Partially Integrable Pj-Symmetric Hierarchies Of The Kdv And Burgers' Equations In (1+1) And (2+1)

Keywords

Extended PT symmetric hierarchies; KdV and Burgers' equations; Partially integrable systems

Abstract

In this paper, we generalize the work of Bender and co-workers to derive new partially-integrable hierarchies of various PJ-symmetric, nonlinear partial differential equations. The possible integrable members are identified employing the Painlevé Test, a necessary but not sufficient, integrabili- ty 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 PJ-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 PJ-symmetric Burgers' equation fails the Painlevé Test for its n = 2 case, but special solutions are nonetheless obtained. Also, a PJ- 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.

Publication Date

1-1-2017

Publication Title

Discontinuity, Nonlinearity, and Complexity

Volume

6

Issue

2

Number of Pages

113-146

Document Type

Article

Personal Identifier

scopus

DOI Link

https://doi.org/10.5890/DNC.2017.06.002

Socpus ID

85020812783 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/85020812783

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