论文标题
二阶PDE的解决方案与一阶商
Solutions of second-order PDEs with first-order quotients
论文作者
论文摘要
我们描述了使用其点对称性的差分不变式求解部分微分方程的方法。通过首先解决其商PDE，该PDE由差分不变的代数中的差分序列给出，我们获得了与所考虑的PDE兼容的新差分约束。将这些约束添加到我们的系统中会使它过度确定，因此更易于解决。我们专注于二阶标量PDE，其商为一阶标量PDE。只有当二阶PDE的对称性的谎言代数是无限维度时，这种情况才会发生。我们将此想法应用于几个不同的PDE，其中之一是Hunter-Saxton方程。
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.
