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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 Camassa–Holm equation ： ウィキペディア英語版 Camassa–Holm equation In fluid dynamics, the Camassa–Holm equation is the integrable, dimensionless and non-linear partial differential equation :$$ u_t + 2\kappa u_x - u_ + 3 u u_x = 2 u_x u_ + u u_. \, The equation was introduced by Camassa and Holm as a bi-Hamiltonian model for waves in shallow water, and in this context the parameter ''κ'' is positive and the solitary wave solutions are smooth solitons. In the special case that ''κ'' is equal to zero, the Camassa–Holm equation has peakon solutions: solitons with a sharp peak, so with a discontinuity at the peak in the wave slope. ==Relation to waves in shallow water== The Camassa–Holm equation can be written as the system of equations: :$$ \begin u_t + u u_x + p_x &= 0, \\ p - p_ &= 2 \kappa u + u^2 + \frac \left( u_x \right)^2, \end with ''p'' the (dimensionless) pressure or surface elevation. This shows that the Camassa–Holm equation is a model for shallow water waves with non-hydrostatic pressure and a water layer on a horizontal bed. The linear dispersion characteristics of the Camassa–Holm equation are: :$\backslash omega\; =\; 2\backslash kappa\; \backslash frac,$ with ''ω'' the angular frequency and ''k'' the wavenumber. Not surprisingly, this is of similar form as the one for the Korteweg–de Vries equation, provided ''κ'' is non-zero. For ''κ'' equal to zero, the Camassa–Holm equation has no frequency dispersion — moreover, the linear phase speed is zero for this case. As a result, ''κ'' is the phase speed for the long-wave limit of ''k'' approaching zero, and the Camassa–Holm equation is (if ''κ'' is non-zero) a model for one-directional wave propagation like the Korteweg–de Vries equation. 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「Camassa–Holm 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.