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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Split-step method ： ウィキペディア英語版 Split-step method In numerical analysis, the split-step (Fourier) method is a pseudo-spectral numerical method used to solve nonlinear partial differential equations like the nonlinear Schrödinger equation. The name arises for two reasons. First, the method relies on computing the solution in small steps, and treating the linear and the nonlinear steps separately (see below). Second, it is necessary to Fourier transform back and forth because the linear step is made in the frequency domain while the nonlinear step is made in the time domain. An example of usage of this method is in the field of light pulse propagation in optical fibers, where the interaction of linear and nonlinear mechanisms makes it difficult to find general analytical solutions. However, the split-step method provides a numerical solution to the problem. ==Description of the method== Consider, for example, the nonlinear Schrödinger equation :$=\; -\; +\; i\; \backslash gamma\; |\; A\; |^2\; A\; =\; (D\; +\; \backslash hat\; N)A,$ where $A(t,z)$ describes the pulse envelope in time $t$ at the spatial position $z$. The equation can be split into a linear part, :$=\; -\; =\; \backslash hat\; D\; A,$ and a nonlinear part, :$=\; i\; \backslash gamma\; |\; A\; |^2\; A\; =\; \backslash hat\; N\; A.$ Both the linear and the nonlinear parts have analytical solutions, but the nonlinear Schrödinger equation containing both parts does not have a general analytical solution. However, if only a 'small' step $h$ is taken along $z$, then the two parts can be treated separately with only a 'small' numerical error. One can therefore first take a small nonlinear step, :$A\_N(t,\; z+h)\; =\; \backslash exp\backslash left(\backslash gamma\; |A|^2\; h\; \backslash right)\; A(t,\; z),$ using the analytical solution. The dispersion step has an analytical solution in the frequency domain, so it is first necessary to Fourier transform $A\_N$ using :$\backslash tilde\; A\_N(\backslash omega,\; z)\; =\; \backslash int\_^\backslash infty\; A\_N(t,z)\; \backslash exp()\; dt$, where $\backslash omega\_0$ is the center frequency of the pulse. It can be shown that using the above definition of the Fourier transform, the analytical solution to the linear step, commuted with the frequency domain solution for the nonlinear step, is : The above shows how to use the method to propagate a solution forward in space; however, many physics applications, such as studying the evolution of a wave packet describing a particle, require one to propagate the solution forward in time rather than in space. The non-linear Schrödinger equation, when used to govern the time evolution of a wave function, takes the form :$i\; \backslash hbar\; =\; -\; +\; \backslash gamma\; |\; \backslash psi|^2\; \backslash psi\; =\; (D\; +\; \backslash hat\; N)\backslash psi,$ where $\backslash psi(x,\; t)$ describes the wave function at position $x$ and time $t$. Note that :$\backslash hat\; D=-$ and $\backslash hat\; N\; =\backslash gamma\; |\; \backslash psi|^2$, and that $m$ is the mass of the particle and $\backslash hbar$ is Planck's constant over $2\backslash pi$. The formal solution to this equation is a complex exponential, so we have that :$\backslash psi(x,\; t)=e^\backslash psi(x,\; 0)$. Since $\backslash hat$ and $\backslash hat$ are operators, they do not in general commute. However, the Baker-Hausdorff formula can be applied to show that the error from treating them as if they do will be of order $dt^2$ if we are taking a small but finite time step $dt$. We therefore can write :$\backslash psi(x,\; t+dt)\; \backslash approx\; e^e^\backslash psi(x,\; t)$. The part of this equation involving $\backslash hat\; N$ can be computed directly using the wave function at time $t$, but to compute the exponential involving $\backslash hat\; D$ we use the fact that in frequency space, the partial derivative operator can be converted into a number by substituting $ik$ for $\backslash part\; \backslash over\; \backslash part\; x$, where $k$ is the frequency (or more properly, wave number, as we are dealing with a spatial variable and thus transforming to a space of spatial frequencies—i.e. wave numbers) associated with the Fourier transform of whatever is being operated on. Thus, we take the Fourier transform of :$e^\backslash psi(x,\; t)$, recover the associated wave number, compute the quantity :$e^$, and use it to find the product of the complex exponentials involving $\backslash hat\; N$ and $\backslash hat\; D$ in frequency space as below: : 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Split-step method」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.