Approximate solutions of vector fields and an application to Denjoy–Carleman regularity of solutions of a nonlinear PDE

In this paper we study microlocal regularity of a (Formula presented.) -solution u of the equation (Formula presented.) where (Formula presented.) is ultradifferentiable in the variables (Formula presented.) and holomorphic in the variables (Formula presented.). We proved that if (Formula presented.) is a regular Denjoy–Carleman class (including the quasianalytic case) then: (Formula presented.) where (Formula presented.) is the Denjoy–Carleman wave-front set of u and (Formula presented.) is the characteristic set of the linearized operator (Formula presented.) : (Formula presented.)