Nonautonomous symmetries of the KdV equation and step-like solutions

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.

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References in corpus (2)

• Phase Shift in the Whitham Zone for the Gurevich-Pitaevskii Special Solution of the Korteweg-de Vries Equation
• KdV equation beyond standard assumptions on initial data

Cited by in corpus (5)

• Negative flows for several integrable models
• Non-autonomous reductions of the KdV equation and multi-component analogs of the Painlevé equations P${}_{34}$ and P${}_3$
• Integrable Abel equation and asymptotics of symmetry solutions of Korteweg-de Vries equation
• Negative flows and non-autonomous reductions of the Volterra lattice
• Meromorphy of solutions for a wide class of ordinary differential equations of Painlevé type