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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 Bretherton equation ： ウィキペディア英語版 Bretherton equation In mathematics, the Bretherton equation is a nonlinear partial differential equation introduced by Francis Bretherton in 1964: :$u\_+u\_+u\_+u\; =\; u^p,$ with $p$ integer and $p\; \backslash ge\; 2.$ While $u\_t,\; u\_x$ and $u\_$ denote partial derivatives of the scalar field $u(x,t).$ The original equation studied by Bretherton has quadratic nonlinearity, $p=2.$ Nayfeh treats the case $p=3$ with two different methods: Whitham's averaged Lagrangian method and the method of multiple scales. The Bretherton equation is a model equation for studying weakly-nonlinear wave dispersion. It has been used to study the interaction of harmonics by nonlinear resonance. Bretherton obtained analytic solutions in terms of Jacobi elliptic functions.〔 ==Variational formulations== The Bretherton equation derives from the Lagrangian density: :$$ \mathcal = \tfrac12 \left( u_t \right)^2 + \tfrac12 \left( u_x \right)^2 -\tfrac12 \left( u_ \right)^2 - \tfrac12 u^2 + \tfrac u^ through the Euler–Lagrange equation: :$$ \frac \left( \frac \right) + \frac \left( \frac \right) - \frac \left( \frac} \right) - \frac = 0. The equation can also be formulated as a Hamiltonian system: :$$ \begin u_t & - \frac = 0, \\ v_t & + \frac = 0, \end in terms of functional derivatives involving the Hamiltonian $H:$ :$H(u,v)\; =\; \backslash int\; \backslash mathcal(u,v;x,t)\backslash ;\; \backslash mathrmx$ and $$ \mathcal(u,v;x,t) = \tfrac12 v^2 - \tfrac12 \left( u_x \right)^2 +\tfrac12 \left( u_ \right)^2 + \tfrac12 u^2 - \tfrac u^ with $\backslash mathcal$ the Hamiltonian density – consequently $v=u\_t.$ The Hamiltonian $H$ is the total energy of the system, and is conserved over time.〔 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「Bretherton equation」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.