Classifying algebraic invariants and algebraically invariant solutions

• Published in: Chaos, solitons and fractals solitons and fractals, 2020-11, Vol.140, p.110219, Article 110219
• Main Author: Demina, Maria V.
• Format: Article
• Language: English
• Subjects: Algebraic invariants; Algebraic traveling waves; Algebraically invariant solutions; Dispersive Kuramoto–Sivashinsky equation; Modified Kuramoto–Sivashinsky equation
• ISSN: 0960-0779; 1873-2887
• DOI: 10.1016/j.chaos.2020.110219

•A novel method for finding and classifying algebraic invariants of autonomous ordinary differential equations.•The complete set of irreducible algebraic invariants for the famous dispersive Kuramoto-Sivashinsky equation and the modified Kuramoto-Sivashinsky equation.•Novel traveling wave solutions of the modified Kuramoto-Sivashinsky equation. A concept of algebraic invariants and algebraically invariant solutions for autonomous ordinary differential equations and systems of autonomous ordinary differential equations is considered. A variety of known exact solutions of autonomous ordinary differential equations and a great number of traveling wave solutions of famous partial differential equations are in fact algebraically invariant solutions. A method, which can be used to find all irreducible algebraic invariants, is introduced. As an example, all irreducible algebraic invariants for the traveling wave reductions of the dispersive Kuramoto–Sivashinsky equation and the modified Kuramoto–Sivashinsky equation are classified. Novel solutions of the modified Kuramoto–Sivashinsky equation are obtained.