Global Well-posedness for the Defocusing, Quintic Nonlinear Schrödinger Equation in One Dimension for Low Regularity Data

2011 ◽  
Vol 2012 (4) ◽  
pp. 870-893 ◽  
Author(s):  
Benjamin G. Dodson
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
One Dimension ◽  
Well Posedness ◽  
Low Regularity ◽  
Nonlinear Schrödinger ◽  
Quintic Nonlinear Schrödinger Equation

scholarly journals Scattering theory below energy space for two-dimensional nonlinear Schrödinger equation

2015 ◽  
Vol 17 (06) ◽  
pp. 1450052
Author(s):  
Changxing Miao ◽  
Jiqiang Zheng
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Well Posedness ◽  
Scale Invariant ◽  
Low Regularity ◽  
Nonlinear Schrödinger ◽  
Low Regularity Solutions ◽  
Appl Math

The purpose of this paper is to illustrate the I-method by studying low-regularity solutions of the nonlinear Schrödinger equation in two space dimensions. By applying this method, together with the interaction Morawetz estimate, (see [J. Colliander, M. Grillakis and N. Tzirakis, Tensor products and correlation estimates with applications to nonlinear Schrödinger equations, Commun. Pure Appl. Math.62 (2009) 920–968; F. Planchon and L. Vega, Bilinear virial identities and applications, Ann. Sci. École Norm. Sup.42 (2009) 261–290]), we establish global well-posedness and scattering for low-regularity solutions of the equation iut+ Δu = λ1|u|p1u + λ2|u|p2u under certain assumptions on parameters. This is the first result of this type for an equation which is not scale-invariant. In the first step, we establish global well-posedness and scattering for low regularity solutions of the equation iut+ Δu = |u|pu, for a suitable range of the exponent p extending the result of Colliander, Grillakis and Tzirakis [Commun. Pure Appl. Math.62 (2009) 920–968].

scholarly journals On the Existence of Bright Solitons in Cubic-Quintic Nonlinear Schrödinger Equation with Inhomogeneous Nonlinearity

2008 ◽  
Vol 2008 ◽  
pp. 1-14 ◽  
Author(s):  
Juan Belmonte-Beitia
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Bifurcation Theory ◽  
Schrodinger Equation ◽  
Chemical Potential ◽  
Soliton Solutions ◽  
Nonlinear Schrodinger Equation ◽  
Nonlinear Schrödinger ◽  
Bose Einstein ◽  
Quintic Nonlinear Schrödinger Equation

We give a proof of the existence of stationary bright soliton solutions of the cubic-quintic nonlinear Schrödinger equation with inhomogeneous nonlinearity. By using bifurcation theory, we prove that the norm of the positive solution goes to zero as the parameterλ, called chemical potential in the Bose-Einstein condensates' literature, tends to zero. Moreover, we solve the time-dependent cubic-quintic nonlinear Schrödinger equation with inhomogeneous nonlinearities by using a numerical method.

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