具有一阶导数的二阶偏微分方程解
Solutions of second-order PDEs with first-order quotients
摘要：
描述了一种利用点对称的微分不变式求解偏微分方程的方法。通过求解由微分不变式代数中的微分合矢给出的商偏微分方程，得到了与所考虑的偏微分方程相容的新的微分约束。在我们的系统中加入这些约束，使它变得超定，从而更容易求解。我们专注于二阶标量偏微分方程，其等价物是一阶标量偏微分方程。这种情况只发生在二阶偏微分方程对称的李代数是无限维的时候。我们将这个想法应用到几个不同的偏微分方程，其中之一是Hunter-Saxton方程。
英文摘要：
We describe a way of solving a partial differential equation using the differential invariants of its point symmetries. By first solving its quotient PDE, which is given by the differential syzygies in the algebra of differential invariants, we obtain new differential constraints which are compatible with the PDE under consideration. Adding these constraints to our system makes it overdetermined, and thus easier to solve. We focus on second-order scalar PDEs whose quotients are first-order scalar PDEs. This situation occurs only when the Lie algebra of symmetries of the second-order PDE is infinite-dimensional. We apply this idea to several different PDEs, one of which is the Hunter-Saxton equation.
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