论文标题
TODA作为多孔培养基方程
The Toda flow as a porous medium equation
论文作者
论文摘要
我们描述了不可压缩的多孔介质（IPM）方程的几何形状：我们证明它是一个梯度动力学系统，在区域保存差异方面，并具有特殊的双支架形式。此外，我们显示出与无分散TODA系统的相似性和差异。 TODA流动描述了几个粒子在邻居之间具有指数势的线上的可集成相互作用，而其连续版本是可集成的PDE，其物理含义是晦涩的。在这里，我们表明这种连续的TODA流可以自然地视为特殊的IPM方程，而TODA的关键双支属性属性由IPM类型的所有方程共享，从而表现出其梯度和非自治的汉密尔顿起源。最后，我们评论QR对角算法的TODA和IPM修改，并在具有任意惯性操作员的一般谎言组的环境中描述了双式支架流动。
We describe the geometry of the incompressible porous medium (IPM) equation: we prove that it is a gradient dynamical system on the group of area-preserving diffeomorphisms and has a special double-bracket form. Furthermore, we show its similarities and differences with the dispersionless Toda system. The Toda flow describes an integrable interaction of several particles on a line with an exponential potential between neighbours, while its continuous version is an integrable PDE, whose physical meaning was obscure. Here we show that this continuous Toda flow can be naturally regarded as a special IPM equation, while the key double-bracket property of Toda is shared by all equations of the IPM type, thus manifesting their gradient and non-autonomous Hamiltonian origin. Finally, we comment on Toda and IPM modifications of the QR diagonalization algorithm, as well as describe double-bracket flows in an invariant setting of general Lie groups with arbitrary inertia operators.
