Probabilistic Approximation of the Solution of the Cauchy Problem for the Higher-Order Schrödinger Equation

2021 ◽  
Vol 65 (4) ◽  
pp. 558-569
Author(s):  
M. V. Platonova ◽  
S. V. Tsykin
Keyword(s):  
Cauchy Problem ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Higher Order ◽  
Probabilistic Approximation ◽  
The Cauchy Problem

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.

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