Quasi-periodic Solutions to the K(−2, −2) Hierarchy

1 Introduction

It is well known that soliton equations have very wide applications in fields of fluid dynamics, plasma physics, optical fibers, biology, and many more. Quasi-periodic solutions of soliton equations are of great importance for it reveals inherent structure of solutions and describes quasi-periodic actions of nonlinear phenomenon, especially can be used to find multi-soliton solutions and elliptic function solutions, and similar ones to these. Since the first research on finite-gap solutions of Korteweg–de Vries equation around 1975, there has been numerous works devoted to constructing quasi-periodic solutions for soliton equations [1–25] and references therein.

In study of the role of nonlinear dispersion in the formation of patterns in liquid drops, Rosenau and Hyman [26] proposed in 1993 the nonlinear dispersive K(m, n) equation

where a is a constant. It was found in [27–33] that nonlinear dispersion can compactify solitary waves and generate compactons, which are solitons with compact support. In addition, the compacton structure of K(m, n) equation was discussed thoroughly. In addition to compatons, the K(m, n) equation has also other kinds of solutions, such as kinks, peakons, and cuspons [29, 34–36]. Furthermore, several generalizations of the K(m, n) equation have also been considered in the literature [37, 38].

In 1991, by introducing a nonconfocal generator of finite-dimensional integrable systems, Cao and Geng [39] obtained a new soliton hierarchy and corresponding Lax pairs. The first nontrivial member in the hierarchy is as follows

which was also found by Olver and Rosenau [27], and Qiao [40] later and can be denoted as K(−2, −2). Subsequently, Geng [41] established the generalised Hamiltonian structure for the K(−2, −2) hierarchy and decomposed it into finite-dimensional Liouville integrable system using the nonlinearization approach, from which its solutions were reduced to solving the compatible Hamiltonian systems of ordinary differential equations. Moreover, Sakovich [42] proposed a transformation, which relates the K(−2, −2) equation with the modified Korteweg–de Vries equation. Based on this transformation, the N-soliton solutions of K(−2, −2) equation were derived applying Darboux transformation [43]. In [44, 45], the authors gave two integrable extensions of K(−2, −2) equation and studied their cuspon and kink wave solutions taking the bifurcation method of dynamical systems.

In this article, we construct the explicit Riemann theta function representations of solutions to the K(−2, −2) hierarchy. Our article is organised as follows. In Section 2, we derive the hierarchy of the K(−2, −2) equations based on the Lenard recursion equations and zero-curvature equation. In Section 3, we introduce the Baker–Akhiezer function and hyperelliptic curve 𝒦_n+1 of arithmetic genus n+1. Then we deduce the associated meromorphic function and the Dubrovin-type differential equations. In Section 4, we present the explicit theta function representation of meromorphic function and, in particular, that of solutions for the entire K(−2, −2) hierarchy.

2 The K(−2, −2) Hierarchy

In this section, we derive the K(−2, −2) hierarchy associated with the spectral problem

where u is a potential and λ a constant spectral parameter. To this end, we introduce a set of Lenard recursion equations

with starting points

and two operators are defined as

K = (∂3−4∂0u∂−∂),   J = (04∂uu∂−∂).

Hence, g_j are uniquely determined by the recursive relation (4), which means to identify constants of integration as zero, for example, the second member reads as

g0 = 14u6(5ux2 − 2uuxx − 3u2u(6ux2 − 2uuxx − 4u2)),

In order to generate a hierarchy of nonlinear evolution equations associated with the spectral problem (3), we solve the stationary zero-curvature equation

which is equivalent to

where each entry V_ij=V_ij(a, b) is a Laurent expansion in λ:

A direct calculation shows that (7) and (8) imply the Lenard equations

Substituting the expansions

into (9) and collecting terms with the same powers of λ, we arrive at the following recursion relation

where G_j=(a_j, b_j)^T. Noticing (4) and (5), then function G_j can be expressed as

where α_j is arbitrary constants.

Let ψ satisfy the spectral problem (3) and an auxiliary problem

with

and a˜j, b˜j determined by

G˜j = (a˜j, b˜j)T = α˜0gj + ⋯ + α˜jg0 + α˜j + 1g−1,   j ≥ −1.

The constants α˜0, …, α˜j + 1 are independently of α₀, …, α_j+1. Then the compatibility condition of (3) and (13) yields the zero-curvature equation, Utm − V˜x(m) + [U, V˜(m)] = 0, which is equivalent to the hierarchy of nonlinear evolution equations

The first nontrivial member in the hierarchy (15) is

which is just the K(−2, −2) equation (2) with ε=1 as α˜0 = 1, t0 = t.

3 The Baker–Akhiezer Function and the Dubrovin-type Equations

In this section, we first introduce the Baker–Akhiezer function and Lax matrix for the K(−2, −2) hierarchy, from which a hyperelliptic curve 𝒦_n+1 and meromorphic function are defined. Then the hierarchy of K(−2, −2) equations are decomposed into the system of solvable differential equations.

Now, we introduce the Baker–Akhiezer function ψ(P, x, x₀, t_m, t_0,m) by

where

V(n) = (λV11(n)λV12(n)V21(n)−λV11(n)), V11(n) = 4b(n),  V12(n) = axx(n) − 2ax(n) − 4λub(n),V21(n) = axx(n) + 2ax(n) − 4λub(n),  a(n) = ∑j = 0naj − 1λn − j,  b(n) = ∑j = 0nbj − 1λn − j.

The compatibility conditions of the three expressions in (17) yield that

A direct calculation shows that yI − V⁽ⁿ⁾ satisfies the (19) and (20). Then the characteristic polynomial of the Lax matrix V⁽ⁿ⁾, ℱ_2n+3(λ, y)=det(yI − V⁽ⁿ⁾), is an independent constant of the variables x and t_m with the expansion

det(yI − V(n)) = y2 − λR2n + 2(λ),

where λR_2n+2(λ) are polynomials with constant coefficients of λ, i.e.

Hence, ℱ_2n+3(λ, y)=0 naturally leads to a hyperelliptic curve of degree 2n+3

For the convenience, we also denote the compactification of the hyperelliptic curve 𝒦_n+1 by the same symbol 𝒦_n+1. Assume that {λj}j = 12n + 2 in (21) are mutually distinct and nonzero, then 𝒦_n+1 becomes nonsingular.

Next we define the meromorphic function ϕ(P, x, t_m) on 𝒦_n+1 as

where P=(λ, y)∈𝒦_n+1, x, x₀, t_m, t₀, _m∈ℂ. It infers from (23) and (17) that

By observing (12) and (17), we know that V12(n) and V21(n) are polynomials with respect to λ of degree n+1, thereby they may be decomposed as

Defining

and P₀=(0, 0), then we have the following results.

Lemma 1.Suppose that {μj(x, tm)}j = 1n + 1 and {νj(x, tm)}j = 1n + 1 remain distinct and nonzero for (x, t_m)∈Ω_μand (x, t_m)∈Ω_ν, respectively, where Ω_μ, Ω_ν⊆ℂ²are open and connected. Then they satisfy the system of Dubrovin-type differential equations

Proof. Equations (19) and (20) imply that

On the other hand, differentiating (25) respect to x and t_m gives rise to

Comparing (32) and (33), we obtain (28) and (29). Similarly, (30) and (31) can be proved.□

4 Quasi-periodic Solutions to the K(−2, −2) Hierarchy

In this section, we derive explicit Riemann theta function representations for the meromorphic function ϕ(P, x, t_m), and in particular, that of potential u, for the entire K(−2, −2) hierarchy.

With the aid of (3) and (23), we arrive at that the meromorphic function ϕ(P, x, t_m) satisfies the Riccati equation

To investigate the property of ϕ(P, x, t_m) near P_∞∈𝒦_n+1, we take the local coordinate ζ = λ− 12, and obtain the Laurent series

with

κ−1 = 1,   κ0 = −1u,   κ1 = u − ux2u3,κj + 1 = −12u[κj,x + 2κj + u∑i = 0jκiκj − i],   (j ≥ −1).

Immediately, one obtains from (24) and (35) that the divisor (ϕ(P, x, t_m)) of ϕ(P, x, t_m) is given by

which means P0, μ^1(x, tm), …, μ^n + 1(x, tm) are the n+2 zeros of ϕ(P, x, t_m) and P∞, ν^1(x, tm), …, ν^n + 1(x, tm) its n+2 poles. In addition, direct computation gives the asymptotic property of y(P) near P_∞

Equip the Riemann surface 𝒦_n+1 with homology basis {𝕒j, 𝕓j}j = 1n + 1, which are independent and have intersection numbers as follows

𝕒j ∘ 𝕓k = δj,k,   𝕒j ∘ 𝕒k = 0,   𝕓j ∘ 𝕓k = 0,   j, k = 1, …, n + 1.

On 𝒦_n+1, we introduce n+1 linearly independent holomorphic differentials

from which the period matrices A and B can be constructed from

It is possible to prove that the matrices A and B are invertible. Defining the matrix C and τ by C=A⁻¹, τ=A⁻¹B, one can show that matrix τ is symmetric (τ_jk=τ_kj) and has a positive-definite imaginary part (Im τ>0) [46, 47]. Let us now normalize ϖ_l(P) into new basis ω_j by

which satisfy ∫𝕒kωj = δjk,∫𝕒kωj = τjk,   j, k = 1, …, n + 1. A straightforward Laurent expansion of (40) near P_∞ yields that

Let ωP∞,2(2)(P) denote the normalised Abelian differential of the second kind defined by

which is holomorphic on 𝒦_n+1\{P_∞} with a pole of order 2 at P_∞, and the constants {γj}j = 1n + 1 are determined by the normalization conditions

The b-periods of the differential ωP∞,2(2) are denoted by

Furthermore, let ωP∞,P0(3)(P) denote the normalised Abelian differential of the third kind defined by

which is holomorphic on 𝒦_n+1\{P_∞, P₀} with simple poles at P_∞ and P₀ with residues ±1, respectively, and the constants {ηj}j = 1n + 1 are determined by the normalization conditions

A direct calculation shows that

which infers that

with Q₀ a chosen base point on 𝒦_n+1\{P_∞, P₀} and e∞(3)  (Q0), e0(3)(Q0) two integration constants.

Let 𝒯_n+1 be the period lattice {ϱ∈ℂⁿ⁺¹|ϱ=N+τL, N, L∈ℤⁿ⁺¹}. The complex torus 𝒥_n+1=ℂⁿ⁺¹/𝒯_n+1 is called the Jacobian variety of 𝒦_n+1. An Abel map 𝒜 : 𝒦_n+1→𝒥_n+1 is defined by

A(P) = (A1(P), ⋯, An + 1(P)) = (∫Q0Pω1, ⋯, ∫Q0Pωn + 1),  (modTn + 1)

with the natural linear extension to the factor group Div(𝒦_n+1)

A(∑nkPk) = ∑nkA(Pk).

Considering the nonspecial divisor Dμ_^(x,tm) = ∑k = 1n + 1μ^k(x, tm) and Dν^_(x,tm) = ∑k = 1n + 1ν^k(x, tm), we define

whose components are ∑k = 1n + 1∫Q0Pk(i)(x,tm)ωj = ρj(i)(x, tm),   i = 1, 2,   j = 1, …, n + 1, where μ^_(x, tm) = (μ^1(x, tm), ⋯, μ^n + 1(x, tm)), ν^_(x, tm) = (ν^1(x, tm), ⋯,ν^n + 1(x, tm)),Pk(1)(x, tm) = μ^k(x, tm), and Pk(2)(x, tm) = ν^k(x, tm).

The Riemann theta function [46, 47] associated with 𝒦_n+1 is defined as

where Q=(Q₁, …, Q_n+1) and M=(M₁, …, M_n+1) is Riemann constant vector. Then

Hence, the theta function representation of ϕ(P, x, t_m) reads as follows.

Theorem 1.Assume the curve 𝒦_n+1to be nonsingular and let P=(λ, y)∈𝒦_n+1\({P_∞}, (x, t_m), (x₀, t_0,m)∈Ω_μ, where Ω_μ⊆ℂ²is open and connected. Suppose also thatDμ_^(x,tm),or equivalently, Dν_^(x,tm)is nonspecial for (x, t_m)∈Ω_μ. Then

Proof. It is easily seen from (36) that ϕ has simple zeros at P₀ and μ^_(x, tm), and simple poles at P_∞ and ν^_(x, tm). Let Φ denote the right-hand side of (52). It follows from (48) that

Applying the Riemann’s vanishing theorem and the Riemann-Roch theorem, one concludes that Φ and ϕ share same simple poles and zeros, which leads to Φϕ = γ, with γ a constant. On the other hand, (53) and (35) show

which implies γ=1 and completes the proof.□

With the above-mentioned results, we obtain the theta function representations of solutions for the entire K(−2, −2) hierarchy.

Theorem 2.Assume the curve 𝒦_n+1to be nonsingular and let (x, t_m)∈Ω_μ, where Ω_μ⊆ℂ²is open and connected. Suppose also that Dμ^_(x,tm), or equivalently, Dν^_(x,tm) is nonspecial for (x, t_m)∈Ω_μ. Then the K(−2, −2) hierarchy (15) admits quasi-periodic solutions