We study solutions of the KdV equation governed by a stationary equation for symmetries from the non-commutative subalgebra, namely, for a linear combination of the master-symmetry and the scaling symmetry. The constraint under study is equivalent to a sixth order nonautonomous ODE possessing two first integrals. Its generic solutions have a singularity on the line t = 0. The regularity condition selects a 3-parameter family of solutions which describe oscillations near u = 1 and satisfy, for t = 0, an equation equivalent to degenerate P5 equation. Numerical experiments show that in this family one can distinguish a two-parameter subfamily of separatrix step-like solutions with power-law approach to different constants for x → ±∞. This gives an example of exact solution for the Gurevich–Pitaevskii problem on decay of the initial discontinuity.

中文翻译：

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KdV 方程的非自治对称性和阶梯解

我们研究了由非交换子代数对称性的平稳方程控制的 KdV 方程的解，即主对称性和标度对称性的线性组合。所研究的约束等效于具有两个一阶积分的六阶非自治常微分方程。它的通用解在 t = 0 线上有一个奇点。正则性条件选择一个 3 参数的解系列，这些解描述 u = 1 附近的振荡，并在 t = 0 时满足等价于退化 P5 方程的方程。数值实验表明，在该族中，对于x → ±∞ 的不同常数，可以通过幂律方法区分分离线阶跃解的双参数子族。这给出了关于初始不连续性衰减的 Gurevich-Pitaevskii 问题的精确解的示例。