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On the Ackerberg–O'Malley Resonance
Author(s) -
Wong R.,
Yang Heping
Publication year - 2003
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1111/1467-9590.00235
Subject(s) - resonance (particle physics) , mathematics , boundary value problem , value (mathematics) , mathematical analysis , work (physics) , constant (computer programming) , boundary (topology) , combinatorics , pure mathematics , physics , thermodynamics , computer science , statistics , atomic physics , programming language
In this paper, we continue our study of the boundary value problemwhere A , B are prescribed constants and 0 < ɛ≪ 1 is a small positive parameter. We assume that the coefficients a ( x ) and b ( x ) are sufficiently smooth functions with the behavior given by a ( x ) ∼α x and b ( x ) ∼β as x → 0 . In our previous work, the problem has been studied for both α > 0 and α < 0 except for the cases β/α= 1,2,3,… when α > 0 and β/α= 0,−1,−2,… when α < 0 . In the present paper, we study these exceptional cases and obtain, by rigorous analysis, uniformly valid asymptotic solutions of the problem. From these solutions, we also show that the conditions in these exceptional cases are exactly the ones which are necessary and sufficient for the Ackerberg–O'Malley resonance.