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The non-linear Schrödinger equation with a periodic δ-interaction

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Abstract

We study the existence and stability of space-periodic standing waves for the space-periodic cubic nonlinear Schrödinger equation with a point defect determined by a space-periodic Dirac distributionat the origin. This equation admits a smooth curve of positive space-periodic solutions with a profile given by the Jacobi elliptic function of dnoidal type. Via a perturbationmethod and continuation argument, we prove that in the case of an attractive defect the standing wave solutions are stable in H 1 per ([−π, π]) with respect to perturbations which have the same space-periodic as the wave itself. In the case of a repulsive defect, the standing wave solutions are stable in the subspace of even functions of H 1 per ([−π, π]) and unstable in H 1 per ([−π, π]) with respect to perturbations which have the same space-periodic as the wave itself.

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Authors and Affiliations

  1. Department of Mathematics, IME-USP Rua do Matão, 1010, Cidade Universitária 05508-090, São Paulo, SP, Brazil

    Jaime Angulo Pava

  2. Department of Mathematics, UCSB Santa, Barbara, CA, 93106, USA

    Gustavo Ponce

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  1. Jaime Angulo Pava
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  2. Gustavo Ponce
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Correspondence to Jaime Angulo Pava.

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Pava, J.A., Ponce, G. The non-linear Schrödinger equation with a periodic δ-interaction. Bull Braz Math Soc, New Series 44, 497–551 (2013). https://doi.org/10.1007/s00574-013-0024-8

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  • DOI: https://doi.org/10.1007/s00574-013-0024-8

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