Geometric singularities for Hamilton‐Jacobi equation
Geometric singularities for Hamilton‐Jacobi equation
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Hamilton-Jacobi 方程的几何奇点
DOI：
10.2969/aspm/02210089
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发表时间：
1991
期刊：
影响因子：
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通讯作者：
S. Izumiya
中科院分区：
文献类型：
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作者：
S. Izumiya
其中Hand是C 00函数。利用特征线方法，显式构造了该问题的解。众所周知，即使对于光滑的初始数据，解也会在有限时间内变为多值。也就是说，奇点出现了。最近Tsuji（[6] [7]）和Nakane（[5]）研究了奇异点附近解的性质。他们假设从多值解到基空间的投影的奇点是折叠或尖点型奇点。但是，其他类型的奇点可能会出现在一般。我们的目的是几何地描述解的奇性分支沿着时间参数。我们将在勒让德开折理论的框架下研究这个问题。在§2中，我们将介绍用于制备的单参数Legendrian展开理论。在§3中，我们将给出Hamilton-Jacobi方程的几何处理，并建立一个与时间参数相关的广义Cauchy问题。定理3.2是我们理论的基础。根据这个定理，我们可以把Arnol 'd-Zakalyukin的波前集和焦散线的单参数变换分类应用到我们的情况（[1]，[2]，[8]）。
where Hand¢ are C 00 -functions. By the method of characteristics, the solution of this problem is explicitly constructed. It is well-known that, even for smooth initial data, the solution becomes multi-valued in finite time. That is, singularities appear. Recently Tsuji ([6] [7]) and Nakane [5] studied the behavior of solutions near the singular point. They assumed that singularities of the projection to the base space from the multi-valued solution are fold or cusp type singularities. But, other type of singularities may be appeared in generic. Our purpose is to describe bifurcations of singularities of solutions along the time parameters geometrically. We will study this problem in the framework of the theory of Legendrian unfoldings. In §2 we will introduce the theory of one-parameter Legendrian unfoldings for preparations. In §3 the geometric treatment of Hamilton-Jacobi equation will be given and we will formulate a generalized Cauchy problem associated with the time parameter. Theorem 3.2 is the base of our theory. By this theorem we can apply Arnol'd-Zakalyukin's classifications of oneparameter perestroika of wave front sets and caustics to our situations ( [l], [2], [8]).