ 2 , x ( x , t r ) = κ 3 , t r ( x , t r ) = 1 3 ( u ( x , t r ) + v ( x , t r ) ) t r = 1 2 b ˜ ¯ r , x x ( r , s ) ( x , t r ) a ˜ ¯ r , x ( r , s ) ( x , t r ) , δ 3 , x ( x , t r ) = κ 4 , t r ( x , t r ) = ( 1 9 u x x ( x , t r ) 1 3 v x ( x , t r ) ) t r = 1 6 b ˜ ¯ r , x x x ( r , s ) ( x , t r ) + a ˜ ¯ r , x x ( r , s ) ( x , t r ) .
(4.29)
That is,
δ 1 ( x , t r ) = γ 1 ( t r ) b ˜ ¯ r ( r , s ) ( x , t r ) , δ 2 ( x , t r ) = γ 2 ( t r ) + 1 2 b ˜ ¯ r , x ( r , s ) ( x , t r ) a ˜ ¯ r ( r , s ) ( x , t r ) , δ 3 ( x , t r ) = γ 3 ( t r ) 1 6 b ˜ ¯ r , x x ( r , s ) ( x , t r ) + a ˜ ¯ r , x ( r , s ) ( x , t r ) ,
(4.30)
with γ 1 ( t r ) , γ 2 ( t r ) , γ 3 ( t r ) being integration constants. From the definition of I ¯ s , the power series for ϕ 1 ( P , x , t r ) and the coefficients of a ˜ ¯ ( ζ , x , t r ) , b ˜ ¯ ( ζ , x , t r ) , we deduce that γ 1 ( t r ) = γ 2 ( t r ) = γ 3 ( t r ) = 0 . Hence one concludes
I ¯ s ( P , x , t r ) = ζ s b ˜ ¯ r ( r , s ) ζ + ( 1 2 b ˜ ¯ r , x ( r , s ) a ˜ ¯ r ( r , s ) ) ζ 2 + ( 1 6 b ˜ ¯ r , x x ( r , s ) + a ˜ ¯ r , x ( r , s ) ) ζ 3 + O ( ζ 4 ) as  P P .
(4.31)
On the other hand, we will get
I ¯ s + 3 ( P , x , t r ) = ζ 3 I ¯ s + ( a ˜ ¯ r ( r + 1 , s + 3 ) 1 2 b ˜ ¯ r , x ( r + 1 , s + 3 ) ) ϕ 1 + b ˜ ¯ r ( r + 1 , s + 3 ) ( ϕ 1 , x + ϕ 1 2 u ) + 1 6 b ˜ ¯ r , x x ( r + 1 , s + 3 ) 1 3 u b ˜ ¯ r ( r + 1 , s + 3 ) a ˜ ¯ r , x ( r + 1 , s + 3 ) = ζ s 3 + O ( ζ ) .
(4.32)

 □

By (3.1) one knows that
I s ( P , x , t r ) = l = 0 r β ˜ r l I ¯ 3 l + 2 ( P , x , t r ) + l = 0 r α ˜ r l I ¯ 3 l + 1 ( P , x , t r ) , s = 3 r + 2  (or  s = 3 r + 1 ) .
(4.33)
Thus
t 0 , r t r I s ( P , x , τ ) d τ = ζ 0 ( t r t 0 , r ) l = 0 r ( β ˜ r l 1 ζ 3 l + 2 + α ˜ r l 1 ζ 3 l + 1 ) + O ( ζ ) as  P P .
(4.34)
Furthermore, integrating (4.23) yields
P 0 P Ω ˜ P , s + 1 ( 2 ) = l = 0 r β ˜ r l ( 3 l + 2 ) ζ 0 ζ ω ˜ P , 3 l + 3 ( 2 ) + l = 0 r α ˜ r l ( 3 l + 1 ) ζ 0 ζ ω ˜ P , 3 l + 2 ( 2 ) = ζ 0 l = 0 r β ˜ r l ( 3 l + 2 ) ζ 0 ζ 1 ζ 3 l + 3 d ζ + l = 0 r α ˜ r l ( 3 l + 1 ) ζ 0 ζ 1 ζ 3 l + 2 d ζ + O ( ζ ) = ζ 0 l = 0 r β ˜ r l 1 ζ 3 l + 2 l = 0 r α ˜ r l 1 ζ 3 l + 1 + e s + 1 ( 2 ) ( P 0 ) + O ( ζ ) as  P P ,
(4.35)
where e s + 1 ( 2 ) ( P 0 ) is a constant. Combing (4.34) and (4.35) indicates
t 0 , r t r I s ( P , x , τ ) d τ = ζ 0 ( t r t 0 , r ) ( e s + 1 ( 2 ) ( P 0 ) P 0 P Ω ˜ P , s + 1 ( 2 ) ) + O ( ζ ) as  P P .
(4.36)

Given these preparations, the theta function representation of ψ 1 ( P , x , x 0 , t r , t 0 , r ) reads as follows.

Theorem 4.5 Let P = ( λ , y ) K m 1 { P } and let ( x , t r ) , ( x 0 , t 0 , r ) Ω μ , where Ω μ C 2 is open and connected. Suppose that D μ ̲ ˆ ( x , t r ) , or equivalently, D ν ̲ ˆ ( x , t r ) is nonspecial for ( x , t r ) Ω μ . Then
ψ 1 ( P , x , x 0 , t r , t 0 , r ) = θ ( λ ̲ ( P , μ ̲ ˆ ( x , t r ) ) ) θ ( λ ̲ ( P , μ ̲ ˆ ( x 0 , t 0 , r ) ) ) θ ( λ ̲ ( P , μ ̲ ˆ ( x , t r ) ) ) θ ( λ ̲ ( P , μ ̲ ˆ ( x 0 , t 0 , r ) ) ) × exp ( ( x x 0 ) ( e 2 ( 2 ) ( P 0 ) P 0 P ω P , 2 ( 2 ) ) + ( t r t 0 , r ) ( e s + 1 ( 2 ) ( P 0 ) P 0 P Ω ˜ P , s + 1 ( 2 ) ) ) .
(4.37)
Proof Let ψ 1 ( P , x , x 0 , t r , t 0 , r ) be defined as in (3.23) and denote the right-hand side of (4.37) by Ψ ( P , x , x 0 , t r , t 0 , r ) . In order to prove that ψ 1 = Ψ , one uses (3.7), (3.12), (3.29), (3.30) and
V 12 ( n ) ϕ 1 + V 13 ( n ) ( ϕ 1 , x + ϕ 1 2 u ) + V 11 ( n ) = y ,
to compute
ϕ 1 ( P , x , t r ) = y 2 V 13 ( n ) y A m + B m ε ( m ) E m 1 ϕ 1 ( P , x , t r ) = y 2 V 13 ( n ) y A m + 2 3 V 13 ( n ) S m 1 3 ε ( m ) E m 1 , x ε ( m ) E m 1 ϕ 1 ( P , x , t r ) = 2 3 V 13 ( n ) 3 y 2 + S m ε ( m ) E m 1 + 1 3 x ln E m 1 + V 13 ( n ) y ( y + A m V 13 ( n ) ) ε ( m ) E m 1 = λ μ j ( x , t r ) μ j , x λ μ j + O ( 1 ) = λ μ j ( x , t r ) x ln ( λ μ j ( x , t r ) ) + O ( 1 ) , I s ( P , x , t r ) = V ˜ 12 ( r ) ϕ 1 + V ˜ 13 ( r ) ( ϕ 1 , x + ϕ 1 2 u ) + V ˜ 11 ( r ) I s ( P , x , t r ) = ( V ˜ 12 ( r ) V ˜ 13 ( r ) V 2 ( n ) V 13 ( n ) ) ϕ 1 + V ˜ 11 ( r ) V ˜ 13 ( r ) V 11 ( n ) V 13 ( n ) + y V ˜ 13 ( r ) V 13 ( n ) I s ( P , x , t r ) = + V ˜ 11 ( r ) V ˜ 13 ( r ) V 11 ( n ) V 13 ( n ) + y V ˜ 13 ( r ) V 13 ( n ) I s ( P , x , t r ) = 1 3 E m 1 , t r E m 1 + ( V ˜ 12 ( r ) V ˜ 13 ( r ) V 12 ( n ) V 13 ( n ) ) y 2 V 13 ( n ) y A m + 2 3 S m V 13 ( n ) ε ( m ) E m 1 + y V ˜ 13 ( r ) V 13 ( n ) = λ μ j ( x , t r ) μ j , t r λ μ j + O ( 1 ) = λ μ j ( x , t r ) t r ln ( λ μ j ( x , t r ) ) + O ( 1 ) as  P μ ˆ j ( x , t r ) .
Hence
ψ 1 ( P , x , x 0 , t r , t 0 , r ) = λ μ j ( x , t r ) λ μ j ( x 0 , t r ) λ μ j ( x 0 , t r ) λ μ j ( x 0 , t 0 , r ) O ( 1 ) = { ( λ μ j ( x , t r ) ) O ( 1 ) for  P  near  μ ˆ j ( x , t r ) μ ˆ j ( x 0 , t 0 , r ) , O ( 1 ) for  P  near  μ ˆ j ( x , t r ) = μ ˆ j ( x 0 , t 0 , r ) ,