Nonautonomous symmetries of the KdV equation and step-like solutions

TL;DR

This paper investigates special solutions of the KdV equation governed by nonautonomous symmetries, revealing a family of step-like solutions that model decay of initial discontinuities, with implications for the Gurevich–Pitaevskii problem.

Contribution

It introduces a new class of solutions to the KdV equation derived from non-commutative symmetry constraints, connecting them to degenerate P5 equations and decay phenomena.

Findings

Identification of a 3-parameter family of solutions with oscillatory behavior near u=1.

Discovery of a two-parameter subfamily of step-like solutions with power-law asymptotics.

Application of these solutions to model decay of initial discontinuities in the Gurevich–Pitaevskii problem.

Abstract

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 $P_5$ 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 $x\to\pm\infty$. This gives an example of exact solution for the Gurevich--Pitaevskii problem on decay of the initial discontinuity.