2006/06/06 by Nikolay A. Kudryashov, Nikolai A. Kudryashov, Kudryashov, Nikolai A. +2
Physics and Astronomy · #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Physical sciences #Nonlinear Waves and Solitons #nlin.SI
paper · pdf · doi:10.48550/arxiv.nlin/0606018
24 pages, 10 figures
arxiv created 2006/06/06 · openalex publication_date 2006/06/06 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A method for finding exact solutions of nonlinear differential equations is presented. Our method is based on the application of the Newton polygons corresponding to nonlinear differential equations. It allows one to express exact solutions of the equation studied through solutions of another equation using properties of the basic equation itself. The ideas of power geometry are used and developed. Our approach has a pictorial rendition, which is is illustrative and effective. The method can be also applied for finding transformations between solutions of the differential equations. To demonstrate the method application exact solutions of several equations are found. These equations are: the Korteveg - de Vries - Burgers equation, the generalized Kuramoto - Sivashinsky equation, the fourth - order nonlinear evolution equation, the fifth - order Korteveg - de Vries equation, the modified Korteveg - de Vries equation of the fifth order and nonlinear evolution equation of the sixth order for the turbulence description. Some new exact solutions of nonlinear evolution equations are given.