标题： 两个函数逐点乘积的短时傅里叶变换及其在非线性薛定谔方程中的应用 显示英文标题

标题： Short-time Fourier transform of the pointwise product of two functions with application to the nonlinear Schrödinger equation

摘要： 我们证明了两个函数$f$和$h$的逐点乘积的短时傅里叶变换可以写成$f$和$h$的短时傅里叶变换的适当乘积。 同样的结果随后被证明对于 Wigner 波包变换也是有效的。 我们研究了新乘积的主要性质。 然后我们利用这些乘积推导出在时频空间中与三次非线性薛定谔方程等价并推广的积分微分方程。 我们还得到了由非线性薛定谔方程解相关的 Wigner-Ville 函数满足的 Weyl-Wigner-Moyal 方程。 新的方程类似于玻尔兹曼方程。 显示英文摘要

摘要： We show that the short-time Fourier transform of the pointwise product of two functions $f$ and $h$ can be written as a suitable product of the short-time Fourier transforms of $f$ and $h$. The same result is then shown to be valid for the Wigner wave-packet transform. We study the main properties of the new products. We then use these products to derive integro-differential equations on the time-frequency space equivalent to, and generalizing, the cubic nonlinear Schr\"odinger equation. We also obtain the Weyl-Wigner-Moyal equation satisfied by the Wigner-Ville function associated with the solution of the nonlinear Schr\"odinger equation. The new equation resembles the Boltzmann equation.