Solutions of the KPI equation with smooth initial data

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Abstract
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 framework of the spectral transform theory and it is shown that u(t,x,y) satisfies a special evolution version of the KPI equation and that, in general, delta tu(t,x,y) has different left and right limits at the initial time t=0. The conditions of the type integral dx u(t,x,y)=0, integral dx xuy(t,x,y)=0 and so on (first, second, etc. 'constraints') are dynamically generated by the evolution equation for t not=0. On the other hand integral dx integral dy u(t,x,y) with prescribed order of integrations is not necessarily equal to zero and gives a non-trivial integral of motion.
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