Stability of periodic waves governed by the modified Kawahara equation (Q1904605)

We investigate the stability of spatially periodic solutions of the modified Kawahara equation \[ {\partial a\over \partial t}= \sigma_3 {\partial^3 a\over \partial x^3}+ \kappa_1 {\partial(a^2)\over \partial x}+ \sigma_2(R- R_c) {\partial^2 a\over \partial x^2}+ \sigma_4 {\partial^4 a\over \partial x^4}+ \kappa_2 {\partial^2(a^2)\over \partial x^2} \] in order to select the stable solutions which can be attractors of the system. We consider the time evolution of small disturbances on the background of a periodic wave. The general formula describing the stability of periodic solutions is obtained. The results of the numerical calculation of the stability region (``Busse balloon'') are presented. The details of the numerical procedure are described in the appendices.