Quasinormal modes in the field of a dyon-like dilatonic black hole corresponding to $A_1+A_1$ Toda chain

We explore the quasinormal modes of a massless test scalar field in the gravitational background of a non-extremal dilatonic dyonic black hole. The dyon-like black hole solution is considered within a four-dimensional gravitational model that includes two scalar fields and two 2-forms. It is governed by two dilatonic coupling vectors ${\vec{\lambda }}_s$ (with $s=1,2$) that satisfy ${\vec{\lambda }}_1={\vec{\lambda }}_2=\vec{\lambda }$ and ${\vec{\lambda }}^2=1\mathrm{/}2$. We derive and examine the quasinormal modes of a massless scalar test field in the eikonal limit. These modes are governed by the solution parameters $\mu >0$, $P_1>0$, and $P_2>0$. In the two limiting scenarios – namely $P_1=P_2=0$ and $P_1=P_2=P>0$ – our results reduce exactly to the known (eikonal) quasinormal spectra for the Schwarzschild and Reissner-Nordström black holes, respectively. Numerical higher-order WKB analysis of quasinormal frequencies for two lower levels (for various values of parameters) is also presented.