Splitting of solutions of weakly nonlinear singularly perturbed equations under regular degeneration

We consider a weakly nonlinear singularly perturbed equation in complex domains. The problem is posed about the possibility of splitting the equation into several components. By introducing new unknown functions, a system of two equations is obtained. Next, the asymptotic behavior of solutions of the resulting equations in complex domains is studied. It has been proven that the solution to each of these equations is dominant in certain parts of the areas under consideration. The solution of one of these equations determines the boundary lines and regions, and the solution of the other system determines the regular region.