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Instability of cnoidal‐peak for the NLS‐δ equation
Author(s) -
Pava Jaime Angulo
Publication year - 2012
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201100209
Subject(s) - cnoidal wave , elliptic function , mathematics , periodic wave , instability , mathematical analysis , mathematical physics , perturbation (astronomy) , cauchy distribution , initial value problem , nonlinear system , continuation , physics , wave equation , traveling wave , quantum mechanics , computer science , programming language
Abstract We study the existence and stability of standing waves for the periodic cubic nonlinear Schrödinger equation with a point defect determined by the periodic Dirac distribution at the origin. We show that this model admits a smooth curve of periodic‐peak standing wave solutions with a profile determined by the Jacobi elliptic function of cnoidal type. Via a perturbation method and continuation argument, we obtain that in the repulsive defect, the cnoidal‐peak standing wave solutions are unstable in \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$H^1_{per}$\end{document} with respect to perturbations which have the same period as the wave itself. Global well‐posedness is verified for the Cauchy problem in \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$H^1_{per}$\end{document} .