TY - JOUR
T1 - Nonlinear fractional Schrödinger equations in one dimension
AU - Ionescu, Alexandru D.
AU - Pusateri, Fabio
N1 - Funding Information:
The first author was partially supported by a Packard Fellowship and NSF grant DMS-1065710 . The second author was partially supported by a Simons Fellowship .
PY - 2014/1/1
Y1 - 2014/1/1
N2 - We consider the question of global existence of small, smooth, and localized solutions of a certain fractional semilinear cubic NLS in one dimension,. i∂tu-Λu=c0|u|2u+c1u3+c2uu-2+c3-3,Λ=-(∂x)=|∂x|12, where c0∈R and c1,c2,c3∈C. This model is motivated by the two-dimensional water wave equation, which has 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.
AB - We consider the question of global existence of small, smooth, and localized solutions of a certain fractional semilinear cubic NLS in one dimension,. i∂tu-Λu=c0|u|2u+c1u3+c2uu-2+c3-3,Λ=-(∂x)=|∂x|12, where c0∈R and c1,c2,c3∈C. This model is motivated by the two-dimensional water wave equation, which has 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.
KW - Global regularity
KW - Modified scattering
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U2 - 10.1016/j.jfa.2013.08.027
DO - 10.1016/j.jfa.2013.08.027
M3 - Article
AN - SCOPUS:84886952212
SN - 0022-1236
VL - 266
SP - 139
EP - 176
JO - Journal of Functional Analysis
JF - Journal of Functional Analysis
IS - 1
ER -