(3+1)-dimensional-modified Kadomtsev–Petviashvili equation and its ∂¯-formalism

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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mn>4</mml:mn> <mml:mo>+</mml:mo> <mml:mn>2</mml:mn> </mml:math> dimensions, but the reduction to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mn>3</mml:mn> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:math> dimensions remained open until 2022 when he introduced a suitable nonlinear Fourier transform to achieve this reduction for the Kadomtsev–Petviashvili (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 ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mn>3</mml:mn> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:math> )-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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mover> <mml:mi mathvariant="normal">∂</mml:mi> <mml:mo stretchy="false">¯</mml:mo> </mml:mover> </mml:mrow> </mml:math> -problem is demonstrated based on the Cauchy–Green formula. Finally, a novel <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mover> <mml:mi mathvariant="normal">∂</mml:mi> <mml:mo stretchy="false">¯</mml:mo> </mml:mover> </mml:mrow> </mml:math> -formalism for the initial value problem of the ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mn>3</mml:mn> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:math> )-dimensional mKP equation is worked out.