Meromorphic traveling wave solutions of the Kuramoto-Sivashinsky equation
Alexandre Eremenko
TL;DR
This paper classifies all meromorphic traveling wave solutions of the Kuramoto-Sivashinsky equation, showing that only known explicit solutions exist, using a method based on Nevanlinna theory.
Contribution
It provides a complete classification of meromorphic solutions for the third order ODE of the Kuramoto-Sivashinsky equation, employing a novel application of Nevanlinna theory.
Findings
No new meromorphic solutions beyond known explicit ones.
Method applicable to a broad class of nonlinear ODEs.
Complete characterization of meromorphic solutions for the equation.
Abstract
We determine all cases when there exists a meromorphic solution of the third order ODE describing traveling waves solutions of the Kuramoto-Sivashinsky equation. It turns out that there are no other meromorphic solutions besides those explicit solutions found by Kuramoto and Kudryashov. The general method used in this paper, based on Nevanlinna theory, is applicable to finding all meromorphic solutions of a wide class of non-linear ODE. Keywords: Kuramoto and Sivashinsky equation, meromorphic functions, elliptic functions, Nevanlinna theory.
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Taxonomy
TopicsNonlinear Waves and Solitons · Differential Equations and Numerical Methods · Nonlinear Photonic Systems