1 Introduction↩︎

The present paper is a continuation of the research started in [1], and focuses on the sine-Gordon equation \[\label{SGEq} \phi_{\mathrm{t}\mathrm{x}} = - 4\sin \phi,\tag{1}\] which is known to be completely integrable, and possess soliton solutions. The equation has various applications in physics, see [2].

In [3], multi-soliton solutions of the sine-Gordon equation are obtained by Hirota’s method; and interaction of two solitons is analyzed in detail. In [4], a similar multi-soliton solution was obtained by the inverse scattering method.

The sine-Gordon equation is associated with a hierarchy of integrable hamiltonian systems. Each system is finite, and produces a finite-gap solution of the equation. Quasi-periodic solutions of 1 , expressed in terms of theta functions, are suggested in [5], where the reality conditions are also addressed. These solutions have the form, see [6], \[\phi(\mathrm{x}, \mathrm{t}) = 2 \imath \log \frac{\theta( \imath (\boldsymbol{U}\mathrm{x}+ \boldsymbol{W} \mathrm{t}) + \boldsymbol{D} + \imath \pi \Delta)}{\theta( \imath(\boldsymbol{U}\mathrm{x}+ \boldsymbol{W} \mathrm{t}) + \boldsymbol{D} )},\] where \(\boldsymbol{U}\), \(\boldsymbol{W}\), \(\boldsymbol{D}\), \(\Delta\) are vectors defined through periods of the corresponding spectral curve. In [6] a derivation of such a solution for an integrable system of the sine-Gordon hierarchy is shown.

Recently, some improvement on computing multi-soliton solutions by the inverse scattering method was suggested in [7]; a review of solutions expressed in elementary and elliptic functions can be found therein. In addition to what mentioned in [7], two- and three-gap elliptic solutions were constructed on the spectral curves which are two- and three-sheeted coverings of the elliptic curve, see [8], [9].

In the present paper we review the method of constructing the sine-Gordon hierarchy on coadjoint orbits of a loop group. This method differs from the one used in [6], and leads straight to algebraic integration in terms of \(\wp\)-functions. As mentioned in [10], a finite-gap solution of the sine-Gordon equation is given by the function \(\wp_{1,2g-1}\), one of \(\wp\)-functions which generalize the Weierstrass \(\wp\)-function. This solution is accurately derived below. Though a similarity in the structure with [1] can be observed, the sine-Gordon hierarchy arises on different coadjoint orbits, and so the seeming repetition presents an essentially different exposition.

The new results are given by the reality conditions for the sine-Gordon equation, which provide real-valued solutions in terms of the particular \(\wp\)-function. The obtained reality conditions are more narrow than the conditions presented in [5]. Actually, hamiltonian systems of the sine-Gordon hierarchy arise on hyperelliptic curves with fixed branch points at \((0,0)\) and infinity, and all other pairwise conjugate branch points. On hyperelliptic curves with all real branch points hamiltonian systems of the sinh-Gordon hierarchy arise.

The obtained real-valued quasi-periodic solutions are easily computed, as illustrated by examples in genera one and two. The proposed method enables to analyze every hamiltonian system in the hierarchy in question by investigating how different regimes of wave propagation depend on values of integrals of motion.

The paper is organized as follows. In Preliminaries we recall basic knowledge on homologies and cohomologies of hyperelliptic curves, definitions of theta, sigma, and \(\wp\)-functions, and also representation of characteristics in terms of partitions of the set of indices of branch points. In Section 3 integrable systems of the sine(sinh)-Gordon hierarchy are constructed on coadjoint orbits in an affine Lie algebra. In Section 4 variables of separation are derived and proved to be quasi-canonical. Section 5 is devoted to algebraic integration, and expressions in terms of \(\wp\)-functions are obtained for all dynamical variables of any hamiltonian system in the hierarchy. Also relations between \(\mathrm{x}\), \(\mathrm{t}\) on the one hand, and coordinates of the Jacobian variety of the spectral curve, which serve as arguments of \(\wp\)-functions, on the other hand, are discovered. In Section 6 behaviour of \(\wp\)-functions on the Jacobian variety is analyzed, and a particular subspace which serves as the domain of the corresponding finite-gap solution is singled out. Finally, in Section 7 examples of systems with spectral curves of genera one and two are presented, with analyses of phase spaces, and computation of non-linear waves corresponding to different values of hamiltonians.

2 Preliminaries↩︎

2.1 Hyperelliptic curves↩︎

We work with the canonical form of a hyperelliptic curve \(\mathcal{V}\) of genus \(g\), namely \[\label{V22g1Eq} f(x,y) \equiv -y^2 + x^{2 g+1} + \sum_{i=0}^{2g} \lambda_{2i+2} x^{2g-i} = 0.\tag{2}\] In terms of branch points \((e_i,0)\), or simply \(e_i\), \(i=1\), …, \(2g+1\), \(f\) has the form \[\label{V22g1EqBP} f(x,y) = -y^2 + \prod_{i=1}^{2g+1} (x- e_i).\tag{3}\] One more branch point \(e_0\) is located at infinity, and serves as the basepoint. We assume that all branch points are distinct, and so the curve is not degenerate, that is, the genus equals \(g\). Finite branch points are enumerated in the ascending order of real and imaginary parts. The Riemann surface of \(\mathcal{V}\) is constructed by means of the monodromy path through all branch points, according to the order; the path starts at infinity and ends at infinity, see the orange line on fig. 1. For more details on constructing the Riemann surfaces of hyperelliptic curves suitable for computation see [11].

A homology basis is defined after H. Baker [12]. Cuts are made between points \(e_{2k-1}\) and \(e_{2k}\) with \(k\) from \(1\) to \(g\). One more cut starts at \(e_{2g+1}\) and ends at infinity. Canonical homology cycles are defined as follows. Each \(\mathfrak{a}_k\)-cycle, \(k=1\), …\(g\), encircles the cut \((e_{2k-1},e_{2k})\) counter-clockwise, and each \(\mathfrak{b}_k\)-cycle emerges from the cut \((e_{2g+1},\infty)\) and enters the cut \((e_{2k-1},e_{2k})\), see fig.1.

Fig. 1 represents the spectral curve of a \(g\)-gap system in the sine-Gordon hierarchy, see Section 6 for more details. Parameters \(\lambda_k\) of the curve \(\mathcal{V}\) are real, and all but one finite branch points are complex conjugate.

Let first kind differentials \(\mathrm{d}u = (\mathrm{d}u_1,\mathrm{d}u_3, \dots, \mathrm{d}u_{2g-1})^t\) and second kind differentials \(\mathrm{d}r = (\mathrm{d}r_1,\mathrm{d}r_3, \dots, \mathrm{d}r_{2g-1})^t\) form a system of associated differentials, see [12], and be defined as in [12]. Actually, \[\tag{4} \begin{align} & \mathrm{d}u_{2n-1} = \frac{x^{g-n} \mathrm{d}x}{\partial_y f(x,y)},\quad n=1,\dots, g,\tag{5} \\ & \mathrm{d}r_{2n-1} = \frac{ \mathrm{d}x}{\partial_y f(x,y)} \sum_{j=1}^{2n-1} (2n-j) \lambda_{2j-2} x^{g+n-j},\; \lambda_0=1,\quad n=1,\dots, g. \tag{6} \end{align}\] Indices of \(\mathrm{d}u_{2n-1}\) display the orders of zeros at infinity, and indices of \(\mathrm{d}r_{2n-1}\) display the orders of poles, which are located at infinity.

The first kind differentials are not normalized. The corresponding periods along the canonical cycles \(\mathfrak{a}_k\), \(\mathfrak{b}_k\), \(k=1\), …, \(g\), are defined as follows \[\begin{gather} \label{FKD} \omega_k = \oint_{\mathfrak{a}_k} \mathrm{d}u,\qquad\qquad \omega'_k = \oint_{\mathfrak{b}_k} \mathrm{d}u. \end{gather}\tag{7}\] The vectors \(\omega_k\), \(\omega'_k\) form first kind period matrices \(\omega\), \(\omega'\), respectively. Similarly, second kind period matrices \(\eta\), \(\eta'\) are composed of columns \[\begin{gather} \label{SKD} \eta_k = \oint_{\mathfrak{a}_k} \mathrm{d}r,\qquad\qquad \eta'_k = \oint_{\mathfrak{b}_k} \mathrm{d}r. \end{gather}\tag{8}\]

The normalized first kind period matrices are \(1_g\), \(\tau\), where \(1_g\) denotes the identity matrix of order \(g\), and \(\tau = \omega^{-1}\omega'\). Matrix \(\tau\) is symmetric with a positive imaginary part: \(\tau^t=\tau\), \(\mathop{\mathrm{\mathrm{Im}}}\tau >0\), that is, \(\tau\) belongs to the Siegel upper half-space. The normalized holomorphic differentials are denoted by \[\begin{gather} \mathrm{d}v = \omega^{-1} \mathrm{d}u. \end{gather}\]

2.2 Abel’s map↩︎

The vectors \(\omega_k\), \(\omega'_k\) form a period lattice \(\{\omega, \omega'\}\), and \(\mathop{\mathrm{\mathrm{Jac}}}(\mathcal{V}) \,{=}\, \mathop{\mathrm{\mathbb{C}}}^g/\{\omega, \omega'\}\) is the Jacobian variety of \(\mathcal{V}\). Let \(u=(u_1\), \(u_3\), …, \(u_{2g-1})^t\) be not normalized coordinates of \(\mathop{\mathrm{\mathrm{Jac}}}(\mathcal{V})\).

The Abel map is defined by \[\begin{gather} \mathcal{A}(P) = \int_{\infty}^P \mathrm{d}u,\qquad P=(x,y)\in \mathcal{V}, \end{gather}\] and on a positive divisor \(D = \sum_{i =1}^n (x_i,y_i)\) by \(\mathcal{A}(D) = \sum_{i =1}^n \mathcal{A}(P_i)\). The Abel map is one-to-one on the \(g\)-th symmetric power of the curve.

2.3 Theta function↩︎

The Riemann theta function is defined by \[\begin{gather} \theta(v;\tau) = \sum_{n\in \mathop{\mathrm{\mathbb{Z}}}^g} \exp \big(\imath \pi n^t \tau n + 2\imath \pi n^t v\big). \end{gather}\] where \(v = \omega^{-1}u\) are normalized coordinates. Theta function with characteristic \([\varepsilon]\) is defined by \[\theta[\varepsilon](v;\tau) = \exp\big(\imath \pi (\varepsilon'{}^t/2) \tau (\varepsilon'/2) + 2\imath \pi (v+\varepsilon/2)^t \varepsilon'/2\big) \theta(v+\varepsilon/2 + \tau \varepsilon'/2;\tau),\] where \([\varepsilon]= (\varepsilon', \varepsilon)^t\) is a \(2\times g\) matrix, all components of \(\varepsilon\), and \(\varepsilon'\) are real values within the interval \([0,2)\). Modulo \(2\) addition is defined on characteristics.

Every point \(u\) within the fundamental domain of \(\mathop{\mathrm{\mathrm{Jac}}}(\mathcal{V})\) can be represented by its characteristic \([\varepsilon]\) as follows \[u = \tfrac{1}{2} \omega \varepsilon + \tfrac{1}{2} \omega' \varepsilon'.\] Characteristics with values \(0\) and \(1\) correspond to half-periods, which are Abel images of divisors composed of branch points. Such a characteristic \([\varepsilon]\) is odd whenever \(\varepsilon^t \varepsilon' = 1\) (\(\mathop{\mathrm{mod}}2\)), and even whenever \(\varepsilon^t \varepsilon' = 0\) (\(\mathop{\mathrm{mod}}2\)). Theta function with characteristic has the same parity as its characteristic.

2.4 Sigma function and \(\wp\)-functions↩︎

The modular invariant entire function on \(\mathop{\mathrm{\mathbb{C}}}^g \,{\supset}\, \mathop{\mathrm{\mathrm{Jac}}}(\mathcal{V})\) is called the sigma function, which we define after [10]: \[\label{SigmaThetaRel} \sigma(u) = C \exp\big({-}\tfrac{1}{2} u^t \varkappa u\big) \theta[K](\omega^{-1} u; \omega^{-1} \omega'),\tag{9}\] where \([K]\) is the characteristic of the vector \(K\) of Riemann constants, and \(\varkappa = \eta \omega^{-1}\) is a symmetric matrix.

In what follows we use multiply periodic \(\wp\)-functions \[\begin{gather} \wp_{i,j}(u) = -\frac{\partial^2 \log \sigma(u) }{\partial u_i \partial u_j },\qquad \wp_{i,j,k}(u) = -\frac{\partial^3 \log \sigma(u) }{\partial u_i \partial u_j \partial u_k}, \end{gather}\] which are defined on \(\mathop{\mathrm{\mathrm{Jac}}}(\mathcal{V}) \backslash \Sigma\), where \(\Sigma = \{u \mid \sigma(u)=0\}\) denotes the theta divisor, see [5], in not normalized coordinates.

2.5 Jacobi inversion problem↩︎

A solution of the Jacobi inversion problem on a hyperelliptic curve is proposed in [12], see also [10]. Let \(u = \mathcal{A}(D )\) be the Abel image of a non-special positive divisor \(D \in \mathcal{V}^g\). Then \(D\) is uniquely defined by the system of equations \[\tag{10} \begin{align} &\mathcal{R}_{2g}(x;u) \equiv x^{g} - \sum_{i=1}^{g} x^{g-i} \wp_{1,2i-1}(u) = 0, \tag{11}\\ &\mathcal{R}_{2g+1}(x,y;u) \equiv 2 y + \sum_{i=1}^{g} x^{g-i} \wp_{1,1,2i-1}(u) = 0. \tag{12} \end{align}\]

2.6 Characteristics and partitions↩︎

Let \(S = \{0,1,2,\dots, 2g+1\}\) be the set of indices of all branch points of a hyperellipitic curve of genus \(g\), and \(0\) stands for the branch point at infinity. According to [12], all characteristics of half-periods are represented by partitions of \(S\) of the form \(\mathcal{I}_\mathfrak{m}\cup \mathcal{J}_\mathfrak{m}\) with \(\mathcal{I}_\mathfrak{m}= \{i_1,\,\dots,\, i_{g+1-2\mathfrak{m}}\}\) and \(\mathcal{J}_\mathfrak{m}= \{j_1,\,\dots,\, j_{g+1+2\mathfrak{m}}\}\), where \(\mathfrak{m}\) runs from \(0\) to \([(g+1)/2]\), and \([\cdot]\) denotes the integer part. Index \(0\) is usually omitted in sets, and also in computation of cardinality of a set.

Denote by \([\varepsilon(\mathcal{I})] = \sum_{i\in\mathcal{I}} [\varepsilon_i]\) \((\mathop{\mathrm{mod}}2)\) the characteristic of \[\begin{gather} \mathcal{A} (\mathcal{I}) = \sum_{i\in\mathcal{I}} \mathcal{A}(e_i) = \omega \Big(\tfrac{1}{2} \varepsilon(\mathcal{I}) + \tfrac{1}{2} \tau \varepsilon'(\mathcal{I}) \Big). \end{gather}\] Characteristics corresponding to \(2g+1\) branch points serve as a basis for constructing all \(2^{2g}\) half-period characteristics. Below, a partition is referred to by the part of less cardinality, denoted by \(\mathcal{I}\). Let \[[\mathcal{I}] = [\varepsilon(\mathcal{I})] + [K] \;(\mathop{\mathrm{mod}}2),\] and \([K]\) equals the sum of \(g\) odd characteristics of branch points. In the basis of canonical cycles introduced by fig. 1 we have \[\begin{gather} [K] = \sum_{k=1}^g [\varepsilon_{2k}]. \end{gather}\]

Let \(\mathcal{I}_\mathfrak{m}\cup \mathcal{J}_\mathfrak{m}\) be a partition introduced above, then \([\mathcal{I}_\mathfrak{m}]=[\mathcal{J}_\mathfrak{m}]\). Characteristics \([\mathcal{I}_\mathfrak{m}]\) of even \(\mathfrak{m}\) are even, and of odd \(\mathfrak{m}\) are odd. According to the Riemann vanishing theorem, \(\theta(v+\mathcal{A} (\mathcal{I}_\mathfrak{m})+K; \tau)\) vanishes to order \(\mathfrak{m}\) at \(v\,{=}\,0\). Number \(\mathfrak{m}\) is called multiplicity. Characteristics of multiplicity \(0\) are called non-singular even characteristics. Characteristics of multiplicity \(1\) are called non-singular odd. All other characteristics are called singular.

3 Integrable systems on coadjoint orbits of a loop group↩︎

The sine-Gordon equation arises within the hierarchy of integrable hamiltonian systems on coadjoint orbits in the loop algebra of \(\mathfrak{su}(2)\). We briefly recall this construction as presented in [13] and revised in [14]. Such a construction is based on the results of [15], [16].

3.1 Affine Lie algebra↩︎

Let \(\widetilde{\mathfrak{g}} = \mathfrak{g} \otimes \mathfrak{L}(z,z^{-1})\), where \(\mathfrak{L}(z,z^{-1})\) denotes the algebra of Laurent series in \(z\), and \(\mathfrak{g}=\mathfrak{sl}(2,\mathop{\mathrm{\mathbb{C}}})\) has the standard basis \[\begin{gather} \textsf{H}= \frac{1}{2} \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix},\quad \textsf{X}= \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix},\quad \textsf{Y}= \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}. \end{gather}\] The algebra \(\widetilde{\mathfrak{g}}\) is principally graded, with the grading operator \[\begin{gather} \label{PrGrad} \mathfrak{d} = 2 z \frac{\mathrm{d}}{\mathrm{d}z} + \mathop{\mathrm{ad}}_\textsf{H}, \end{gather}\tag{13}\] where \(\mathop{\mathrm{ad}}\) denotes the adjoint operator in \(\mathfrak{g}\), and \(\forall\, \textsf{Z}\in \mathfrak{g}\) \(\mathop{\mathrm{ad}}_\textsf{H}\textsf{Z}= [\textsf{H},\textsf{Z}]\).

Let \(\{\textsf{X}_{2m-1} \,{=}\, z^{m-1} \textsf{X},\, \textsf{Y}_{2m-1} \,{=}\, z^{m} \textsf{Y},\, \textsf{H}_{2m} \,{=}\, z^m \textsf{H}\mid m \in \mathop{\mathrm{\mathbb{Z}}}\}\) form a basis of \(\widetilde{\mathfrak{g}}\). Let \(\mathfrak{g}_{\ell}\) be the eigenspace of \(\mathfrak{d}\) of degree \(\ell\). Actually, \(\mathfrak{g}_{2m-1} = \mathop{\mathrm{span}}\{\textsf{X}_{2m-1},\, \textsf{Y}_{2m-1}\}\), and \(\mathfrak{g}_{2m} = \mathop{\mathrm{span}}\{\textsf{H}_{2m}\}\). We also denote the basis elements in \(\widetilde{\mathfrak{g}}\) by \(\textsf{Z}_{a;\ell}\), \(a=1\), \(2\), \(3\), \(\ell \in \mathop{\mathrm{\mathbb{Z}}}\), where \(\mathfrak{d} \textsf{Z}_{a;\ell} = \ell \textsf{Z}_{a;\ell}\), and \(\textsf{Z}_{1;\ell}= \textsf{H}_\ell\), \(\textsf{Z}_{2;\ell}=\textsf{Y}_\ell\), \(\textsf{Z}_{3;\ell}=\textsf{X}_\ell\).

With a fixed positive integer \(N\), let the bilinear form be defined by \[\begin{gather} \label{BiLinFSinG} \forall A(z), B(z) \in \widetilde{\mathfrak{g}} \qquad \langle A(z), B(z) \rangle = \mathop{\mathrm{res}}_{z=0} z^{-N} \mathop{\mathrm{tr}}A(z) B(z). \end{gather}\tag{14}\] By means of \(\langle \textsf{Z}_{a;\ell}, \textsf{Z}^{\ast}_{b;\ell'} \rangle = \delta_{a,b} \delta_{\ell,\ell'}\) the basis \(\{\textsf{Z}^{\ast}_{a;\ell} \mid a=1,2,3,\, \ell\in \mathop{\mathrm{\mathbb{Z}}}\}\) of the dual algebra \(\widetilde{\mathfrak{g}}^\ast\) is introduced. In more detail, \[\begin{gather} \textsf{Z}_{1;2m}^\ast = 2 \textsf{H}_{2N-2m-2},\quad \textsf{Z}_{2;2m-1}^\ast = \textsf{X}_{2N-2m-1},\quad \textsf{Z}_{3;2m-1}^\ast = \textsf{Y}_{2N-2m-1}. \end{gather}\]

According to the Adler—Kostant—Symes scheme, see [16], \(\widetilde{\mathfrak{g}}\) falls into two subalgebars \[\begin{gather} \widetilde{\mathfrak{g}}_+ = \oplus \sum_{\ell \geqslant 0} \mathfrak{g}_{\ell},\qquad \widetilde{\mathfrak{g}}_- = \oplus \sum_{\ell \leqslant -1} \mathfrak{g}_{\ell}. \end{gather}\] The dual spaces \(\widetilde{\mathfrak{g}}_+^\ast\) and \(\widetilde{\mathfrak{g}}_-^\ast\) with respect to the bilinear form 14 are \[\begin{gather} \widetilde{\mathfrak{g}}_+^\ast = \oplus \sum_{\ell \leqslant 2N-2} \mathfrak{g}_{\ell},\qquad \widetilde{\mathfrak{g}}_-^\ast = \oplus \sum_{\ell > 2N-2} \mathfrak{g}_{\ell}. \end{gather}\] Note, that \(\mathfrak{g}_{N-1}^\ast = \mathfrak{g}_{N-1}\), and \(\mathfrak{g}_{\ell}^\ast\) is dual to \(\mathfrak{g}_{2N-\ell-2}\).

3.2 Phase space of sine-Gordon hierarchy↩︎

The sine-Gordon hierarchy of hamiltonian systems arises on coadjoint orbits of the group \(\widetilde{G}_+ = \exp(\widetilde{\mathfrak{g}}_+)\). The subspace \(\mathcal{M}_N = \widetilde{\mathfrak{g}}^\ast_+ / \big( \sum_{\ell < -2} \mathfrak{g}_\ell \big)\) is \(\mathop{\mathrm{ad}}^\ast\)-invariant under the coadjoint action of \(\widetilde{\mathfrak{g}}_+\). Actually, \[\begin{gather} \mathcal{M}_N = \bigg\{\mathsf{L}= \sum_{\ell=-2}^{2N-2} \sum_{a=1,2,3} L_{a;\ell} \textsf{Z}_{a;2N-\ell-2}^\ast \bigg\}, \end{gather}\] where \(L_{a;\ell}\) are coordinates on \(\mathcal{M}_N\): \[L_{a;\ell} = \langle \mathsf{L}, \textsf{Z}_{a;2N-\ell-2}\rangle,\] and also serve as dynamic variables of the \(N\)-gap system of the hierarchy, Let \(L_{1;\ell}=\alpha_{\ell}\), \(L_{2;\ell} = \beta_{\ell}\), \(L_{3;\ell} = \gamma_{\ell}\), then every element \(\mathsf{L}\in \mathcal{M}_N\) has the form \[\label{SinGPhSp} \begin{align} \mathsf{L}(z) = \begin{pmatrix} \alpha(z) & \beta(z) \\ \gamma(z) & -\alpha(z) \end{pmatrix},\\\alpha(z) = \sum_{m=0}^{N} &\alpha_{2m} z^m,\quad \beta(z) = \sum_{m=0}^{N} \beta_{2m-1} z^{m-1},\quad \gamma(z) = \sum_{m=0}^{N} \gamma_{2m-1} z^{m},\\ &\alpha_{2N} = 0,\qquad \beta_{2N-1} = \gamma_{2N-1} = \mathrm{b}= \mathop{\mathrm{const}}.\end{align}\tag{15}\] The coordinates of \(\mathcal{M}_N\), of number \(3N\), are ordered as follows: \[\label{SinGDynVar} \{\beta_{2m -1}, \gamma_{2m - 1}, \alpha_{2m}\}_{m=0}^{N-1}.\tag{16}\]

The symplectic manifold \(\mathcal{M}_N\) is equipped with the Lie-Poisson bracket \[\label{LiePoiBraSinG} \begin{gather} \forall \mathcal{F}, \mathcal{H}\in \mathcal{C}^1(\mathcal{M}_N) \qquad \{\mathcal{F},\mathcal{H}\} = \sum_{i,j =-2}^{2N-2} \sum_{a, b=1,2,3} W_{i,j}^{a,b} \frac{\partial \mathcal{F}}{\partial L_{a;i}} \frac{\partial \mathcal{H}}{\partial L_{b;j}}, \\ W_{i,j}^{a,b} = \langle \mathsf{L}, [\textsf{Z}_{a,2N-i-2},\textsf{Z}_{a,2N-j-2}] \rangle, \end{gather}\tag{17}\] or in terms of the dynamic variables: \[\begin{align} \label{SinGPoiBra}&\{\beta_{2m-1}, \alpha_{2n}\} = \beta_{2(n+m-N)+1},\\ &\{\gamma_{2m-1}, \alpha_{2n}\} = - \gamma_{2(n+m-N)+1},\quad N-1 \leqslant m+n, \\ &\{\beta_{2m-1}, \gamma_{2n-1}\} = - 2 \alpha_{2(m+n-N)},\quad N \leqslant m+n.\end{align}\tag{18}\] The Lie-Poisson structure \(\textsf{W}= (W_{i,j}^{a,b})\) has the form \[\begin{align} &\textsf{W} = \begin{pmatrix} 0 & 0 & \dots & 0 & \textsf{w}_{0} \\ \vdots & \vdots & \dots & \textsf{w}_{0} & \textsf{w}_1\\ 0 & 0 & \iddots & \textsf{w}_1 & \textsf{w}_2 \\ 0 & \textsf{w}_{0} & \iddots & \vdots & \vdots \\ \textsf{w}_{0} & \textsf{w}_1 & \dots & \textsf{w}_{N-2} & \textsf{w}_{N-1} \end{pmatrix},\\ &\textsf{w}_{0} = \begin{pmatrix} 0 & 0 & \beta_{-1} \\ 0 & 0 & - \gamma_{-1} \\ - \beta_{-1} & \gamma_{-1} & 0 \\ \end{pmatrix}, \\ &\textsf{w}_n = \begin{pmatrix} 0 & - 2\alpha_{2n-2} & \beta_{2n-1} \\ 2\alpha_{2n-2}& 0 & -\gamma_{2n-1} \\ - \beta_{2n-1} & \gamma_{2n-1} & 0 \\ \end{pmatrix},\quad n = 1,\,\dots,\, N-1. \end{align}\] This Poisson bracket is degenerate, and not canonical.

The action of \(\widetilde{G}_+\) splits \(\mathcal{M}_N\) into orbits \[\mathcal{O} = \{\mathsf{L}= \mathop{\mathrm{Ad}}^\ast_{g} \mathsf{L}^{\text{in}} \mid g\in \widetilde{G}_+\},\qquad \Phi \in \mathcal{M}_N.\] An initial point \(\mathsf{L}^{\text{in}}\) belongs to the Weyl chamber of \(\widetilde{G}_+\), and is represented by a diagonal matrix, that is, spanned by \(H^{\ast}_{2m}\), \(m = 0\), …, \(N\). Orbits are of dimension \(2N\), and serve as phase spaces of hamiltonian systems.

Physically meaningful hamiltonian systems arise when \(\mathfrak{g}\) is one of the real forms of \(\mathfrak{sl}(2, \mathop{\mathrm{\mathbb{C}}})\), namely \(\mathfrak{sl}(2, \mathop{\mathrm{\mathbb{R}}})\) or \(\mathfrak{su}(2)\). In the case of \(\mathfrak{su}(2)\) the sine-Gordon hierarchy is obtained, and in the case of \(\mathfrak{sl}(2, \mathop{\mathrm{\mathbb{R}}})\) the \(\sinh\)-Gordon hierarchy.

3.3 Integrals of motion↩︎

Let \[\label{InvF} \begin{align} H(z) &= \tfrac{1}{2} \mathop{\mathrm{tr}}\mathsf{L}^2(z) = \alpha(z)^2 + \beta(z) \gamma(z)\\ &= h_{2N-1} z^{2N-1} + \cdots + h_1 z + h_0 + h_{-1} z^{-1}, \end{align}\tag{19}\] where \[\begin{align} \label{hExprs}& h_{2N-1} = \mathrm{b}^2,\\ & h_{2N-2} = \alpha_{2N-2}^2 + \mathrm{b}(\beta_{2N-3} + \gamma_{2N-3}),\\ &\dots \\ & h_{0} = \alpha_0^2 + \beta_1 \gamma_{-1} + \beta_{-1} \gamma_{1},\\ & h_{-1} = \beta_{-1} \gamma_{-1}.\end{align}\tag{20}\]

Evidently, \(H(z)\) is invariant under the action of \(\widetilde{G}_{+}\). Therefore, every \(h_n\) is an integral of motion, and \(h_{2N-1} = \mathrm{b}^2\) is an absolute constant. With respect to the symplectic structure 18 , \(h_{N-1}\), …, \(h_{2N-2}\) give rise to non-trivial hamiltonian flows, we call them hamiltonians.

On the other hand, \(h_{-1}\), \(h_0\), …, \(h_{N-2}\) annihilate the Poisson bracket 17 , since \[\sum_{i =-1}^{2N-2} \sum_{a =1,2,3} W_{i,j}^{a,b} \frac{\partial h_n}{\partial L_{a;i}} = 0, \quad n=-1, 0,\dots, N-2,\] and so \(\{h_n, \mathcal{F}\} = 0\) for any \(\mathcal{F} \in \mathcal{C}^1 (\mathcal{M}_N)\). Thus, \[\begin{gather} \label{SinGConstr} h_{-1} = \mathrm{r}_{-1},\quad h_{0} = \mathrm{r}_{0},\quad \ldots,\quad h_{N-2} = \mathrm{r}_{N-2}, \end{gather}\tag{21}\] with constant \(\mathrm{r}_{-1}\), \(\mathrm{r}_{0}\), …, \(\mathrm{r}_{N-2}\), serve as constraints on the symplectic manifold \(\mathcal{M}_N\). They fix an orbit \(\mathcal{O}\) of dimension \(2N\), which serves as the phase space of the \(N\)-gap hamiltonian system in the sine(sinh)-Gordon hierarchy.

3.4 Real forms of \(\mathfrak{sl}(2,\mathop{\mathrm{\mathbb{C}}})\)↩︎

The algebra \(\mathfrak{sl}(2,\mathop{\mathrm{\mathbb{C}}})\) has two real forms: \(\mathfrak{su}(2)\), and \(\mathfrak{sl}(2,\mathop{\mathrm{\mathbb{R}}})\). In the both cases, \(h_{-1}\), \(h_0\), …, \(h_{2N-2}\), and \(h_{2N-1} = \mathrm{b}^2\) are real.

If \(\mathfrak{g} = \mathfrak{sl}(2,\mathop{\mathrm{\mathbb{R}}})\), then all coordinates \(\beta_{2m -1}\), \(\gamma_{2m - 1}\), \(\alpha_{2m}\), \(m=0\), …\(N-1\), and also \(\mathrm{b}\) are real. Thus, \(h_{2N-1} = \mathrm{b}^2 > 0\).

If \(\mathfrak{g} = \mathfrak{su}(2)\), then \(\alpha_{2m} = \imath a_{2m}\), \(a_{2m} \in \mathop{\mathrm{\mathbb{R}}}\), and \(\beta_{2m-1} = - \bar{\gamma}_{2m-1}\), \(\mathrm{b}= \imath b\), where \(b \in \mathop{\mathrm{\mathbb{R}}}\). Thus, \(h_{2N-1} = \mathrm{b}^2 = -b^2 < 0\). We also have \[\label{hm1Rel} h_{-1} = \beta_{-1} \gamma_{-1} = - |\gamma_{-1}|^2 < 0.\tag{22}\]

3.5 Sine(sinh)-Gordon equation↩︎

Let \(h_{2N-2}\) give rise to a stationary flow with a parameter \(\mathrm{x}\), and \(h_{N-1}\) give rise to an evolutionary flow with a parameter \(\mathrm{t}\): \[\begin{gather} \label{TwoFlowsEq} \frac{\mathrm{d}L_{a;\ell}}{\mathrm{d}\mathrm{x}} = \{h_{2N-2}, L_{a;\ell}\},\qquad \frac{\mathrm{d}L_{a,\ell}}{\mathrm{d}\mathrm{t}} = \{h_{N-1}, L_{a,\ell}\}. \end{gather}\tag{23}\] In more detail, the stationary flow is \[\begin{align} &\frac{\mathrm{d}\beta_{-1}}{\mathrm{d}\mathrm{x}} = 2 \alpha_{2N-2} \beta_{-1},\qquad \frac{\mathrm{d}\gamma_{-1}}{\mathrm{d}\mathrm{x}} = - 2 \alpha_{2N-2} \gamma_{-1}, \label{BetaGammaM1EqSt} \\ &\frac{\mathrm{d}\alpha_{2m}}{\mathrm{d}\mathrm{x}} = \mathrm{b}(\gamma_{2m-1} - \beta_{2m-1}),\quad m=0,\dots, N-1,\\ &\frac{\mathrm{d}\beta_{2m-1}}{\mathrm{d}\mathrm{x}} = 2 \alpha_{2N-2} \beta_{2m-1} - 2 \mathrm{b}\alpha_{2m-2},\\ &\frac{\mathrm{d}\gamma_{2m-1}}{\mathrm{d}\mathrm{x}} = - 2 \alpha_{2N-2} \gamma_{2m-1} + 2 \mathrm{b}\alpha_{2m-2}, \quad m=1,\dots, N-1. \end{align}\tag{24}\] The evolutionary flow is \[\begin{align} &\frac{\mathrm{d}\alpha_{2m-2}}{\mathrm{d}\mathrm{t}} = \gamma_{-1} \beta_{2m-1} - \beta_{-1} \gamma_{2m-1},\quad m=1,\dots, N-1, \tag{25}\\ &\frac{\mathrm{d}\beta_{2m-1}}{\mathrm{d}\mathrm{t}} = 2 \alpha_{2m} \beta_{-1}, \qquad \frac{\mathrm{d}\gamma_{2m-1}}{\mathrm{d}\mathrm{t}} = - 2 \alpha_{2m} \gamma_{-1},\quad m=0,\dots, N-1, \tag{26} \\ &\frac{\mathrm{d}\alpha_{2N-2}}{\mathrm{d}\mathrm{t}} = \mathrm{b}(\gamma_{-1} - \beta_{-1}). \tag{27} \end{align}\]

The sine(sinh)-Gordon equation is obtained from the equality \[\begin{gather} \label{SinGEqZeroCurv} \frac{\mathrm{d}\alpha_{2N-2}}{\mathrm{d}\mathrm{t}} = \frac{\mathrm{d}\alpha_{0}}{\mathrm{d}\mathrm{x}}. \end{gather}\tag{28}\] Indeed, from 24 one can see that \(\alpha_{2N-2} = \frac{1}{2} \mathrm{d}\ln \beta_{-1} / \mathrm{d}\mathrm{x}= - \frac{1}{2} \mathrm{d}\ln \gamma_{-1} / \mathrm{d}\mathrm{x}\). On the other hand, 26 implies \(\alpha_{0} = \frac{1}{2} \mathrm{d}\ln \beta_{-1} / \mathrm{d}\mathrm{t}= - \frac{1}{2} \mathrm{d}\ln \gamma_{-1} / \mathrm{d}\mathrm{t}\).

In the case of \(\mathfrak{g} = \mathfrak{su}(2)\), we have \(\beta_{-1} = - \bar{\gamma}_{-1}\), and so \(- |\gamma_{-1}|^2 = \mathrm{r}_{-1} = \mathop{\mathrm{const}}\), cf. 22 . This means that \(\gamma_{-1}\) runs along a circle of radius \(-\mathrm{r}_{-1}\) if \(\mathrm{r}_{-1}<0\). Let \(\gamma_{-1} = \imath r \exp (\imath \phi)\), and \(\beta_{-1} = \imath r \exp (- \imath \phi)\), where \(r = \sqrt{-\mathrm{r}_{-1}}\), and \(\phi\) is a function of \(\mathrm{x}\) and \(\mathrm{t}\). Then 24 and 26 imply, respectively, \[\begin{gather} \alpha_{2N-2} = \frac{\imath}{2} \frac{\mathrm{d}\phi}{\mathrm{d}\mathrm{x}},\qquad \alpha_{0} = \frac{\imath}{2} \frac{\mathrm{d}\phi}{\mathrm{d}\mathrm{t}}. \end{gather}\] Finally, 27 gives the sine-Gordon equation (\(\mathrm{b}= \imath b\), \(b \in \mathop{\mathrm{\mathbb{R}}}\)) \[\label{SinGEqBeta} \frac{\mathrm{d}^2 \phi}{\mathrm{d}\mathrm{t}\mathrm{d}\mathrm{x}} = 4 b r \sin \phi.\tag{29}\]

In the case of \(\mathfrak{g} = \mathfrak{sl}(2,\mathop{\mathrm{\mathbb{R}}})\), let \(\gamma_{-1} = r \exp ( \phi)\), \(\beta_{-1} = r \exp (- \phi)\), where \(\phi\) is a function of \(\mathrm{x}\) and \(\mathrm{t}\), and \(r = \sqrt{\mathrm{r}_{-1}} = \mathop{\mathrm{const}}\), \(\mathrm{r}_{-1}\) is supposed to be positive. Then 27 gives the sinh-Gordon equation (\(\mathrm{b}\in \mathop{\mathrm{\mathbb{R}}}\)): \[\label{SinhGEqBeta} \frac{\mathrm{d}^2 \phi}{\mathrm{d}\mathrm{t}\mathrm{d}\mathrm{x}} = - 4 \mathrm{b}r \sinh \phi.\tag{30}\]

3.6 Zero curvature representation↩︎

The system of dynamic equations 23 admits the matrix form \[\begin{gather} \frac{\mathrm{d}\mathsf{L}}{\partial \mathrm{x}} = [\mathsf{L}, \nabla h_{2N-2}], \qquad \frac{\mathrm{d}\mathsf{L}}{\partial \mathrm{t}} = [\mathsf{L}, \nabla h_{N-1}], \end{gather}\] where \(\nabla h_n\) denotes the matrix gradient of \(h_n\), namely, \[\begin{gather} \nabla h_n = \sum_{i =-1}^{2N-1} \sum_{a =1,2,3} \frac{\partial h_n}{\partial L_{a;i}} \textsf{Z}_{a,2N-i-2}. \end{gather}\] The matrix gradient of each flow has a complementary matrix \(\mathrm{A}\), such that \[[\mathsf{L}, \nabla h_{n}] = [\mathsf{L}, \mathrm{A}].\] Unlike \(\nabla h_n\), the complementary matrix \(\mathrm{A}\) is defined uniquely for all \(N\), that is for all hamiltonian systems in the hierarchy. Actually, \[\begin{align} \label{PsiMatrEqs} \frac{\mathrm{d}\mathsf{L}}{\partial \mathrm{x}} = [\mathsf{L}, \mathrm{A}_\text{st}], \qquad \frac{\mathrm{d}\mathsf{L}}{\partial \mathrm{t}} = [\mathsf{L}, \mathrm{A}_\text{ev}], \\& \mathrm{A}_\text{st} = - \begin{pmatrix} \alpha_{2N-2} & \mathrm{b}\\ \mathrm{b}z & -\alpha_{2N-2} \end{pmatrix},\qquad \mathrm{A}_\text{ev} = \begin{pmatrix} 0 & \frac{1}{z} \beta_{-1} \\ \gamma_{-1} & 0 \end{pmatrix}.\notag \end{align}\tag{31}\] The zero curvature representation for the sine(sinh)-Gordon hierarchy has the form \[\begin{gather} \frac{\mathrm{d}\mathrm{A}_\text{st}}{\partial \mathrm{t}} - \frac{\mathrm{d}\mathrm{A}_\text{ev}}{\partial \mathrm{x}} = [ \mathrm{A}_\text{st}, \mathrm{A}_\text{ev}]. \end{gather}\]

3.7 Summary↩︎

The affine algebra \(\widetilde{\mathfrak{g}} \,{=}\, \mathfrak{su}(2) \otimes \mathcal{L}(z,z^{-1})\) with the principal grading 13 and the bilinear form 14 is associated with the sine-Gordon hierarchy. Let \(\mathcal{M}_N\) be the manifold \(\widetilde{\mathfrak{g}}^\ast_+ / \big( \sum_{\ell < -2} \mathfrak{g}_\ell \big)\), \(\dim \mathcal{M}_N \,{=}\, 3N\), \(N\,{\in}\, \mathop{\mathrm{\mathbb{N}}}\). Evidently, \(\mathcal{M}_1 \subset \mathcal{M}_2 \subset \cdots \subset \mathcal{M}_N \subset \mathcal{M}_{N+1} \subset \cdots\). Each manifold \(\mathcal{M}_N\) is equipped with the symplectic structure 18 . Under the action of the loop group \(\widetilde{G}_+ \,{=}\, \exp(\widetilde{\mathfrak{g}}_+)\) a manifold \(\mathcal{M}_N\) splits into orbits \(\mathcal{O}\), each defined by the system of \(N\) constraints 21 . Each orbit serves as a phase space of dimension \(2N\) for a hamiltonian system integrable in the Liouville sense. On orbits within \(\mathcal{M}_N\), \(N \geqslant 2\), there exist two hamiltonians whose flows give rise to the sine-Gordon equation 29 . We call these flows stationary and evolutionary with parameters \(\mathrm{x}\) and \(\mathrm{t}\), correspondingly.

In a similar way, the affine algebra \(\widetilde{\mathfrak{g}} = \mathfrak{sl}(2,\mathop{\mathrm{\mathbb{R}}}) \otimes \mathcal{L}(z,z^{-1})\) is associated with the sinh-Gordon hierarchy, cf. 30 .

4 Separation of variables↩︎

4.1 Spectral curve↩︎

The sine(sinh)-Gordon hierarchy is associated with the family of hyperelliptic curves \[\label{SpectrCurve} - w^2 + z^2 H(z) = 0,\tag{32}\] where \(H\) is defined by 19 . This equation of a genus \(N\) curve is obtained from the characteristic polynomial of \(\mathsf{L}\), namely \(\det \big(\mathsf{L}(z) - (w/z)\big) = 0\). All parameters of the curve are integrals of motion: hamiltonians \(h_{N-1}\), …, \(h_{2N-2}\), constraints \(h_{-1}\), …, \(h_{N-2}\), and \(h_{2N-1} = \mathrm{b}^2 = \mathop{\mathrm{const}}\).

4.2 Canonical coordinates↩︎

As shown in [17], variables of separation in the sine-Gordon hierarchy are given by a positive non-special¹ divisor of degree equal to the genus \(N\) of the spectral curve. In fact, pairs of coordinates of \(N\) points from the support of such a divisor serve as quasi-canonical variables, and so lead to separation of variables. Below, we briefly explain how the required points can be obtained from the dynamic variables of the \(N\)-gap system in the sine-Gordon hierarchy, and prove that pairs of coordinates of the \(N\) points serve as quasi-canonical variables.

Recall, that the symplectic manifold \(\mathcal{M}_N\), with \(3N\) coordinates 16 on it, splits into orbits \(\mathcal{O}\) fixed by \(N\) constraints, and so \(\dim \mathcal{O}= 2N\). Then, \(\beta_{2m-1}\), \(m=0\), …, \(N-1\), are eliminated with the help of the constraints. Since the expressions 20 are linear with respect to \(\beta_{2m-1}\), it is convenient to present them in the matrix form. The constraints are written as \[\begin{gather} \label{Constr} \Gamma_{\text{c}} \boldsymbol{\beta} + \textsf{A}_{\text{c}} = \boldsymbol{r},\\ \intertext{where} \Gamma_{\text{c}} = \begin{pmatrix} \mathrm{b}& 0 & 0 &\dots & 0 \\ 0& \gamma_{-1} & \gamma_{1} & \dots & \gamma_{2N-3} \\ \vdots & 0 & \gamma_{-1} & \ddots & \vdots \\ 0 & \vdots & \ddots & \ddots & \gamma_{1} \\ 0 & 0& \dots & 0 & \gamma_{-1} \end{pmatrix},\qquad \boldsymbol{\beta} = \begin{pmatrix} \mathrm{b}\\ \beta_{2N-3} \\ \vdots \\ \beta_1 \\ \beta_{-1} \end{pmatrix}, \notag \\ \textsf{A}_{\text{c}} = \begin{pmatrix} 0 \\ \sum_{k=0}^{N-2} \alpha_{2(N-2-k)} \alpha_{2k} \\ \vdots \\ \sum_{k=0}^{n} \alpha_{2(n-k)} \alpha_{2k} \\ \vdots\\ \alpha_0^2 \\ 0 \end{pmatrix},\qquad \boldsymbol{r} = \begin{pmatrix} \mathrm{b}^2 \\ \mathrm{r}_{N-2} \\ \vdots \\ \mathrm{r}_{n} \\ \vdots \\ \mathrm{r}_{0} \\ \mathrm{r}_{-1} \end{pmatrix}. \notag \end{gather}\tag{33}\] The first equation is an identity. From 33 we find \[\label{GammaFromConstr} \boldsymbol{\beta} = \Gamma_{\text{c}}^{-1}(\boldsymbol{r} - \textsf{A}_{\text{c}}).\tag{34}\]

Let values of the hamiltonians be denoted by \(\mathrm{h}_{N-1}\), \(\mathrm{h}_N\), …, \(\mathrm{h}_{2N-2}\). Expressions for the hamiltonians admit the matrix form \[\begin{gather} \label{Hams} \Gamma_{\text{h}} \boldsymbol{\beta} + \textsf{A}_{\text{h}} = \boldsymbol{h},\\ \intertext{where} \Gamma_{\text{h}} = \begin{pmatrix} \gamma_{2N-3} & \mathrm{b}& 0 & \dots & 0 \\ \vdots & \gamma_{2N-3} & \ddots & \ddots & \vdots \\ \gamma_1 & \vdots & \ddots & \mathrm{b}& 0 \\ \gamma_{-1} & \gamma_1 & \dots & \gamma_{2N-3} & \mathrm{b} \end{pmatrix}, \notag \\ \textsf{A}_{\text{h}} = \begin{pmatrix} \alpha_{2N-2}^2 \\ \vdots \\ \sum_{k=n}^{N-1} \alpha_{2(N-1-k+n)} \alpha_{2k} \\ \vdots \\ \sum_{k=0}^{N-1} \alpha_{2(N-1-k)} \alpha_{2k} \end{pmatrix},\qquad \boldsymbol{h} = \begin{pmatrix} \mathrm{h}_{2N-2} \\ \vdots \\ \mathrm{h}_{N-1+n} \\ \vdots \\ \mathrm{h}_{N-1} \end{pmatrix}. \notag \end{gather}\tag{35}\] Substituting 34 into 35 , we obtain \[\label{HamsBeta} \boldsymbol{h} = \Gamma_{\text{h}} \Gamma_{\text{c}}^{-1}(\boldsymbol{r} - \textsf{A}_{\text{c}})+ \textsf{A}_{\text{h}}.\tag{36}\]

On the other hand, values of hamiltonians \(\mathrm{h}_{2N-2}\), …, \(\mathrm{h}_{N-1}\) can be found from the equation 32 of the spectral curve taken at points \(\{(z_i,w_i)\}_{i=1}^N\) which form a non-special divisor. Namely, with \(i=1\), …, \(N\), \[- w_i^2 + \mathrm{b}^2 z_i^{2N+1} + \mathrm{h}_{2N-2} z_i^{2N} + \cdots + \mathrm{h}_{N-1} z_i^{N+1} + \mathrm{r}_{N-2} z_i^{N} + \dots + \mathrm{r}_{-1} z_i = 0,\] or in the matrix form \[- \boldsymbol{w} + \mathrm{Z}_{\text{h}} \boldsymbol{h} + \mathrm{Z}_{\text{c}} \boldsymbol{r} = 0,\] where \[\begin{gather} \mathrm{Z}_{\text{c}} = \begin{pmatrix} z_1^{2N+1} & z_1^{N} & \dots & z_1 \\ z_2^{2N+1} & z_2^{N} & \dots & z_2 \\ \vdots & \vdots & \ddots & \vdots \\ z_N^{2N+1} & z_N^{N} & \dots & z_N \end{pmatrix},\quad \mathrm{Z}_{\text{h}} = \begin{pmatrix} z_1^{2N} & \dots & z_1^{N+2} & z_1^{N+1} \\ z_2^{2N} & \dots & z_2^{N+2} & z_2^{N+1} \\ \vdots & \ddots & \vdots & \vdots \\ z_N^{2N} & \dots & z_N^{N+2} & z_N^{N+1} \end{pmatrix},\quad \boldsymbol{w} = \begin{pmatrix} w_1^2 \\ w_2^2 \\ \vdots \\ w_N^2 \end{pmatrix}. \end{gather}\] The matrix \(\mathrm{Z}_{\text{h}}\) is square and invertible. Thus, \[\label{HamsSpectrC} \boldsymbol{h} = \mathrm{Z}_{\text{h}}^{-1} \big(\boldsymbol{w} - \mathrm{Z}_{\text{c}} \boldsymbol{r}\big).\tag{37}\]

Equations 36 and 37 define the same hamiltonians. Therefore, \[\begin{gather} \Gamma_{\text{h}} \Gamma_{\text{c}}^{-1}(\boldsymbol{r} - \textsf{A}_{\text{c}})+ \textsf{A}_{\text{h}} = \mathrm{Z}_{\text{h}}^{-1} \big(\boldsymbol{w} - \mathrm{Z}_{\text{c}} \boldsymbol{r}\big). \end{gather}\] Moreover, constants \(\boldsymbol{c}\) can be taken arbitrarily, and so the corresponding coefficients coincide, as well as the remaining terms: \[\begin{gather} \Gamma_{\text{h}} \Gamma_{\text{c}}^{-1} = - \mathrm{Z}_{\text{h}}^{-1} \mathrm{Z}_{\text{c}}, \tag{38} \\ -\Gamma_{\text{h}} \Gamma_{\text{c}}^{-1}\textsf{A}_{\text{c}} + \textsf{A}_{\text{h}} = \mathrm{Z}_{\text{h}}^{-1} \boldsymbol{w}. \tag{39} \end{gather}\] From 38 we find \[\mathrm{Z}_{\text{h}} \Gamma_{\text{h}} + \mathrm{Z}_{\text{c}} \Gamma_{\text{c}} = 0,\] which is equivalent to \(\gamma(z_i) = 0\). Then from 39 we obtain \[\mathrm{Z}_{\text{c}} \textsf{A}_{\text{c}} + \mathrm{Z}_{\text{h}}\textsf{A}_{\text{h}} = \boldsymbol{w},\] which produces \(w_i^2 - z_i^2 \alpha(z_i)^2 = 0\). Thus, the points \((z_i,w_i)\) are defined by \[\label{PointsDef2} \gamma(z_i) = 0,\qquad w_i^2 - z_i^2 \alpha(z_i)^2 = 0, \quad i=1,\dots, N.\tag{40}\] This result was firstly discovered in [17].

In what follows we assign \(\epsilon = 1\).

4.3 Summary↩︎

An orbit \(\mathcal{O} \subset \mathcal{M}_N\), which serves as a phase space of dimension \(2N\), is completely parameterized by non-canonical variables \(\gamma_{2m-1}\), \(\alpha_{2m}\), \(m=0\), …, \(N-1\). Points \(\{(z_i,w_i)\}_{i=1}^N\) of the spectral curve 32 serve as variables of separation, which are quasi-canonical, see Theorem 1. The relation ?? between non-canonical and quasi-canonical variables is closely connected to the solution 10 of the Jacobi inversion problem. Indeed, the both systems of equations define the same point in the fundamental domain of the Jacobian variety of the spectral curve, as we see below.

5 Algebro-geometric integration↩︎

5.1 Uniformization of the spectral curve↩︎

Separation of variables provides a solution of the Jacobi inversion problem on the spectral curve 32 , which is a hyperelliptic curve of genus \(N\): \[\begin{gather} \label{CurveGN} 0 = F(z,w) \equiv - w^2 + \mathrm{b}^2 z^{2N+1} + \mathrm{h}_{2N-2} z^{2N} + \cdots + \mathrm{h}_{N-1} z^{N+1} \\ + \mathrm{r}_{N-2} z^{N} + \dots + \mathrm{r}_0 z^2 + \mathrm{r}_{-1} z. \end{gather}\tag{41}\]

The spectral curve is reduced to the canonical form 2 , which we continue to denote by \(\mathcal{V}\), by applying the transformation (\(N\equiv g\)) \[\label{SectrCToCanonC} \begin{align} z &\mapsto x,\qquad w \mapsto y = w/\mathrm{b},\qquad F(x, \mathrm{b}y) = \mathrm{b}^2 f(x,y),\\ &\mathrm{h}_{n}\; (\text{or } \mathrm{r}_n)= \mathrm{b}^2 \lambda_{4g-2-2n},\;\;n=-1,\,\dots,\, 2N-2. \end{align}\tag{42}\] Note, that one of the branch points of 41 is fixed at \(0\), and so the corresponding canonical form has \(\lambda_{4g+2}=0\).

We will work with \(\wp\)-functions associated with \(\mathcal{V}\), and use the coordinates \(u = (u_1,\dots,u_{2g-1})^t\) of \(\mathop{\mathrm{\mathrm{Jac}}}(\mathcal{V})\). The not normalized differentials of the first and second kinds defined by 4 are expressed in terms of coordinates of 41 as follows \[\begin{align} & \mathrm{d}u_{2n-1} = \frac{\mathrm{b}z^{N-n} \mathrm{d}z}{\partial_w F(z,w)},\quad n=1,\dots, N,\tag{43} \\ & \mathrm{d}r_{2n-1} = \frac{ \mathrm{d}z}{\mathrm{b}\partial_w F(z,w)} \sum_{j=1}^{2n-1} (2n-j) \mathrm{h}_{2N-j} z^{N+n-j}, \quad n=1,\dots, N, \tag{44} \end{align}\] where the notation \(\mathrm{h}_n\) is used for \(\mathrm{h}_n\) and \(\mathrm{r}_n\), and \(\mathrm{h}_{2N-1} = \mathrm{b}^2\).

Applying the transformation 42 to 10 , we are able to find the pre-image \(D = \sum_{i=1}^N (z_i,w_i)\) of any \(u\in \mathop{\mathrm{\mathrm{Jac}}}(\mathcal{V}) \backslash \Sigma\). On the other hand, the \(N\) values \(z_i\) are zeros of the polynomial \(\gamma(z)\), and the \(N\) values \(w_i\) are obtained from \(w_i = z_i \alpha(z_i)\), cf. ?? . Thus, \[\tag{45} \begin{align} &\gamma_{2(N-k)-1} = - \mathrm{b}\wp_{1,2k-1}(u), \quad k=1,\dots, N; \tag{46}\\ &\alpha_{2N-2} = - \frac{\mathrm{b}\wp_{1,1,2N-1}(u)}{2 \wp_{1,2N-1}(u)}, \\ &\alpha_{2(N-k)} = - \frac{\mathrm{b}}{2} \Big(\wp_{1,1,2k-3}(u) - \frac{\wp_{1,2k-3}(u) \wp_{1,1,2N-1}(u)}{\wp_{1,2N-1}(u)} \Big), \;\; k=2,\dots, N. \tag{47} \end{align} Taking into account the relations between \wp-functions obtained from \eqref{BetaGammaM1EqEv}, we find \begin{aligned}\tag{48} &\alpha_{2(N-k)} = - \frac{\mathrm{b}\wp_{1,2k-1,2N-1}(u)}{2 \wp_{1,2N-1}(u)}, \quad k=1,\dots, N, '} \end{aligned} and also \begin{align}\tag{49} &\beta_{-1} = \frac{- c_{-1}}{\mathrm{b}\wp_{1,2N-1}(u)},\\ &\beta_{2(N-k)-1} = \mathrm{b}\frac{ \wp_{2k-1,2N-1}(u)}{\wp_{1,2N-1}(u)}, \quad k=1,\dots, N-1. \end{align}\]

Finally, we obtain \[\label{SinGSolGamma} \gamma_{-1} = \imath r \exp (-\imath \phi) = - \mathrm{b}\wp_{1,2N-1}(u),\tag{50}\] which produces the \(N\)-gap solution \(\phi\) of the sine-Gordon equation 29 in the \(2N\)-dimensional phase space. This solution coincides with the one proposed in [10].

5.2 Equation of motion in variables of separation↩︎

From 31 we find \[\begin{align*} &\frac{\mathrm{d}}{\mathrm{d}\mathrm{x}} \gamma(z) = - 2 \alpha_{2N-2} \gamma(z) + 2 \mathrm{b}z \alpha(z),\\ &\frac{\mathrm{d}}{\mathrm{d}\mathrm{t}} \gamma(z) = - 2 \gamma_{-1} \alpha(z), \end{align*}\] where all dynamic variables are functions of \(\mathrm{x}\) and \(\mathrm{t}\). Therefore, zeros of \(\gamma(z)\) are functions of \(\mathrm{x}\) and \(\mathrm{t}\) as well, namely \(\gamma(z) = \mathrm{b}\prod_{i=1}^N (z-z_i(\mathrm{x},\mathrm{t}))\). Then \[\begin{align*} &\frac{\mathrm{d}}{\mathrm{d}\mathrm{x}} \log \gamma(z) = - \sum_{j=1}^N \frac{1}{z-z_j}\frac{\mathrm{d}z_j}{\mathrm{d}\mathrm{x}} = - 2 \alpha_{2N-2} + 2 \mathrm{b}z \frac{\alpha(z)}{\gamma(z)},\\ &\frac{\mathrm{d}}{\mathrm{d}\mathrm{t}} \log \gamma(z) = - \sum_{j=1}^N \frac{1}{z-z_j}\frac{\mathrm{d}z_j}{\mathrm{d}\mathrm{t}} = - 2 \gamma_{-1} \frac{\alpha(z)}{\gamma(z)}. \end{align*}\] Taking into account ?? , we find as \(z\to z_i\), \(i=1\), …, \(N\), \[\begin{gather} \label{DzDxEqs} \frac{\mathrm{d}z_i}{\mathrm{d}\mathrm{x}} = - \frac{2 w_i}{\prod_{j\neq i}^N (z_i - z_j)},\qquad \frac{\mathrm{d}z_i}{\mathrm{d}\mathrm{t}} = - \frac{2 w_i \prod_{j\neq i} (-z_j)}{\prod_{j\neq i} (z_i - z_j) }. \end{gather}\tag{53}\]

Let \(D = \sum_{i=1}^N (z_i,w_i)\), and the \(N\) points satisfy ?? . The Abel image of \(D\) \[u = \mathcal{A}(D) = \sum_{i=1}^N \int_{\infty}^{(z_i,w_i)} \mathrm{d}u = \sum_{i=1}^N \int_{\infty}^{(z_i,w_i)} \begin{pmatrix} z^{N-1} \\ \vdots \\ z \\ 1 \end{pmatrix} \frac{\mathrm{b}\mathrm{d}z}{-2w}\] depends on \(\mathrm{x}\) and \(\mathrm{t}\). Applying 53 , we find \[\begin{align*} &\frac{\mathrm{d}u_{2n-1}}{\mathrm{d}\mathrm{x}} = \sum_{i=1}^N \frac{\mathrm{b}z_i^{N-n}}{-2w_i} \frac{\mathrm{d}z_i}{\mathrm{d}\mathrm{x}} = \sum_{i=1}^N \frac{\mathrm{b}z_i^{N-n}}{\prod_{j\neq i}^N (z_i - z_j)} = \mathrm{b}\delta_{n,1}, \\ & \frac{\mathrm{d}u_{2n-1}}{\mathrm{d}\mathrm{t}} = \sum_{i=1}^N \frac{\mathrm{b}z_i^{N-n}}{-2w_i} \frac{\mathrm{d}z_i}{\mathrm{d}\mathrm{t}} = \sum_{i=1}^N \frac{\mathrm{b}z_i^{N-n}\prod_{j\neq i} (-z_j)}{\prod_{j\neq i} (z_i - z_j) } = \mathrm{b}\delta_{n,N}. \end{align*}\] And so, we obtain \[\begin{align}u_{1} &= \mathrm{b}\mathrm{x}+ C_1,\\ u_{2n-1} &= \mathrm{b}\mathrm{c}_{2n-1} + C_{2n-1} = \mathop{\mathrm{const}},\quad n=2, \ldots, N-1, \\ u_{2N-1} &= \mathrm{b}\mathrm{t}+ C_{2N-1}.\end{align}\] Here \(\mathrm{c}_{2n-1}\), \(n=2, \ldots, N-1\), are arbitrary real constants, and \(\boldsymbol{C} = (C_1\), \(C_3\), …, \(C_{2N-1})\) is a constant vector, chosen in such a way that \(\gamma_{-1} \,{=}\, {-} \mathrm{b}\wp_{1,2N-1}(u)\) has the desired property, see subsection 3.5. Thus, 50 acquires the form \[\label{KdVSolRealCond} \gamma_{-1}(\mathrm{x}, \mathrm{t}) = - \mathrm{b}\wp_{1,2N-1}\big(\mathrm{b}(\mathrm{x}, \mathrm{c}_{3}, \dots, \mathrm{c}_{2N-3}, \mathrm{t})^t + \boldsymbol{C}\big),\tag{54}\] which produces a finite-gap solution of the sine(sinh)-Gordon equation.

In the sine-Gordon hierarchy, we have \(\mathrm{b}= \imath b\), \(b \in \mathop{\mathrm{\mathbb{R}}}\), and \(r=\sqrt{-\mathrm{r}_{-1}}\), and so \(r/b = \sqrt{\lambda_{4N}}\). According to 22 , values of \(\gamma_{-1}\) draw a circle of radius \(\sqrt{\lambda_{4N}}\). Recalling the substitution for \(\gamma_{-1}\), we find \[\tag{55} \begin{equation}\tag{56} \phi(\mathrm{x},\mathrm{t}) = \imath \log \Big({-} \frac{b}{r} \wp_{1,2N-1}\big(\imath b (\mathrm{x}, \mathrm{c}_{3}, \dots, \mathrm{c}_{2N-3}, \mathrm{t})^t + \boldsymbol{C}\big) \Big). \end{equation} In the \sinh-Gordon hierarchy, \mathrm{b}\in \mathop{\mathrm{\mathbb{R}}}, and r=\sqrt{\mathrm{r}_{-1}}, and so r/ \mathrm{b}\,{=}\, \sqrt{\lambda_{4N}}. Recalling the substitution for \gamma_{-1}, we find \begin{equation}\tag{57} \phi(\mathrm{x},\mathrm{t}) = \log \Big({-} \frac{\mathrm{b}}{r} \wp_{1,2N-1}\big( \mathrm{b}(\mathrm{x}, \mathrm{c}_{3}, \dots, \mathrm{c}_{2N-3}, \mathrm{t})^t + \boldsymbol{C}\big) \Big). \end{equation}\]

5.3 Summary↩︎

The spectral curve 41 is uniformized by means of \(\wp\)-functions associated with the corresponding canonical curve \(\mathcal{V}\), obtained by the transformation 42 . Explicit expressions 45 for the dynamic variables \(\gamma_{2n-1}\), \(\alpha_{2n}\), \(\beta_{2n-1}\), \(n=0\), …, \(N-1\), in terms of \(\wp\)-functions on \(\mathop{\mathrm{\mathrm{Jac}}}(\mathcal{V}) \backslash \Sigma\) are obtained. In particular, the finite-gap solution 55 of the sine(sinh)-Gordon equation expressed through \(\wp_{1,2N-1}\) is found as a function of \(\mathrm{x}\) and \(\mathrm{t}\). Coordinates \(u_1\) and \(u_{2N-1}\) of \(\mathop{\mathrm{\mathrm{Jac}}}(\mathcal{V})\), up to the constant multiple \(\mathrm{b}\), serve as the parameters \(\mathrm{x}\) and \(\mathrm{t}\) of the stationary and evolutionary flows, correspondingly, in the \(N\)-gap hamiltonian system of the sine(sinh)-Gordon hierarchy.

6 Reality conditions↩︎

In what follows, we work with the canonical curve \(\mathcal{V}\) of the form 2 , corresponding to the spectral curve 41 of the \(N\)-gap hamiltonian system of the sine(sinh)-Gordon hierarchy. All parameters \(\lambda_k\) of 2 are real, since integrals of motion \(h_n\), \(c_n\) in 41 are required to be real. Two of branch points of the curve \(\mathcal{V}\) in question are fixed at \((0,0)\) and infinity, and so \(\lambda_{4g+2}\equiv 0\). Moreover,

The reality conditions for the sinh-Gordon hierarchy, associated with the affine algebra \(\mathfrak{sl}(2,\mathop{\mathrm{\mathbb{R}}}) \otimes \mathfrak{L}(z,z^{-1})\), are similar to the ones for the KdV hierarchy, see [1]. The reality conditions for the sine-Gordon hierarchy are discussed in detail below.

In this section, we will find the constant vector \(\boldsymbol{C}\) such that \(\wp_{1,2N-1}\) is bounded, and with the arguments as in 56 runs along the circle of radius \(\sqrt{\lambda_{4g}}\). In the sine-Gordon hierarchy, the parametrization of \(\gamma_{-1}\) introduced in subsection 3.5 implies \(\lambda_{4g}\,{>}\,0\).

6.1 Complex conjugate branch points↩︎

We assume that all non-zero branch points of \(\mathcal{V}\) are complex conjugate, and enumerate branch points in the ascending order of real and imaginary parts, as shown on fig. 1.

In the sine-Gordon hierarchy, the parameter \(\lambda_{4g} = c_{-1}/ \mathrm{b}^2\) is positive, since \(\mathfrak{su}(2)\)-algebra implies \(c_{-1}= -r^2\) and \(\mathrm{b}^2 = -b^2\), \(r\), \(b\in \mathop{\mathrm{\mathbb{R}}}\). Then \[\label{SqLambda4g} \sqrt{\lambda_{4g}} = \prod_{i=1}^g |e_{2i}|.\tag{58}\] At the same time, each \(e_{2i}\) in the product on the right hand side can be replaced with its complex conjugate counterpart. Thus, we obtain \(2^g\) collections of \(g\) non-zero roots which compose the same product.

The branch point divisors singled out by Proposition 2, by necessity, belong to the domain of \(\wp_{1,2g-1}\) in 56 . Indeed, a \(g\)-gap hamiltonian system in terms of variables of separation \((z_i,w_i)\) splits into \(g\) independent systems, each with the coordinate \(z_i\), the momentum \(w_i\), and the hamiltonian given by the spectral curve equation 41 . Finite trajectories contain branch points as turning points.

6.2 Singularities of \(\wp\)-functions↩︎

Each collection of \(g\) branch points form a divisor whose Abel image is an even non-singular half-period. Odd non-singular and singular half-periods are Abel images of divisors of degrees less than \(g\). As explained in subsection 2.6, each half-period is associated with a partition \(\mathcal{I}\cup \mathcal{J}\) of indices of all branch points, where cardinality of \(\mathcal{I}\) is equal to or less than \(g\), that is \(|\mathcal{I}|\leqslant g\). \(\wp\)-Functions have singularities at all half-periods with \(|\mathcal{I}| < g\). Therefore, at half-periods corresponding to collections of \(g\) distinct branch points \(\wp\)-functions have finite values.

Let ‘\(\sim\)’ denote the congruence relation on \(\mathop{\mathrm{\mathrm{Jac}}}(\mathcal{V})\).

Let \(\mathfrak{J}= \mathop{\mathrm{\mathbb{C}}}^g\) be the vector space where \(\mathop{\mathrm{\mathrm{Jac}}}(\mathcal{V})\) is embedded. Let \(\mathop{\mathrm{\mathrm{Re}}}\mathfrak{J}\sim \mathop{\mathrm{\mathbb{R}}}^g\) be the span of real axes of \(\mathfrak{J}\) over \(\mathop{\mathrm{\mathbb{R}}}\), and \(\mathop{\mathrm{\mathrm{Im}}}\mathfrak{J}\sim \mathop{\mathrm{\mathbb{R}}}^g\) be the span of imaginary axes of \(\mathfrak{J}\) over \(\mathop{\mathrm{\mathbb{R}}}\). Then \(\mathfrak{J}= \mathop{\mathrm{\mathrm{Re}}}\mathfrak{J}\oplus \mathop{\mathrm{\mathrm{Im}}}\mathfrak{J}\). As seen from 56 , the domain of \(\wp_{1,2N-1}\) is a subspace of \(\mathfrak{J}\) parallel to \(\mathop{\mathrm{\mathrm{Im}}}\mathfrak{J}\) where \(\wp\)-functions are bounded.

6.3 Hyperelliptic addition law↩︎

Below, we briefly recall the addition laws on hyperelliptic curves, formulated in [19].

The addition operation on the Jacobian variety of a plane algebraic curve is defined by the polynomial function \(\mathcal{R}_{3g}\) of weight \(3g\). On a hyperelliptic curve \(\mathcal{V}\) of genus \(g\) it has the form \[\begin{align} \mathcal{R}_{3g}(x,y) \equiv y \nu_y(x) + \nu_x(x), \\&\nu_y(x) \equiv \sum_{i=1}^{\mathfrak{k}} \nu_{2i-2} x^{\mathfrak{k}-i},\quad \nu_x(x) \equiv \sum_{i=1}^{3\mathfrak{k}-1} \nu_{2i-1} x^{3\mathfrak{k}- 1 - i},\quad \text{if } g = 2\mathfrak{k}-1,\;\;\text{or} \\ &\nu_y(x) \equiv \sum_{i=1}^{\mathfrak{k}} \nu_{2i-1} x^{\mathfrak{k}-i},\quad \nu_x(x) \equiv \sum_{i=1}^{3\mathfrak{k}+1} \nu_{2i-2} x^{3\mathfrak{k}+1-i},\quad \text{if } g = 2\mathfrak{k},\end{align}\] where \(\nu_0=1\) and \(\mathfrak{k}\in \mathop{\mathrm{\mathbb{N}}}\). The polynomial function \(\mathcal{R}_{3g}\) contains \(2g\) arbitrary coefficients, and so is uniquely defined by fixing \(2g\) points in its divisor of zeros. Let \(2g\) points of \(\mathcal{V}\) with no pairs of points in involution form two non-special divisors \(D_\text{I}\) and \(D_\text{II}\), and \(\mathcal{A}(D_\text{I}) = u_\text{I}\), \(\mathcal{A}(D_\text{II}) = u_\text{II}\). Let \(D_\text{III}\) be such a divisor that \((\mathcal{R}_{3g})_0 = D_\text{I} + D_\text{II} + D_\text{III}\), and \(\mathcal{A}(D_\text{III}) = u_\text{III}\). This implies \(u_\text{I}+u_\text{II}+u_\text{III}=0\).

Equations 10 allow to reduce \(\mathcal{R}_{3g}\) to a degree \(g-1\) polynomial in \(x\) with coefficients expressed in terms of \(\nu_k\) and basis \(\wp\)-functions, namely: \(\wp_{1,2i-1}\), \(\wp_{1,1,2i-1}\), \(i=1\), …, \(g\). Since the pre-images of \(u_\text{I}\), \(u_\text{II}\), \(u_\text{III}\) form the divisor of zeros of \(\mathcal{R}_{3g}\), we have \[\begin{gather} \label{AddLawEqs} \begin{pmatrix} 1_g & \textsf{Q}(u_\text{I}) \\ 1_g & \textsf{Q}(u_\text{II}) \\ 1_g & \textsf{Q}(u_\text{III}) \end{pmatrix} \begin{pmatrix} \bar{\nu} \\ \nu \end{pmatrix} = - \begin{pmatrix} \mathbf{q}(u_\text{I}) \\ \mathbf{q}(u_\text{II}) \\ \mathbf{q}(u_\text{III}) \end{pmatrix}, \end{gather}\tag{63}\] where \(\bar{\nu} = (\nu_{g+2}, \nu_{g+4}, \dots, \nu_{3g} )^t\), \(\nu = (\nu_g, \dots, \nu_2, \nu_1)^t\), and \(\textsf{Q}\), \(\mathbf{q}\) are defined as in [1].

Let \(u_\text{I}\) and \(u_\text{II}\) be given, then \(u_\text{III} = - u_\text{I} - u_\text{II}\). Thus, equations for \(u_\text{I}\) and \(u_\text{II}\) in 63 are used to find \(\nu\) and \(\bar{\nu}\) by means of Cramer’s rule. Values \(\wp_{1,2i-1}(u_\text{III})\) are obtained from the equality \[\nu_y^2 f(x,- \nu_x / \nu_y) = \mathcal{R}_{2g}(x,y;u_\text{I}) \mathcal{R}_{2g}(x,y;u_\text{II}) \mathcal{R}_{2g}(x,y;u_\text{III}),\] explicit expressions can be found in [1]. Next, values \(\wp_{1,1,2i-1}(u_\text{III})\) are computed from the equations for \(u_\text{III}\) in 63 . A more general approach to the addition laws can be found in [19], with hyperelliptic addition laws as an illustration.

For our purpose, we need an expression for \(\wp_{1,2g-1}\)-function of the form \[\label{wp12gm1} \wp_{1,2g-1}(u_\text{III}) = (-1)^{g-1} \frac{\nu_{3g}^2 - \lambda_{4g+2} \nu_{g-1}^2 }{\wp_{1,2g-1}(u_\text{I})\wp_{1,2g-1}(u_\text{II})}.\tag{64}\] Note, that \(\lambda_{4g+2} = 0\) in the sine(sinh)-Gordon hierarchy.

6.4 Real-valued \(\wp\)-functions↩︎

Recall, that the function 54 in the sine-Gordon hierarchy satisfies the constraint 22 , which is equivalent to \[\label{AbsWPLambda4g} |\wp_{1,2g-1}\big(\imath b (\mathrm{x}, \mathrm{c}_{3}, \dots, \mathrm{c}_{2g-3}, \mathrm{t})^t + \boldsymbol{C}\big)|^2 = \lambda_{4g},\tag{65}\] where \(g\,{\equiv}\, N\). Below we prove, that 65 holds if the spectral curve of the system in question has the canonical form \(\mathcal{V}\) described in Proposition 1, and \(\boldsymbol{C}\) is defined by ?? . We work with \(\wp\)-functions associated with \(\mathcal{V}\). Then, \(\wp_{i,j}(s)\), \(\wp_{i,j,k}(s)\), \(\wp_{i,j}(\imath s)\) as \(s \in \mathop{\mathrm{\mathbb{R}}}^g\) are real-valued, and \(\wp_{i,j,k}(\imath s)\) have purely imaginary values. Values of \(\wp_{1,2i-1}(\boldsymbol{C})\) are symmetric functions in \(\{e_{2i}\}_{i=1}^g\) computed from 11 , and \(\wp_{1,1,2i-1}(\boldsymbol{C})=0\) as follows from 12 . Let \(u_\text{I} \equiv \boldsymbol{C}\), and \(u_\text{II} = \imath b (\mathrm{x}, \mathrm{c}_{3}, \dots, \mathrm{c}_{2N-3}, \mathrm{t})^t \equiv \imath s\), where \(s \in \mathop{\mathrm{\mathbb{R}}}^g\).

6.5 Summary↩︎

Quasi-periodic solutions of the sine-Gordon equation arise on the canonical curve \(\mathcal{V}\) with \(\lambda_{4g+2}=0\), positive \(\lambda_{4g}\), and all finite non-zero branch points pairwise conjugate. In \(\bar{\mathfrak{J}} = \mathop{\mathrm{\mathbb{C}}}^g \supset \mathop{\mathrm{\mathrm{Jac}}}(\mathcal{V})\) all half-periods split, up to congruence, between \(2^g\) affine subspaces parallel to the imaginary axes subspace \(\mathop{\mathrm{\mathrm{Im}}}\mathfrak{J}\). There exists one subspace \(\boldsymbol{C} + \mathop{\mathrm{\mathrm{Im}}}\mathfrak{J}\), with \(\boldsymbol{C}\) defined by ?? , where \(\wp_{1,2g-1}\)-function is bounded, and has the range on the circle of radius \(\sqrt{\lambda_{4g}}\). The subspace \(\boldsymbol{C} + \mathop{\mathrm{\mathrm{Im}}}\mathfrak{J}\) serves as the domain of the finite-gap solution of the sine-Gordon equation in the \(2g\)-dimensional phase space, namely \[\label{SinGSol} \phi(\mathrm{x},\mathrm{t}) = \imath \log \Big({-} \lambda_{4g}^{-1/2} \wp_{1,2g-1} \big(\imath b (\mathrm{x}, \mathrm{c}_{3}, \dots, \mathrm{c}_{2N-3}, \mathrm{t})^t + \boldsymbol{C} \big) \Big).\tag{73}\]

Quasi-periodic solutions of the \(\sinh\)-Gordon equation arise on the canonical curve \(\mathcal{V}\) with all real branch points, \(\lambda_{4g+2}=0\). Reality conditions in this case are the same as in the KdV hierarchy, see [1].

7 Non-linear waves↩︎

In this section we present effective computation of quasi-periodic finite-gap solutions of the sine-Gordon equation.

7.1 Numerical computation↩︎

We will use the explicit formula 73 . All computations are made in Wolfram Mathematica 12. Integrals between branch points are computed with the help of NIntegrate with WorkingPrecision equal to, or greater than \(30\). The canonical curve 2 is defined through its parameters \(\lambda_k\), and roots of the polynomial \(\Lambda\) are computed by means of NSolve, with the same WorkingPrecision.

\(\wp\)-Functions are computed by the formulas \[\begin{align} \label{WPdefComp}&\wp_{i,j}(u) = \varkappa_{i,j} - \frac{\partial^2}{\partial u_i \partial u_j } \theta[K](\omega^{-1} u; \omega^{-1} \omega'),\\ & \wp_{i,j,k}(u) =- \frac{\partial^3}{\partial u_i \partial u_j \partial u_k} \theta[K](\omega^{-1} u; \omega^{-1} \omega'),\end{align}\tag{74}\] where \(\varkappa_{i,j}\) are entries of the symmetric matrix \(\varkappa = \eta \omega^{-1}\), and the period matrices \(\omega\), \(\omega'\), \(\eta\) are obtained from the differentials 5 , 6 along the canonical cycles, see fig. 1. Actually, columns of the matrices \(\omega\), \(\omega'\), \(\eta\) are computed as follows \[\begin{align}&\omega_k = 2\int_{e_{2k-1}}^{e_{2k}} \mathrm{d}u,\qquad\quad \eta_k = 2\int_{e_{2k-1}}^{e_{2k}} \mathrm{d}r,\\ &\omega'_k = - 2 \sum_{i=1}^k \int_{e_{2i-2}}^{e_{2i-1}} \mathrm{d}u = 2 \sum_{i=k}^g \int_{e_{2i}}^{e_{2i+1}} \mathrm{d}u.\end{align}\]

7.2 Genus \(1\)↩︎

The hamiltonian system of the sine-Gordon equation in \(\mathcal{M}_1\) with variables \(\gamma_{-1}\), \(\beta_{-1}\), \(\alpha_0\), (\(\beta_{-1} \,{=}\,{-} \bar{\gamma}_{-1}\), \(\alpha_0 \,{=}\, \imath a_0\)) lives on the submanifold defined by the constraint \[|\gamma_{-1}|^2 = r^2 = - \mathrm{r}_{-1}.\] The system is governed by the hamiltonian \[h_0 = \alpha_0^2 + \mathrm{b}(\gamma_{-1} - \bar{\gamma}_{-1}) = - a_0^2 - 2 b \mathop{\mathrm{\mathrm{Im}}}\gamma_{-1},\] and possesses the spectral curve \[\label{G1HamCanon} -w^2 - b^2 z^3 + \mathrm{h}_0 z^2 + \mathrm{r}_{-1} z = 0,\tag{75}\] which has complex conjugate branch points if \(|\mathrm{h}_0| < 2 b r\).

On fig.2 (a)a several trajectories of the sine-Gordon equation in \(\mathcal{M}_1\) are displayed, \(b=1/2\), \(r=1/2\). In the \(3\)-dimensional space with \(\mathop{\mathrm{\mathrm{Re}}}\gamma_{-1}\), \(\mathop{\mathrm{\mathrm{Im}}}\gamma_{-1}\), \(a_0\)-axes, the reader can see a vertical circular cylinder \(|\gamma_{-1}| = r\), and several parabolic cylinders which represent different level surfaces \(h_0=\mathrm{h}_0\) of the hamiltonian \(h_0\). Dark lines show trajectories, which arise in the intersection. When \(|\mathrm{h}_0| < 2 b r\), each trajectory is a one loop (marked by green on fig.2 (a)b). At \(\mathrm{h}_0 \,{=}\, 2 b r\) the intersection collapses to the point \(\gamma_{-1} \,{=}\, {-}\imath r\), \(a_0 \,{=}\, 0\). When \(\mathrm{h}_0 \,{>}\, 2 b r\), there is no intersection, that is, no solution exists. When \(\mathrm{h}_0\) approaches \(-2 b r\), the trajectory twists so that two points with \(\mathop{\mathrm{\mathrm{Im}}}\gamma_{-1} \,{=}\, {-}\mathrm{h}_0/(2b)\), \(a_0=0\) coincide at \(\gamma_{-1} = \imath r\), \(a_0 = 0\) (marked by blue on fig.2 (a)b). At \(\mathrm{h}_0 \,{<}\, {-}2 b r\) one loop splits into two (marked by orange on fig.2 (a)b).

Recall that \(\gamma_{-1} \,{=}\, \imath r \exp(\imath \phi)\), where \(\phi\) obeys the sine-Gordon equation, which, in this case, coincides with the equation of motion of a simple pendulum. On fig. 2 (a)b the reader can see a phase portrait of the \(1\)-gap system from the sine-Gordon hierarchy, with the hamiltonian \[h_0 = - a_0^2 - 2 b r \cos \phi.\]

According to 45 , we have \[\begin{gather} \label{UniformG1} \gamma_{-1} = - \imath b \wp_{1,1}(u),\qquad a_0 = - \frac{b}{2} \frac{\wp_{1,1,1}(u)}{\wp_{1,1}(u)}, \end{gather}\tag{76}\] where \(u = \imath b \mathrm{x}+ \frac{1}{2} \omega + \frac{1}{2} \omega'\) as we combine 56 and ?? . Therefore, the phase \(\phi\) is defined by \[\label{SinGSolN1} \phi(\mathrm{x}) = \imath \log \big({-}(b / r) \wp_{1,1}( \imath b \mathrm{x}+ \tfrac{1}{2} \omega + \tfrac{1}{2} \omega')\big).\tag{77}\]

As an example, we choose five values of \(\mathrm{h}_0\), which produce different mutual positions of branch points:

The corresponding fundamental domains and shapes of \(\phi\) with \(r = 1/2\) , \(b = 1/2\) are shown on fig.3 (a).

7.3 Genus \(2\)↩︎

The \(2\)-gap hamiltonian system of the sine-Gordon hierarchy arises in \(\mathcal{M}_2\) with variables \(\gamma_{-1}\), \(\beta_{-1}\), \(\alpha_0\), \(\gamma_{1}\), \(\beta_{1}\), \(\alpha_2\). In fact, the system lives in the submanifold defined by two constraints \[\begin{align} &\gamma_{-1} \beta_{-1} = \mathrm{r}_{-1},\\ &\alpha_0^2 + \gamma_{-1} \beta_{1} + \beta_{-1} \gamma_{1} = \mathrm{r}_0, \end{align}\] which we solve for \(\beta_{-1}\) and \(\beta_{1}\). The system is governed by the two hamiltonians \[\begin{align} &h_1 = 2 \alpha_0 \alpha_2 + \mathrm{b}\gamma_{-1} + (\mathrm{r}_0 - \alpha_0^2) \frac{\gamma_1}{\gamma_{-1}} + \mathrm{r}_{-1} \Big( \frac{\mathrm{b}}{\gamma_{-1}} - \frac{\gamma_1^2}{\gamma_{-1}^2} \Big),\\ &h_2 = \alpha_2^2 + \mathrm{b}\gamma_{1} + (\mathrm{r}_0 - \alpha_0^2) \frac{\mathrm{b}}{\gamma_{-1}} - \mathrm{r}_{-1} \frac{\mathrm{b}\gamma_1}{\gamma_{-1}^2} . \end{align}\] After the substitution \[\begin{align} &\gamma_{-1} = \mathrm{b}z_1 z_2,& &\gamma_1 = - \mathrm{b}(z_1+z_2),& \\ &\alpha_0 = - \frac{w_1 z_2^2 - w_2 z_1^2}{z_1 z_2 (z_1 z_2)},& &\alpha_2 = \frac{w_1 z_2 - w_2 z_1}{z_1 z_2 (z_1 z_2)},& \end{align}\] induced by ?? , the hamiltonians acquire the form \[\begin{align} &h_1 = - \frac{z_2}{z_1^3(z_1-z_2)}\big(w_1^2 - \mathrm{b}^2 z_1^5 - \mathrm{r}_0 z_1^2 - \mathrm{r}_{-1} z_1 \big) \\ &\qquad + \frac{z_1}{z_2^3(z_1-z_2)}\big(w_2^2 - \mathrm{b}^2 z_2^5 - \mathrm{r}_0 z_2^2 - \mathrm{r}_{-1} z_2 \big),\\ &h_2 = \frac{1}{z_1^3(z_1-z_2)}\big(w_1^2 - \mathrm{b}^2 z_1^5 - \mathrm{r}_0 z_1^2 - \mathrm{r}_{-1} z_1 \big) \\ &\qquad - \frac{1}{z_2^3(z_1-z_2)}\big(w_2^2 - \mathrm{b}^2 z_2^5 - \mathrm{r}_0 z_2^2 - \mathrm{r}_{-1} z_2 \big). \end{align}\] In the new variables \(z_1\), \(z_2\), \(w_1\), \(w_2\), separation into two samples of the spectral curve is achieved: \[\begin{gather} - w_i^2 + \mathrm{b}^2 z_i^5 + \mathrm{h}_2 z_i^4 + \mathrm{h}_1 z_i^3 + \mathrm{r}_0 z_i^2 + \mathrm{r}_{-1} z_i = 0,\qquad i=1,\,2, \end{gather}\] where \(\mathrm{h}_1\), \(\mathrm{h}_2\) denote values of the hamiltonians \(h_1\), \(h_2\).

Taking into account, that \(\beta_{2i-1} = - \bar{\gamma}_{2i-1}\), \(\alpha_{2i} = \imath a_{2i}\), \(i=0\), \(1\), and \(\mathrm{b}= \imath b\), we obtain the constraints in the form \[\begin{align} \label{ConstrSinG2}&|\gamma_{-1}|^2 = r^2 = - \mathrm{r}_{-1},\\ &a_0^2 + 2 \mathop{\mathrm{\mathrm{Im}}}\gamma_{1} \mathop{\mathrm{\mathrm{Im}}}\gamma_{-1} + 2 \mathop{\mathrm{\mathrm{Re}}}\gamma_{1} \mathop{\mathrm{\mathrm{Re}}}\gamma_{-1} = - \mathrm{r}_0.\end{align}\tag{78}\] In the projection to the plane \((\mathop{\mathrm{\mathrm{Re}}}\gamma_{-1}) O (\mathop{\mathrm{\mathrm{Im}}}\gamma_{-1})\) the first equation is a circle of radius \(r = \sqrt{-\mathrm{r}_{-1}}\); the second one is a line with the normal \(2(\mathop{\mathrm{\mathrm{Re}}}\gamma_{1}, \mathop{\mathrm{\mathrm{Im}}}\gamma_{1})\) at the distance \(-\frac{1}{2}(\mathrm{r}_0 + a_0^2)/|\gamma_1|\) from the origin. Therefore, an intersection exists if \[-\mathrm{r}_{-1} |\gamma_1|^2 \geqslant \tfrac{1}{4} |\mathrm{r}_0 + a_0^2|^2.\]

The hamiltonians acquire the form \[\begin{align} \label{HamSinG2}&h_1 = - 2a_0 a_2 - 2 b \mathop{\mathrm{\mathrm{Im}}}\gamma_{-1} - |\gamma_1|^2,\\ &h_2 = -a_2^2 - 2 b \mathop{\mathrm{\mathrm{Im}}}\gamma_{1}.\end{align}\tag{79}\] Equations 79 together with 78 define a \(2\)-dimensional surface parametrized by real variables \((\mathrm{x},\mathrm{t})^t = s\). Let \(u = \imath b s + \boldsymbol{C}\) with \(\boldsymbol{C} = \frac{1}{2} \omega_2 + \frac{1}{2} \omega'_1 + \frac{1}{2} \omega'_2\) as follows from 56 and ?? . Then 45 gives a parametrization of the surface: \[\begin{align} &\gamma_{-1} = - \imath b \wp_{1,3}(u),& &\gamma_{1} = - \imath b \wp_{1,1}(u),& \\ &a_0 = - \frac{b}{2} \frac{\wp_{1,3,3}(u)}{\wp_{1,3}(u)},& &a_2 = - \frac{b}{2} \frac{\wp_{1,1,3}(u)}{\wp_{1,3}(u)}.& \end{align}\]

The spectral curve of the system has the canonical form \[\label{G2CanonC} y^2 + x^5 + \lambda_{2} x^4 + \lambda_{4} x^3 + \lambda_{6} x^2 + \lambda_{8} x = 0,\tag{80}\] where \(\lambda_8 \,{=}\, {-}\mathrm{r}_{-1}/ b^2\), \(\lambda_6 \,{=}\, {-}\mathrm{r}_{0}/ b^2\), \(\lambda_4 \,{=}\, {-}\mathrm{h}_{1}/ b^2\), \(\lambda_2 \,{=}\, {-}\mathrm{h}_{2}/ b^2\). By fixing \(\mathrm{r}_{-1}\), \(\mathrm{r}_0\) we choose an orbit where a \(2\)-gap hamiltonian system lives. The hamiltonians \(h_1\), \(h_2\) acquire such values \(\mathrm{h}_1\), \(\mathrm{h}_2\) that all non-zero branch points of 80 are complex conjugate. Each pair \(\mathrm{h}_1\), \(\mathrm{h}_2\) defines a phase trajectory of the hamiltonian system on the fixed orbit. The corresponding \(2\)-gap solution of the sine-Gordon equation is given by \[\label{SinGSolN2} \phi(\mathrm{x}, \mathrm{t}) = \imath \log \Big({-} \lambda_8^{-1/2} \wp_{1,3} \big( \imath b (\mathrm{x},\mathrm{t})^t + \tfrac{1}{2} \omega_2 + \tfrac{1}{2} \omega'_1 + \tfrac{1}{2} \omega'_2 \big) \Big).\tag{81}\]

Let \(b=1/2\), and an orbit be fixed by \(\mathrm{r}_{-1} \,{=}\, {-}1/4\), \(\mathrm{r}_0\,{=}\,{-}3\). The values of \(h_1\) and \(h_2\) permitted within this hamiltonian system are shown as a marigold region on fig.4 (a).

At the several points marked on the region, \(2\)-gap solutions of the sine Grodon equation are computed below.

A surface of \(\phi\) is displayed over \(\mathrm{x}\,{\in}\, [0,8]\), \(\mathrm{t}\,{\in}\, [0,8]\), with black contours representing cuts \(\phi(x, t_k)\) at \(t_k = 0.5 k\), \(k=0\), …, \(17\).

Analyzing the plots at different values of \(\mathrm{h}_1\) and \(\mathrm{h}_2\) within the region shown on fig.4 (a), we find the following. As \(\mathrm{h}_2\) decreases at a fixed \(\mathrm{h}_1\), the length of waves (described by \(\phi(\mathrm{x},0)\)), the swing along \(\mathrm{t}= 0\), and the swing along \(\mathrm{x}= 0\) increase, while the period of waves (described by \(\phi(0,\mathrm{t})\)) remains almost unchanged. When a point \((\mathrm{h}_1,\mathrm{h}_2)\) moves from the right to the left near the upper boundary, or near the lower boundary of the region, the length and the period of waves decrease, and the swing shrinks slowly.

8 Proof of Lemmas 1↩︎

Genus 1. Let an elliptic curve \(\mathcal{V}\) of the form 2 have branch points located at \(e_2\), \(\bar{e}_2\), \(0\), and \(\wp\) be associated with this curve. That is, \(\wp_{1,1}(u) \equiv \wp(u)\), \(\wp_{1,1,1}(u) \equiv \wp'(u)\) satisfy the cubic relation \[\forall u\in \mathop{\mathrm{\mathbb{C}}}\backslash\{0\} \qquad \tfrac{1}{4} \wp_{1,1,1}(u)^2 = \wp_{1,1}(u)^3 + \lambda_2 \wp_{1,1}(u)^2 + \lambda_4 \wp_{1,1}(u).\] Let \(\nu_3\) be defined by 67 with \[\begin{gather} \label{NDG1} |\textsf{D}| = \wp_{1,1}(s) - e_2,\qquad |\textsf{N}| = -\tfrac{1}{2} e_2 \wp_{1,1,1}(s), \quad s\in\mathop{\mathrm{\mathbb{R}}}. \end{gather}\tag{82}\] Then 70 acquires the form \[\begin{gather} \label{nuRelG1} \tfrac{1}{4} e_2 \bar{e}_2 \wp_{1,1,1}(s)^2 - \lambda_{4} \wp_{1,1}(s) (\wp_{1,1}(s) - e_2)(\wp_{1,1}(s) - \bar{e}_2) \\ = (e_2 \bar{e}_2 - \lambda_{4}) \wp_{1,1}(s)^3 + \big(e_2 \bar{e}_2 \lambda_2 + \lambda_{4}(e_2+\bar{e}_2) \big) \wp_{1,1}(s)^2 = 0, \end{gather}\tag{83}\] where the cubic relation is applied, and the equalities \(\lambda_4 = e_2 \bar{e}_2\), \(\lambda_2 = - (e_2+\bar{e}_2)\) are taken into account.

Genus 2. Let a genus two curve \(\mathcal{V}\) of the form 2 have branch points located at \(e_2\), \(\bar{e}_2\), \(e_4\), \(\bar{e}_4\), \(0\). Functions \(\wp_{1,2i-1}\), \(\wp_{1,1,2i-1}\), \(i=1\), \(2\), associated with this curve satisfy the fundamental cubic relations: \(\forall u\in \mathop{\mathrm{\mathrm{Jac}}}(\mathcal{V}) \backslash \Sigma\) \[\begin{align} &\tfrac{1}{4} \wp_{1,1,1}(u)^2 = \wp_{1,1}(u)^2 (\wp_{1,1}(u) +\lambda_2) + \wp_{1,1}(u) \wp_{1,3}(u) \notag \\ &\phantom{mmmmmmmmmmmmma} + \wp_{3,3}(u) + \lambda_4 \wp_{1,1}(u) + \lambda_6, \notag\\ &\tfrac{1}{4} \wp_{1,1,1}(u) \wp_{1,1,3}(u) = \wp_{1,1}(u) \wp_{1,3}(u) (\wp_{1,1}(u) +\lambda_2) \label{FundRelsG2} \\ &\phantom{mmmm} + \tfrac{1}{2} \big(\wp_{1,3}(u)^2 - \wp_{1,1}(u) \wp_{3,3}(u) + \lambda_4 \wp_{1,3}(u) + \lambda_8\big), \notag\\ &\tfrac{1}{4} \wp_{1,1,3}(u)^2 = \wp_{1,3}(u)^2 (\wp_{1,1}(u) +\lambda_2) - \wp_{1,3}(u) \wp_{3,3}(u). \notag \end{align}\tag{84}\]

Recalling that \(\nu_6\) is defined by 67 with \[\begin{align} &|\textsf{D}| = \tfrac{1}{2} \wp_{1,1,1}(s) (\wp_{1,3}(s) + e_2 e_4) - \tfrac{1}{2} \wp_{1,1,3}(s) (\wp_{1,1}(s) - e_2 - e_4),&\\ &|\textsf{N}| = - \tfrac{1}{2} \wp_{1,1,1}(s) e_2 e_4 \wp_{1,3}(s) (\wp_{1,1}(s) - e_2 - e_4)\\ &\qquad\qquad\quad + \tfrac{1}{2} \wp_{1,1,3}(s) e_2 e_4 \big( \wp_{1,1}(s)(\wp_{1,1}(s) - e_2 - e_4)+ \wp_{1,3}(s) + e_2 e_4 \big), & \end{align}\] \(s\in \mathop{\mathrm{\mathbb{R}}}^2\), and substituting into 70 , we obtain \[\begin{gather} \lambda_{8}^{-1} |\textsf{N}| |\bar{\textsf{N}}| + \wp_{1,3} |\textsf{D}| |\bar{\textsf{D}}| = \tfrac{1}{4} \wp_{1,1,1}^2 \big( 2 \wp_{1,3} \mathrm{a}_{1,3} - \wp_{1,1} \mathrm{a}_{3,3} \big) \\ - \tfrac{1}{2} \wp_{1,1,1} \wp_{1,1,3} \big( \wp_{1,3} \mathrm{a}_{1,1} + \mathrm{a}_{3,3} \big) + \tfrac{1}{4} \wp_{1,1,3}^2 \big( \wp_{1,1} \mathrm{a}_{1,1} + 2 \mathrm{a}_{1,3} \big) = 0, \end{gather}\] where the argument \(s\) of \(\wp\)-functions and \(\mathrm{a}_{i,j}\) is omitted for brevity, and \(\mathrm{a}_{i,j}(u) \equiv \tfrac{1}{4} \wp_{1,1,2i-1}(u) \wp_{1,1,2j-1}(u)\).

Genus 3. Let a genus three curve \(\mathcal{V}\) of the form 2 possess the branch points \(e_2\), \(\bar{e}_2\), \(e_4\), \(\bar{e}_4\), \(e_6\), \(\bar{e}_6\), \(0\), and \(\wp\)-functions be associated with this curve. Then, the fundamental cubic relations hold: \(\forall u\in \mathop{\mathrm{\mathrm{Jac}}}(\mathcal{V}) \backslash \Sigma\) \[\begin{align} &\tfrac{1}{4} \wp_{1,1,1}(u)^2 = \wp_{1,1}(u)^2 (\wp_{1,1}(u) +\lambda_2) + \wp_{1,1}(u) \wp_{1,3}(u) \notag \\ &\phantom{mmmmmmmmmmmmm} + \wp_{3,3}(u) - \wp_{1,5}(u) + \lambda_4 \wp_{1,1}(u) + \lambda_6, \notag\\ &\tfrac{1}{4} \wp_{1,1,1}(u) \wp_{1,1,3}(u) = \wp_{1,1}(u) \wp_{1,3}(u) (\wp_{1,1}(u) +\lambda_2) + \wp_{3,5}(u) \notag \\ &\phantom{mmmm} + \wp_{1,1}(u) \wp_{1,5}(u) + \tfrac{1}{2} \big( \wp_{1,3}(u)^2 - \wp_{1,1}(u) \wp_{3,3}(u) + \lambda_4 \wp_{1,3}(u) + \lambda_8\big), \notag\\ &\tfrac{1}{4} \wp_{1,1,1}(u) \wp_{1,1,5}(u) = \wp_{1,1}(u) \wp_{1,5}(u) (\wp_{1,1}(u) +\lambda_2) \notag \\ &\phantom{mmmm} + \tfrac{1}{2} \big( \wp_{1,3}(u) \wp_{1,5}(u) - \wp_{1,1}(u) \wp_{3,5}(u) - \wp_{5,5}(u) + \lambda_4 \wp_{1,5}(u)\big), \label{FundRelsG3} \\ &\tfrac{1}{4} \wp_{1,1,3}(u)^2 = \wp_{1,3}(u)^2 (\wp_{1,1}(u) +\lambda_2) + 2 \wp_{1,3}(u) \wp_{1,5}(u) \notag \\ &\phantom{mmmmmmmmmmmmmm} - \wp_{1,3}(u) \wp_{3,3}(u) + \wp_{5,5}(u) + \lambda_{10}, \notag \\ &\tfrac{1}{4} \wp_{1,1,3}(u) \wp_{1,1,5}(u) = \wp_{1,3}(u) \wp_{1,5}(u) (\wp_{1,1}(u) +\lambda_2) \notag\\ &\phantom{mmmm} + \wp_{1,5}(u)^2 + \tfrac{1}{2} \big( {-}\wp_{1,5}(u) \wp_{3,3}(u) - \wp_{1,3}(u) \wp_{3,5}(u) + \lambda_{12}\big), \notag \\ &\tfrac{1}{4} \wp_{1,1,5}(u)^2 = \wp_{1,5}(u)^2 (\wp_{1,1}(u) +\lambda_2) - \wp_{1,5}(u) \wp_{3,5}(u). \notag \end{align}\tag{85}\]

In the definition 67 of \(\nu_9\) we have, \(s\in \mathop{\mathrm{\mathbb{R}}}^3\), \[\begin{align} &\textsf{P}(s) = \begin{pmatrix} \wp_{1,1}(s) & \wp_{1,1}(s)^2 + \wp_{1,3}(s) \\ \wp_{1,3}(s) & \wp_{1,1}(s)\wp_{1,3}(s) + \wp_{1,5}(s) \\ \wp_{1,5}(s) & \wp_{1,1}(s) \wp_{1,5}(s) \end{pmatrix},\quad \textsf{R}(s) = \begin{pmatrix} -\tfrac{1}{2} \wp_{1,1,1}(s)\\ -\tfrac{1}{2} \wp_{1,1,3}(s) \\ -\tfrac{1}{2} \wp_{1,1,5}(s) \end{pmatrix} \\ &{-}\boldsymbol{q}(s) = \begin{pmatrix} \tfrac{1}{2} \wp_{1,1,3}(s) + \tfrac{1}{2} \wp_{1,1,1}(s) \wp_{1,1}(s) \\ \tfrac{1}{2} \wp_{1,1,5}(s) + \tfrac{1}{2} \wp_{1,1,1}(s) \wp_{1,3}(s) \\ \tfrac{1}{2} \wp_{1,1,1}(s) \wp_{1,5}(s) \end{pmatrix}, \quad \begin{array}{l} \wp_{1,1}(\boldsymbol{C}) = e_2 + e_4 + e_6,\\ \wp_{1,3}(\boldsymbol{C}) = - e_2 e_4 - e_2 e_6 - e_4 e_6, \\ \wp_{1,5}(\boldsymbol{C}) = e_2 e_4 e_6, \end{array} \end{align}\] and \(\wp_{1,1,2i-1}(\boldsymbol{C}) = 0\), \(i=1\), \(2\), \(3\). By means of \(\mathrm{a}_{i,j}(u) \equiv \tfrac{1}{4} \wp_{1,1,2i-1}(u) \wp_{1,1,2j-1}(u)\) we eliminate all \(3\)-index \(\wp\)-functions in 70 , and expand in \(\mathrm{a}_{i,j} \mathrm{a}_{k,l} - \mathrm{a}_{i,k} \mathrm{a}_{j,l}\), which vanish. Actually, \[\begin{gather} \lambda_{12}^{-1} |\textsf{N}| |\bar{\textsf{N}}| - \wp_{1,5} |\textsf{D}| |\bar{\textsf{D}}| = C_{1,1;3,3}(\mathrm{a}_{1,1} \mathrm{a}_{3,3} - \mathrm{a}_{1,3}^2 ) \\ + C_{1,1;3,5}(\mathrm{a}_{1,1} \mathrm{a}_{3,5} - \mathrm{a}_{1,3} \mathrm{a}_{1,5}) + C_{1,1;5,5}(\mathrm{a}_{1,1} \mathrm{a}_{5,5} - \mathrm{a}_{1,5}^2 ) \\ + C_{1,3;3,5}(\mathrm{a}_{1,3} \mathrm{a}_{3,5} - \mathrm{a}_{1,5} \mathrm{a}_{3,3}) + C_{1,3;5,5}(\mathrm{a}_{1,3} \mathrm{a}_{5,5} - \mathrm{a}_{1,5} \mathrm{a}_{3,5} )\\ + C_{3,3;5,5}(\mathrm{a}_{3,3} \mathrm{a}_{5,5} - \mathrm{a}_{3,5}^2 ) = 0, \end{gather}\] where \[\begin{align} C_{1,1;3,3} =&\; \wp_{1,5}(s)^3 (2\wp_{1,1}(s) +\lambda_2) \\ &+ \wp_{1,5}(s)^2 \big( \wp_{1,3}(\boldsymbol{C}) \wp_{1,3}(\bar{\boldsymbol{C}}) - (\wp_{1,5}(\boldsymbol{C}) + \wp_{1,5}(\bar{\boldsymbol{C}}))\wp_{1,1}(u) + \lambda_8 \big),\\ C_{1,1;3,5} = &\;{-}2 \wp_{1,5}(s)^3 - 2\wp_{1,5}(s)^2 \big( \wp_{1,3}(s) (2\wp_{1,1}(s) +\lambda_2) + \wp_{1,5}(\boldsymbol{C}) + \wp_{1,5}(\bar{\boldsymbol{C}})\\ &+ (\wp_{1,3}(\boldsymbol{C}) + \wp_{1,3}(\bar{\boldsymbol{C}}) ) \wp_{1,1}(s) - \lambda_6 \big) + \wp_{1,5}(s) \wp_{1,3}(s) \big(\wp_{1,3}(\boldsymbol{C}) \wp_{1,3}(\bar{\boldsymbol{C}}) \\ & +(\wp_{1,5}(\boldsymbol{C}) + \wp_{1,5}(\bar{\boldsymbol{C}}) ) \wp_{1,1}(s) - \lambda_8 \big) - 2 \lambda_{12} \wp_{1,5}(s),\\ C_{1,1;5,5} = &\; \wp_{1,5}(s)^2 \big( 2\wp_{1,3}(s) - \wp_{1,3}(\boldsymbol{C}) - \wp_{1,3}(\bar{\boldsymbol{C}})\big) + \wp_{1,5}(s) \big( \wp_{1,3}(s)^2 (2\wp_{1,1}(s) +\lambda_2) \\ & + \wp_{1,3}(s) \big({-} (\wp_{1,3}(\boldsymbol{C}) + \wp_{1,3}(\bar{\boldsymbol{C}}))\wp_{1,1}(s) + \lambda_6 \big) + \lambda_{10} \big),\\ C_{1,3;3,5} = &\; \wp_{1,5}(s)^2 \big(2 \wp_{1,3}(s)+ \wp_{1,1}(s) (4\wp_{1,1}(s) + 3\lambda_2) + \wp_{1,1}(\boldsymbol{C}) \wp_{1,1}(\bar{\boldsymbol{C}}) + \lambda_4\big) \\ &+ \wp_{1,5}(s) \big({-} (\wp_{1,5}(\boldsymbol{C}) + \wp_{1,5}(\bar{\boldsymbol{C}})) ( \wp_{1,1}(s)^2 + \wp_{1,3}(s) ) \\ & + (\wp_{1,3}(\boldsymbol{C}) \wp_{1,3}(\bar{\boldsymbol{C}}) + \lambda_8) \wp_{1,1}(s) \big),\\ C_{1,3;5,5} =&\; {-} \wp_{1,5}(s)^2 (2\wp_{1,1}(s) +\lambda_2) - \wp_{1,5}(s) \wp_{1,3}(s) \big(2 \wp_{1,3}(s) \\ &+ \wp_{1,1}(s) (4\wp_{1,1}(s) + 3\lambda_2) + 2 \lambda_4\big) - \wp_{1,5}(s) \big((\wp_{1,3}(\boldsymbol{C}) + \wp_{1,3}(\bar{\boldsymbol{C}}))\wp_{1,1}(s)^2 \\ & + \lambda_6 \wp_{1,1}(s) + \wp_{1,3}(\boldsymbol{C}) \wp_{1,3}(\bar{\boldsymbol{C}}) + \lambda_8\big),\\ C_{3,3;5,5} = &\; \wp_{1,5}(s) \wp_{1,3}(s) (2\wp_{1,1}(s) +\lambda_2) + \wp_{1,5}(s) \big(2 \wp_{1,1}(s)^3 + 2 \lambda_2 \wp_{1,1}(s)^2 \\ & + (\wp_{1,1}(\boldsymbol{C}) \wp_{1,1}(\bar{\boldsymbol{C}}) + \lambda_4 ) \wp_{1,1}(s) + \wp_{1,5}(\boldsymbol{C}) + \wp_{1,5}(\bar{\boldsymbol{C}}) + \lambda_6 \big). \end{align}\] The identities \(\mathrm{a}_{i,j} \mathrm{a}_{k,l} - \mathrm{a}_{i,k} \mathrm{a}_{j,l} = 0\) expressed in terms of \(2\)-index \(\wp\)-functions define the Kummer variety of \(\mathcal{V}\).

This completes the proof of Lemma 1.

References↩︎