Coupled discrete Sawada–Kotera equations and their explicit quasi-periodic solutions

Abstract

In this paper, we propose a hierarchy of coupled discrete Sawada–Kotera equations associated with a $3\times 3$ matrix spectral problem. Using the characteristic polynomial of Lax matrix for the hierarchy of coupled discrete Sawada–Kotera equations, we introduce a trigonal curve, a Baker–Akhiezer function and a meromorphic function. We study the asymptotic properties of the Baker–Akhiezer function and the meromorphic function near two infinite points and two zero points on the trigonal curve. The straightening out of various flows is exactly given by means of the Abel map and the meromorphic differential. On the basis of these results and the theory of theory of algebraic curves, we obtain explicit quasi-periodic solutions of the entire coupled discrete Sawada–Kotera hierarchy.

Appendix: The proof of the locality

Our goal in this appendix is to show that ${\hat{g}}_{j,+}$ and ${\hat{g}}_{j,-}, j\ge 0$ are local functions. In the continuous case [47, 49], the locality means pure differential equations. In the discrete case [48], we call a function $f(n)=f(v, w)$ is a local function, if f involves only a finite number of $E^kw, E^lv, k, l\in {\mathbb {Z}}$. In [47, 48], a unified method was proposed to prove the local property for $2\times 2$ matrix cases. According to the idea of Ref. [48], it is first proved that ${\hat{g}}_{j,+}, j\ge 0$ are local functions.

For simplicity, we shall use $a_{j}, b_{j}, c_{j}, d_{j}$ instead of ${\hat{a}}_{j,+}, {\hat{b}}_{j,+}, {\hat{c}}_{j,+}, {\hat{d}}_{j,+}$. Noticing the assumption of $\Delta \Delta ^{-1}=\Delta ^{-1}\Delta =1$, Eq. (2.10) is equivalent to

$$\begin{aligned}&wa_{j} + Ec_{j} + (E+1)d_{j+1} = 0, \end{aligned}$$

(A.1)

$$\begin{aligned}&(E+1)c_{j} + wb_{j+1} + d_{j+1} = 0, \end{aligned}$$

(A.2)

$$\begin{aligned}&(wE^{-1} - Ew)a_{j+1} + (E^2v - vE^{-1})b_{j+1} - w\Delta d_{j+1} = 0, \end{aligned}$$

(A.3)

$$\begin{aligned}&(Ev - vE^{-2})a_{j+1} + (vE^{-1}w - wEv)b_{j+1} - w\Delta c_{j+1} = 0. \end{aligned}$$

(A.4)

Following [34], the explicit expression of $(E+1)^{-1}f(n)$ is given by

$$\begin{aligned} (E+1)^{-1} f(n)={\left\{ \begin{array}{ll} (-1)^{n-n_{0}} \alpha -\sum \limits _{n^{\prime }=n_{0}}^{n-1}(-1)^{n-n^{\prime }} f\left( n^{\prime }\right) , &{} n \ge n_{0}+1, \\ \alpha , &{} n=n_{0}, \\ (-1)^{n-n_{0}} \alpha +\sum \limits _{n^{\prime }=n}^{n_{0}-1}(-1)^{n-n^{\prime }} f\left( n^{\prime }\right) , &{} n \le n_{0}-1, \end{array}\right. } \end{aligned}$$

(A.5)

where $\alpha = (E+1)^{-1}f(n)|_{n=n_0}$. For some fixed integer $n_0$, $d_{j+1}(n_0)=\delta _{j+1}$ is a constant. Then from (A.1) we have

$$\begin{aligned} d_{j+1}(n) = {\left\{ \begin{array}{ll} (-1)^{n-n_{0}} \delta _{j+1} + \sum \limits _{n^{\prime }=n_{0}}^{n-1}(-1)^{n-n^{\prime }} (w(n^\prime )a_{j}(n^\prime ) + c_{j}(n^\prime +1)), &{} n> n_0, \\ \delta _{j+1}, &{} n=n_0, \\ (-1)^{n-n_{0}} \delta _{j+1} - \sum \limits _{n^{\prime }=n}^{n_0-1}(-1)^{n-n^{\prime }} (w(n^\prime )a_{j}(n^\prime ) + c_{j}(n^\prime +1)), &{} n< n_0. \end{array}\right. } \end{aligned}$$

(A.6)

Eqs. (A.6) and (A.2) mean that $b_{j+1}$ and $d_{j+1}$ are polynomials of $a_{j}, c_{j}$ and their translations.

Next, we prove that $a_{j+1}$ and $c_{j+1}$ are polynomials of $b_{j+1}, d_{j+1}, a_{j}, b_{j}, c_{j}, d_{j}, \cdots$ and their translations.

Proposition A.1

Assume $v(n, t)\ne 0, \text {for all }(n, t)\in {\mathbb {Z}}\times {\mathbb {R}}$. Suppose that the Lax matrix V satisfies (2.3). Then the traces $\text {tr}V^k$ of V, $k\ge 1$, are independent of n.

Proof

Noting $v\ne 0$ implies that $\det U\ne 0$. Then using (2.3), it is easy to see that

$$\begin{aligned} E\text {tr}V^k = \text {tr}(UVU^{-1})^k = \text {tr}V^k,\ k\in {\mathbb {N}}, \end{aligned}$$

(A.7)

in terms of $\text {tr}(AB)=\text {tr}(BA)$, which shows that $\text {tr}V^k$ is independent of n. $\square$

From (2.3), (2.5), (2.9) and (2.14), we arrive at

$$\begin{aligned} \begin{aligned} \text {tr}V^2=&\lambda ^2\big [\Delta _1 d^2+E^{-1}(wb+d)^2\big ] + 2\lambda \big [ \Delta _1 c ^{-} d ^{-} + w^- b^- c^- + a ^{-}(v ^{+} b ^{+} - w a )\\&+ v(a b ^{-} + a ^{--}b - w ^{-} b ^{-} b) + (w a + c ^{+})d ^{+} \big ] + \Delta _1 (c^-)^2 + (wa+c^+)^2\\ =&\sum _{m\ge 0}C_{m}\lambda ^{-m}, \end{aligned} \end{aligned}$$

(A.8)

where the constant $C_m$ given by

$$\begin{aligned} \begin{aligned} C_m=&\sum ^{m+2}_{\begin{array}{c} i,j\ge 0,\\ i+j=m+2 \end{array}}\big [\Delta _1 d_{i}d_{j}+E^{-1}(wb_{i}+d_{i})(wb_{j}+d_{j})\big ] + 2\sum ^{m+1}_{\begin{array}{c} i,j\ge 0,\\ i+j=m+1 \end{array}}\big [ \Delta _1 c ^{-}_{i} d ^{-}_{j} + w^- b^-_{i} c^-_{j} \\&+ a ^{-}_{i}(v ^{+} b ^{+}_{j} - w a_{j} )+ v(a_{i} b ^{-}_{j} + a ^{--}_{i}b_{j} - w ^{-} b ^{-}_{i} b_{j}) + (w a_{i} + c ^{+}_{i})d^{+}_{j} \big ] \\&+ \sum ^{m}_{\begin{array}{c} i,j\ge 0,\\ i+j=m \end{array}}\big [\Delta _1 c^-_{i}c^-_{j} + (wa_{i}+c^+_{i})(wa_{j}+c^+_{j}) \big ]\\ =&\sum ^{m+2}_{\begin{array}{c} i,j\ge 1,\\ i+j=m+2 \end{array}}\big [\Delta _1 d_{i}d_{j}+E^{-1}(wb_{i}+d_{i})(wb_{j}+d_{j})\big ] + 2\sum ^{m+1}_{\begin{array}{c} i,j\ge 1,\\ i+j=m+1 \end{array}}\big [ \Delta _1 c ^{-}_{i} d ^{-}_{j} + w^- b^-_{i} c^-_{j}\\&+ a ^{-}_{i}(v ^{+} b ^{+}_{j} - w a_{j} )+ v(a_{i} b ^{-}_{j} + a ^{--}_{i}b_{j} - w ^{-} b ^{-}_{i} b_{j}) + (w a_{i} + c ^{+}_{i})d^{+}_{j} \big ] \\&+ \sum ^{m}_{\begin{array}{c} i,j=0,\\ i+j=m \end{array}}\big [\Delta _1 c^-_{i}c^-_{j} + (wa_{i}+c^+_{i})(wa_{j}+c^+_{j})\big ]\\&+ 2\big [ \Delta _1 c ^{-}_{0} d ^{-}_{m+1} + v ^{+} a ^{-}_{0}b ^{+}_{m+1} + v(a_{0} b ^{-}_{m+1} + a ^{--}_{0}b_{m+1} - b_{m+1}) + (w a_{0} + c ^{+}_{0})d^{+}_{m+1} \big ] \\&+ 2(w^- c^-_{0}b^-_{m+1} - w ^{-}v b_{0} b ^{-}_{m+1}) + 2w^{-1}va_{m+1}^{--} -2c_{m+1}. \end{aligned} \end{aligned}$$

(A.9)

This shows that

$$\begin{aligned} w^{-1}va_{j+1}^{--} -c_{j+1} = f_1( b_{j+1}, d_{j+1}, a_{j}, b_{j}, c_{j}, d_{j}, \cdots ), \end{aligned}$$

(A.10)

where $f_1$ is a polynomial of $b_{j+1}, d_{j+1}, a_{j}, b_{j}, c_{j}, d_{j}, \cdots$ and their translations (the same hereinafter). Similarly, through tedious calculations, we obtain from (A.3), (A.4) and $\text {tr}V^2$ that

$$\begin{aligned} w^{-1}va_{j+1}^{--} + c_{j+1}^- = f_2( b_{j+1}, d_{j+1}, a_{j}, b_{j}, c_{j}, d_{j}, \cdots ). \end{aligned}$$

(A.11)

Combining (A.3), (A.4), (A.10) and (A.11), we have

$$\begin{aligned} \begin{aligned} a_{j+1} =&E^2v^{-1}\Big [(w^+)^{-1}v^+\big ((E^2v - vE^{-1})b_{j+1} - w\Delta d_{j+1}\big ) +{\frac{1}{2}}wE(f_1 + f_2)\\&- (wEv - vE^{-1}w)b_{j+1}\Big ]\\ =&f_3(b_{j+1}, d_{j+1}, a_j, b_j, c_j, d_j, \cdots ), \\ c_{j+1} =&w^{-1}vf_3^{--} - f_1 = f_4(b_{j+1}, d_{j+1}, a_j, b_j, c_j, d_j, \cdots ). \end{aligned} \end{aligned}$$

(A.12)

Finally, we see that ${\hat{a}}_{0,+}$, ${\hat{b}}_{0,+}$, ${\hat{c}}_{0,+}$, ${\hat{d}}_{0,+}$ and ${\hat{a}}_{1,+}$, ${\hat{b}}_{1,+}$, ${\hat{c}}_{1,+}$, ${\hat{d}}_{1,+}$ are local functions. Assume that ${\hat{a}}_{l, +}$, ${\hat{b}}_{l, +}$, ${\hat{c}}_{l, +}$, ${\hat{d}}_{l, +}, 0\le l\le j$ are local functions. Then Eqs. (A.6), (A.2) and (A.12) implies that ${\hat{a}}_{j+1, +}$, ${\hat{b}}_{j+1, +}$, ${\hat{c}}_{j+1, +}$, ${\hat{d}}_{j+1, +}$ are also local functions. By induction, we conclude that ${\hat{g}}_{j, +}=({\hat{a}}_{j,+}, {\hat{b}}_{j,+}, {\hat{c}}_{j,+}, {\hat{d}}_{j,+})^T, j\ge 0$, are local functions. The locality of ${\hat{g}}_{j,-}=({\hat{a}}_{j,-}, {\hat{b}}_{j,-}, {\hat{c}}_{j,-}, {\hat{d}}_{j,-})^T, j\ge 0$ can be proved similarly. In particular, the CDSK equations (2.23) are pure difference differential equations.