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Abstract
We consider the cubicquintic nonlinear Schrödinger equation:
i∂_t u = −∆u − u^2u + u^4u.
In the first part of the paper, we analyze the oneparameter family of groundstate solitons associated to this equation with particular attention to the shape of the associated mass/energy curve. Additionally, we are able to characterize the kernel of the linearized operator about such solitons and to demonstrate that they occur as optimizers for a oneparameter family of inequalities of Gagliardo–Nirenberg type. Building on this work, in the latter part of the paper we prove that scattering holds for solutions belonging to the region R of the mass/energy plane where the virial is positive. We show this region is partially bounded by solitons but also by rescalings of solitons (which are not soliton solutions in their own right). The discovery of rescaled solitons in this context is new and highlights an unexpected limitation of any virialbased methodology.
i∂_t u = −∆u − u^2u + u^4u.
In the first part of the paper, we analyze the oneparameter family of groundstate solitons associated to this equation with particular attention to the shape of the associated mass/energy curve. Additionally, we are able to characterize the kernel of the linearized operator about such solitons and to demonstrate that they occur as optimizers for a oneparameter family of inequalities of Gagliardo–Nirenberg type. Building on this work, in the latter part of the paper we prove that scattering holds for solutions belonging to the region R of the mass/energy plane where the virial is positive. We show this region is partially bounded by solitons but also by rescalings of solitons (which are not soliton solutions in their own right). The discovery of rescaled solitons in this context is new and highlights an unexpected limitation of any virialbased methodology.
Original language  English 

Pages (fromto)  469548 
Number of pages  80 
Journal  Archive for Rational Mechanics and Analysis 
Volume  225 
Issue number  1 
Early online date  24 Mar 2017 
DOIs  
Publication status  Published  Jul 2017 
Keywords
 Energycritical NLS
 cubicquintic NLS
 scattering
 soliton
 Gagliardo–Nirenberg inequality
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 1 Finished

ProbDynDispEq  Probabilistic and Dynamical Study of Nonlinear Dispersive Equations
1/03/15 → 29/02/20
Project: Research