On the Schrödinger equations with isotropic and anisotropic fourth-order dispersion

This paper deals with the Cauchy problem associated to the nonlinear fourth-order Schrödinger equation with isotropic and anisotropic mixed dispersion. This model is given by the equation $i\partial _{t}u+εΔu+δA u+λ|u|^αu=0,$ $x\in \mathbb{R}^{n},$ $t\in \mathbb{R},$ where $A$ represents either the operator $Δ^2$ (isotropic dispersion) or $\sum_{i=1}^d\partial_{x_ix_ix_ix_i},\ 1\leq d<n$ (anisotropic dispersion), and $α, ε, λ$ are given real parameters. We obtain local and global well-posedness results in spaces of initial data with low regularity, such as weak-$L^p$ spaces. Our analysis also includes the biharmonic and anisotropic biharmonic equation $(ε=0)$ for which, the existence of self-similar solutions is obtained as consequence of his scaling invariance. In a second part, we investigate the vanishing second order dispersion limit in the framework of weak-$L^p$ spaces. We also analyze the convergence of the solutions for the nonlinear fourth-order Schrödinger equation $i\partial _{t}u+εΔu+δΔ^2 u+λ|u|^αu=0$, as $ε$ goes to zero, in $H^2$-norm, to the solutions of the corresponding biharmonic equation $i\partial _{t}u+δΔ^2 u+λ|u|^αu=0$.

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