IST for KPI

The cauchy problem of the Kadomtsev-Petviashvili I (KPI) equation (ut + 6uux + uxxx)x = 3uxx can be solved by the inverse scattering method with the following Lax pair iψy + ψxx + uψ = 0, ψt + 4ψxxx + 6uψx + 3(ux - i/2∂-1xuy)ψ = 0. This problem has been formally solved by Zakharov and Manakov, Fokas and Ablowitz, and Boiti, Leon and Pempinelli. The work of Zakharov and Manakov contains important ideas such as triangular factorizations and the positivity of the scattering operators. The work of Fokas and Ablowitz includes the lump solutions into the inverse scattering scheme. The work of Boiti, Leon and Pempinelli derives systematically the relations among different scattering operators. Ablowitz and Fokas also pointed out recently that the operator ∂-1x should have the symmetric form 1/2 ∫x∞ -1/2 ∫∞x. The author summarizes this work and other work on the inverse scattering transform for the KPI equation