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Eliminating Indeterminacy in Singularly Perturbed Boundary Value Problems with Translation Invariant Potentials
Author(s) -
Ward Michael J.
Publication year - 1992
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 - 10.1002/sapm199287295
Subject(s) - mathematics , eigenfunction , eigenvalues and eigenvectors , mathematical analysis , method of matched asymptotic expansions , boundary value problem , nonlinear system , invariant subspace , linearization , exponential function , invariant (physics) , asymptotic expansion , linear subspace , pure mathematics , physics , mathematical physics , quantum mechanics
Solutions exhibiting an internal layer structure are constructed for a class of nonlinear singularly perturbed boundary value problems with translation invariant potentials. For these problems, a routine application of the method of matched asymptotic expansions fails to determine the locations of the internal layer positions. To overcome this difficulty, we present an analytical method that is motivated by the work of Kath, Knessl and Matkowsky [4]. To construct a solution having n internal layers, we first linearize the boundary value problem about the composite expansion provided by the method of matched asymptotic expansions. The eigenvalue problem associated with the homogeneous form of this linearization is shown to have n exponentially small eigenvalues. The condition that the solution to the linearized problem has no component in the subspace spanned by the eigenfunctions corresponding to these exponentially small eigenvalues determines the internal layer positions. These “near” solvability conditions yield algebraic equations for the internal layer positions, which are analyzed for various classes of nonlinearities.