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 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.

Keywords:

• Gurevich–Pitaevskii problem
• master-symmetry
• Painlevé type equations

2000 Mathematics Subject Classification:

• 35Q53
• 35G25
• 35C06
• 37B55
• 34M55