In this talk, we shall consider generalizations of the work of Bender and co-workers to derive new partially-integrable hierarchies of various PT -symmetric, nonlinear partial differential equations. The possible integrable members are identified employing the Painleve Test, a necessary but not sufficient integrability condition, and are indexed by the integer n, corresponding to the negative of the order of the dominant pole in the singular part of the Painleve expansion for the solution.
For the PT -symmetric Korteweg-de Vries (KdV) family, as with some other hierarchies, the first or n = 1 equation fails the test, the n = 2 member corresponds to the regular KdV equation, while the remainder form an entirely new, possibly integrable, hierarchy. Typical integrability properties of the n = 3 and n = 4 members, including Backlund Transformations, a 'near-Lax
Pair', and analytic solutions are derived. The solutions prove to be algebraic in form, and the extended homogeneous balance technique appears to be the most efficient in exposing the near-Lax Pair.
If time permits, analogous results will be discussed for the PT symmetric (2+1) Burgers' and Kadomtsev-Petviashvili equations.
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