论文标题
非线性schrödinger方程的系数测定。
Coefficient Determination for Non-Linear Schrödinger Equations on manifolds
论文作者
论文摘要
我们考虑恢复未知系数$β（t，x）$和$ v（t，x）$出现在时间依赖时间的非线性schrödinger方程$（\ mathrm {i} \ partial_t +δ +δ +Δ +v）u +βu^2 = 0 in $（n y in $ n y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y eucean n o eucean， 几何学。我们考虑$ m $边界的$ω\子集M $的测量值和源至解决方案的映射$ l_ {β，v} $映射$ω\ times（0，t）$支撑的源$ f $的限制，以限制解决方案$ u $ in $ $ u $ in $ω\ times（0，t）$。我们表明，对于上述非线性schrödinger方程以及Gross-Pitaevskii方程，地图$ l_ {β，v} $唯一地确定了非线性的时间相关潜力和数字系数，并用立方体的非线性项$β| u | U | U | U | u $ $，unum of Quantics of Quantics of Quantics of Quantics of Quantics of Quantics of Quantic of Quintic of。
We consider an inverse problem of recovering the unknown coefficients $β(t,x)$ and $V(t,x)$ appearing in a time-dependent nonlinear Schrödinger equation $ (\mathrm{i} \partial_t +Δ+V)u + βu^2=0$ in $(0,T) \times M$, on Euclidean geometry as well as on Riemannian geometry. We consider measurements in $Ω\subset M$ that is a neighborhood of the boundary of $M$ and the source-to-solution map $ L_{β, V}$ that maps a source $f$ supported in $ Ω\times (0,T) $ to the restriction of the solution $u$ in $ Ω\times (0,T) $. We show that the map $L_{β, V}$ uniquely determines the time-dependent potential and the coefficient of the non-linearity, for the above non-linear Schrödinger equation and for the Gross-Pitaevskii equation, with a cubic non-linear term $β|u|^2 \, u$, that is encountered in quantum physics.
