ON THE SCHRODINGER EQUATIONS WITH ISOTROPIC AND ANISOTROPIC FOURTH-ORDER DISPERSION

This article concerns the Cauchy problem associated with the non-linear fourth-order Schrodinger equation with isotropic and anisotropic mixed dispersion. This model is given by the equation i partial derivative(t)u + epsilon Delta u + delta Au + lambda vertical bar u vertical bar(alpha)u - 0, x is an element of R-n, t is an element of R, where A is either the operator Delta(2) (isotropic dispersion) or Sigma(d)(i)=1 partial derivative x(i) x(i) x(i) x(i), 1 <= d < n (anisotropic dispersion), and alpha, epsilon, lambda are real parameters. We obtain local and global well-posedness results in spaces of initial data with low regularity, based on weak-L-p spaces. Our analysis also includes the biharmonic and anisotropic biharmonic equation (epsilon - 0); in this case, we obtain the existence of self-similar solutions because of their scaling invariance property. In a second part, we analyze the convergence of solutions for the nonlinear fourth-order Schrodinger equation i partial derivative(t)u + epsilon Delta u + delta Delta(2)u + lambda vertical bar u vertical bar(alpha)u = 0, x is an element of R-n, t is an element of R, as epsilon approches zero, in the H-2 -norm, to the solutions of the corresponding biharmonic equation i partial derivative(t)u + delta Delta(2)u + lambda vertical bar u vertical bar(alpha)u = 0.