Fokas A S
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, UK.
Viterbi School of Engineering, University of Southern California,Los Angeles, CA, USA.
Proc Math Phys Eng Sci. 2022 Jul;478(2263):20220074. doi: 10.1098/rspa.2022.0074. Epub 2022 Jul 27.
There are integrable nonlinear evolution equations in two spatial variables. The solution of the initial value problem of these equations necessitated the introduction of novel mathematical formalisms. Indeed, the classical Riemann-Hilbert problem used for the solution of integrable equations in one spatial variable was replaced by a non-local Riemann-Hilbert problem or, more importantly, by the so-called -bar formalism. The construction of integrable nonlinear evolution equations in three spatial dimensions has remained the key open problem in the area of integrability. For example, the two versions of the Kadomtsev-Petviashvili (KP) equation constitute two-dimensional generalizations of the celebrated Korteweg-de Vries equation. Are there three-dimensional generalizations of the KP equations? Here, we present such equations. Furthermore, we introduce a novel non-local -bar formalism for solving the associated initial value problem.
存在含两个空间变量的可积非线性演化方程。求解这些方程的初值问题需要引入新的数学形式。实际上，用于求解含一个空间变量的可积方程的经典黎曼 - 希尔伯特问题已被非局部黎曼 - 希尔伯特问题所取代，或者更重要的是，被所谓的(\bar{\partial})形式所取代。构建含三个空间维度的可积非线性演化方程一直是可积性领域的关键开放性问题。例如，卡多姆采夫 - 彼得维谢夫利（KP）方程的两个版本构成了著名的科特韦格 - 德弗里斯方程的二维推广。是否存在KP方程的三维推广呢？在此，我们给出这样的方程。此外，我们引入一种新的非局部(\bar{\partial})形式来求解相关的初值问题。
