标题： (3+1)维改进的Kadomtsev-Petviashvili方程及其\bar{\partial}形式化 显示英文标题

标题： (3+1)-dimensional modified Kadomtsev-Petviashvili equation and its \bar{\partial}-formalism

摘要： 在三维空间维度中构建可积的非线性偏微分方程是可积性领域中最重要的开放问题之一。 Fokas 在 2006 年通过构建 4+2 维的可积非线性方程取得了进展，但将其约化到 3+1 维的问题一直未被解决，直到 2022 年他引入了一种合适的非线性傅里叶变换，从而实现了对 Kadomtsev-Petiashvili（KP）方程的约化。 在这里，提出了修改后的 KP（mKP）方程的可积推广，其新颖之处在于它涉及复时间并保持拉普拉斯方程。 (3+1)-维 mKP 方程是通过施加实时间的要求得到的。 然后给出了特征值方程的谱分析，并基于柯西-格林公式展示了$\bar{\partial}$-问题的解。 最后，推导出了 (3+1)-维 mKP 方程初值问题的新$\bar{\partial}$-形式。 显示英文摘要

摘要： Constructing integrable evolution nonlinear PDEs in three spatial dimensions is one of the most important open problems in the area of integrability. Fokas achieved progress in 2006 by constructing integrable nonlinear equations in 4+2 dimensions, but the reduction to 3+1 dimensions remained open until 2022 when he introduced a suitable nonlinear Fourier transform to achieve this reduction for the Kadomtsev-Petiashvili (KP) equation. Here, the integrable generalization of the modified KP (mKP) equation has been presented which has the novelty that it involves complex time and preserves Laplace's equation. The (3+1)-dimensional mKP equation is obtained by imposing the requirement of real time. Then, the spectral analysis of the eigenvalue equation is given and the solution of the $\bar{\partial}$-problem is demonstrated based on Cauchy-Green formula. Finally, a novel $\bar{\partial}$-formalism for the initial value problem of the (3+1)-dimensional mKP equation is worked out.