Exact Criteria for the Existence of a Moving Singular Point in a Complex Domain for a Nonlinear Differential Third-Degree Equation with a Polynomial Seventh-Degree Right-Hand Side

Abstract

Earlier, the authors formulated and proved interval and point criteria for the existence of moving singular points of a third-degree nonlinear differential equation with a polynomial seventh-degree right-hand side for a real domain. For the complex domain, these criteria are associated with specificity of transition to phase spaces. Necessary as well as necessary and sufficient conditions for the existence of moving singular points are obtained. The results of these studies are the basis of the algorithm for obtaining moving singular points of nonlinear differential equations in the complex domain. The results of this paper allow us to expand the classes of the studied nonlinear differential equations with moving peculiarities.

Technology for Obtaining the Approximate Value of Moving Singular Points for a Class of Nonlinear Differential Equations in a Complex Domain

“…Function w(z) is represented in the form w(z) = u(x, y) + i • v(x, y), where functions u(x, y) and v(x, y) are characterized accordingly by the phase spaces Φ 1 (x, y, u(x, y)) and Φ 2 (x, y, v(x, y)). Let us use terminology [17] to determine the correct and incorrect lines, necessary to facilitate the presentation of the following provisions. Theorem 1.…”

“…Let us note that, for nonlinear differential equations, there is a possibility for an asymptotic approach [6,7] in single cases of solvability in quadratures [8][9][10][11]. Let us present the development of the authors' approach in the method of analytical approximate solution for a certain class of nonlinear equations [12][13][14][15][16][17][18][19].…”