SOLUTIONS OF THE KPI EQUATION WITH SMOOTH INITIAL DATA

The solution u(t, x, y) of the Kadomtsev-Petviashvili I (KPI) equation with given initial data u(0, x, y) belonging to the Schwartz space is considered. No additional special constraints, usually considered in the literature, such as integral dx u(0, x, y) = 0 are required to be satisfied by the initial data. The problem is completely solved in the frame-work of the spectral transform theory and it is shown that u(t, x, y) satisfies a special evolution version of the KPI equation and t hat, in general, partial derivative(t)u(t, x, y) has different left an d right limits at the initial time t = 0. The conditions of the type i ntegral dx u(t, x, y) = 0, integral dx x u(y)(1, x, y) = 0 and so on ( first, second, etc. 'constraints') are dynamically generated by the ev olution equation for t not-equal 0. On the other hand integral dx inte gral dy u(t, x, y) with prescribed order of integrations is not necess arily equal to zero and gives a non-trivial integral of motion.