Quasi-periodic solutions of three-component Burgers hierarchy

1.   Introduction

It is well known that quasi-periodic solutions of integrable dynamical systems can not only describe periodic nonlinear behavior, but they can also show characteristics of Liouville integrability ^[1,2,3,4,5]. Therefore, constructing quasi-periodic solutions for integrable nonlinear equations is a hot topic in the field of modern mathematical and theoretical physics. In the past decades, several systematic approaches have been used to study quasi-periodic solutions for nonlinear integrable models ^{[6,7,8,9,10,11,12]}. The successful idea is to use the hyperelliptic curves, and finite-genus solutions of nonlinear equations related to $2\times2$ matrix spectral problems have been derived ^{[2,3,6,7,8,9,10,11,12,13,14,15,16,17]}. But, this method is invalid to solve higher-order matrix spectral problems. In Refs. ^[18,19], a unified work was given to study the algebro-geometric solutions to the Boussinesq equations associated with a third-order spectral problem. After that, according to the characteristic polynomial of the Lax matrix, Geng and colleagues ^[20,21] developed a general approach to handle the case of the $3\times3$ matrix spectral problem by introducing trigonal curves. Using this method, quasi-periodic solutions for some famous equations, such as the Manakov, the cmKdV, the modified Boussinesq, the coupled Sasa-Satsuma, the Dym-type equations and so on ^{[22,23,24,25,26]}, were successfully generalized.

Recently, in Ref. ^[27], three-component Burgers hierarchy was derived, and its bi-Hamiltonian structures were discussed. The first member in the whole hierarchy is

which can be reduced to the well-known Burgers equation ^[28]

During the past few decades, there have been some remarkable works on the Burgers equation due to its prominent mathematical structures and physical properties. The Darboux transformation for the generalized Burgers equation was constructed based on the Lax pairs ^[29]. Quasi-periodic solutions for the 2+1 dimensional discrete Burgers equation were proposed in view of the Jacobi inversion ^[30]. The Cole-Hopf transformation were used to generate the multiple-front solutions for the coupled Burgers equation ^[31].

The purpose of this paper is to study quasi-periodic solutions of three-component Burgers flows. By means of the asymptotic expansions and Abelian differentials, we deduce the theta function expressions of the potential functions. In Section 2, we first list the recursion relations of the three-component Burgers equation; then, we define a trigonal curve $\mathcal{K}_{m-2}$ based on the characteristic polynomial of the Lax matrix. By adding two different infinite points, the curve $\mathcal{K}_{m-2}$ is compactified. In Section 3, we discuss the asymptotic behaviors of the meromorphic functions $\phi_2$ and $\phi_3$. Section 4 is devoted to discussing the divisors of $\phi_2, \phi_3$, which can reveal essential singularities. In Section 5, we derive the theta function solutions of $\psi_1, \phi_2, \phi_3$ by using the Abelian differentials inferred in Section 3. Especially, we generate the quasi-periodic solutions of the three-component Burgers hierarchy.

2.   The trigonal curve

The main idea of this section is to define a trigonal curve $\mathcal{K}_{m-2}$ related to the three-component Burgers hierarchy. We first list some necessary results from Ref. ^[27] to make this paper easier to read. We define the Lenard recurrence relations

and $K, J$ are two $3\times3$ operators

where

The recursion relations given by (2.1) can be uniquely solved. So, we have

with

Now, we discuss the zero-curvature equation

where

with $\lambda$ as a constant spectral parameter; also, $u(x, t), v(x, t)$ and $w(x, t)$ are three potentials. Each entry $V_{ij}(a, b, c)$ is defined as follows:

From (2.5) and (2.6), we obtain

Set

then,

satisfies

for $\bar{g}_0 = (0, 0, 1)^T\in \mathrm{ker} K\cap \mathrm{ker} J$; also, $\alpha_j, \beta_j$ and $\gamma_j$ are arbitrary constants.

In order to generate the three-component Burgers hierarchy, we set

and give the following Lax pairs

where

with $\tilde{\alpha}_j, \tilde{\beta}_j$ as arbitrary constants independent of $\alpha_j, \beta_j$. The compatible condition of (2.11) implies that $U_{t_r}-\widetilde{V}^{(r)}_x+[U, \widetilde{V}^{(r)}] = 0$ is equivalent to the following three-component Burgers equations:

Next, we define a Lax matrix $V^{(n)} = (\lambda^n V)_+ = (V^{(n)}_{ij}(a_j, b_j, c_j))_{3\times3}$, and

So, the characteristic polynomial of $V^{n}$ is independent of $(x, t_r)$. Moreover, we have

where $R_m, S_m$, $T_m$ are polynomials of $\lambda$:

Hence, (2.16) leads to a trigonal curve $\mathcal{K}_{m-2}$:

The discriminant of (2.18) is

here, we assume that $\alpha_0\beta_0\gamma_0(\alpha_0+\gamma_0)\neq0$ and $\mathcal{K}_{m-2}$ is nonsingular. So $\mathcal{K}_{m-2}$ can be compactified by adding the infinite point $P_{\infty_1}$ and the double branch point $P_{\infty_2}$. In our paper, the compactification of $\mathcal{K}_{m-2}$ is still expressed by this symbol. Then, $\mathcal{K}_{m-2}$ becomes a three-sheeted Riemann surface, and its genus is $3n+1$. Each point $P = (\lambda, y)$$\in\mathcal{K}_{m-2}$ satisfies (2.18), along with $P_{\infty_1}$ and $P_{\infty_2}$.

We asumme that $P, P^{\ast}, P^{\ast\ast}$ are three points on three different sheets of $\mathcal{K}_{m-2}$. Set $y_i(\lambda)$, $i = 0, 1, 2$ to satisfy

Using (2.19), we obtain

3.   Asymptotic expansions

We shall discuss the asymptotic expansions of two associated meromorphic functions. Meanwhile, we introduce Abelian differentials to represent quasi-periodic solutions of the three-component Burgers hierarchy. First, the Baker-Akhiezer function $\psi(P, x, x_0, t_r, t_{0, r})$ satisfies

And the meromorphic functions $\phi_2(P, x, t_r)$, $\phi_3(P, x, t_r)$ on ${\mathcal K}_{m-2}$ are respectively introduced as follows:

Expressions (3.1)–(3.3) imply that $\phi_2$ and $\phi_3$ satisfy the following Riccati-type equations

So, we can get the following lemma.

Lemma 3.1. Let $P\in \mathcal {K}_{m-2}\setminus \{P_{\infty_1}, P_{\infty_2}\}$; then,

where

Proof. Substituting the hypotheses

into (3.4) and (3.5), and by analyzing the coefficients of $\zeta$, we prove the lemma. □

Next, we study the asymptotic properties of $\psi_1$ near $P_{\infty_1}$ and $P_{\infty_2}$. By means of the first two expressions of (3.1), we obtain

with

which can imply the essential singularities of $\psi_1$. We use the following symbols to represent the corresponding homogeneous case of $\widetilde{V}^{(r)}_{ij}$, that is to say,

So, we have

Especially,

In fact, consider that $(r, \epsilon) = (0, 1),$ $\bar{\tilde{a}}^{(0, 1)} = -w, \bar{\tilde{b}}^{(0, 1)} = 1, \bar{\tilde{c}}^{(0, 1)} = 0,$ $\lambda = \zeta^{-1}$ denotes the local coordinate near $P_{\infty_1}$ and $\lambda = \zeta^{-2}$ denotes the local coordinate near $P_{\infty_2}$. When $P\to P_{\infty_1}$,

and, when $P\to P_{\infty_2}$,

So, as $P\to P_{\infty_1}$, one infers the following:

as $P\to P_{\infty_1}$, we assume

and $\{\sigma_j(x, t_r)\}, \{\delta_j(x, t_r)\}, {j\in\mathbb{N}}$ are some coefficients. Using (3.1), we obtain

Furthermore, we have

When $\tilde{\alpha}_0 = 1, \tilde{\alpha}_1 = \dots = \tilde{\alpha}_r = 0$ and $\tilde{\beta}_0 = \dots = \tilde{\beta}_r = 0$, we arrive at

Combining (2.5), (2.13) and Lemma 3.1, one gets

from which it can be inferred that

Furthermore,

so (3.13) holds. Equation (3.14) can be proved by using the same method.

By considering (3.13), (3.14) and (2.12), one infers the following:

Given those above preparations, the asymptotic expansions of $\psi_1$ near $P_{\infty_1}$ and $P_{\infty_2}$ read as follows.

Lemma 3.2. Let $P\in \mathcal {K}_{m-2}\setminus \{P_{\infty_1},$ $P_{\infty_2}\}$ and let $(x, x_0, t_r, t_{0, r}) \in \mathbb{C}^4$; so,

where

are two functions independent of variable $x$.

Regarding $\mathcal {K}_{m-2}$, one can choose the basis $\{\mathbb{a}_j, \mathbb{b}_j\}_{j = 1}^{m-2}$ with the intersection numbers

Our basis is as follows

Let us construct the matrices $A = (A_{jk})$ and $B = (B_{jk})$ by respectively applying the following:

Then, $\tau = A^{-1}B$ is a symmetric matrix ^[32,33,34] and we denote $C = A^{-1}$. If $\varpi_l(P)$ takes the normalized basis $\underline{\omega} = (\omega_1, \dots, \omega_{m-2})$ with

then $\int_{\mathbb{a}_k}\omega_j = \delta_{jk}$ and $\int_{\mathbb{b}_k}\omega_j = \tau_{jk}, \ \ j, k = 1, 2, \dots, m-2.$ Utilizing (3.1) and (3.6)–(3.7), one gets the asymptotic properties of $y(P)$:

So,

where

Moreover, $\omega_j$ can be rewritten as

where $\varrho_{j, l}(P_{\infty_s})$ represents constants.

Let $\omega_{P_{\infty_2}, j}^{(2)}(P) \ (j\geq2)$ be the normalized differential of the second kind, satisfying

with

We introduce

Then, we have

for some constants $e_1^{(2)}, e_2^{(2)}, \tilde{e}_1^{(2)}, \tilde{e}_2^{(2)}$ depending on the appropriately chosen point $Q_0$. The associated $\mathbb{b}$-periods are defined by

By means of the relations between the second kind of differential and holomorphic differential $\underline{\omega}$, we can respectively express the members $U_{2, k}^{(2)}$ of $U_{2}^{(2)}$ and $\widetilde{U}_{2r+3, k}^{(2)}$ of $\widetilde{U}_{2r+3}^{(2)}$ as follows:

We define the Abelian differential of the third kind, $\omega^{(3)}_{Q_1, Q_2}$, on ${\mathcal K}_{m-2}\setminus\{Q_1, Q_2\}$, i.e.,

In particular,

with integration constants $e_{1, \infty_s}(Q_0),$ $e_{2, \infty_s}(Q_0), e_{1, \hat{\nu}_0}(Q_0), e_{2, P_0}(Q_0)$ and $s = 1, 2$.

4.   Divisors of $\phi_2$ and $\phi_3$

We will propose the divisors of two meromorphic functions $\phi_2$ and $\phi_3$. The definitions of the two meromorphic functions in (3.2) and (3.3) respectively imply that

where

By complex computation, we obtain

Due to the observation of (4.5), one infers that ${E}_{m-2}, {F}_{m-2}$ and $\mathcal{F}_{m-2}$ are polynomials of $\lambda$:

As $\lambda = \mu_j(x, t_r),$ we get

so we can define

In fact, for $\lambda = \mu_j(x, t_r)$, combining (4.7) and (4.13), we have

that is,

which means that

so, the first definition of (4.16) is reasonable. Similarly, we can prove the others.

From (3.6), (3.7), (4.1) and (4.2), one infers that the divisors of $\phi_2$ and $\phi_3$ have the following respective forms:

Next, our main purpose is to discuss the poles and zeros of $\psi_1$ on $\mathcal{K}_{m-2}\setminus\{P_{\infty_1}, P_{\infty_2}\}$. Observing (4.1) and (4.2), one gets

Lemma 4.1. We suppose that (3.1) holds. Let $(\lambda, x, t_r)\in\mathbb{C}^3$; then,

Proof. Observing the compatibility condition given by (3.21), we have

That is to say,

so we know that the first expression in (4.21) holds.

Furthermore, since

differentiating the above equation with respect to the variable $t_r$, one gets

which can yield the second expression in (4.21). By the same method, the third expression can be obtained. □

By using (4.1), (4.2), (4.12), (4.16) and (4.21), one can compute

On the other hand,

Consequently,

where $O(1)\neq0$. So $\hat{\mu}_1(x, t_r), \dots, \hat{\mu}_{m-2}(x, t_r)$ are $m-2$ zeros of $\psi_1(P, x, x_0, t_r, t_{0, r})$, and $\hat{\mu}_1(x_0, t_{0, r}),$ $\dots, \hat{\mu}_{m-2}(x_0, t_{0, r})$ are $m-2$ poles of $\psi_1$ on $\mathcal{K}_{m-2}$.

5.   Quasi-periodic solutions

We will study the solutions for the three-component Burgers hierarchy in this section. The period lattice $\mathcal {T}_{m-2}$ = $\{\underline{z}\in\mathbb{C}^{m-2}\mid\underline{z} = \underline{N}+\underline{M}\tau, \quad \underline{N}, \underline{M}\in\mathbb{Z}^{m-2}\}$. The Jacobian variety ${\mathcal J}_{m-2}$ of ${\mathcal K}_{m-2}$ is defined by $\mathbb{C}^{m-2}/\mathcal {T}_{m-2}$. The Abel map $\mathcal {\underline{A}}: {\mathcal K}_{m-2}\rightarrow {\mathcal J}_{m-2}$ is as follows:

and it can be extended to $\hbox{Div}({\mathcal K}_{m-2})$:

Define

Then, the Riemann theta function

where $\langle\cdot, \cdot\rangle$ is the Euclidean scalar product and $\underline{z} = (z_1, \dots, z_{m-2})\in\mathbb{C}^{m-2}$.

For simplicity, we introduce a function $\underline{z}: {\mathcal K}_{m-2}\times\sigma^{m-2}{\mathcal K}_{m-2}\rightarrow \mathbb{C}^{m-2}$,

in which the vector of the Riemann constant $\underline{\Lambda} = (\Lambda_1, \dots, \Lambda_{m-2})$ only depends on $Q_0$, and

then,

Using the above preparations, we can give the solutions for the three-component Burgers hierarchy.

Theorem 5.1. Suppose that ${\mathcal K}_{m-2}$ is nonsingular and $\Omega_\mu\subseteq\mathbb{C}^2$ is connected and open. Let $(x, t_r), (x_0, t_{0, r})\in\Omega_\mu$ and $P\in{\mathcal K}_{m-2}\setminus\{P_{\infty_1}, P_{\infty_2}\}$. If $\mathcal{D}_{\hat{\underline{\mu}}(x, t_r)}$, $\mathcal{D}_{\hat{\underline{\xi}}(x, t_r)}$ or $\mathcal{D}_{\hat{\underline{\nu}}(x, t_r)}$ is nonspecial, then

(i)

(ii)

(iii)

Proof. (ⅰ) From (4.17), we can know that $\hat{\mu}_1, \ldots, \hat{\mu}_{m-2}$, $P_{\infty_2}$ are simple poles of $\phi_2$ and $\hat{\nu}_0, \hat{\nu}_1, \ldots, \hat{\nu}_{m-2}$ are simple poles of $\phi_2$. So, one infers the following:

then, we need to determine the expression of $N(x, t_r)$.

From (3.6), combining the expressions of $\phi_2$ near $P_{\infty_2}$, we have

so (5.4) holds. Equation (5.5) can be proved by using a similar method.

(ⅱ) By (5.5), as $P\rightarrow P_{\infty_1},$

as $P\rightarrow P_{\infty_2}$,

Hence, for $P\rightarrow P_{\infty_1},$

for $P\rightarrow P_{\infty_2},$

with $\theta_{s1} = \theta(\underline{z}(P_{\infty_s}, \hat{\underline{\mu}}(x, t_r)))$ and $\theta_{s3} = \theta(\underline{z}(P_{\infty_s}, \hat{\underline{\xi}}(x, t_r))), s = 1, 2.$ Combining (3.6), (3.7), (5.14) and (5.15), we can get (5.6)–(5.8).

(ⅲ) Let $\Psi_1$ be the right hand of (5.9). We will prove that $\psi_1 = \Psi_1$. Applying Proposition 4.2, we find that $\psi_1$ and $\Psi_1$ have the same zeros and poles. Based on the Riemann-Roch theorem, we infer that $\frac{\Psi_1}{\psi_1} = \gamma$ for some constant $\gamma$. From (3.27), we get

Hence, $\gamma = 1$, and (5.9) holds. □

Theorem 5.2. Let $(x_0, t_{0, r}), (x, t_r)\in\mathbb{C}^2.$ Then

Proof. Set

By (5.9), we have

where for some $e_j\in\mathbb{C}$, $\widetilde{\omega} = \sum_{j = 1}^{m-2}e_j\omega_j$. Since $\psi_1$ is single-valued function, all $\mathbb{a}$- and $\mathbb{b}$-periods of $\Omega$ are integer multiples of $2\pi i$; so, for some $M_k, N_k\in\mathbb{Z}$,

So,

where $\underline{N} = (N_1, \dots, N_{m-2})$ and $\underline{M} = (M_1, \dots, M_{m-2})\in\mathbb{Z}^{m-2}$.

Thus, (5.24) has the equivalent form of (5.17). The other two equations in (5.18) and (5.19) can be obtained in the same way. □

Use of AI tools declaration

The authors declare that they have not used artificial intelligence tools in the creation of this article.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant Nos. 12101418, 11931017, 11871440, 11971442).

Conflict of interest

All authors declare no conflict of interest that may affect the publication of this paper.