Mathematical Analysis of the Gurevich-Pitaevskii Solution in Korteweg-de Vries Equation
Researchers have analyzed the Gurevich-Pitaevskii solution, a universal solution of the Korteweg-de Vries (KdV) equation used to describe dispersive shock waves. The solution is known to satisfy both the KdV equation and a self-similar reduction of the next member in the KdV hierarchy. The study demonstrates that if this solution obeys a lower-order partial differential equation, that equation must be first-order, and provides its local representation as a converging Laurent series.
This theoretical physics paper examines the Gurevich-Pitaevskii solution, a fundamental solution introduced to model the onset of dispersive shock waves in the Korteweg-de Vries equation. The researchers establish that this universal solution simultaneously satisfies both the KdV equation and a self-similar reduction of the next member in the KdV hierarchy. Through mathematical analysis, they prove that if this common solution obeys any lower-order partial differential equation, the differential order of that equation must be exactly one. The work provides an explicit local representation of the solution as a converging Laurent series that depends on both spatial and temporal variables, contributing to the theoretical understanding of nonlinear wave phenomena.
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- arXiv physicsCenter
On the Gurevich-Pitaevskii solution of KdV
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