IST for KPI

The Cauchy problem of the Kadomtsev-Petviashvili I (KPI) equation $${\left( {{u_t} + 6u{u_x} + {u_{xxx}}} \right)_x} = 3{u_{xx}}$$ (1) can be solved by the inverse scattering method with the following Lax pair $$i{\psi _y} + {\psi _{xx}} + u\psi = 0$$ (2) $${\psi _t} + 4{\psi _{xxx}} + 6u{\psi _x} + 3\left( {{u_x} - \frac{i}{2}\partial _y^{ - 1}{u_y}} \right)\psi = 0$$ (3) .