Algebro-Geometric Solutions of a ( 2 + 1 )-Dimensional Integrable Equation Associated with the Ablowitz-Kaup-Newell-Segur Soliton Hierarchy

The ( 2 + 1 )-dimensional Lax integrable equation is decomposed into solvable ordinary di ﬀ erential equations with the help of known ( 1 + 1 )-dimensional soliton equations associated with the Ablowitz-Kaup-Newell-Segur soliton hierarchy. Then, based on the ﬁ nite-order expansion of the Lax matrix, a hyperelliptic Riemann surface and Abel-Jacobi coordinates are introduced to straighten out the associated ﬂ ows, from which the algebro-geometric solutions of the ( 2 + 1 )-dimensional integrable equation are proposed by means of the Riemann θ functions.


Introduction
Algebro-geometric solutions are an important class among exact solutions of soliton equations, which can be regarded as explicit solutions of the nonlinear integrable evolution equation and used to approximate more general solutions. Based on the nonlinearization technique of Lax pairs and direct method, many of algebro-geometric solutions of (1 + 1)-dimensional [1][2][3], (2 + 1)-dimensional [4,5], and differential-difference [5,6] soliton equations have been obtained, such as the Gerdjikov-Ivanov, modified Kadomtsev-Petviashvili, and Toda lattice equations [7][8][9]. The existence of infinitely many exact solutions is a reflection of this complete integrability.
Ablowitz-Kaup-Newell-Segur (AKNS) soliton hierarchy is an important class of integrable equations, which can be reduced to Korteweg-de Vries (KdV), modified Kortewegde Vries (mKdV), sine-Gordon equation hierarchies, etc. The purpose of the paper is to further develop the direct method for constructing algebro-geometric solution of the following (2 + 1)-dimensional integrable equation [15] which concerns with the AKNS soliton hierarchy [17].
In fact, system (1) is the Lax integrable equations from the AKNS soliton hierarchy, which has nonisospectral zero curvature representation. B€ acklund transformation for a splitting of slð2Þ and a soliton exact solution for it was obtained [18].
The whole paper is organized as follows: in Section 2, we use Lenard operator pairs to briefly derive (1 + 1)-dimensional AKNS soliton hierarchy and give the (2 + 1)-dimensional integrable equation (1). Then, in Section 3, based on the solutions of the (1 + 1)-dimensional soliton equations and the elliptic coordinates, the solution of the (2 + 1)-dimensional integrable equation is reduced to solving ordinary differential equations. In Section 4, a hyperelliptic Riemann surface and Abel-Jacobi coordinates are introduced to straighten the associated flows. The Jacobi's inversion problem is discussed, from which the algebro-geometric solution of the (2 + 1)-dimensional integrable equation is obtained in terms of the Riemann theta functions. A short summary is in Section 5.

The (2 + 1)-Dimensional Soliton Equation
It is well known that the AKNS soliton hierarchy is isospectral evolution equation hierarchy associated with the spectral problem [17].
Consider the Lenard gradient sequence fS j g ∞ j=0 by where It is easy to see that S j is uniquely determined by the recursion relation. A direct calculation gives that The auxiliary spectral of (2) is The compatibility condition between (2) and (6) is the zero curvature equation: which is equivalent to the hierarchy of soliton equations The first two nontrivial members in the hierarchy are Let t 2 = y, t 3 = t, uðx, y, tÞ = qðx, y, tÞ, and vðx, y, tÞ = rðx , y, tÞ in (9) and (10); then, we can obtain the (2 + 1 )-dimensional equation (1) by the use of the following equation: Therefore, if q and r are the compatible solutions of (9) and (10), then we can get that u = q and v = r are also the solutions of the (2 + 1)-dimensional equation (1).

Variable Separation
In this section, we shall show how the (1 + 1)-dimensional (9) and (10) are reduced to solvable ordinary differential 2 Advances in Mathematical Physics equations. Assume that (2) and (6) have two basic solutions ψ = ðψ 1 , ψ 2 Þ T and ϕ = ðϕ 1 , ϕ 2 Þ T . We define a matrix W of three functions f , g, h by It is easy to verify by (2) and (6) that which imply that the functions detW is a constant independent of x and t m . Equation (13) can be written as Now, suppose that the functions f , g, and h are finiteorder polynomials in λ: Substituting (16) into (14) yields It is easy to see that JG 0 = 0 has the general solution: where α 0 is constant of integration. So, KerJ = fcS 0 j∀cg. Acting with the operator ðJ −1 KÞ K+1 upon (18), we can obtain from (3) and (17) that where α 0 , …, α k are integral constants. Substituting (19) into (17) obtains the following stationary evolution equation: This means that expression (16) is existent.

Algebro-Geometric Solution
We first introduce the hyperelliptic Riemann surface with genus g = N. On Γ, there are two infinite points ∞ 1 and ∞ 2 , which are not branch points of Γ. Equip Γ with the canonical basis of cycles a 1 , ⋯, a N , b 1 , ⋯, b N , and the holomorphic differentials Then, the period matrices A and B are defined by Using A and B, we can define the matrices C and τ, where Then, matrix τ can be shown to be symmetric, and it has positive define imaginary part. We normalize f ω j into the new basis ω j : For a fixed point p 0 , then we introduce Abel-Jacobi coordinate as follows: whose components are From (47) and the fist expression of (36), we get with the help of the following equality In a similar way, we obtain from (36)-(39), (47), and (48) that On the basis of these results, we get the following: Advances in Mathematical Physics An Abel map on Γ is defined as Consider two special divisors ∑ N k=1 p ðkÞ m ðm = 1, 2Þ, and we have where p ðkÞ 1 = ð e μ k , ξð e μ k ÞÞ, p ðkÞ 2 = ð e μ k , ξð e μ k ÞÞ. The Riemann theta function of Γ is defined as where ζ = ðζ 1 ,⋯,ζ N Þ T , hζ, zi = ∑ N j=1 ζ j z j . According to the Riemann theorem, there exist two constant vector M 1 , M 2 ∈ C N such that has exactly zeros at μ 1 , ⋯, μ N for m = 1 or ν 1 , ⋯, ν N for m = 2 and m = 3. 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. Notice the fact that the integrals are constants independent of ρ 1 , ρ 2 with I = IðΓÞ = ∑ N j=1 Ð a j λ k ω j . By the residue theorem, we have Re s λ=∞ s λ k dlnF 1 λ ð Þ, ð59Þ Re s λ=∞ s λ k dlnF 2 λ ð Þ: Here, we only need to compute the residues in (59) for k = 1, 2, 3. In the way similar to calculations in [1,2,4], we obtain where θ ð1Þ s = θðΩ 0 x + Ω 1 y + Ω 2 t + π s Þ, θ ð2Þ s = θð−Ω 0 x − Ω 1 y − Ω 2 t + η s Þ, and π s , η s are constants. Thus from, we arrive at Similarly, we obtain where c 1 and c 2 are constants.

Summary
The nonisospectral (2 + 1)-dimensional breaking soliton system is given by the Lenard gradient sequence for a classical (1 + 1)-dimensional AKNS spectral problem. Then, the (2 + 1)-dimensional Lax integrable equation associated with the AKNS soliton hierarchy (1) is decomposed into solvable ordinary differential equations with the help of known (1 + 1 )-dimensional soliton equations. With introducing the hyperelliptic Riemann surface and the Abel-Jacobi coordinates, the flow can be straighten out, and the algebrogeometric solutions of the (2 + 1)-dimensional soliton system (1) are presented by means of the Riemann θ functions.

Data Availability
All data and models generated or used during this study appear in the article.

Conflicts of Interest
The author declares that there are no conflicts of interest.