Partial degeneration of finite gap solutions to the Korteweg–de Vries equation: soliton gas and scattering on elliptic backgrounds
Partial degeneration of finite gap solutions to the Korteweg–de Vries equation: soliton gas and scattering on elliptic backgrounds
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DOI：
10.1088/1361-6544/accfdf
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发表时间：
2022-10
期刊：
影响因子：
1.7
通讯作者：
Marco Bertola;Robert Jenkins;A. Tovbis
中科院分区：
文献类型：
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作者：
Marco Bertola;Robert Jenkins;A. Tovbis
我们得到了函数在任意格（不可约）节点曲线上的部分退化的Fredholm型公式，重点讨论了1格的节点曲线。一个应用是研究Korteweg-de Vries （KdV）方程在椭圆（余弦）背景驻波上的“多孤子”解，从一个让人想起经典的n孤子的Kay-Moses公式的公式开始。特别是，我们将这样的解表示为以下两项的和：一个“移位”的椭圆（余弦）背景波和一个包含Jacobi θ函数的Kay-Moses型行列式，它可以被看作是余弦背景上的孤立干扰的集合。这些孤立扰动的传播（群）速度表达式，以及描述这些孤立扰动对散射的相互作用核，用雅可比函数得到了明确的表达式。我们还证明了当N个带退化为1的节点曲线时，具有随机初始相的N + 1类有限间隙解在概率上收敛于确定性余弦波解。最后，导出了残馀椭圆背景下KdV孤子气体的非线性色散关系和状态方程。
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 travelling (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.