The second Painlevé equation in the electrostatic probe theory: Numerical solutions for the partial absorption of charged particles by the surface
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We continue the study of the second Painlevé equation within the framework of the electrostatic probe theory. The integrability conditions for the equation are found for the partial absorption of charged particles by the probe surface. A sets of solutions with the asymptotics y ∼ ν/x for x → +∞ is constructed numerically in a wide range of the free parameter ν. Also, solutions (related to those mentioned above) for half-integer and integer ν, including solutions representable in asymptotic form at x → +∞ through the Airy function y ∼ cAi(x) in the limit ν → 0, are found. The results are discussed from the standpoint of the isomonodromic deformation method.
KeywordsCharged Particle Probe Theory Free Parameter Probe Surface Integrability Condition
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