Nonlinear fractional Schrödinger equations in one dimension

Abstract

We consider the question of global existence of small, smooth, and localized solutions of a certain fractional semilinear cubic NLS in one dimension, $$i{\partial }_{t}u-\Lambda u=c_0{|u|}^{2}u+c_1{u}^3+c_{2}u{\bar{u}}^2+c_3{\bar{u}}^3,\Lambda =\Lambda ({\partial }_x)={|{\partial }_x|}^{(}1/2)$$, where $c_0\in \mathbb{R}$ and $c_1,c_2,c_3\in \mathbb{C}$. This model is motivated by the two-dimensional water waves equations, which have a somewhat similar structure in the Eulerian formulation, in the case of irrotational flows. We show that one cannot expect linear scattering, even in this simplified model. More precisely, we identify a suitable nonlinear logarithmic correction, and prove global existence and modified scattering of solutions.

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• Numerical study of fractional Nonlinear Schrödinger equations
• Sharp Criteria of Scattering for the Fractional NLS
• On the orbital stability of fractional Schrödinger equations
• Existence, non-existence and blow-up behavior of minimizers for the mass-critical fractional nonlinear Schrödinger equations with periodic potentials
• Bound state solutions for non-autonomous fractional Schrödinger-Poisson equations with critical exponent
• The cubic nonlinear fractional Schrödinger equation on the half-line
• Global well-posedness of cubic fractional Schrödinger equations in one dimension