Ellipticity Conditions for the Lax Operator of the KP Equations

Details Ellipticity Conditions for the Lax Operator of the KP Equations Ellipticity Conditions for the Lax Operator of the KP Equations

2000-04-07

arxiv.org/abs/nlin/0004011v1

J.Phys.A33:7123-7135,2000

Nonlinear Sciences

Exactly Solvable and Integrable Systems

21 pages, plain Tex

Scientific paper

10.1088/0305-4470/33/40/309

The Lax pseudo-differential operator plays a key role in studying the general set of KP equations, although it is normally treated in a formal way, without worrying about a complete characterization of its mathematical properties. The aim of the present paper is therefore to investigate the ellipticity condition. For this purpose, after a careful evaluation of the kernel with the associated symbol, the majorization ensuring ellipticity is studied in detail. This leads to non-trivial restrictions on the admissible set of potentials in the Lax operator. When their time evolution is also considered, the ellipticity conditions turn out to involve derivatives of the logarithm of the tau-function.

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