2011/10/16 by Andrey Melnikov, Melnikov, Andrey · 1 citation
Mathematics · Physics and Astronomy · #35Q53 #37K10 #37K35 #46C20 #93A10 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Mathematical Physics (math-ph) #Nonlinear Photonic Systems #Nonlinear Waves and Solitons #Spectral Theory (math.SP) #math-ph #math.AP #math.FA #math.MP #math.SP #msc:35Q53 #msc:37K10 #msc:37K35 #msc:46C20 #msc:93A10
paper · pdf · doi:10.48550/arxiv.1110.3495
openalex publication_date 2011/10/16 · arxiv created 2011/11/09 · arxiv updated 2011/11/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this work we present a new method for solving of the Korteweg-de Vries (KdV) equation q't = - \dfrac32 q q'x + \dfrac14 q"'xxx. The proposed method is a particular case of the theory of evolutionary vessels, developed in this work. Inverse scattering of the Sturm-Liouville operator and evolution of its potential are the basic ingredients, similar to the existing methods developed by Gardner-Greene-Kruskal-Miura (1967), Zacharov-Shabbath (1974) and Peter Lax (1977). Evolutionary KdV vessel may be considered as a generalization of these previous works. The advantage of the new method is that it produces a unified approach to existing solutions of the KdV equation. For example, odd or even analytic, periodic, almost periodic solutions are shown to be particular cases of this theory. Generalizing this method we can also produce many PDEs, associated with integrable systems, in an arbitrary number of variables (in the spirit of Zakarov-Shabat).