Integrability criteria for systems of nonlinear partial differential equations
Integrability criteria for systems of nonlinear partial differential equations
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非线性偏微分方程组的可积性准则
DOI：
10.4310/jdg/1214428094
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发表时间：
1967
影响因子：
2.5
通讯作者：
H. Goldschmidt
中科院分区：
文献类型：
--
作者：
H. Goldschmidt
在[5]中，我们在本文中证明了解析线性偏微分方程局部解的存在性，我们将之前的结果推广到任意解析（非线性）偏微分方程组，并证明了用 Ehresmann 引入的射流束表示的此类系统的嘉当-卡勒定理。强烈建议读者阅读[5]的线性Cartan-Kahler定理（定理4.1）的证明作为本文的介绍，尽管本文本质上独立于[5]。我们考虑纤维流形上的偏微分方程，并且不像线性方程那样根据微分算子定义非线性方程（参见[5]）。在非线性情况下，这种观点过于严格（参见第 7 节），因此我们让射流束的任意纤维子流形为偏微分方程，因为微分几何中出现的许多方程都是这种类型（参见 E. Cartan [3]）。为了概括[5]的方法，其中所考虑的射流丛是矢量丛，我们定义仿射丛并分析射流丛所拥有的仿射丛的结构（§§3和5）。如果 k 阶偏微分方程 Rk 的每个 k 阶解都可以推广到 k + 1 阶解，并且如果 k + 1 阶解满足正则性条件，则 k + 1 阶解形成 k + 1 阶方程 Rk+1，即 Rk 的延拓。这个延拓Rk+1实际上是方程Rk上的仿射丛，其仿射丛结构是由射流丛的仿射丛结构导出的。在 Rk 的这些条件下，Rk+1 在 JR* 上的仿射丛结构，以及某些射流丛的仿射丛结构（参见命题 5.3），允许我们定义偏微分方程 Rk 的曲率 K。曲率 K 是将 k + 1 阶解扩展到 k + 2 阶解的障碍。我们对 A 的定义：将齐次线性方程简化为 Quillen [7] 给出的 K 的定义，通过改变符号，它等价于一个 Bott
In [5], we showed the existence of local solutions of analytic linear partial differential equations in this paper, we generalize our previous result to an arbitrary analytic (nonlinear) system of partial differential equations and prove the Cartan-Kahler theorem for such a system formulated in terms of the jet bundles introduced by Ehresmann. The reader is strongly advised to read the proof of the linear Cartan-Kahler theorem (Theorem 4.1) of [5] as an introduction to the present paper, although this paper is essentially independent of [5]. We consider partial differential equations on fibered manifolds and do not define nonlinear equations in terms of differential operators as one can for linear equations (see [5]). In the nonlinear case this point of view would be too restrictive (see § 7) and so we let arbitrary fibered submanifolds of jet bundles be partial differential equations, since many equations occurring in differential geometry are of this type (see E. Cartan [3]). To generalize the methods of [5], where the jet bundles considered are vector bundles, we define affine bundles and analyse the structure of affine bundles which the jet bundles possess (§§3 and 5). If every solution of order k of a partial differential equation Rk of order k can be extended to a solution of order k + 1, and if the solutions of order k + 1 satisfy a regularity condition, the solutions of order k + 1 form an equation Rk+1 of order k + 1, the prolongation of Rk. This prolongation Rk+1 is actually an affine bundle over the equation Rk whose affine bundle structure is induced by the affine bundle structure of a jet bundle. Under these conditions on Rk, this affine bundle structure of Rk+1 over JR*, together with the affine bundle structure of certain jet bundles (see Proposition 5.3), permits us to define the curvature K of the partial differential equation Rk. The curvature K is the obstruction to extending a solution of order k + 1 to a solution of order k + 2. Our definition of A: reduces for homogeneous linear equations to the definition of K given by Quillen [7], which is equivalent, with a change in sign, to the one Bott