Symmetric Approach for Integrable Nonlinear Eolution Equations in 1-Space and 1-Time Dimensions

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タイトル別名
  • ジクウ 1 + 1 ジゲン ニ オケル ヒセンケイ ハッテン ホウテイシキ ノ
  • 時空(1+1)次元における非線型発展方式のシンメトリック・アプローチ

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説明

Since the famous investigation of the KdV equation b y GGKM, both existence of infinitely many conservation laws and the hierarchy of nonlinear evolution equations (NLLEs) had been understood as essential for a given NLEE to be integrable. This problem was studied by many authors, but we remark the contribution by Magri, who had explained these properties from the view of geometrical point. Starting from symmetries (contravariant quantity), he introduced a potential operator (covariant quantity) and a sympletic operator which maps the covariant to the contravariant. The conseration laws were simply derived and he proposed a "bi-H amiltonian structure" for integrable systems. Fuchsteiner deeply considered symmetries and introduced both concepts of strong symmetries and hereditary symmetries. Fokas had used a Lie-Backlund transformation and also arrived at the hereditary symmetry. Their idear was further developed and connections with the Backlund transformation and with the cannonical structures were made clear. The iso-spectral problem is essential for the inverse spectral method (ISM) and its relation with the symmetric approach is very interesting. Such relations were first treated by Lax for the case of KdV equation and extended to other cases. The motiation of this issue is to detail with the review of the symmetric approach. We specially consider a certain linear integra-differential operator K_± and make clear the role of squared eigenfunctions which is closely related with the 2 X 2-matrix isospectral problem. We inspect that this operator is both strong and hereditary symmetries and make clear the associated cannonical structures. Considering that K_± are also obtained by the compatibility condition of NLLEs, we propose a direct and simple method for developing the hereditary symmetries with a N x N-matrix formula.

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