Group Analysis of Differential Equations and Integrable Systems − 2018


Olena Vaneeva (Institute of Mathematics of NAS of Ukraine, Kyiv, Ukraine)

Classification of Lie and nonclassical reduction operators of a class of generalized Newell-Whitehead-Segel equations

Abstract:
The reduction method is an efficient tool for seeking exact solutions of nonlinear partial differential equations as the general theory of integration of such equations does not exist. Among the most known reduction techniques are prominent Lie reduction method that originates from works by S. Lie and the nonclassical reduction method suggested by G. Bluman and J. Cole (the well-developed theory of the nonclassical reduction method can be found in [Boyko V.M., Kunzinger M., Popovych R.O., Singular reduction modules of differential equations, J. Math. Phys. 57 (2016), 101503]). Therefore an important problem arises: to classify reduction operators for those classes of partial differential equations that are of interest for applications.
We classify Lie reduction operators and regular nonclassical reduction operators for a class of variable coefficient Newell-Whitehead-Segel equations. The set of admissible transformations of the class is described exhaustively. The criterion of reducibility of variable coefficient Newell-Whitehead-Segel equations to their constant coefficient counterparts is derived. We also present a list of exact solutions for variable coefficient equations from the class.

The talk is based on joint work with V. Boyko, C. Sophocleous and A. Zhalij.