%0 Unpublished work
%T A new class of Traveling Solitons for cubic Fractional Nonlinear Schrodinger equations
%+ Departement of Mathematics [Austin]
%+ Institut de Mathématiques de Marseille (I2M)
%A Hong, Younghun
%A Sire, Yannick
%Z Y.H. would like to thank IHE ́S for their hospitality and support while he visited in the summer of 2014. Y.S. would like to thank the hospitality of the Department of Mathematics at University of Texas at Austin where part of the work was initiated.
%8 2015-09-13
%D 2015
%Z 1501.01415
%Z Mathematics [math]/Analysis of PDEs [math.AP]Preprints, Working Papers, ...
%X We consider the one-dimensional cubic fractional nonlinear Schr\"odinger equation $$i\partial_tu-(-\Delta)^\sigma u+|u|^{2}u=0,$$ where $\sigma \in (\frac12,1)$ and the operator $(-\Delta)^\sigma$ is the fractional Laplacian of symbol $|\xi|^{2\sigma}$. Despite of lack of any Galilean-type invariance, we construct a new class of traveling soliton solutions of the form $$u(t,x)=e^{-it(|k|^{2\sigma}-\omega^{2\sigma})}Q_{\omega,k}(x-2t\sigma|k|^{2\sigma-2}k),\quad k\in\mathbb{R},\ \omega>0$$ by a rather involved variational argument.
%G English
%L hal-01341878
%U https://hal.science/hal-01341878
%~ CNRS
%~ UNIV-AMU
%~ EC-MARSEILLE
%~ I2M
%~ I2M-2014-
%~ TDS-MACS
%~ ANR