Random data Cauchy problem for the quadratic nonlinear Schrödinger equation without gauge invariance

2020 ◽  
Author(s):  
Mamoru Okamoto
Keyword(s):  
Cauchy Problem ◽  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Gauge Invariance ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Random Data ◽  
Nonlinear Schrödinger

scholarly journals Phenomena of Blowup and Global Existence of the Solution to a Nonlinear Schrödinger Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-14
Author(s):  
Xiaowei An ◽  
Desheng Li ◽  
Xianfa Song
Keyword(s):  
Cauchy Problem ◽  
Global Existence ◽  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Sufficient Conditions ◽  
Nonlinear Schrodinger Equation ◽  
Nonlinear Schrödinger ◽  
Complex Valued ◽  
Sharp Thresholds

We consider the following Cauchy problem:-iut=Δu-V(x)u+f(x,|u|2)u+(W(x)⋆|u|2)u,x∈ℝN,t>0,u(x,0)=u0(x),x∈ℝN,whereV(x)andW(x)are real-valued potentials andV(x)≥0andW(x)is even,f(x,|u|2)is measurable inxand continuous in|u|2, andu0(x)is a complex-valued function ofx. We obtain some sufficient conditions and establish two sharp thresholds for the blowup and global existence of the solution to the problem.

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