Solutions of the Kpi Equation with Smooth Initial Data

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 literature, as $\int\!dx\,u(0,x,y)=0$ 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, $\partial_t u(t,x,y)$ has different left and right limits at the initial time $t=0$. The conditions of the type $\int\!dx\,u(t,x,y)=0$, $\int\!dx\,xu_y(t,x,y)=0$ and so on (first, second, etc. `constraints') are dynamically generated by the evolution equation for $t\not=0$. On the other side $\int\!dx\!\!\int\!dy\,u(t,x,y)$ with prescribed order of integrations is not necessarily equal to zero and gives a nontrivial integral of motion.