Hyperelliptic Riemann surfaces of infinite genus and solutions of the KdV equation
The authors study hyperelliptic Riemann surfaces of infinite genus associated with infinite-gap solutions to the KdV equation \[ u_t= 6uu_x -u_{xxx},\qquad u|_{t=0}=q \tag{1} \] where \(q\) is a function of \(x\in\mathbb R.\) Their approach is based on an extension of the Its-Matveev formula: Let \({\mathcal M}\) be a hyperelliptic Riemann surface of genus \(g\) (= number of gaps) associated with the fixed gap structure. Let \(\Omega\) be the period matrix of \({\mathcal M},\) and let \(\vartheta_g(z|\Omega)\) be the corresponding Riemann theta function. Then there exist \(U, V\in\mathbb R^g\) and a constant \(C\) such that for every \(W\in\mathbb R^g,\) \[ u(x,t) =-2\frac{d^2}{dx^2}\log\vartheta_g(Ut +Vx + W|\Omega) +C \tag{2} \] is a solution of the KdV equation. The vectors \(U\) and \(V\) are given by periods of certain normalized meromorphic \(1\)-forms on \({\mathcal M}\) with a unique pole at \(\infty\) of order \(2\) and \(4\), respectively, and \(C\) is also determined by \({\mathcal M}.\) The solutions obtained in this way are quasi-periodic. To extend the above formula the authors make use of a Riemann theta function that exists for infinite-genus surfaces whose definition is due to \textit{K. Tahara} [Nagoya Math. J. 33, 57-73 (1968; Zbl 0167.19102)]. The authors consider in the present paper surfaces \({\mathcal M}={\mathcal M}({\Lambda})\) determined by a sequence of ramification points of the form \[ \Lambda :\lambda_0=0<\lambda_1 <\lambda_2 <\cdots\rightarrow \infty \] where they assume that \(\Lambda\) has no finite points of accumulation. Let \(A_1, B_1, A_2, B_2,\ldots \) be a certain canonical homology basis of \({\mathcal M}.\) They construct explicitly a basis \(\phi_1, \phi_2,\ldots \) for the space of holomorphic square integrable \(1\)-forms on \({\mathcal M},\) which is normalized by \[ \int_{A_i}\phi_j =\delta_{ij}. \] Then they prove that such a basis is uniquely determined and, the period matrix of \({\mathcal M}\) is defined by \[ \Omega =\left (\int_{B_i}\phi_j\right). \] They prove the following theorem: Given a diagonal matrix \(T= \text{diag}(\theta_1, \theta_2,\ldots)\) with \(\theta_j >0\), \(j\geq 1,\) and real numbers \(p > 1\), \(C >0,\) there exists a sequence \(\Lambda\) such that the period matrix \(\Omega\) of \({\mathcal M}(\Lambda)\) is purely imaginary and satisfies \[ |(\Omega -\sqrt{-1}T)_{ij}|\leq \frac{C}{i^pj^p}\theta_i^{1/2}\theta_j^{1/2}\tag{3} \] for all \( i, j\geq 1.\) In a previous work the authors have introduced an infinite-dimensional analogue \(H_T^p\) of Siegel upper half-space, and the above theorem means that \(\Omega\in H_T^p.\) Let \[ \ell_{p, T}= \left\{(z_i)_{i\in\mathbb N} \in \mathbb C^{\infty}\mid \|z\|^2_{p,T}= \sum_{i\geq 1}|z_i|^2 \theta_i^{-1} i^p< \infty \right\}. \] The authors have also previously defined renormalized theta functions \(\vartheta_T(z|\Omega)\) that depends on \(z\in \ell_{p, t}\) and \(\Omega\in H_T^p.\) Using these theta functions they extend the Its-Matveev formula \((2)\), and construct new solutions to the KdV equation. More precisely, they prove the following theorem: Let \(p >3\) and \(T=\text{diag}(\theta_1,\theta_2,\ldots)\) be given. Let \(\Omega\) be the period matrix of a hyperelliptic surface \({\mathcal M}\) such that \((3)\) holds. Then \(\Omega\in H_T^p,\) and there exists \(U, V\in \ell_{p, T}\) and \(C\in\mathbb C\) such that for every real \(W\in \ell_{p, T},\) \[ u(x, t) = -2\frac{\text{ d}^2}{\text{ d}x^2}\log\vartheta_g(Ut +Vx + W|\Omega) +C,\quad x, t\in\mathbb R, \] is a nontrivial solution to the KdV equation \((1).\)
holomorphic square integrable \(1\)-forms
scientific article; zbMATH DE number 1425080
scientific article