Dear Daniel, thank you for your response!

Firstly, the most interesting area for me is 0<y<4, 0<x<5.

Secondly, there is a physical reason to expect real-valued zeros because this equation describes waves in a dispersive media WITHOUT any absorption, damping. X is a wave vector, y is frequency. Moreover, I know asymptotic behavior of the solution of this equation when parameters a->inb, b is finite and when b->inf a is finite. These solutions are described by other equations which are simply solved in Mathematica (using ContourPlot).

But I can't solve this equation using ContourPlot (in some cases ContourPlot gives a solution, but it is wrong). Also, when I use FindRoot I receive very strange (sometimes even unphysical) solutions. As I know after looking through literature, it is possible to find solutions by Nelder-Mead minimization procedure "While no exact solution exists for the dispersion relations... in the real frequency domain...To obtain the complex wave vectors, numerical solution of ... was accomplished through implementation of a two-dimensional unconstrained Nelder-Mead minimization algorithm". In that article authors have considered absorptive media, but I firstly would like to consider non absorptive media, that's why I expect real-valued zeros.