Abstract
Starting from the zero-curvature equation and Lenard recurrence relations, we derive a hierarchy of generalized Toda lattices. The trigonal curve is introduced through the Lax pair characteristic polynomial for the discrete hierarchy, from which a Dubrovin-type equation is established. Then the asymptotic behavior of the Baker–Akhiezer function and the meromorphic function is analyzed, and the divisors of the two functions are also discussed. Moreover, the Abel map is defined and the corresponding flows are straightened out on the Jacobian variety, such that the final algebro-geometric solutions of the hierarchy are calculated in terms of the Riemann theta function.
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1. Introduction
is an absolutely integrable equation with exponential interaction that was discovered in the course of seeking a system with rigorous periodic solutions; its exponential interaction was used to explain the nonergodic character of the famous Fermi–Pasta–Ulam problem [3]. It has abundant mathematical structures and is regarded as a model of physical phenomena, with the well-known equations such as the nonlinear Schrödinger (NLS) and Korteweg–de Vries equations being closely related to it or deduced from it by appropriate limit procedures [4], [5]. In addition, it can describe the motion of a chain of particles with nearest-neighbor interaction in constructing different mathematical models; the Toda lattice model of DNA is also a typical representative in biology [6].
It is worth mentioning that by the variable transformation \(\varpi=-\exp(y-y^{+})\) and \(x=y_t\), the Toda lattice can be rewritten in the form
With the increase in scholars’ attention to the Toda lattice, a variety of important methods were applied to it and numerous results have been achieved since it was proposed [7]–[12]. As one of the most effective research tools, the algebro-geometric methods were extensively applied to the Toda lattices. With the development of the finite-gap integration method in the works of Novikov, Matveev, Its, and others, the mathematical theory of the algebro-geometric method has been systematically developed since the early 1970s [13], [14]. The solutions can not only describe the integrability properties of the equations but also reveal the internal structure of the solutions for soliton equations [15]–[18].
The algebro-geometric solutions for numerous soliton equations related to a \(2\times 2\) matrix spectral problem have been obtained using the theta functions of hyperelliptic curves in a series of studies [19]–[21]. However, the studies of algebro-geometric solutions of 3rd-order soliton equations are very few. In the course of studying the algebro-geometric solutions of the 3rd-order soliton equations, the most classical findings originates in the Boussinesq equation, whose 3rd-order differential operators were studied in terms of the reduction theory of Riemann theta functions [22]; finite-gap solutions of the NLS equation were also confirmed smoothly by means of a special algorithm [23]. In 1999, based on the algebro-geometric method, Dickson, Gesztesy, and Unterkofler proposed a unified framework that yields all algebro-geometric solutions of the entire Boussinesq hierarchy related to a 3rd-order differential operator [24]. Based on the framework proposed previously, a systematic method for introducing a trigonal curve was developed with the help of the characteristic polynomial of the Lax matrix associated with the higher-order matrix spectral problem, from which the algebro-geometric method was successfully generalized to yield algebro-geometric solutions of the continuous hierarchies related to \(3\times 3\) matrices [25], [26]. Then the algebro-geometric method was further extended to 3rd-order discrete hierarchies [27], [28].
In this paper, we introduce the trigonal curve to define the Baker–Akhiezer function \(\Xi\) and the corresponding meromorphic function \(\Theta\). The soliton equations can then be separated into solvable Dubrovin-type ordinary differential equations. Based on the above step, the characteristics of the functions can be further analyzed. With the systematic algebro-geometric theory as support, we discuss the application of the algebro-geometric methods to the discrete hierarchy of a 3rd-order generalized Toda lattice
which becomes is the Toda lattice mentioned above (1.2) if \(q=0 \), \(r=\varpi^{+}\), \(x=s/s^{-}\), and \(v=0\). The Hamiltonian system for (1.3) was constructed in [29].
The paper is organized as follows. In Sec. 2, the difference operators \(K_n\) and \(J_n\) are deduced in accordance with the Lenard recurrence relations and hierarchy (1.3) is then derived from the zero-curvature equation. In Sec. 3, the trigonal curve \(\mathcal K_{l-1}\) is defined for the characteristic polynomial of the Lax pair for hierarchy (1.3), whence the functions \(\Xi\) and \(\Theta\) can be defined. In Sec. 4, in the stationary case, we analyze the characteristics of the functions and introduce the Abel differentials; the potentials of the Lax pair are then expressed in terms of the Riemann theta function. In Sec. 5, we apply the analysis in last two sections to the time-dependent case and separate hierarchy (1.3) into solvable Dubrovin-type ordinary differential equations. Then we straighten out the flows and obtain the Riemann theta representation. On the whole, the algebro-geometric solutions of hierarchy (1.3) are obtained and we rewrite the Riemann theta representation of the potentials for low genera. We summarize and conclude in Sec. 6.
2. The hierarchy of a generalized Toda lattice
We suppose that \(q\), \(r\), \(s\), and \(v\) satisfy the following conditions: in the stationary case,
and in the time-dependent case,
where \(\mathbb{C}^{\,\mathbb{Z}}\) is the set of all complex-valued functions of a variable in \(\mathbb{Z}\).
On the complex-valued sequence \(\hbar=\{\hbar(n)\}_{n\in\mathbb{Z}}\), we define the shift operators \(E^{\pm}\) as
and write \(\hbar^{\pm}=E^{\pm}\hbar\) with \( \hbar\in\mathbb{C}^{\,\mathbb{Z}}\).
We consider the discrete \(3\times 3\) matrix spectral problem [29]
where \(q\), \(r\), \(s\), and \(v\) are potentials and \(\lambda\) is a constant. The Lenard recurrence relations are
We then introduce the starting points
and define two difference operators \(J_n\) and \(K_n\) as
where
Hence, \(\tilde g_j\) and \(\bar g_j\) can be found using the operators \(K_n\) and \(J_n\); the first two members are given by
To deduce the hierarchy related to spectral problem (2.1), we introduce the stationary zero-curvature equation
which is equivalent to
where each element \(\Gamma_{ij}=\Gamma_{ij}(a,b,c,d)\) is a Laurent expansion in \(\lambda\),
with
We can show by direct calculation that Eqs. (2.6) and (2.7) imply the Lenard equation
We substitute (2.8) in (2.9) and compare the powers of \(\lambda\) to deduce the recurrence relations
where \(G_j=(a_j,b_j,c_j,d_j)^{\mathrm T}\). It is evident that \(\operatorname{ker}{J_n}=\{\alpha_0 \tilde g_0+\beta_0\bar g_0\mid\alpha_0,\beta_0\in\mathbb{R}\}\) and \(G_j\) has the expansion
where \(\alpha_j\) and \(\beta_j\) are constants.
Assuming that \(\Xi\) satisfies the discrete matrix spectral problem (2.1), we have
where \(\widehat\Gamma_{ij}^{(m)}=\widehat\Gamma_{ij}(\hat a^{(m)},\hat b^{(m)},\hat c^{(m)},\hat d^{(m)})\) and
Similarly, the elements \(\hat a_j\), \(\hat b_j\), \(\hat c_j\), and \(\hat d_j\) can be determined as
where \(\widehat G_j\) are also solutions of (2.10). We note, importantly, that \(\hat\alpha_j\), \(\hat\beta_j\) and \(\alpha_j\), \(\beta_j\) in (2.12) are absolute of each other. The zero-curvature equation \(U_{t_m}=(E\widehat\Gamma^m)U-U\widehat\Gamma^m\) is generated by the compatibility condition of Eqs. (2.1) and (2.12), which is equivalent to discrete hierarchy (1.3),
and the vector can be represented as
where \( \underline {\hat\alpha}^{(j)}=(\hat\alpha_0,\ldots,\hat\alpha_j)\), and \( \underline {\hat\beta}^{(j)}=(\hat\beta_0,\ldots,\hat\beta_j)\) for \(j\ge 0\).
The first nontrivial member of hierarchy (2.16) is given by as
whence, with \(\alpha_0=1\) and \(\beta_0=0\), we have
Similarly, for \(j=2\) and \(\widehat{\mathcal I}_1=K_n(\alpha_0\tilde g_1+\alpha_1\tilde g_0+\beta_0\bar g_1+\beta_1\bar g_0)\) with \(\alpha_0=1\), \(\beta_0=1\), and \(t_0=t\), we obtain the hierarchy that we study in what follows:
If \(q=0 \), \(r=\varpi^{+}\), \(x=s/s^{-}\), and \(v=0\), Eqs. (2.18) become the Toda lattice (1.2).
3. The stationary meromorphic function
We consider hierarchy (1.3) in the stationary case \(\mathcal I_p=\mathcal I(q,r,s,v; \underline {\alpha}^{(p)}\), \( \underline {\beta}^{(p)})=0\), \( \underline {\alpha}^{(p)}=(\alpha_0\ldots\alpha_p)\), and \( \underline {\beta}^{(p)}=(\beta_0\ldots\beta_p)\). It is then equivalent to the stationary zero-curvature equation
with \(\Gamma_{ij}^{(p)}=\Gamma_{ij}(a^{(p)},b^{(p)},c^{(p)},d^{(p)})\),
Direct calculation indicates that the characteristic polynomial \(\digamma_l(\lambda,f)=\det( fI-\Gamma^{(p)})\) of \(\Gamma^{(p)}\) also satisfies zero-curvature equation (3.1) and is a constant independent of \(n\). It has the expansion
where \(X_l(\lambda),Y_l(\lambda)\) and \(Z_l(\lambda)\) are constant-coefficient polynomials in \(\lambda\),
Then the trigonal curve \(\digamma_l(\lambda,f)=0\) whose degree is \(l=3p\) for \(\alpha_0\beta_0\neq 0\) can be introduced as
Under the condition \(l=3p\), it is obvious that the trigonal curve \(\mathcal K_{l-1}\) can be compactified by adding different infinite points \(u_{\infty'}\) and \(u_{\infty''}\) based on (3.2) and (3.4), where we choose \(u_{\infty''}\) as a double branch point. We still use \(\mathcal K_{l-1}\) to denote the compactified curve. The discriminant of (3.5) is
By the Riemann–Hurwitz formula, we can obtain that the arithmetic genus of \(\mathcal K_{l-1}\) is \(l-1\). Therefore, \(\mathcal K_{l-1}\) turns into a three-sheet Riemann surface of genus \(l-1\) if the curve is irreducible and
for any \(u_0=(\lambda_0,f_0)\in\mathcal K_{l-1}\).
We introduce the stationary Baker–Akhiezer function \(\Xi\) as
Based on the function \(\Xi\), the meromorphic function \(\Theta\) on \(\mathcal K_{l-1}\) is defined as
whence we have
The meromorphic function \(\Theta(u,n)\) obtained in accordance with (3.7) and (3.8) is
where
Moreover, we introduce two other elements
Obviously, we can find various relations among polynomials (3.11), (3.12) and \(X_l\), \(Y_l\), \(Z_l\). We list some of them:
Using (3.1), (3.2), (3.11), and (3.13), we find that \(F_{l-1}\) and \(E_{l-1}\) are polynomials of degree \(l-1\) and can therefore be represented as
On the trigonal curve \(\mathcal K_{l-1}\), we define \(\{\tilde\mu_j(n)\}_{j=\overline{1,l-1}}\) and \(\{\tilde\mu_j^{+}(n)\}_{j=\overline{1,l-1}}\) as
For convenience, we let \(u\), \(u^*\), and \(u^{**}\) denote points on each of the three different sheets of the Riemann surface \(\mathcal K_{l-1}\) and suppose that \(f_i(\lambda)\) (\(i=1,2,3\)) are three roots of \(\digamma_l(\lambda,f)=0\):
Then the three points \((\lambda,f_1(\lambda))\), \((\lambda, f_2(\lambda))\), and \((\lambda,f_3(\lambda))\) are also on the Riemann surface \(\mathcal K_{l-1}\). Let \(\{u,u^*,u^{**}\}=\{(\lambda,f_i(\lambda)),\,i=1,2,3\}\) be any one of the three points. From (3.16), the following system can easily be obtained:
The function \(\Theta(u,n)\) then satisfies the relations
4. Algebro-geometric solutions of the stationary hierarchy
We analyze the asymptotic behavior of the functions \(\Theta(u,n)\) and \(\Xi(u,n)\), and then introduce the Abel differential and the Riemann theta function. As a result, we obtain algebro-geometric solutions in the stationary case, whereby the potentials \(q\), \(r\), \(s\), and \(v\) can be expressed as in terms of the Riemann theta function.
First, it follows by direct calculation that \(\Theta(u,n)\) satisfies the Riccati-type equation
Introducing the local coordinate \(\varsigma=\lambda^{-1}\) near \(u_{\infty'}\) and comparing the powers of \(\varsigma\), we have the formula
with
As at the preceding step, we introduce the local coordinate \(\lambda=\eta^{-2}\) near \(u_{\infty''}\) and compare the powers of \(\eta\), which yields
with
The divisors [16] of the meromorphic function are
whence it follows that \(\Theta(u,n)\) has \(l\) zeros, \(u_{\infty'}\), \(\tilde\mu_1^{+}(n),\ldots,\tilde\mu_{l-1}^{+}(n)\), and \(l\) poles \(u_{\infty''}\), \(\tilde\mu_1(n),\ldots,\tilde\mu_{l-1}(n)\). Besides, according to (3.8), (4.2) and (4.3), we have
where
and
The divisors of the Baker–Akhiezer function \(\Xi_1(u,n,n_0)\) are
The Riemann surface \(\mathcal K_{l-1}\) has a canonical basis of cycles \(\mathbf w_1,\ldots,\mathbf w_{l-1}\) and \(\mathbf o_1,\ldots,\mathbf o_{l-1}\) whose intersection numbers are
On \(\mathcal K_{l-1}\), we define
and set
where the matrices \(\mathbb{O}\) and \(\mathbb{P}\) are invertible. Now, we introduce new matrices \(\mathbb{Q}\) and \(\tau\) such that \(\mathbb{Q}=\mathbb{O}^{-1}\) and \(\tau=\mathbb{O}^{-1}\mathbb{P}\). It is easy to see that \(\tau\) is symmetric \((\tau_{ij}=\tau_{ji})\) and its imaginary part is positive definite (\(\operatorname{Im}\tau>0\)).
Transforming \(\tilde\omega_h\) into the new basis \(\omega_j\),
we have
We define the third kind holomorphic differential on \(\mathcal K_{l-1}\backslash\{Q',Q''\}\) as \(\omega_{Q',Q''}^{(3)}\). It has poles at \(Q_k\) with the residues \((-1)^{k+1}\), \(k=1,2\). In particular,
For \(\omega_{u_{\infty'},u_{\infty''}}^{(3)}\), we have
whence
where \(Q_0\) is a variable base point on \(\mathcal K_{l-1}\{u_{\infty'},u_{\infty''}\}\), and \(\ell_1(Q_0)\), \(\ell_2(Q_0)\), \(\omega_{0}^{\infty'}\), and \(\omega_{0}^{\infty''}\) are constants.
Let \(\mathcal T_{l-1}\) be the period lattice \(\{ \underline {z}\in\mathbb{C}^{l-1}| \underline {z}= \underline {\mathcal F}+ \underline {\mathcal H}\tau, \underline {\mathcal F}, \underline {\mathcal H}\in\mathbb{Z}^{l-1}\}\). On \(\mathcal K_{l-1}\), we regard \(\mathcal J_{l-1}=\mathbb{C}^{l-1}/\mathcal T_{l-1}\) as the Jacobian variety. We can then introduce the Abel map \( \underline {\mathcal A}\colon\mathcal K_{l-1}\to\mathcal J_{l-1}\),
We define the divisors group \(\operatorname{Div}(\mathcal K_{l-1})\) and continue the above equation to it by linearity:
We consider the nonspecial divisor \(\mathcal D_{ \underline {\tilde\mu}(n)}=\sum_{\sigma=1}^{l-1}\tilde\mu_\sigma(n)\) and define
where \( \underline {\rho}=(\rho_1(n),\ldots,\rho_{l-1}(n))\) and \( \underline {\omega}=(\omega_1,\ldots,\omega_{l-1})\). We define the Riemann theta function \(\theta( \underline {z})\) on \(\mathcal K_{l-1}\) as
We then introduce the function
where \(\sigma^{l-1}\mathcal K_{l-1}\) is the \((l-1)\)th symmetric power of \(\mathcal K_{l-1}\), and the expression of the vector \(\Lambda\) depending on the base point \(Q_0\) is
Theorem 1.
Let \(u=(\lambda,f)\in\mathcal K_{l-1}\backslash\{u_{\infty'},u_{\infty''}\}\) , \((n,n_0)\in\mathbb{Z}^2\) , and \(\mathcal D_{ \underline {\tilde\mu}(n)}\) be a nonspecial divisor. Then
The divisor \(\mathcal D_{ \underline {\tilde\mu}}\) can be linearized as follows under the Abel map:
Proof.
Using the Abel theorem, we can obtain (4.17) from (4.7) and deduce the equations by (4.10),
Letting \(\phi\) denote the right-hand side of (4.16), we find that \(\phi\) and \(\Theta\) have the same zeros and poles. According to the Riemann–Roch theorem and Eqs. (4.3) and (4.18), we have
Hence, the Riemann theta representation of \(\Theta(u,n)\) can be proved and the representation of \(\Xi_1(u,n,n_0)\) can also be proved by (3.8) and the representation of \(\Theta\).
Theorem 2.
Let \(n\in\mathbb{Z}\) , \(\mathcal D_{ \underline {\tilde\mu}(n)}\) be a nonspecial divisor. Then the potentials \(q\) , \(r\) , \(s\) , and \(v\) can be expressed in terms of the Riemann theta function as
Proof.
According to the Abel theorem and (4.6), we have
so
whence using \(f=\Gamma_{11}^{(p)}+\Gamma_{12}^{(p)}+\Gamma_{13}^{(p)}/\Theta^{-1}\) we have
Using (4.9) and (4.10), we deduce the equality
The expression of \(\omega_j\) can then be obtained by direct calculation:
With the Riemann theta representation of \(\Theta(u,n)\) in (4.16), we have
where \(u\to u_{\infty'}\) and \(\theta'=\theta( \underline {z}(u_{\infty'}, \underline {\tilde\mu}(n)))\), \(\theta_{+}^{'}=\theta( \underline {z}(u_{\infty'}, \underline {\tilde\mu}^{+}(n)))\).
Similarly to the previous steps, we obtain
where \(\eta\to 0\), \(\theta''=\theta( \underline {z}(u_{\infty''}, \underline {\tilde\mu}(n)))\), and \(\theta_{+}^{''}=\theta( \underline {z}(u_{\infty''}, \underline {\tilde\mu}^{+}(n)))\). Hence, we have the following formulas as \(\eta\to 0\) and \(\varsigma\to 0\):
In addition, in accordance with (4.2) and (4.3), we have
Formulas (4.19) are thus proved.
We let the \(\mathbf o\)-periods of \(\omega_{u_{\infty'},u_{\infty''}}^{(3)}\) be denoted as
Combining (4.11), (4.17), (4.19), and (4.20) shows that the Riemann theta representation for \(q(n)\), \(r(n)\), \(s(n)\), and \(v(n)\) has a remarkable linearity in \(n\in\mathbb{Z}\times\mathbb{R}\). As a matter of fact, Eqs. (4.19) can be rewritten as
where
5. Algebro-geometric solutions of the hierarchy in the time-dependent case
In this section, we discuss the algebro-geometric solutions of (1.3) in the time-dependent case. We first define the time-dependent Baker–Akhiezer function
From the compatibility condition for Eqs. (5.1), we have
Direct calculation shows that \(\digamma_l(\lambda,f)=\det(f I-\Gamma^{(p)})\) satisfies the stationary zero-curvature equation. The Lax pair \(\Gamma^{(p)}\) characteristic polynomial is a constant independent of \(n\) and \(t_m\), and we have
Then the trigonal curve \(\mathcal K_{l-1}\) is naturally defined in the time-dependent case as
The meromorphic function \(\Theta (u,n,t_m)\) on \(\mathcal K_{l-1}\) is defined as
whence we have
From (5.3), we have
where \(u=(\lambda,f)\) and the elements such as \(A_l(\lambda,n,t_m)\) are defined the same as in the stationary case. Similarly,
We give the expressions for \(\{\tilde\mu_j(n,t_m)\}_{j=1,\ldots,l-1}\subset\mathcal K_{l-1}\) and \(\{\tilde\mu_j^{+}(n,t_m)\}_{j=1,\ldots,l-1}\subset\mathcal K_{l-1}\) in the form
From (5.4), the divisor of \(\Theta(u,n,t_m)\) can be expressed as
and hence \(\Theta(u,n,t_m)\) still has \(l\) zeros, \(u_{\infty'},\tilde\mu_1^{+}(n,t_m),\ldots,\tilde\mu_{l-1}^{+}(n,t_m)\), and \(l\) poles \(u_{\infty''},\tilde\mu_1(n,t_m),\ldots, [0]\tilde\mu_{l-1}(n,t_m)\).
By the same calculation, it is clear that \(\Theta(u,n,t_m)\) satisfies the Riccati-type equation
Also similarly to the preceding subsection, it can be shown that the function \(\Theta(u,n,t_m)\) satisfies the system of equations
Differentiating the meromorphic function with respect to \(t_m\), we have
whence
where \(\Delta\) is the difference operator and \(\Delta=E-1\).
The dynamics of \(\mu_j(n,t_m)\) of \(F_{l-1}(\lambda,n,t_m)\) can be described by Dubrovin-type equations in accordance with the following lemma.
Lemma 1.
Let \((n,t_m)\in\mathbb{Z}\times\mathbb{R}\) . The zeros \(\{\mu_j(n,t_m)\}_{j=\overline{1,l-1}}\) of \(F_{l-1}(\lambda,n,t_m)\) satisfy the equations
where \(1\le j\le l-1\) .
Proof.
From (3.10), (3.11), and (5.2), we have
With (3.12) and (5.5), we then have
whence
Therefore,
and Eq. (5.10) is thus proved.
Moreover, in accordance with (5.1), we have
and
where \(u=(\lambda,f)\in\mathcal K_{l-1}\backslash\{u_{\infty'},u_{\infty''}\}\), and \((n,t_m),(n_0,t_{0m})\in\mathbb{Z}\times\mathbb{R}\). Using the function in (5.11), we define \(\Pi_m(u,n,t_m)\) as
whence
where
and \(\hat\alpha_1=\cdots=\hat\alpha_m=\hat\beta_1=\cdots=\hat\beta_m=0\). Hence,
Lemma 2.
Let \((n,t_m)\in\mathbb{Z}\times\mathbb{R}\) , and let \(\varsigma=\lambda^{-1}\) and \(\eta=\lambda^{1/2}\) be local coordinates near \(u_{\infty'}\) and \(u_{\infty''}\) . Then
Proof.
We set \(\tilde\alpha^m=\hat\alpha^m|_{\hat\alpha_0=1,\hat\beta_0=0}\). From (5.1) and (5.13), we then have
Using (4.2) and (4.3), we have the following result as \(m=0\):
We now suppose that
where the coefficients of \(\{\delta_j(n,t_m)\}_{j\in\mathbb{Q}_0}\) and \(\{\kappa_j(n,t_m)\}_{j\in\mathbb{Q}_0}\) can be determined. In accordance with (5.9) and (5.13), we obtain
Comparing the coefficients of \(\varsigma\) and \(\eta\), we have
whence
It then follows that
Therefore, in view of \(\Delta\Delta^{-1}=\Delta^{-1}\Delta=1\), the following results can be deduced:
We have proved (5.15) for \(\widetilde\Pi_{m+1}^{(k)}\) for \(k=1\); the proof for \(k=2\) is similar.
Let \(\omega_{u_{\infty k},j}^{(2)}\), \(j\ge 2\), be the normalized differential of the second kind that is holomorphic on \(\mathcal K_{l-1}\backslash\{u_{\infty k}\}\) and has a \(j\)th-order pole at \(u_{\infty k}\) (\(k=1,2\)),
and has the \(\mathbf w\)-periods \(\int_{\mathbf w_\sigma}\omega_{u_{\infty k},j}^{(2)}=0\), \(\sigma=1,\ldots,l-1\). Let
Integrating (5.16), we obtain
We next find the explicit Riemann theta function representations for the functions \(\Theta(u,n,t_m)\) and \(\Xi_1(u,n,n_0,t_m,t_{0m})\).
Theorem 3.
Let \(u=(\lambda,f)\in\mathcal K_{l-1}\backslash\{u_{\infty'},u_{\infty''}\}\) , \((n,n_0,t_m,t_{0m})\in\mathbb{Z}^2\times\mathbb{R}^2\) . If \(\mathcal D_{ \underline {\tilde\mu}(n,t_m)}\) is nonspecial and \((n,t_m)\in\mathbb{Z}\times\mathbb{R}\) , then \(\Theta(u,n,t_m)\) and \(\Xi_1(u,n,n_0,t_m,t_{0m})\) can be represented as
and
Proof.
For \(t_{0m}=t_m\), \(\Xi_1(u,n,n_0,t_m,t_m)\) has the form
We also need to verify that
We let \(\mathcal W_1(u,n_0,n_0,t_m,t_m)\) denote the right-hand side of (5.18). Then
Next, we prove that
First, we use (3.12), (5.4), and (5.13) to obtain the formula
where \(O(1)\neq 0\). Consequently,
Hence, \(\Xi_1(u,n_0,n_0,t_m,t_{0m})\) and \(\mathcal W_1(u,n_0,n_0,t_m,t_{0m})\) have the same poles and zeros on \(\mathcal K_{l-1}\). In addition, we can find that \(\mathcal K_{l-1}\), \(\Xi_1(u,n_0,n_0,t_m,t_{0m})\) and \(\mathcal W_1(u,n_0,n_0,t_m,t_{0m})\) have the identical essential singularities. Because of \(\mathcal D_{\tilde{ \underline {\mu}}(n,t_m)}\) is nonspecial, Eqs. (5.17) and (5.18) have been proved.
We let the \(\mathbf o\)-periods of \(\widehat{\mho}_m^{(2)}\) be denoted as
Theorem 4 (straightening out of the flows).
The following equality holds :
Proof.
Introducing the meromorphic differential
we use (5.18) to obtain
where \(\check{\ell}\in\mathbb{C}\), \(j=1,\ldots,l-1\). On \(\mathcal K_{l-1}\), any of the \(\mathbf w\)-periods and \(\mathbf o\)-periods is an integer multiple of \(2\pi i\) because \(\Xi_1(u,n,n_0,t_m,t_{0m})\) is single-valued, and hence
where \(\mathcal B_\sigma\in\mathbb{Z}\). Similarly, for \(\mathcal{C}_\sigma\in\mathbb{Z}\) (\(\sigma=1,\ldots,l-1\)), we have
whence
where \( \underline {\mathcal{C}}=(\mathcal{C}_1,\ldots,\mathcal{C}_{l-1})\in\mathbb{Z}^{l-1}\) and \( \underline {\mathcal B}=(\mathcal B_1,\ldots,\mathcal B_{l-1})\in\mathbb{Z}^{l-1}\). Therefore, we have proved (5.20) because (5.21) is equivalent to (5.20).
From Theorem 4, we have
where
Theorem 5.
Let \((n,t_m)\in\mathbb{Z}\times\mathbb{R}\) and let the divisor \(\mathcal D_{\tilde{ \underline {\mu}}(n,t_m)}\) be nonspecial. Then
Combining (5.20) and (5.23) shows that the Riemann theta representation for \(q(n,t_m)\), \(r(n,t_m)\), \(s(n,t_m)\) and \(v(n,t_m)\) has a remarkable linearity in \((n,t_m)\in\mathbb{Z}\times\mathbb{R}\). Expressions (5.23) can then be rewritten as
Hence, formulas (5.23) and (5.24). give algebro-geometric solutions of the discrete hierarchy of the generalized Toda lattice (1.3).
To clarify the algebro-geometric solutions, we consider a simpler example of the Riemann theta representation under the condition \(p=1\). The genus of \(\mathcal K_2\) is therefore equal to 2, and we can obtain the following results by direct calculation:
The trigonal curve \(\digamma_3(\lambda,f)=0\), whose degree is \(l=3\) (\(\alpha_0\beta_0\neq 0\)), can then be defined as
where
and \(\imath_1\), \(\imath_2\), and \(\imath_3\) are arbitrary constants. Therefore, the polynomials of \(F_2\) and \(E_2\) can be reexpressed as
The Riemann theta representations of the potentials in the case of genus is 2 can therefore be rewritten as
where
These formulas define algebro-geometric solutions of the discrete hierarchy of the generalized Toda lattice (1.3) in the case of genus 2.
6. Conclusions and Remarks
We have found algebro-geometric solutions of the hierarchy of generalized Toda lattices. The hierarchy was generated using the zero-curvature equation, and the functions \(\Xi\) and \(\Theta\) were introduced on the trigonal curve. Based on the Abel differential, the Riemann theta representations of the potentials were constructed in the stationary and time-dependent cases, and solutions of the hierarchy were obtained. Currently, increasingly many researchers focus on trigonal curves and the application of these methods is gaining in popularity. Discussing the algebro-geometric solutions of the 4th-order soliton equations is also interesting, and we plan to address this problem in the future. Equally important is the study of soliton solutions beyond the algebro-geometric solutions, such as the lump–soliton and breather solutions.
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Acknowledgments
The authors are very grateful to Professor Xian-Guo Geng for his kind assistance and constructive suggestions.
Funding
Partial financial support was received from the National Nature Science Foundation of China (grant No. 11701334) and the “Jingying” Project of Shandong University of Science and Technology.
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Prepared from an English manuscript submitted by the author; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, 2023, Vol. 215, pp. 47–73 https://doi.org/10.4213/tmf10388.
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Zhao, Q., Li, C. & Li, X. Application of the trigonal curve to a hierarchy of generalized Toda lattices. Theor Math Phys 215, 495–519 (2023). https://doi.org/10.1134/S0040577923040037
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DOI: https://doi.org/10.1134/S0040577923040037