Group Analysis of Differential Equations and Integrable Systems − 2018

Roman O. Popovych (Faculty of Mathematics, University of Vienna, Austria & Institute of Mathematics of NAS of Ukraine, Kyiv, Ukraine)

Darboux transformations of linear evolution equations

We revisit the framework of Darboux transformations of (1+1)-dimensional linear evolution equations [1,2] from the global point of view. For this purpose, we begin with the study of solutions of ordinary differential equations with the independent variable $x$ and the variable $t$ playing the role of a parameter on open sets of the $(t,x)$-plane. We show that the existence of global fundamental sets of solutions and tuples of solutions with nonzero Wronskians with respect to~$x$ for such an equation depends on the connectedness and the $x$-simplicity degree of the domain where the equation is considered. Then various results on factorization of Darboux operators are proved within the global settings and applied to extended symmetry analysis of linear evolution equations of arbitrary order greater than one, which properly generalizes results of [2-5] to potential conservation laws, potential generalized symmetries and potential reduction operators that involve linear potentials. In particular, we show that any differential substitution relating linear evolution equations $\mathcal L$ and $\tilde{\mathcal L}$ can be represented as a composition of a Darboux transformations with respect to a tuple of solutions of $\mathcal L$, of multiplication by a smooth functions and of linear superposition with a solution of $\tilde{\mathcal L}$.

The talk is based on joint work with Vyacheslav Boyko and Michael Kunzinger.

[1] Matveev V.B. and Salle M.A., Darboux Transformations and Solitons, Springer-Verlag, Berlin, 1991.
[2] Popovych R.O., Kunzinger M. and Ivanova N.M., Conservation laws and potential symmetries of linear parabolic equations, Acta. Appl. Math. 100 (2008), 113-185, arXiv:0706.0443.
[3] Popovych R.O. and Sergyeyev A., Conservation laws and normal forms of evolution equations, Phys. Lett. A 374 (2010), 2210-2217, arXiv:1003.1648.
[4] Boyko V.M. and Popovych R.O., Simplest potential conservation laws of linear evolution equations, Proceedings of 5th Workshop "Group Analysis of Differential Equations and Integrable Systems" (June 6-10, 2010, Protaras, Cyprus), 2011, 28-39, arXiv:1008.4851.
[5] Bihlo A. and Popovych R.O., Group classification of linear evolution equations, J. Math. Anal. Appl. 448 (2017), 982-1005, arXiv:1605.09251.