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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 Gibbons–Tsarev equation ： ウィキペディア英語版 Gibbons–Tsarev equation The Gibbons–Tsarev equation is a second order nonlinear partial differential equation.〔Andrei D. Polyanin,Valentin F. Zaitsev, HANDBOOK OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS, SECOND EDITION p764 CRC PRESS〕 In its simplest form, in two dimensions, it may be written as follows: : $u\_t\; u\_-u\_x\; u\_+u\_+1=0\; \backslash qquad\; (1)$ The equation arises in the theory of dispersionless integrable systems, as the condition that solutions of the Benney moment equations may be parametrised by only finitely many of their dependent variables, in this case 2 of them. It was first introduced by John Gibbons and Serguei Tsarev in.〔J. Gibbons and S.P. Tsarev, Reductions of the Benney Equations, Physics Letters A, Vol. 211, Issue 1, Pages 19–24, 1996.〕 This system was also derived,〔E. Ferapontov, A.P. Fordy J. Geom. Phys., 21 (1997), p. 169〕〔E.V Ferapontov, A.P Fordy, Physica D 108 (1997) 350-364〕 as a condition that two quadratic Hamiltonians should have vanishing Poisson bracket. ==Relationship to families of slit maps== The theory of this equation was subsequently developed in.〔J. Gibbons and S.P. Tsarev, Conformal Maps and the reduction of Benney equations, Phys Letters A, vol 258, No4-6, pp 263–271, 1999.〕 In $N$ independent variables, one looks for solutions of the Benney hierarchy in which only $N$ of the moments $A^n$ are independent. The resulting system may always be put in Riemann invariant form. Taking the characteristic speeds to be $p\_i$ and the corresponding Riemann invariants to be $\backslash lambda\_i$, they are related to the zeroth moment $A^0$ by: :$\backslash frac\; =\; -\backslash frac\},\backslash qquad\; (2a)$ :$\backslash frac\; =\; 2\; \backslash frac\; \backslash frac\}.\backslash qquad\; (2b)$ Both these equations hold for all pairs $i\backslash neq\; j$. This system has solutions parametrised by N functions of a single variable. A class of these may be constructed in terms of N-parameter families of conformal maps from a fixed domain D, normally the complex half $p$-plane, to a similar domain in the $\backslash lambda$-plane but with N slits. Each slit is taken along a fixed curve with one end fixed on the boundary of $D$ and one variable end point $\backslash lambda\_i$; the preimage of $\backslash lambda\_i$ is $p\_i$. The system can then be understood as the consistency condition between the set of N Loewner equations describing the growth of each slit: :$\backslash frac\; =\; -\backslash frac\}.\backslash qquad\; (3)$ 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「Gibbons–Tsarev 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.