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

TL;DR

This paper introduces a new (3+1)-dimensional integrable modified KP equation using a arormalism, involving complex time and Laplace's equation, advancing the understanding of high-dimensional integrable PDEs.

Contribution

It presents the first integrable (3+1)-dimensional mKP equation with a novel arormalism and spectral analysis, extending the reduction of KP equations to higher dimensions.

Findings

Derived the (3+1)-dimensional mKP equation from the KP framework.

Developed a arormalism for solving the initial value problem.

Provided spectral analysis and solution techniques for the new PDE.

Abstract

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.