二维托达晶格和戴维 - 斯图尔特森方程的组合几何
Combinatorial geometry of the 2D Toda lattice and Davey Stewartson equation
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中文总结 AI 辅助
研究二维托达晶格和戴维 - 斯图尔特森方程，借助Kodama和Williams受热带几何启发的算法方法，通过全非负格拉斯曼流形元素生成等高线图，推广其方法并在渐近情况完善前人工作。
中文摘要 AI 辅助
KP方程是典型的(2 + 1)维可积偏微分方程，其孤子解由佐藤格拉斯曼流形参数化。Kodama和Williams发现孤子解的组合学与全正格拉斯曼流形的组合学密切相关，并引入了受热带几何多面体结构启发的新算法方法。二维托达晶格和戴维 - 斯图尔特森方程的孤子解也由佐藤格拉斯曼流形分类，本文表明Kodama和Williams的方法可推广到这两个方程，还推导了从全非负格拉斯曼流形元素生成等高线图的算法，在渐近情况下恢复并完善了前人工作。
英文摘要
The KP equation is a prototypical $(2+1)$-dimensional integrable PDE. Its soliton solutions are famously parametrized by the Sato Grassmannian. In seminal work, Kodama and Williams made the surprising discovery that the combinatorics of soliton solutions are intimately related to the combinatorics of the totally positive Grassmannian as pioneered by Postnikov. They introduced novel algorithmic methods inspired by polyhedral structures arising from tropical geometry. Soliton solutions to the 2D Toda lattice and the Davey--Stewartson equation, two closely related integrable systems with soliton solutions, are also classified by the Sato Grassmannian. Kodama suggested that the methods of his work with Williams could generalize to these two integrable equations. In this work, we show that this is indeed the case. We derive algorithms to produce contour plots from elements in the totally nonnegative Grassmannian in both cases. In the asymptotic setting, we recover and refine previous work of Biondini and Wang; as well as Biondini, Kireyev and Maruno.
