On the Gurevich-Pitaevskii solution of KdV

2026/06/06 by Robert Conte · 1 voice
Physics and Astronomy · #nlin.SI #physics.flu-dyn

paper · pdf · doi:10.1016/j.wavemoti.2026.103784

Abstract

The universal solution of the Korteweg-de Vries equation (KdV) introduced by Gurevich and Pitaevskii in order to describe the onset of dispersive shock waves is known to also obey the self-similar reduction of the next member in the KdV hierarchy. We show that, if this common solution obeys some lower order partial differential equation, its differential order must be one, and we provide its local representation as a converging Laurent series depending on both space and time.

Citations

• Linearisable Abel equations and the Gurevich--Pitaevskii problem
• Phase Shift in the Whitham Zone for the Gurevich-Pitaevskii Special Solution of the Korteweg-de Vries Equation
• Sur les équations différentielles du second ordre et du premier degré dont l'intégrale générale est a points critiques fixes

Discussions

• Robert Conte (ENS Paris-Saclay, France,,Dept of mathematics, The University of Hong Kong): On the Gurevich-Pitaevskii solution of KdV https://arxiv.org/abs/2606.08074 https://arxiv.org/pdf/2606.08074 [bsky, 0 points, 0 comments]

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• Phase Shift in the Whitham Zone for the Gurevich-Pitaevskii Special Solution of the Korteweg-de Vries Equation