2011 Fiscal Year Final Research Report
Analysis on Nonlinear Schrodinger Equations with nonlocal nonlinearity growing at the spatial infinity
| Project/Area Number |
22840039
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| Research Category |
Grant-in-Aid for Research Activity Start-up
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| Allocation Type | Single-year Grants |
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| Research Field |
Global analysis
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| Research Institution | Gakushuin University |
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Principal Investigator |
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| Project Period (FY) |
2010 – 2011
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| Keywords | 函数方程式 / シュレディンガー方程式 / 非線型シュレディンガー方程式 / シュレディンガー・ポアッソン方程式系 / ニュートンポテンシャル / 励起状態解 |
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| Research Abstract |
Two dimensional version of Schrodinger-Poisson system, which is a model equation for semiconductor devices, has a nonlinear potential growing at the spatial infinity due to the fact that the Newtonian kernel, which is a fundamental solution of the Poisson equation in two dimensions, becomes a logarithmic function. Our research concerns Schrodinger equations with a nonlocal nonlinearity given by more general kernels growing at the spatial infinity. Although a derivation is rather simple, this type of equations requires quite different mathematical treatment. We first establish a way to treat this class of nonlinear Schrodinger equations rigorously by introducing a novel transform of equation which is based on conservative quantities of Schrodinger equations. In particular, it turns out that nonlocal nonlinearities of this type contain an effect like a linear potential. Based on this fact, we are able to prove time-global well-posedness result in an energy class. Moreover, when integral kernel is a quadratic function, it turns out that the solutions are written explicitly nevertheless the equation is fully nonlinear. The behavior of this explicit solution, which can be analyzed completely, helps up to understand the effects of nonlinearities which we concern. This example also reveals that there exists a stable excited state.
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