Complete integrability of the Kadomtsev-Petviashvili equations in \(2+1\) dimensions (Q1820935)

The authors discuss the complete integrability of the equations \[ (u_ t+6uu_ x+u_{xxx})_ x=-3\alpha^ 2 u_{yy}, \] where \(\alpha =i\) or \(\alpha =-1\), which appear to be an analogue in \(2+1\) dimensions of the Korteweg-de Vries equation. It is stated that the case \(\alpha =-1\) is solved by the ''D-bar'' method, and the case \(\alpha =i\) by means of a nonlocal Riemann-Hilbert problem.