Abstract We degenerate the finite gap solutions of the KdV equation from the general formulation given in terms of abelian functions when the gaps tends to points, to get solutions to the KdV equation given in terms of Fredholm determinants and wronskians. For this we establish a link between Riemann theta functions, Fredholm determinants and wronskians. This gives the bridge between the algebro-geometric approach and the Darboux dressing method. We construct also multi-parametric degenerate rational solutions of this equation.

中文翻译：

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KdV 方程和退化有理情况的解的退化黎曼 theta 函数、Fredholm 和 wronskian 表示

摘要 当间隙趋向于指向时，我们从阿贝尔函数给出的一般公式中退化 KdV 方程的有限间隙解，得到用 Fredholm 行列式和 wronskian 给出的 KdV 方程的解。为此，我们在 Riemann theta 函数、Fredholm 行列式和 wronskians 之间建立了联系。这提供了代数几何方法和 Darboux 修整方法之间的桥梁。我们还构造了这个方程的多参数退化有理解。