Critical exponents and collapse of nonlinear Schrödinger equations with anisotropic fourth-order dispersion

We calculate the critical exponent of nonlinear Schrödinger (NLS) equations with anisotropic negative fourth-order dispersion using an anisotropic Gagliardo–Nirenberg inequality. We also prove global existence, and in some cases uniqueness, for subcritical solutions and for critical solutions with small L2 norm, without making use of Strichartz-type estimates for the linear operator. At exponents equal to or above critical, the blowup profile is anisotropic. Our results imply, in particular, that negative fourth-order temporal dispersion arrests spatio-temporal collapse in Kerr media with anomalous time-dispersion in one transverse dimension but not in two transverse dimensions. We also show that a small negative anisotropic fourth-order dispersion stabilizes the (otherwise unstable) waveguide solutions of the two-dimensional critical NLS. Mathematics Subject Classification: 35Q55