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The Asymptotic Solution of a Connection Problem of a Second Order Ordinary Differential Equation

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The solutions of the equation are discussed in the limit ρ → 0. The solutions which oscillate about − |t| as t → ∞ have asymptotic expansions whose leading terms are where à +, , à , and are constants. The connection problem is to determine the asymptotic expansion at + ∞. In other words, we wish to find (à +, ) as functions of à and The nonlinear solutions with à ± not small are analyzed by using the method of averaging. It is shown that this method breaks down for small amplitudes. In this case a solution can be obtained on [0, ∞) as a small amplitude perturbation about the exact nonoscillating solution W(t) whose asymptotic expansion is This is a solution of (1) which corresponds to à + ≡ 0 in (2). A quantity which determines the scale of the small amplitude response is −W'(0). This quantity is found to be exponentially small. The determination of this constant is shown to reduce to a solution of the equation for the first Painlevé transcendent. The asymptotic behavior of the required solution is determined by solving an integral equation.
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Document Type: Research Article

Affiliations: University of Edinburgh

Publication date: April 1, 1989

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