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The application of trigonal curve theory to the second-order Benjamin-Ono hierarchy
Advances in Difference Equations volume 2014, Article number: 195 (2014)
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 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.
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 [1–3]. 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]
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. [6–10].
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 matrix spectral problems, such as the KdV, nonlinear Schrödinger, sine-Gordon, Toda equations and so on [11–15]. Unfortunately, the situation is not so good for soliton equations associated with matrix spectral problems, which are more complicated and more difficult. In , 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 matrix spectral problems, we have developed the method in  to deal with some important soliton equations by introducing the trigonal curves of arithmetic genus and deriving the explicit Riemann theta function representations of the entire hierarchies, such as the modified Boussinesq, the Kaup-Kupershmidt hierarchies and others [17–19].
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 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 matrix spectral problem
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
with two starting points
where the initial conditions mean to identify constants of integration as zero, and two operators are defined as follows:
Hence and are uniquely determined, for example, the first two members read as
In order to generate a hierarchy of evolution equations associated with the spectral problem (2.1), we solve the stationary zero-curvature equation
which is equivalent to
where each entry is a Laurent expansion in λ:
A direct calculation shows that (2.5) and (2.6) imply the Lenard equation
Substituting (2.7) into (2.8) and collecting terms with the same powers of λ, we arrive at the following recursion relation:
where . Since the equation has the general solution
then can be expressed as
where and are arbitrary constants.
Let ψ satisfy the spectral problem (2.1) and its auxiliary problem
where each entry ,
Then the compatibility condition of (2.1) and (2.12) yields the zero-curvature equation, , which is equivalent to the hierarchy of nonlinear evolution equations
where the vector fields , and , . The first nontrivial member in the hierarchy (2.13) is as follows:
For , (), equation (2.14) is reduced to the second-order Benjamin-Ono equation by canceling the variable v
The second one in the hierarchy (2.13) (as , ) can be written as
For , (), equation (2.16) is reduced to a 5-order coupled equation
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 by
where and ,
with , determined by (2.11). The compatibility conditions of the first three expressions in (3.1) yield that
Through a direct calculation we can show that satisfies equations (3.3) and (3.4). So is an independent constant of the variables x and , from which we can define a trigonal curve with the expansion
Immediately, from (2.10) if we choose , an arbitrary constant or , , we shall know that the corresponding values of m in (3.5) are or , respectively. For the convenience, the compactification of the curve is denoted by the same symbol . Thus becomes a three-sheeted Riemann surface of arithmetic genus when it is nonsingular or smooth.
Next we shall introduce the meromorphic function , which is closely related to , by
which implies from (3.1) that
where , ,
which is introduced to ensure that , are both monic polynomials. It is easy to see that there exist various interrelationships between polynomials , , , , , and , , some of which are summarized as follows:
For displaying the properties of exactly, we introduce the holomorphic map ∗, changing sheets, as
where , , denote the three branches of satisfying . Then it is easy to show the properties of immediately:
After tedious calculations, we have the following lemma.
Lemma 3.1 Assume (3.1), (3.2), and let . Then
Moreover, by institute of (3.2), (3.6), (3.16), and (3.19), we arrive at the properties of immediately.
Lemma 3.2 Assume (3.1), (3.6), , and let . Then
By inspection of (3.9), one shall know that and are both monic polynomials with respect to λ of degree and m, respectively. Hence we may decompose them into
The dynamics of the zeros and of and are then described in terms of Dubrovin-type equations as follows.
Lemma 3.3 (i) Suppose that the zeros of remain distinct for , where is open and connected. Then satisfy the system of differential equations
Suppose that the zeros of remain distinct for , where is open and connected. Then satisfy the system of differential equations(3.31)(3.32)
Proof Using (3.10), we have ()
After substituting into (3.12), we get
On the other hand, derivatives of the expression in (3.25) with respect to x and respectively, are
Comparing (3.34) and (3.35), we can obtain (3.29). From (3.19), one can know
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 is nonsingular.
For investigating the asymptotic expansion of near , we choose the local coordinate , then we get the following lemma.
Lemma 4.1 Let , near , we have
Proof In terms of the local coordinate , (3.13) reads
Then, by inserting the power series ansatz of in ζ as follows:
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 () of is given by
That is, are the m zeros of and , are its m poles.
A straightforward calculation reveals that the asymptotic behaviors of and near are
Next we will introduce the three kinds of holomorphic differentials and show some properties of them. The holomorphic differentials on are defined by
To construct the theta function and normalize the holomorphic differentials, we choose a homology basis on so that they satisfy
Introducing an invertible matrix and , where
and the normalized holomorphic differentials for ,
Let denote the normalized second Abel differential defined by
which is holomorphic on with a pole of order 2 at , and the constants are determined by the normalization condition
The -periods of the differential are denoted by
On the other hand, denotes the normalized third Abel differential which is holomorphic on with a pole of order 3 at
and the -periods of it are defined by
Furthermore, the normalized third Abel differential is holomorphic on with simple poles at and with residues ±1, respectively, that is,
with being an integration constant.
A straightforward Laurent expansion of (4.9), (4.10), and (4.11) near yields the following results.
Lemma 4.2 Near in the local coordinate , the differentials and have the Laurent series
From Lemma 4.2 we infer
where is an appropriate constant, and
where is the vector of Riemann constants, and the Abel maps and are defined by (period lattice )
In view of these preparations, we give the theta function representation of our fundamental object .
Theorem 4.3 Let , and let , where is open and connected. Suppose also that , or equivalently, is nonspecial for . Then
Proof Let Φ denote the right-hand side of (4.20). From (4.15) it follows that
Using (4.6) we immediately know that has simple poles at and , and simple zeros at , . 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 , where γ is a constant. Using (4.21) and Lemma 4.1, we have
from which we conclude . □
Let , (or ), , be the normalized differential of the second kind holomorphic on , with a pole of order s at ,
Then we define the normalized differentials as
In addition, we define the vector of -periods of them as
Motivated by the second integration in (3.23), one defines the function , meromorphic on , by
Denote by the associated homogeneous one replacing by , where
Lemma 4.4 Let (or ), , , and be the local coordinate near . Then
Proof For the sake of convenience, we introduce the notation , . From (2.12) and (4.25), one easily gets
From (4.1), we can see
So (4.26) is correct for and . Then one may rewrite (4.26) as
for some coefficients . From (3.20) and (4.25), we can see
Using (3.2), (4.2), and comparing coefficients of ζ in (4.28), we should obtain
with , , being integration constants. From the definition of , the power series for and the coefficients of , , we deduce that . Hence one concludes
On the other hand, we will get
By (3.1) one knows that
Furthermore, integrating (4.23) yields
where is a constant. Combing (4.34) and (4.35) indicates
Given these preparations, the theta function representation of reads as follows.
Theorem 4.5 Let and let , where is open and connected. Suppose that , or equivalently, is nonspecial for . Then