Nonautonomous symmetries of the KdV equation and step-like solutions

DOI

10.1080/14029251.2020.1757236How to use a DOI?

Keywords

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

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.

Copyright

© 2020 The Authors. Published by Atlantis and Taylor & Francis

Open Access

This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

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TY  - JOUR
AU  - V.E. Adler
PY  - 2020
DA  - 2020/05/04
TI  - Nonautonomous symmetries of the KdV equation and step-like solutions
JO  - Journal of Nonlinear Mathematical Physics
SP  - 478
EP  - 493
VL  - 27
IS  - 3
SN  - 1776-0852
UR  - https://doi.org/10.1080/14029251.2020.1757236
DO  - 10.1080/14029251.2020.1757236
ID  - Adler2020
ER  -