scholarly journals Local existence, global existence, and scattering for the nonlinear Schrödinger equation

2017 ◽  
Vol 19 (02) ◽  
pp. 1650038 ◽  
Author(s):  
Thierry Cazenave ◽  
Ivan Naumkin

In this paper, we construct for every [Formula: see text] and [Formula: see text] a class of initial values [Formula: see text] for which there exists a local solution of the nonlinear Schrödinger equation [Formula: see text] on [Formula: see text] with the initial condition [Formula: see text]. Moreover, we construct for every [Formula: see text] a class of (arbitrarily large) initial values for which there exists a global solution that scatters as [Formula: see text].

2013 ◽  
Vol 2013 ◽  
pp. 1-14
Author(s):  
Xiaowei An ◽  
Desheng Li ◽  
Xianfa Song

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.


1995 ◽  
Vol 09 (17) ◽  
pp. 1045-1052 ◽  
Author(s):  
Y. YANG ◽  
Y. TAN ◽  
W.Y. ZHANG ◽  
C.Y. ZHENG

The nonlinear Schrödinger equation with spatially periodic boundary conditions is numerically solved by means of the spectrum method. It is found that with the initial condition carefully chosen, the phase recurrence just appears when the amplitudes have the nineteenth recurrences to the initial condition. This phenomenon is called as the phase super-recurrence. Using a simple perturbation model, the amplitude recurrence period Ta and the phase change ∆φa in the period Ta are estimated, and a good agreement between this estimation and the numerical results of the nonlinear Schrödinger equation is shown.


2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Chenglin Wang ◽  
Jian Zhang

<p style='text-indent:20px;'>In this paper, we study the nonlinear Schrödinger equation with a partial confinement. By applying the cross-constrained variational arguments and invariant manifolds of the evolution flow, the sharp condition for global existence and blowup of the solution is derived.</p>


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