The author studies limits of quasi-periodic solutions of the Kadomtsev-Petviashvili (KP) equation \[ 3u_{t_2t_2} + (-4u_{t_3} + 6uu_{t_1} + u_{t_1t_1t_1})_{t_1} = 0, \] where \((t_1, t_2)\) are space variables and \(t_3\) is the time variable, and the KP hierarchy, an infinite system of differential equations which contains the KP equation as its first member.
Quasi-periodic solutions (or algebro-geometric solutions) form a class of solutions which can be expressed explicitly using theta functions on algebraic curves with positive genus. Soliton solutions can be considered as limits of quasi-periodic solutions when periods go to infinity. Y. Kodama [KP solitons and the Grassmannians. Combinatorics and geometry of two-dimensional wave patterns. Singapore: Springer (2017; Zbl 1372.35266)] recently discovered that the shapes of soliton solutions form web patterns and are related to the geometry of Grassmann manifolds. So how does the structure of these solitons compare with those arising as limits of the quasi-periodic solution?
It turns out that the limits to positive genus solutions are more fundamental. To overcome the difficulty of taking a limit of a theta function (respectively a limit of the period matrix of an algebraic curve), the author further develops the Sato Grassmannian approach, which utilizes a one-to-one correspondence between points of the Sato Grassmannian method and solutions of the KP-hierarchy (up to constants). Using this method, an algebro-geometric solution can be described as a series whose coefficients are constructed from certain rational functions on an algebraic curve. In this way the difficult problem of taking limits of period matrices reduces to the much easier problem of taking limits of rational functions. In the obtained solution, the mixing of solitons and quasi-periodic solutions is visible.
For the entire collection see [Zbl 1472.53006].