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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 Toda lattice ： ウィキペディア英語版 Toda lattice The Toda lattice, introduced by , is a simple model for a one-dimensional crystal in solid state physics. It is given by a chain of particles with nearest neighbor interaction described by the equations of motion :$\backslash begin$ \frac p(n,t) &= e^ - e^, \\ \frac q(n,t) &= p(n,t), \end where $q(n,t)$ is the displacement of the $n$-th particle from its equilibrium position, and $p(n,t)$ is its momentum (mass $m=1$). The Toda lattice is a prototypical example of a completely integrable system with soliton solutions. To see this one uses Flaschka's variables :$a(n,t)\; =\; \backslash frac\; ^,\; \backslash qquad\; b(n,t)\; =\; -\backslash frac\; p(n,t)$ such that the Toda lattice reads :$\backslash begin$ \dot(n,t) &= a(n,t) \Big(b(n+1,t)-b(n,t)\Big), \\ \dot(n,t) &= 2 \Big(a(n,t)^2-a(n-1,t)^2\Big). \end Then one can verify that the Toda lattice is equivalent to the Lax equation :$\backslash frac\; L(t)\; =\; (L(t))$ where () = ''LP'' - ''PL'' is the commutator of two operators. The operators ''L'' and ''P'', the Lax pair, are linear operators in the Hilbert space of square summable sequences $\backslash ell^2(\backslash mathbb)$ given by :$\backslash begin$ L(t) f(n) &= a(n,t) f(n+1) + a(n-1,t) f(n-1) + b(n,t) f(n), \\ P(t) f(n) &= a(n,t) f(n+1) - a(n-1,t) f(n-1). \end The matrix $L(t)$ has the property that its eigenvalues are invariant in time. These eigenvalues constitute independent integrals of motion, therefore the Toda lattice is completely integrable. In particular, the Toda lattice can be solved by virtue of the inverse scattering transform for the Jacobi operator ''L''. The main result implies that arbitrary (sufficiently fast) decaying initial conditions asymptotically for large ''t'' split into a sum of solitons and a decaying dispersive part. ==References== * * * *Integrable Hamiltonians with Exponential Potential, Eugene Gutkin, Physica 16D (1985) 398-404. * * 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「Toda lattice」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.