A Reverse Infinite-Period Bifurcation for the Nonlinear Schrodinger Equation in 2+1 Dimensions with a Parametric Excitation

Attilio Maccari


We consider the nonlinear Schrodinger equation in 2+1 dimensions and an external periodic excitation ns parametric resonance with the frequency of a generic mode. Using an adequate perturbation method we get two coupled equations for the amplitude and phase. We show frequency-response curves and demonstrate the existence for the focusing case of a reverse infinite-period bifurcation when the parametric excitation increases its value. The same bifurcation is possible even in the defocusing case but for a different excitation amplitude value.


nonlinear Schrodinger equation in 2+1 dimensions; parametric excitation; infinite-period bifurcation; vibration control

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B. Malomed, Nonlinear Schrodinger equations, (Scott A. editor),

Encyplopedya of Nonlinear Science, Rutledge, New york, (2005)

L. Pitaevskii and S. Stringari, Bose-Einstein condensation, Clarendon, Oxford (2003)

R. Balakrishnan, Soliton propagation in non uniform media, Physical Review A, 32(2), 1144-1149(1985)

V. E. Zakharov and S. V. Manakov, On the complete integrability of a nonlinear Schrodinger equation, Journal of Theoretical and Mathematical Physics 19(3), 551-559, (1974)

M. J. Ablowitz, Nonlinear dispersive waves. Asymptotic analysis and solitons, Cambridge University Press, (2011)

G. B. Whitham, Linear and nonlinear waves, Wiley-Interscience, (1974)

K. Dysthe, H. E. Krogstad, P. Muller, Oceanic rogue waves, Annual Review of Fluid Mechanics 40(1) 287-310.

E. B. Davies, Symmetry breaking for a nonlinear Schrodinger equation, Communications in Mathematical Physics, 64(3) 191-210, (1979)

K. W. Mahmud, J. N. Kutz, and W. P. Reinhardt, Bose-Einstein condensates in a one-dimensional double square well: Analyrical solutionsof the nonlinear Schrodinger equation, Physical Review A, 66(6), 063607, (2002)

J. L. Marzuola and M. J. Weinstein, Long timedynamics near the symmetrybreaking bifurcation for nonlinear Schrodinger/Gross-Pitaevskii equations, Discrete & Continous Dynamical Systems A, 28(4), 1505-1554, (2010)

R. K. Jackson and mI weinstein, Geometric analysis of bifurcation and symmetry breaking in a Gross-Pitaevskii equation, Journal of Statistical Physics, 116(1-4)881-905, (2004)

H. Susanto, J. Cuevas, P. Kruger, Josephson tunneling of dark solitons in a double-well potential, Journal of Physics B:Atomic, Molecular and OpticalPhysics 44(9), 095003, (2011)

H. Susanto and J. Cuevas, Josephson tunneling of excited states in a double-well potential, in Sponaneous Symmetry Breaking,in Self-Trapping, and Josephson Oscillations, Springer, Berlin, Heidelberg, 2012, 583-599

R. Marangell, C.K. R. T. Jones, H. Susanto, Localized standing wavesin inhomogenousShrodinger equations, Nonlinearity, 23(9), 2059, (2010)

R. Marangell, C.K. R. T. Jones, H. Susanto, Instability of standing wavesfor nonlinear Schrodinger-type equations, Journal of Differential Equations, 253(4), 1191-1205, (2012).

G. Falkovich, Fluid Mechanics, a short course for phisicists, Cambridge University Press, (2011)

A. Maccari, Coherent solutions for the fundamental resonance of the Boussinesq equation, Chaos, Solitons and Fractals 54, 57-64, (2013)


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