容许无限维李代数的非线性偏微分方程组及其与里奇流的联系。第二部分：二维空间情形
Nonlinear systems of PDEs admitting infinite-dimensional Lie algebras and their connection with Ricci flows. II: The two-dimensional space case
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中文总结 AI 辅助
该文为前期研究的延续，构造了容许无限维李代数的（1+2）维二阶偏微分方程二分量演化方程组，确定了其与里奇流相关的特殊情形，得到径向对称下的稳态与含时精确解，证明相关李代数的可约化性质。
中文摘要 AI 辅助
本工作是发表于《Stud Appl Math. 2024; 153:e12737》的论文的自然延续。构造了所有容许无限维李代数的（1+2）维二阶偏微分方程的二分量演化方程组。研究表明，将该李代数自然推广到高维情形不会得到更一般的结果，因为无限维对称性会被破坏。新近推导的与里奇流（Ricci flows）相关的方程组，被确认为所得到的演化方程组中的一个非常特殊的情形。利用约化常微分方程组（ODEs）的惊人丰富的李代数，构造了该方程组在径向对称情形下的所有可能稳态解。此外，证明了该李代数可约化为最简单的二分量二阶常微分方程组的十五维代数。还构造了径向对称情形下的若干含时精确解，表明只要正确指定任意参数，所得到的解是有界且光滑的。
英文摘要
The work is a natural continuation of that published in Stud Appl Math. 2024; 153:e12737. All possible two-components evolutions systems of (1+2)-dimensional second-order PDEs admitting an infinite-dimensional Lie algebra are constructed. It is shown that a natural generalisation of this Lie algebra to the higher-dimensional case does not lead to a more general result because the infinite-dimensional symmetry is broken. The recently derived system, which is related to Ricci flows, is identified as a very particular case among the evolution systems obtained. All possible stationary solutions of this system in the radially symmetric case are constructed using the surprisingly rich Lie algebra of the reduced system of ODEs. Moreover, it is proved that this Lie algebra is reducible to the fifteen-dimensional algebra of the simplest system of two second-order ODEs. Several time-dependent exact solutions in the radially symmetric case are constructed as well. It is shown that the solutions obtained are bounded and smooth provided arbitrary parameters are correctly specified.
