The Lax pseudo-differential operator plays a key role in studying the generalset of KP equations, although it is normally treated in a formal way, withoutworrying about a complete characterization of its mathematical properties. Theaim of the present paper is therefore to investigate the ellipticity condition.For this purpose, after a careful evaluation of the kernel with the associatedsymbol, the majorization ensuring ellipticity is studied in detail. This leadsto non-trivial restrictions on the admissible set of potentials in the Laxoperator. When their time evolution is also considered, the ellipticityconditions turn out to involve derivatives of the logarithm of thetau-function.
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