On some class of non-Lie solutions of nonlinear d'Alambert equation
type: Item
instance of: scholarly article
zbMATH Open document ID: 0957.35093
author: R. Z. Zhdanov
author: O. A. Panchak
publication date: 2001-03-01T00:00:00Z
Mathematics Subject Classification ID: 35L70
Mathematics Subject Classification ID: 35C05
zbMATH DE Number: 1571636
zbMATH Keywords: exact solution
zbMATH Keywords: multidimensional complex nonlinear d'Alembert equation
MaRDI profile type: Publication
title: On some class of non-Lie solutions of nonlinear d'Alambert equation
published in: Visnyk. Seriya: Fizyko-Matematychni Nauky. Kyïvs'kyĭ Universytet Imeni Tarasa Shevchenka
review text: The authors construct a new class of the exact solutions of the multidimensional complex nonlinear d'Alambert equation \((\partial^{2}/\partial x_{0}^{2} -\Delta)u=F(|u|)u\), where \(F\) is some continuous function; \(u=u(x_{0},x_{1},x_{2},x_{3})\). This class can not be obtained within the framework of the traditional Lie approach. For the equation \((\partial^{2}/\partial x_{0}^{2}-\Delta)u=\lambda|u|^{2}u\) this class of solutions has the form \(u(x)=(1/\lambda)w_{1}(\omega)\exp\{iw_{1}(\omega)x_{3}+ iw_{2}(\omega)\}\), \(\omega=(x_{1}^{2}+x_{2}^{2})^{-1} (x_{0}x_{1}\pm x_{2}\sqrt{x_{1}^{2}+x_{2}^{2}-x_{0}^{2}})\), where \(w_{1},w_{2}\) are arbitrary functions.
reviewed by: Aleksandr D. Borisenko
Recommended article: Q4518509
Recommended article: Q2935725
Recommended article: Nonlinear d'Alambert equation in the pseudo-Euclidean space \(\mathbb R_{2,n}\) and its solutions
Recommended article: Q4283753
Recommended article: On nontrivial solutions to some nonlinear ordinary differential equations
Recommended article: On the solutions of the nonlinear Liouville hierarchy
Recommended article: Q3799224
Recommended article: Q5398494
Recommended article: On the Solvability of a Certain Class of Non-Pellian Equations.
description: scientific article