The Hamiltonian Structure and Algebrogeometric Solution of a 1 + 1-Dimensional Coupled Equations

Abstract

A 1 + 1-dimensional coupled soliton equations are decomposed into two systems of ordinary differential equations. The Abel-Jacobi coordinates are introduced to straighten the flows, from which the algebrogeometric solutions of the coupled 1 + 1-dimensional equations are obtained in terms of the Riemann theta functions.

1. Introduction

The study of explicit solutions for soliton equations is very important in modern mathematics, physics, and other sciences. There are several systematic approaches to obtain explicit solutions of the soliton equations, such as the inverse scattering transformation, the algebrogeometric method, and the Hirota bilinear method (see, e.g., [1–4] and references therein). Algebrogeometric (or quasiperiodic) solutions are very important explicit solutions for soliton equations, recently, based on the nonlinearization technique of Lax pairs and direct method proposed by Cao [5]. This new scheme is further shown to be a very powerful tool, through which algebrogeometric solutions of (1+1)-dimensional and (2+1)-dimensional continuous and discrete soliton equations can be obtained; see Cao et al. [6, 7], Geng et al. [8, 9], and Dai and Fan [10].

In this paper, we will construct the Hamiltonian structure and search for the algebrogeometric solution of the following coupled 1+1-dimensional soliton equations: 𝑞_𝑡=𝑟_𝑥𝑥−3𝑞²𝑟_𝑥+𝑞_𝑥𝑟²+2𝑞𝑟𝑟_𝑥,𝑟_𝑡=𝑞_𝑥𝑥+3𝑟²𝑟_𝑥−𝑟_𝑥𝑞²−2𝑞𝑟𝑞_𝑥. (1)

Our purpose is to construct the Hamiltonian structure and give the algebrogeometric solutions of the coupled 1+1-dimensional soliton equations based on its obtained Lax pairs. The paper is organized as follows. In Section 2, we use Lenard operator pairs to derive another form of the coupled 1+1-dimensional soliton equations. In Section 3, based on the trace identity [11, 12], we construct the Hamiltonian structure of the coupled 1+1-dimensional soliton equations. In Section 4, based on the Lax pairs of the coupled 1+1-dimensional soliton equations, variable separation technique is used to translate the solution of the coupled 1+1-dimensional soliton equations to solve ordinary differential equations. In Section 5, a hyperelliptic Riemann surface of genus 𝑁 and Abel-Jacobi coordinates are defined to straighten the associated flows. Jacobi's inverse problem is discussed, from which the algebrogeometric solutions of the coupled 1+1-dimensional soliton equations are constructed in terms of the Riemann theta functions.

2. The Hierarchy and Lax Pairs of the Coupled 1 + 1-Dimensional Soliton Equations

In this section, we introduce the Lenard gradient sequence {𝑆_𝑗}_{𝑗=0,1,2…} to derive the hierarchy and its stationary hierarchy associated with (1) by the recursion relation: 𝐾𝑆_𝑗−1=𝐽𝑆_𝑗,𝑗=1,2,3,….𝑆_{𝑗(𝑢,𝑣)=0}𝑆=0,₀=𝑞−𝑟,𝑞+𝑟,1^𝑇, (2) where 𝑆_𝑗=(𝑆_𝑗⁽¹⁾,𝑆_𝑗⁽²⁾,𝑆_𝑗⁽³⁾) and operators (𝜕=𝜕/𝜕𝑥) −,−.𝐾=𝜕000𝜕0𝑞+𝑟𝑞−𝑟𝜕𝐽=−202𝑞−𝑟02−2𝑞+𝑟𝑞+𝑟𝑞−𝑟𝜕 (2) A direct calculation gives from the recursion relation (2) that𝑆₁=−12𝑞+𝑟𝑞−𝑟²−12𝑞_𝑥−𝑟_𝑥−12𝑞+𝑟²+1𝑞−𝑟2𝑞_𝑥+𝑟_𝑥−12,𝑆𝑞+𝑟𝑞−𝑟₂=38𝑞+𝑟²𝑞−𝑟³+12𝑟𝑞−𝑟𝑞_𝑥−𝑟_𝑥+12𝑞+𝑟𝑞−𝑟𝑞_𝑥−𝑟_𝑥+14𝑞_𝑥𝑥−𝑟_𝑥𝑥+14𝑞_𝑥+𝑟_𝑥𝑞−𝑟²−12𝑞−𝑟𝜕⁻¹𝑞_𝑥−𝑟_𝑥𝑟_𝑥38𝑞+𝑟³𝑞−𝑟²+12𝑟𝑞+𝑟𝑞_𝑥−𝑟_𝑥−12𝑞+𝑟𝑞−𝑟𝑞_𝑥+𝑟_𝑥−14𝑞_𝑥𝑥−𝑟_𝑥𝑥−14𝑞_𝑥−𝑟_𝑥𝑞+𝑟²−12𝑞+𝑟𝜕⁻¹𝑞_𝑥−𝑟_𝑥𝑟_𝑥38𝑞+𝑟²𝑞−𝑟²+12𝑟𝑞_𝑥−𝑟_𝑥−12𝜕⁻¹𝑞_𝑥−𝑟_𝑥𝑟_𝑥. (4)Consider the spectral problem 𝜓_𝑥𝜆=𝑈𝜓,𝑈=𝜆𝑞+𝑟𝑞−𝑟−𝜆, (5) and the auxiliary problem 𝜓_𝑡𝑚=𝑉^(𝑚)𝜓,𝑉^(𝑚)=𝑉^(𝑚)₁₁𝑉^(𝑚)₁₂𝑉^(𝑚)₂₁−𝑉^(𝑚)₁₁, (6) where 𝑉^(𝑚)₁₁=^𝑚_𝑗=0𝑆_𝑗⁽³⁾𝜆^{𝑚+1−𝑗},𝑉^(𝑚)₁₂=^𝑚_𝑗=0𝑆_𝑗⁽²⁾𝜆^𝑚−𝑗,𝑉^(𝑚)₂₁=^𝑚_𝑗=0𝑆_𝑗⁽¹⁾𝜆^{𝑚+1−𝑗}. (7)

Then the compatibility condition of (5) and (6) is 𝑈_𝑡𝑚−𝑉_𝑥^(𝑚)+[𝑈,𝑉^(𝑚)]=0, which is equivalent to the hierarchy of nonlinear evolution equations 𝑞_𝑡𝑚=12𝑆⁽²⁾_𝑚𝑥+𝑆⁽¹⁾_𝑚𝑥,𝑟_𝑡𝑚=12𝑆⁽²⁾_𝑚𝑥−𝑆⁽¹⁾_𝑚𝑥. (8)

In brief, 𝑞_𝑡𝑚,𝑟_𝑡𝑚^𝑇=𝑋_𝑚𝑋,𝑚≥0,_𝑚=𝜕𝜕𝑆𝜕−𝜕𝑆^𝑚(2)_𝑚⁽¹⁾. (9)(10)

The first two nontrivial equations are 𝑞_𝑡0=𝑞_𝑥,𝑟_𝑡0=𝑟_𝑥,𝑞_𝑡1=𝑟_𝑥𝑥−3𝑞²𝑟_𝑥+𝑞_𝑥𝑟²+2𝑞𝑟𝑟_𝑥,𝑟_𝑡1=𝑞_𝑥𝑥+3𝑟²𝑟_𝑥−𝑟_𝑥𝑞²−2𝑞𝑟𝑞_𝑥. (11)

The second system is our coupled 1+1-dimensional soliton equations (1).

Let 𝜓=(𝜓₁,𝜓₂)^𝑇 and 𝜙=(𝜙₁,𝜙₂)^𝑇 be two basic solutions of the spectral equations (5) and (6). We define a matrix 𝑊 by 1𝑊=2𝜙𝜓^𝑇+𝜓𝜙^𝑇𝜎=𝑓𝑔ℎ−𝑓,𝜎=0−110, (12) in which 𝑓,𝑔, and ℎ are three functions. It is easy to calculate by (5) and (6) that 𝑊_𝑥=[𝑈,𝑊],𝑊_𝑡𝑚=𝑉^(𝑚),𝑊, (13) which implies that 𝜕_𝑥det𝑊=0,𝜕_𝑡𝑚det𝑊=0. Equation (13) can be written as 𝑓_𝑥𝑔=−𝜆𝑞−𝑟𝑔+𝑞+𝑟ℎ,_𝑥ℎ=−2𝑞+𝑟𝑓+2𝜆𝑔,_𝑥𝑓=2𝜆𝑞−𝑟𝑓−2𝜆ℎ,_𝑡𝑚=ℎ𝑉^(𝑚)₁₂−𝑔𝑉^(𝑚)₂₁,𝑔_𝑡𝑚=2𝑔𝑉^(𝑚)₁₁−2𝑓𝑉^(𝑚)₁₂,ℎ_𝑡𝑚=2𝑓𝑉^(𝑚)₂₁−2ℎ𝑉^(𝑚)₁₁. (14)(15)

We suppose that the functions 𝑓,𝑔, and ℎ are finite-order polynomials in 𝜆: 𝑓=^𝑁_𝑗=0𝑓_𝑗𝜆^{𝑁−𝑗+1},𝑔=^𝑁_𝑗=0𝑔_𝑗𝜆^𝑁−𝑗,ℎ=^𝑁_𝑗=0ℎ_𝑗𝜆^𝑁−𝑗. (16)

Substituting (16) into (14) yields 𝐾𝐺_𝑗−1=𝐽𝐺_𝑗𝑗=1,2,…,𝑁,𝐽𝐺₀=0,𝐾𝐺_𝑁=0,𝐺_𝑗=ℎ_𝑗,𝑔_𝑗,𝑓_𝑗^𝑇. (17)

It is easy to see that (17) implies −𝑞+𝑟ℎ_𝑗+𝑞−𝑟𝑔_𝑗+𝑓_𝑗𝑥=0 (18) and the equation 𝐽𝐺₀=0 has the general solution 𝐺₀=𝛼₀𝑆₀, (19) where 𝛼₀ is constant of integration. Therefore, if we take (19) as a starting point, then 𝐺_𝑗 can be determined recursively by relation (17). In fact, noticing ker𝐽={𝑐𝑆₀|∀𝑐∈ℝ} and acting with the operator (𝐽⁻¹𝐾)^𝑘 upon (19), we obtain from (2) and (17) that 𝐺_𝑘=^𝑘_𝑗=0𝛼_𝑗𝑆_𝑘−𝑗,𝑘=0,1,…,𝑁, (20) where 𝛼₀,𝛼₁,…,𝛼_𝑘 are integral constants. Substituting (20) into (17) yields a certain stationary evolution equation: 𝛼₀𝑋_𝑁+𝛼₁𝑋_𝑁−1+⋯+𝛼_𝑁𝑋₀=0, (21) where 𝑋_𝑗=𝜕0𝑆0𝜕𝑆^𝑗(1)_𝑗⁽²⁾. (22)

This means that expression (16) is existent.

3. Hamiltonian Structure

Let 𝑉𝑉=𝑉=¹¹𝑉₁₂𝑉₂₁−𝑉₁₁, (23) where 𝑉₁₁=_𝑗≥0𝑆_𝑗⁽³⁾𝜆^−𝑗+1,𝑉₁₂=_𝑗≥0𝑆_𝑗⁽²⁾𝜆^−𝑗,𝑉₂₁=_𝑗≥0𝑆_𝑗⁽¹⁾𝜆^−𝑗+1. (24)

It is easy to calculate tr𝑉𝜕𝑈𝜕𝜆=2𝑉₁₁+𝑞−𝑟𝑉₁₂=_𝑗≥12𝑆_𝑗⁽³⁾+𝑞−𝑟𝑆⁽²⁾_𝑗−1𝜆^−𝑗+1+2𝑆₀⁽³⁾𝑉𝜆,tr𝜕𝑈𝜕𝑞=𝜆𝑉₁₂+𝑉₂₁=_𝑗≥0𝑆_𝑗⁽²⁾+𝑆_𝑗⁽¹⁾𝜆^−𝑗+1,tr𝑉𝜕𝑈𝜕𝑟=−𝜆𝑉₁₂+𝑉₂₁=_𝑗≥0−𝑆_𝑗⁽²⁾+𝑆_𝑗⁽¹⁾𝜆^−𝑗+1. (25)

According to the trace identity [11, 12], we have 𝛿𝛿𝛿𝑞𝛿𝑟2𝑉₁₁+𝑞−𝑟𝑉₁₂=𝜆^−𝑠𝜕𝜆𝜕𝜆^𝑠𝜆𝑉₁₂+𝑉₂₁−𝜆𝑉₁₂+𝑉₂₁. (26)

Comparing the coefficients of 𝜆^−𝑗+1, we obtain 𝛿𝛿𝛿𝑞𝛿𝑟2𝑆_𝑗⁽³⁾+𝑞−𝑟𝑆⁽²⁾_𝑗−1=𝑆−𝑗+2+𝑠^(2)𝑗−1+𝑆⁽¹⁾_𝑗−1−𝑆⁽²⁾_𝑗−1+𝑆⁽¹⁾_𝑗−1, (27) we set 𝑗=1 and then get 𝑠=−1 and 𝛿𝛿𝛿𝑞ℋ𝛿𝑟^𝑗=𝑆⁽²⁾_𝑗−1+𝑆⁽¹⁾_𝑗−1−𝑆⁽²⁾_𝑗−1+𝑆⁽¹⁾_𝑗−1=11𝑆−11𝑆^{(2)𝑗−1(1)}_𝑗−1, (28) where ℋ_𝑗=(2𝑆_𝑗⁽³⁾+(𝑞−𝑟)𝑆⁽²⁾_𝑗−1)/(−𝑗+1).

Thus the soliton equation (9) has a Hamiltonian structure: 𝑞𝑟_𝑡𝑚=12𝜕𝜕𝑆𝜕−𝜕𝑆^𝑚(2)_𝑚⁽¹⁾=𝐽̃𝛿𝛿𝛿𝑞ℋ𝛿𝑟^𝑚+1,𝑚≥0, (29) where 𝐽̃=12𝜕00−𝜕. (30)

In speciality, the Hamiltonian structure of (1) is (𝑚=1): ℋ₂=18𝑞+𝑟²𝑞−𝑟²−𝑞𝑞_𝑥−2𝑟𝑟_𝑥+𝑞𝑟_𝑥2+𝜕⁻¹𝑞_𝑥−𝑟_𝑥𝑟_𝑥2. (31)

4. Ordinary Differential Equations

In this section, (1) will be decomposed into two systems of solvable ordinary differential equations. Without loss of generality, let 𝛼₀=1. From (2), (17), and (20), we have 𝑓₀=1,𝑔₀=𝑞+𝑟,ℎ₀𝑓=𝑞−𝑟,₁1=−2𝑞+𝑟𝑞−𝑟+𝛼₁,𝑔₁1=−2𝑞+𝑟²+1𝑞−𝑟2𝑞_𝑥+𝑟_𝑥+𝛼₁,ℎ𝑞+𝑟₁1=−2𝑞+𝑟𝑞−𝑟²−12𝑞_𝑥−𝑟_𝑥+𝛼₁𝑞−𝑟. (32)

By using (16), we can write 𝑔 and ℎ as the following finite products: 𝑔=−𝑞+𝑟^𝑁_𝑗=1𝜆−𝑢_𝑗,ℎ=𝑞−𝑟^𝑁_𝑗=1𝜆−𝑣_𝑗. (33)

Equation (33) implies by comparing the coefficients of 𝜆^𝑁−1 that 𝑔₁=−𝑞+𝑟^𝑁_𝑗=1𝑢_𝑗,ℎ₁=−𝑞+𝑟^𝑁_𝑗=1𝑣_𝑗. (34) Thus from (32) and (34), we obtain −12𝑞+𝑟²+1𝑞−𝑟2𝑞_𝑥+𝑟_𝑥+𝛼₁𝑞+𝑟=−𝑞+𝑟^𝑁_𝑗=1𝑢_𝑗,−12𝑞+𝑟𝑞−𝑟²−12𝑞_𝑥−𝑟_𝑥+𝛼₁𝑞−𝑟=−𝑞−𝑟^𝑁_𝑗=1𝑣_𝑗. (35)

Let us consider the function det𝑊 which is a (2𝑁+2)th-order polynomial in 𝜆 with constant coefficients of the 𝑥-flow and 𝑡_𝑚-flow: −det𝑊=𝑓²+𝑔ℎ=^2𝑁+2_𝑗=1𝜆−𝜆_𝑗=𝑅(𝜆). (36) Substituting (16) into (36) and comparing the coefficients of 𝜆^2𝑁+1 yield 2𝑓₀𝑓₁+𝑔₀ℎ₀=−^2𝑁+2_𝑗=1𝜆_𝑗, (37) which together with (32) gives 𝛼₁1=−2^2𝑁+2_𝑗=1𝜆_𝑗. (38)

From (36) we see that 𝑓|_{𝜆=𝑢𝑘}=𝑅𝑢_𝑘,𝑓|_{𝜆=𝑣𝑘}=𝑅𝑣_𝑘. (39)

Again by using (14) and (33), we obtain 𝑔_{𝑥𝜆=𝑢𝑘}=−𝑞+𝑟𝑢^𝑁_{𝑘𝑥𝑗=1,𝑗≠𝑘}𝑢_𝑘−𝑢_𝑗=−2(𝑞+𝑟)𝑓_{𝜆=𝑢𝑘},ℎ_{𝑥𝜆=𝑣𝑘}=−𝑞−𝑟𝑣^𝑁_{𝑘𝑥𝑗=1,𝑗≠𝑘}𝑣_𝑘−𝑣_𝑗=2𝑣_𝑘(𝑞−𝑟)𝑓_{𝜆=𝑣𝑘}, (40) which together with (39) gives 𝑢_𝑘𝑥=2𝑅𝑢_𝑘∏^𝑁_{𝑗=1,𝑗≠𝑘}𝑢_𝑘−𝑢_𝑗𝑣,1≤𝑘≤𝑁,_𝑘,𝑥=−2𝑅𝑣_𝑘∏^𝑁_{𝑗=1,𝑗≠𝑘}𝑣_𝑘−𝑣_𝑗,1≤𝑘≤𝑁. (41)

In a way similar to the above expression, by using (6) (𝑚=1,𝑡₁=𝑡), (15), and (39), we arrive at the evolution of {𝑢_𝑘} and {𝑣_𝑘} along the 𝑡_𝑚-flow: 𝑢_𝑘,𝑡=2𝑓|_{𝜆=𝑢𝑘}𝑉⁽¹⁾₁₂|_{𝜆=𝑢𝑘}𝑞+𝑟∏^𝑁_{𝑗=1,𝑗≠𝑘}𝑢_𝑘−𝑢_𝑗=2𝑅𝑢_𝑘𝑢_𝑘−(1/2)𝑞+𝑟𝑞−𝑟+(1/2)𝜕ln𝑞−𝑟∏^𝑁_{𝑗=1,𝑗≠𝑘}𝑢_𝑘−𝑢_𝑗,𝑣_𝑘,𝑡=−2𝑓|_{𝜆=𝑣𝑘}𝑉⁽¹⁾₂₁|_{𝜆=𝑣𝑘}𝑞−𝑟∏^𝑁_{𝑗=1,𝑗≠𝑘}𝑣_𝑘−𝑣_𝑗=−2𝑅𝑣_𝑘𝑣²_𝑘−12𝑞+𝑟𝑞−𝑟𝑣_𝑘+12𝜕ln𝑞−𝑟𝑣_𝑘^𝑁_{𝑗=1,𝑗≠𝑘}𝑣_𝑘−𝑣_𝑗⁻¹. (42)(43)

Therefore, if the (2𝑁+2) distinct parameters 𝜆₁,𝜆₂,…,𝜆_2𝑁+2 are given and let 𝑢_𝑘(𝑥,𝑡) and 𝑣_𝑘(𝑥,𝑡) be distinct solutions of ordinary differential equations (41), (42), and (43), then (𝑞,𝑟) determined by (35) is a solution of the coupled 1+1-dimensional equations (1).

5. Algebrogeometric Solutions

In this section, we will give the algebrogeometric solutions of the coupled 1+1-dimensional equation (1). To this end, we first introduce the Riemann surface Γ of the hyperelliptic curve Γ∶𝜁²=𝑅(𝜆),𝑅(𝜆)=^2𝑁+2_𝑗=1𝜆−𝜆_𝑗, (44) with genus 𝑁 on Γ. On Γ there are two infinite points ∞₁ and ∞₂, which are not branch points of Γ. We equip Γ with a canonical basis of cycles: 𝑎₁,𝑎₂,…,𝑎_𝑁;𝑏₁,𝑏₂,…,𝑏_𝑁 which are independent and have intersection numbers as follows: 𝑎_𝑖∘𝑎_𝑗=0,𝑏_𝑖∘𝑏_𝑗𝑎=0,_𝑖∘𝑏_𝑗=𝛿_𝑖𝑗,𝑖,𝑗=1,2,…,𝑁. (45)

We will choose the following set as our basis: 𝜔_𝑙=𝜆^𝑙−1𝑑𝜆𝑅(𝜆),𝑙=1,2,…,𝑁, (46) which are linearly independent of each other on Γ, and let 𝐴_𝑖𝑗=_𝑎𝑗𝜔_𝑖,𝐵_𝑖𝑗=_𝑏𝑗𝜔_𝑖. (47)

It is possible to show that the matrices 𝐴=(𝐴_𝑖𝑗) and 𝐵=(𝐵_𝑖𝑗) are 𝑁×𝑁 invertible matrices [13, 14]. Now we define the matrices 𝐶 and 𝜏 by 𝐶=(𝐶_𝑖𝑗)=𝐴⁻¹,𝜏=(𝜏_𝑖𝑗)=𝐴⁻¹𝐵. Then the matrix 𝜏 can be shown to be symmetric (𝜏_𝑖𝑗=𝜏_𝑗𝑖) and it has a positive-definite imaginary part (Im 𝜏>0). If we normalize 𝜔_𝑗 into the new basis 𝜔_𝑗𝜔_𝑗=^𝑁_𝑙=1𝐶_𝑗𝑙𝜔_𝑙,𝑙=1,2,…,𝑁, (48) then we have _𝑎𝑗𝜔_𝑗=^𝑁_𝑙=1𝐶_{𝑗𝑙𝑎𝑗}𝜔_𝑙=^𝑁_𝑙=1𝐶_𝑗𝑙𝐴_𝑙𝑖=𝛿_𝑗𝑖,_𝑏𝑗𝜔_𝑖=^𝑁_𝑙=1𝐶_{𝑗𝑙𝑏𝑗}𝜔_𝑙=^𝑁_𝑙=1𝐶_𝑗𝑙𝐵_𝑙𝑖=𝜏_𝑗𝑖. (49)

Now we introduce the Abel-Jacobi coordinates as follows: 𝜌_𝑗⁽¹⁾(𝑥,𝑡)=^𝑁_𝑘=1^{𝑝(𝑢𝑘}_𝑝^(𝑥,𝑡))₀𝜔_𝑗=^𝑁𝑁_𝑘=1𝑙=1^𝑢𝑘_𝜆(𝑝0)𝐶_𝑗𝑙𝜆^𝑙−1𝑑𝜆𝑅(𝜆),𝜌_𝑗⁽²⁾(𝑥,𝑡)=^𝑁_𝑘=1^{𝑝(𝑣𝑘}_𝑝^(𝑥,𝑡))₀𝜔_𝑗=^𝑁𝑁_𝑘=1𝑙=1^{𝑣𝑘(𝑥,𝑡)}_𝜆(𝑝0)𝐶_𝑗𝑙𝜆^𝑙−1𝑑𝜆𝑅(𝜆), (50)(51) where 𝑝(𝑢_𝑘(𝑥,𝑡))=(𝑢_𝑘,𝑅(𝑢_𝑘)),𝑝(𝑣_𝑘(𝑥,𝑡))=(𝑣_𝑘,𝑅(𝑣_𝑘)), and 𝜆(𝑝₀) is the local coordinate of 𝑝₀. From (42) and (50), we get 𝜕_𝑥𝜌_𝑗⁽¹⁾=^𝑁𝑁_𝑘=1𝑙=1𝐶_𝑗𝑙𝑢_𝑘^𝑙−1𝑢_𝑘𝑥𝑅𝑢_𝑘=^𝑁𝑁_𝑘=1𝑙=1𝐶_𝑗𝑙𝑢_𝑘^𝑙−1∏^𝑁_{𝑗=1,𝑗≠𝑘}𝑢_𝑘−𝑢_𝑗, (52) which implies 𝜕_𝑥𝜌_𝑗⁽¹⁾=2𝐶_𝑗𝑁=Ω_𝑗⁽¹⁾,𝑗=1,2,…,𝑁. (53)

With the help of the following equality ^𝑁_𝑘=1𝑢_𝑘^𝑙−1∏^𝑁_{𝑗=1,𝑗≠𝑘}𝑢_𝑘−𝑢_𝑗=𝛿_𝑙𝑁,𝑙=1,2,…,𝑁, (54) in a similar way, we obtain from (50), (51), (41), (42), and (43) that 𝜕_𝑡𝜌_𝑗⁽¹⁾=2^𝑁_𝑙=1𝐶_𝑗𝑙^𝑁_𝑖=1𝑢_𝑖−𝐶_𝑗𝑁𝑞+𝑟𝑞−𝑟+𝐶_𝑗𝑁𝜕ln𝑞−𝑟=Ω_𝑗⁽²⁾,𝜕_𝑥𝜌_𝑗⁽²⁾=−Ω_𝑗⁽¹⁾𝜕,𝑗=1,2,…,𝑁,_𝑡𝜌_𝑗⁽²⁾=−Ω_𝑗⁽²⁾,𝑗=1,2,…,𝑁. (55)

On the basis of these results, we obtain the following: 𝜌_𝑗⁽¹⁾(𝑥,𝑡)=Ω_𝑗⁽¹⁾𝑥+Ω_𝑗⁽²⁾𝑡+𝛾_𝑗⁽¹⁾,𝜌_𝑗⁽²⁾(𝑥,𝑡)=−Ω_𝑗⁽¹⁾𝑥−Ω_𝑗⁽²⁾𝑡+𝛾_𝑗⁽²⁾, (56) where 𝛾_𝑗^(𝑖)(𝑖=1,2) are constants and 𝛾_𝑗⁽¹⁾=^𝑁_𝑘=1^𝑝(_𝑘𝑝𝑢_(0,0))0𝜔_𝑗,𝛾_𝑗⁽²⁾=^𝑁_𝑘=1^𝑝(_𝑘𝑝𝑣_(0,0))0̃𝜔_𝑗,𝜌⁽¹⁾=𝜌₁⁽¹⁾,𝜌₂⁽¹⁾,…,𝜌_𝑁^(1)𝑇,𝜌⁽²⁾=𝜌₁⁽²⁾,𝜌₂⁽²⁾,…,𝜌_𝑁^(2)𝑇,Ω^(𝑚)=Ω₁^(𝑚),Ω₂^(𝑚),…,Ω_𝑁^(𝑚)𝑇,𝛾^(𝑚)=𝛾₁^(𝑚),𝛾₂^(𝑚),…,𝛾_𝑁^(𝑚)𝑇,𝑚=1,2. (57)

Now we introduce the Abel map 𝒜(𝑝): 𝒜𝑝=^𝑝_𝑝0𝜔,𝜔=𝜔₁,𝜔₂,…,𝜔_𝑁^𝑇,𝒜_𝑘𝑝_𝑘=𝑛_𝑘𝒜𝑝_𝑘 (58) and Abel-Jacobi coordinates: 𝜌⁽¹⁾=𝒜^𝑁_𝑘=1𝑝𝑢_𝑘=^𝑁_𝑘=1^{𝑝(𝑢𝑘)}_𝑝0𝜌𝜔,₍₂₎=𝒜^𝑁_𝑘=1𝑝𝑣_𝑘=^𝑁_𝑘=1^{𝑝(𝑣𝑘)}_𝑝0𝜔. (59) According to the Riemann theorem [13, 14], there exists a Riemann constant vector 𝑀∈ℂ^𝑁 such that the function 𝐹^(𝑚)(𝜆)=𝜃𝒜𝑝(𝜆)−𝜌^(𝑚)−𝑀^(𝑚),𝑚=1,2, (60) has exactly 𝑁 zeros at 𝑢₁,𝑢₂,…,𝑢_𝑁 for 𝑚=1 or 𝑣₁,𝑣₂,…,𝑣_𝑁 for 𝑚=2. To make the function single valued, the surface Γ is cut along all 𝑎_𝑘,𝑏_𝑘 to form a simple connected region, whose boundary is denoted by 𝛾. By [13, 14], the integrals 𝐼(Γ)=12𝜋𝑖_𝛾𝜆𝑑ln𝐹^(𝑚),𝑚=1,2, (61) are constants independent of 𝜌⁽¹⁾ and 𝜌⁽²⁾ with 𝐼=𝐼(Γ)=^𝑁_{𝑗=1𝑎𝑗}𝜆𝜔_𝑗. (62)

By the residue theorem, we have ^𝑁_𝑗=1𝑢_𝑗=𝐼−²_𝑠=1Res_𝜆=∞𝑠𝜆𝑑ln𝐹⁽¹⁾(𝜆),_𝑁𝑗=1𝑣_𝑗=𝐼−²_𝑠=1Res_𝜆=∞𝑠𝜆𝑑ln𝐹⁽²⁾(𝜆). (63)

Here we need only to compute the residues in (63). In a way similar to calculations in [10], we arrive at Res_𝜆=∞𝑠𝜆𝑑ln𝐹^(𝑚)(𝜆)=(−1)^𝑠+𝑚𝜕ln𝜃_𝑠^(𝑚),𝑚=1,2;𝑠=1,2, (64) where 𝜃_𝑠⁽¹⁾=𝜃Ω⁽¹⁾𝑥+Ω⁽²⁾𝑡+𝜉_𝑠,𝜃_𝑠⁽²⁾=𝜃−Ω⁽¹⁾𝑥−Ω⁽²⁾𝑡+𝜂_𝑠, (65) and 𝜉_𝑠 and 𝜂_𝑠 are constants. Thus from (63) and (64), we arrive at ^𝑁_𝑗=1𝑢_𝑗𝜃=𝐼−𝜕ln¹⁽¹⁾𝜃₂⁽¹⁾,^𝑁_𝑗=1𝑣_𝑗𝜃=𝐼−𝜕ln²⁽²⁾𝜃₁⁽²⁾. (66)

Substituting (66) into (35), we get an algebrogeometric solution for the coupled 1+1-dimensional soliton equations (1): 𝐴(𝑡)𝑞=2exp−𝜕^−12𝑁+2_𝑗=1𝜆_𝑗+2𝜕⁻¹𝜃𝐼−2ln²⁽²⁾𝜃₁⁽²⁾−𝜕⁻¹𝜃exp−2ln²⁽²⁾𝜃₂⁽¹⁾𝜃₁⁽²⁾𝜃₁⁽¹⁾+𝐴(𝑡)2exp𝜕^−12𝑁+2_𝑗=1𝜆_𝑗+2𝜕⁻¹𝜃𝐼−2ln¹⁽²⁾𝜃₂⁽¹⁾+𝜕⁻¹𝜃exp−2ln²⁽²⁾𝜃₂⁽¹⁾𝜃₁⁽²⁾𝜃₁⁽¹⁾,𝐴(𝑡)𝑟=2exp𝜕^−12𝑁+2_𝑗=1𝜆_𝑗+2𝜕⁻¹𝜃𝐼−2ln¹⁽²⁾𝜃₂⁽¹⁾+𝜕⁻¹𝜃exp−2ln²⁽²⁾𝜃₂⁽¹⁾𝜃₁⁽²⁾𝜃₁⁽¹⁾−𝐴(𝑡)2exp−𝜕^−12𝑁+2_𝑗=1𝜆_𝑗+2𝜕⁻¹𝜃𝐼−2ln²⁽²⁾𝜃₁⁽²⁾−𝜕⁻¹𝜃exp−2ln²⁽²⁾𝜃₂⁽¹⁾𝜃₁⁽²⁾𝜃₁⁽¹⁾, (67)(68) where 𝐴(𝑡) is arbitrary complex functions about variable 𝑡.

Acknowledgments

This work is in part supported by the Natural Science Foundation of China (Grant nos. 11271008, 61072147, and 11071159) and the Shanghai University Leading Academic Discipline Project (A.13-0101-12-004).