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
Keyword(s):  
Global Existence ◽  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Local Existence ◽  
Nonlinear Schrodinger Equation ◽  
Local Solution ◽  
Initial Condition ◽  
Nonlinear Schrödinger ◽  
Initial Values

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].

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.

SUPER-RECURRENCE PHENOMENON RELEVANT TO THE PHASE IN THE NONLINEAR SCHRÖDINGER EQUATION

1995 ◽  
Vol 09 (17) ◽  
pp. 1045-1052 ◽  
Author(s):  
Y. YANG ◽  
Y. TAN ◽  
W.Y. ZHANG ◽  
C.Y. ZHENG
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Periodic Boundary Conditions ◽  
Nonlinear Schrodinger Equation ◽  
Initial Condition ◽  
Nonlinear Schrödinger ◽  
Recurrence Period ◽  
Spatially Periodic ◽  
Good Agreement

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.

scholarly journals Cross-constrained variational method and nonlinear Schrödinger equation with partial confinement

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Chenglin Wang ◽  
Jian Zhang
Keyword(s):  
Global Existence ◽  
Schrödinger Equation ◽  
Variational Method ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Invariant Manifolds ◽  
Nonlinear Schrodinger Equation ◽  
Sharp Condition ◽  
Nonlinear Schrödinger ◽  
Partial Confinement

<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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