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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 Degasperis–Procesi equation ： ウィキペディア英語版 Degasperis–Procesi equation In mathematical physics, the Degasperis–Procesi equation : $\backslash displaystyle\; u\_t\; -\; u\_\; +\; 2\backslash kappa\; u\_x\; +\; 4u\; u\_x\; =\; 3\; u\_x\; u\_\; +\; u\; u\_$ is one of only two exactly solvable equations in the following family of third-order, non-linear, dispersive PDEs: :$\backslash displaystyle\; u\_t\; -\; u\_\; +\; 2\backslash kappa\; u\_x\; +\; (b+1)u\; u\_x\; =\; b\; u\_x\; u\_\; +\; u\; u\_,$ where $\backslash kappa$ and ''b'' are real parameters (''b''=3 for the Degasperis–Procesi equation). It was discovered by Degasperis and Procesi in a search for integrable equations similar in form to the Camassa–Holm equation, which is the other integrable equation in this family (corresponding to ''b''=2); that those two equations are the only integrable cases has been verified using a variety of different integrability tests.〔Degasperis & Procesi 1999; Degasperis, Holm & Hone 2002; Mikhailov & Novikov 2002; Hone & Wang 2003; Ivanov 2005〕 Although discovered solely because of its mathematical properties, the Degasperis–Procesi equation (with $\backslash kappa\; >\; 0$) has later been found to play a similar role in water wave theory as the Camassa–Holm equation.〔Johnson 2003; Dullin, Gottwald & Holm 2004; Constantin & Lannes 2007; Ivanov 2007〕 == Soliton solutions == (詳細はmultipeakon solutions, which are functions of the form :$\backslash displaystyle\; u(x,t)=\backslash sum\_^n\; m\_i(t)\; e^$ where the functions $m\_i$ and $x\_i$ satisfy〔Degasperis, Holm & Hone 2002〕 :$\backslash dot\_i\; =\; \backslash sum\_^n\; m\_j\; e^,\backslash qquad\; \backslash dot\_i\; =\; 2\; m\_i\; \backslash sum\_^n\; m\_j\backslash ,\; \backslash sgn\; e^.$ These ODEs can be solved explicitly in terms of elementary functions, using inverse spectral methods.〔Lundmark & Szmigielski 2003, 2005〕 When $\backslash kappa\; >\; 0$ the soliton solutions of the Degasperis–Procesi equation are smooth; they converge to peakons in the limit as $\backslash kappa$ tends to zero.〔Matsuno 2005a, 2005b〕 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「Degasperis–Procesi 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.