Partially Integrable ℘ T -Symmetric Hierarchies of the KdV and Burgers’ Equations in (1+1) and (2+1) - Pecora, Keri; Choudhury, S. Roy (Discontinuity, Nonlinearity, and Complexity, 2026)
In this paper, we generalize the work of Bender and co-workers to derive new partially-integrable hierarchies of various ℘ T -symmetric, nonlinear partial differential equations. The possible integrable members are identified employing the Painlevé 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 Painlevé expansion for the solution. For the ℘ T -symmetric Korteweg-de Vries (KdV) equation, 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. Bäcklund Transformations and analytic solutions of the n = 3 and n = 4 members are derived. The solutions, or solitary waves, prove to be algebraic in form. The ℘ T -symmetric Burgers’ equation fails the Painlevé Test for its n = 2 case, but special solutions are nonetheless obtained. Also, a ℘ T - Symmetric hierarchy of the (2+1) Burgers’ equation is analyzed. The Painlevé Test and invariant Painlevé analysis in (2+1) dimensions are utilized, and BTs and special solutions are found for those cases that pass the Painlevé Test.