scholarly journals Gevrey regularizing effect for a nonlinear Schrödinger equation in one space dimension

1998 ◽  
Vol 50 (4) ◽  
pp. 1015-1026
Author(s):  
Kazuo TANIGUCHI
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Space Dimension ◽  
Nonlinear Schrodinger Equation ◽  
Nonlinear Schrödinger ◽  
One Space Dimension

Large Time Behavior for the Cubic Nonlinear Schrödinger Equation

2002 ◽  
Vol 54 (5) ◽  
pp. 1065-1085 ◽  
Author(s):  
Nakao Hayashi ◽  
Pavel I. Naumkin
Keyword(s):  
Cauchy Problem ◽  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Large Time ◽  
Space Dimension ◽  
Nonlinear Schrodinger Equation ◽  
The Cauchy Problem ◽  
Cubic Nonlinear Schrödinger Equation ◽  
One Space Dimension

AbstractWe consider the Cauchy problem for the cubic nonlinear Schrödinger equation in one space dimensionCubic type nonlinearities in one space dimension heuristically appear to be critical for large time. We study the global existence and large time asymptotic behavior of solutions to the Cauchy problem (1). We prove that if the initial data are small and such that for some n ∈ Z, and , then the solution has an additional logarithmic timedecay in the short range region . In the far region the asymptotics have a quasilinear character.

scholarly journals On a nonlinear Schrödinger equation for nucleons in one space dimension

2021 ◽  
Vol 55 (2) ◽  
pp. 409-427
Author(s):  
Christian Klein ◽  
Simona Rota Nodari
Keyword(s):  
Schrödinger Equation ◽  
Time Evolution ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Space Dimension ◽  
Ground States ◽  
Nonlinear Schrodinger Equation ◽  
Nonlinear Schrödinger ◽  
Long Time ◽  
Localized Initial Data

We study a 1D nonlinear Schrödinger equation appearing in the description of a particle inside an atomic nucleus. For various nonlinearities, the ground states are discussed and given in explicit form. Their stability is studied numerically via the time evolution of perturbed ground states. In the time evolution of general localized initial data, they are shown to appear in the long time behaviour of certain cases.

scholarly journals CRITICAL NONLINEAR SCHRÖDINGER EQUATIONS WITH AND WITHOUT HARMONIC POTENTIAL

2002 ◽  
Vol 12 (10) ◽  
pp. 1513-1523 ◽  
Author(s):  
RÉMI CARLES
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Space Dimension ◽  
Einstein Condensate ◽  
Nonlinear Schrodinger Equation ◽  
Harmonic Potential ◽  
Nonlinear Schrödinger ◽  
Change Of Variables ◽  
Bose Einstein Condensate

We use a change of variables that turns the critical nonlinear Schrödinger equation into the critical nonlinear Schrödinger equation with isotropic harmonic potential, in any space dimension. This change of variables is isometric on L2, and bijective on some time intervals. Using the known results for the critical nonlinear Schrödinger equation, this provides information for the properties of Bose–Einstein condensate in space dimension one and two. We discuss in particular the wave collapse phenomenon.

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