The Second Painlevé Equation in the Large-Parameter Limit I: Local Asymptotic Analysis

Details The Second Painlevé Equation in the Large-Parameter Limit I: Local Asymptotic Analysis The Second Painlevé Equation in the Large-Parameter Limit I: Local Asymptotic Analysis

1997-10-24

arxiv.org/abs/solv-int/9710022v1

Nonlinear Sciences

Exactly Solvable and Integrable Systems

30 pages in LaTeX2e. Submitted

Scientific paper

In this paper, we find all possible asymptotic behaviours of the solutions of the second Painlev\'e equation $y''=2y^3+xy +\alpha$ as the parameter $\alpha\to\infty$ in the local region $x\ll\alpha^{2/3}$. We prove that these are asymptotic behaviours by finding explicit error bounds. Moreover, we show that they are connected and complete in the sense that they correspond to all possible values of initial data given at a point in the local region.

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