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Spectral Stability for Classical Periodic Waves of the Ostrovsky and Short Pulse Models
Author(s) -
Hakkaev S.,
Stanislavova M.,
Stefanov A.
Publication year - 2017
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/sapm.12166
Subject(s) - eigenvalues and eigenvectors , pulse (music) , periodic boundary conditions , traveling wave , mathematics , mathematical analysis , operator (biology) , boundary value problem , stability (learning theory) , interval (graph theory) , boundary (topology) , elliptic function , pulse wave , physics , optics , quantum mechanics , computer science , laser , biochemistry , chemistry , repressor , combinatorics , machine learning , detector , transcription factor , gene
We consider the Ostrovsky and short pulse models in a symmetric spatial interval, subject to periodic boundary conditions. For the Ostrovsky case, we rederive the formulas for the classical periodic traveling waves, while for the short pulse model, we explicitly construct traveling waves in terms of Jacobi elliptic functions. In both cases, we show spectral stability, for all values of the parameters. This is achieved by studying the nonstandard eigenvalue problems in the form L [ u ] = λ u ′ , where L is a Hill operator.