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TMF, 2014, Volume 179, Number 1, Pages 78–89 (Mi tmf8568)  

This article is cited in 9 scientific papers (total in 9 papers)

Short-wave transverse instabilities of line solitons of the two-dimensional hyperbolic nonlinear Schrödinger equation

D. E. Pelinovskyab, E. A. Ruvinskayaa, O. E. Kurkinaac, B. Deconinckd

a Nizhny Novgorod State Technical University, Nizhny Novgorod, Russia
b Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada
c Higher School of Economics, Nizhny Novgorod, Russia
d Department of Applied Mathematics, University of Washington, Seattle, WA, USA

Abstract: We prove that line solitons of the two-dimensional hyperbolic nonlinear Schrödinger equation are unstable under transverse perturbations of arbitrarily small periods, i.e., short waves. The analysis is based on the construction of Jost functions for the continuous spectrum of Schrödinger operators, the Sommerfeld radiation conditions, and the Lyapunov–Schmidt decomposition. We derive precise asymptotic expressions for the instability growth rate in the limit of short periods.

Keywords: nonlinear Schrödinger equation, soliton, transverse instability, Lyapunov–Schmidt decomposition, Fermi's golden rule

DOI: https://doi.org/10.4213/tmf8568

Full text: PDF file (734 kB)
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English version:
Theoretical and Mathematical Physics, 2014, 179:1, 452–461

Bibliographic databases:

Received: 24.06.2013

Citation: D. E. Pelinovsky, E. A. Ruvinskaya, O. E. Kurkina, B. Deconinck, “Short-wave transverse instabilities of line solitons of the two-dimensional hyperbolic nonlinear Schrödinger equation”, TMF, 179:1 (2014), 78–89; Theoret. and Math. Phys., 179:1 (2014), 452–461

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. J. T. Cole, Z. H. Musslimani, “Spectral transverse instabilities and soliton dynamics in the higher-order multidimensional nonlinear Schrödinger equation”, Physica D, 313 (2015), 26–36  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    2. Ch. Klein, J.-C. Saut, “A numerical approach to blow-up issues for Davey-Stewartson II systems”, Commun. Pure Appl. Anal, 14:4 (2015), 1443–1467  crossref  mathscinet  zmath  isi  scopus
    3. J. T. Cole, K. G. Makris, Z. H. Musslimani, D. N. Christodoulides, S. Rotter, “Modulational instability in PT-symmetric vector nonlinear Schrödinger system”, Physica D, 336 (2016), 53–61  crossref  mathscinet  zmath  isi  elib  scopus
    4. D. Pelinovsky, Yu. Shimabukuro, “Transverse instability of line solitary waves in massive Dirac equations”, J. Nonlinear Sci., 26:2 (2016), 365–403  crossref  mathscinet  zmath  isi  elib  scopus
    5. M. J. Ablowitz, Y.-P. Ma, I. Rumanov, “A universal asymptotic regime in the hyperbolic nonlinear Schrödinger equation”, SIAM J. Appl. Math., 77:4 (2017), 1248–1268  crossref  mathscinet  zmath  isi  scopus
    6. M. Stanislavova, A. Stefanov, “On the stability of standing waves for $\mathcal{PT}$ symmetric Schrödinger and Klein-Gordon equations in higher space dimensions”, Proc. Amer. Math. Soc., 145:12 (2017), 5273–5285  crossref  mathscinet  zmath  isi  scopus
    7. N. V. Alexeeva, I. V. Barashenkov, Y. S. Kivshar, “Solitons in $\mathcal{PT}$ symmetric ladders of optical waveguides”, New J. Phys., 19 (2017), 113032  crossref  mathscinet  isi  scopus
    8. D. Pelinovsky, “Normal form for transverse instability of the line soliton with a nearly critical speed of propagation”, Math. Model. Nat. Phenom., 13:2 (2018), UNSP 23  crossref  mathscinet  isi  scopus
    9. Cisneros-Ake L.A., Carretero-Gonzalez R., Kevrekidis P.G., Malomed B.A., “Dynamics and Stabilization of Bright Soliton Stripes in the Hyperbolic-Dispersion Nonlinear Schrodinger Equation”, Commun. Nonlinear Sci. Numer. Simul., 74 (2019), 268–281  crossref  mathscinet  isi  scopus
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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