On the spectral stability of periodic traveling waves for the critical Korteweg-de Vries and Gardner equations

In this paper, we determine spectral stability results of periodic waves for the critical Korteweg-de Vries and Gardner equations. For first equation, show that both positive zero mean traveling wave solutions possess a threshold value which may provides us rupture in stability. Concerning second establish existence using Galilean transformation on cnoidal solution modified equation equations, values are same. The main advantage presented our paper concerns solving some auxiliary initial problems to obtain