We consider the quasilinear Cauchy problem (CP) P(t,x,u(t,x),D_t,D_x)u(t,x) = f(t,x), with (t,x)∈[0,T]xR, and initial condition u(0,x) = g(x), x∈R, where P(t,x,u,D_t,D_x) = D_t + a_3(t)D_x^3 + a_2(t,x,u)D_x^2 + a_1(t,x,u)D_x + a_0(t,x,u), a_j(t,x,w), (0≤j≤2), are continuous functions of time t, projective Gevrey regular with respect to the space variable x and holomorphic in the complex parameter w. The coefficient a_3(t) is assumed to be a real-valued continuous function which never vanishes. In this talk we shall discuss how to apply the Nash-Moser inversion theorem in order to obtain local in time well-posedness in projective Gevrey classes for the Cauchy problem (CP).