A numerical representation of hyperelliptic KdV solutions. (English) Zbl 07899964
The present paper considers a numerical representation of hyperelliptic Korteweg-de Vries (KdV) solutions. The periodic and quasi-periodic solutions of the integrable system have been studied for four decades based on the Riemann theta functions. However, there is a fundamental difficulty in representing the solutions numerically because the Riemann theta function requires several transcendental parameters. This paper presents a novel method for the numerical representation of such solutions from the algebraic treatment of the periodic and quasi-periodic solutions of the Baker-Weierstrass hyperelliptic \(\wp\) functions. The author proves the numerical representation of the hyperelliptic \(\wp\) functions of genus two. The method presented is by limiting the genus to two but it can be easily extended to general genus. It can be provided for various curves and allow the numerical investigations of periodic and quasi-periodic solutions of integrable equation. It is extremely simple and clear, which is based on the well known Euler’s numerical quadrature method. This paper is organized as follows. Section 1 is an introduction to the subject and summarizes the main results. Section 2 deals with KdV equation and the Baker-Weierstrass hyperelliptic \(\wp\) functions of genus two. The author shows the relationship between the KdV equation and the hyperelliptic \(\wp\) function algebraically. Section 3 is devoted to graphical representation of \(\wp\) function. It provides the novel algorithm to obtain the numerical representation of the hyperelliptic \(\wp\) function and its demonstrations. Section 4 is devoted to the discussion and conclusion.
Reviewer: Ahmed Lesfari (El Jadida)
MSC:
| 14H45 | Special algebraic curves and curves of low genus |
| 14H40 | Jacobians, Prym varieties |
| 14H70 | Relationships between algebraic curves and integrable systems |
| 35Q53 | KdV equations (Korteweg-de Vries equations) |
Keywords:
hyperelliptic \(\mathcal{P}\) functions; numerical representation; periodic and quasi-periodic solutions; KdV equation; integrable system; algebro-geometrical method; solitonReferences:
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