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Normal form for the symmetry-breaking bifurcation in the nonlinear Schrödinger equation. (English) Zbl 1251.35153

Summary: We derive and justify a normal form reduction of the nonlinear Schrödinger equation for a general pitchfork bifurcation of the symmetric bound state that occurs in a double-well symmetric potential. We prove persistence of normal form dynamics for both supercritical and subcritical pitchfork bifurcations in the time-dependent solutions of the nonlinear Schrödinger equation over long but finite time intervals.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
35B32 Bifurcations in context of PDEs
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[1] Albiez, M.; Gati, R.; Fölling, J.; Hunsmann, S.; Cristiani, M.; Oberthaler, M.K., Direct observation of tunneling and nonlinear self-trapping in a single bosonic Josephson junction, Phys. rev. lett., 95, 010402, (2005)
[2] Cuccagna, S., On asymptotic stability in energy space of ground states of NLS in 1D, J. differential equations, 245, 653-691, (2008) · Zbl 1185.35251
[3] Fukuizumi, R.; Sacchetti, A., Bifurcation and stability for nonlinear Schrödinger equations with double well potential in the semiclassical limit, (2011) · Zbl 1252.82017
[4] Grillakis, M., Analysis of the linearization around a critical point of an infinite dimensional Hamiltonian system, Comm. pure appl. math., 43, 299-333, (1990) · Zbl 0731.35010
[5] Grillakis, M.; Shatah, J.; Strauss, W., Stability theory of solitary waves in the presence of symmetry. I, J. funct. anal., 74, 160-197, (1987) · Zbl 0656.35122
[6] Jeanjean, H.; Stuart, C., Nonlinear eigenvalue problems having an unbounded branch of symmetric bound states, Adv. differential equations, 4, 639-670, (1999) · Zbl 0958.34017
[7] Kato, T., Perturbation theory for linear operators, (1976), Springer-Verlag
[8] Kevrekidis, P.G.; Chen, Z.; Malomed, B.A.; Frantzeskakis, D.J.; Weinstein, M.I., Spontaneous symmetry breaking in photonic lattices: theory and experiment, Phys. lett. A, 340, 275-280, (2005) · Zbl 1145.78310
[9] Kirr, E.W.; Kevrekidis, P.G.; Shlizerman, E.; Weinstein, M.I., Symmetry-breaking bifurcation in nonlinear Schrödinger/Gross-Pitaevskii equations, SIAM J. math. anal., 40, 566-604, (2008) · Zbl 1157.35479
[10] Kirr, E.W.; Kevrekidis, P.G.; Pelinovsky, D.E., Symmetry-breaking bifurcation in the nonlinear Schrödinger equation with symmetric potentials, Comm. math. phys., 308, 795-844, (2011) · Zbl 1235.34128
[11] Marzuola, J.L.; Weinstein, M.I., Long time dynamics near the symmetry breaking bifurcation for nonlinear Schrödinger/Gross-Pitaevskii equations, Discrete contin. dyn. syst. ser. A, 28, 1505-1554, (2010) · Zbl 1223.35288
[12] Mizumachi, T., Asymptotic stability of small solitary waves to 1D nonlinear Schrödinger equations with potential, J. math. Kyoto univ., 48, 471-497, (2008) · Zbl 1175.35138
[13] Ornigotti, M.; Della Valle, G.; Gatti, D.; Longhi, S., Topological suppression of optical tunneling in a twisted annular fiber, Phys. rev. A, 76, 023833, (2007)
[14] Sacchetti, A., Nonlinear double well Schrödinger equations in the semiclassical limit, J. stat. phys., 119, 1347-1382, (2005) · Zbl 1096.82014
[15] Sacchetti, A., Universal critical power for nonlinear Schrödinger equations with a symmetric double well potential, Phys. rev. lett., 103, 194101, (2009), 4 pp
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