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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 Invariant factorization of LPDOs ： ウィキペディア英語版 Invariant factorization of LPDOs ==Introduction== The factorization of a linear partial differential operator (LPDO) is an important issue in the theory of integrability, due to the Laplace-Darboux transformations,〔Weiss (1986)〕 which allow to construct integrable LPDEs. Laplace solved factorization problem for a bivariate hyperbolic operator of the second order (see Hyperbolic partial differential equation), constructing two Laplace invariants. Each Laplace invariant is an explicit polynomial condition of factorization; coefficients of this polynomial are explicit functions of the coefficients of the initial LPDO. The polynomial conditions of factorization are called invariants because they have the same form for equivalent (i.e. self-adjoint) operators. Beals-Kartashova-factorization (also called BK-factorization) is a constructive procedure to factorize a bivariate operator of the arbitrary order and arbitrary form. Correspondingly, the factorization conditions in this case also have polynomial form, are invariants and coincide with Laplace invariants for bivariate hyperbolic operator of the second order. The factorization procedure is purely algebraic, the number of possible factirzations depends on the number of simple roots of the Characteristic polynomial (also called symbol) of the initial LPDO and reduced LPDOs appearing at each factorization step. Below the factorization procedure is described for a bivariate operator of the arbitrary form, of the order 2 and 3. Explicit factorization formulas for an operator of the order $n$ can be found in〔R. Beals, E. Kartashova. Constructively factoring linear partial differential operators in two variables. (Theor. Math. Phys. 145(2), pp. 1510-1523 (2005) ) 〕 General invariants are defined in〔 E. Kartashova. A Hierarchy of Generalized Invariants for Linear Partial Differential Operators. (Theor. Math. Phys. 147(3), pp. 839-846 (2006) ) 〕 and invariant formulation of the Beals-Kartashova factorization is given in〔 E. Kartashova, O. Rudenko. Invariant Form of BK-factorization and its Applications. Proc. GIFT-2006, pp.225-241, Eds.: J. Calmet, R. W. Tucker, Karlsruhe University Press (2006); (arXiv )〕 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「Invariant factorization of LPDOs」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.