Geometric-optics for nonlinear concentrating waves in focusing and non-focusing two geometries

With the methods used in Bahouri and Gerard (1999) and Gallagher and Gerard (2001), we prove that, in absence of focus, nonlinear geometrical optics of the critical wave equation with variable coefficients, is reduced to linear geometrical optics associated to wave operators for the critical wave equation with coefficients fixed on concentrating points. On the sphere $S^d$, we prove that passing through a focus is generated by a modified scattering operator.