论文标题
对Korteweg-De Vries方程的有限间隙解的部分变性：椭圆背景上的孤子气和散射
Partial degeneration of finite gap solutions to the Korteweg-de Vries equation: soliton gas and scattering on elliptic background
论文作者
论文摘要
我们获得了fredholm型公式，用于theta在任意属的（不可可理）淋巴结曲线上的部分变性，重点是One属的淋巴结曲线。应用程序是对Korteweg-de Vries（KDV）方程的椭圆形（cNoidal）背景的“多索顿”溶液的研究，从使人想起$ n $ solitons的古典kay-moses公式开始。特别是，我们将这种解决方案表示为以下两个术语的总和：``移动的椭圆形（cNoidal）背景波和kay-moses类型的决定因素，这些决定因素含有jacobi theta的函数，可以将其视为cnoidal systerions for systrive for systrive for the sypersions for the systerive for the systerive for the sypertion sypersions的速度。对这种孤立的干扰，我们还明确地从雅各布theta函数方面获得了$ n+1 $的缝隙属，随机初始阶段有限的间隙解决方案概率与确定性的cNoidal波动溶液作为$ n $ bresss and derive of non dention and of sore and the and of nonse and s and s and s and s and the and s and the and the s and the and。残留椭圆背景。
We obtain Fredholm type formulas for partial degenerations of Theta functions on (irreducible) nodal curves of arbitrary genus, with emphasis on nodal curves of genus one. An application is the study of "many-soliton" solutions on an elliptic (cnoidal) background standing wave for the Korteweg-de Vries (KdV) equation starting from a formula that is reminiscent of the classical Kay-Moses formula for $N$-solitons. In particular, we represent such a solution as a sum of the following two terms: a ``shifted" elliptic (cnoidal) background wave and a Kay-Moses type determinant containing Jacobi theta functions for the solitonic content, which can be viewed as a collection of solitary disturbances on the cnoidal background. The expressions for the traveling (group) speed of these solitary disturbances, as well as for the interaction kernel describing the scattering of pairs of such solitary disturbances, are obtained explicitly in terms of Jacobi theta functions. We also show that genus $N+1$ finite gap solutions with random initial phases converge in probability to the deterministic cnoidal wave solution as $N$ bands degenerate to a nodal curve of genus one. Finally, we derive the nonlinear dispersion relations and the equation of states for the KdV soliton gas on the residual elliptic background.
