The second Painlevé equation in the electrostatic-probe theory: Numerical solutions (Q1571178)

The second Painleve equation \(y{{_x}{_x}}=2y^3+xy-\nu\) is investigated on the basis of the asymptotic analysis of a boundary value problem for singularly perturbed nonlinear ordinary differential equations describing the operation of an electrostatic probe in a collisional plasma. The behavior of certain properties of the equation and its regular solutions with the asymptotics \(y\rightarrow\nu/x\) as \(x\rightarrow+\infty\) is considered within the framework of electrostatic-probe theory. Conditions satisfied by these solutions at a certain point \(x_0\) are indicated, which makes it possible to calculate the solutions numerically.