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Parametric normal forms for Bogdanov--Takens singularity; the generalized saddle-node case
Self-trapping and Josephson tunneling solutions to the nonlinear Schrödinger / Gross-Pitaevskii equation
1. | Department of Mathematical Sciences, New Jersey Institute of Technology, University Heights, Newark, NJ 07102, United States |
2. | Mathematics Department, University of North Carolina, Phillips Hall, CB#3250, Chapel Hill, NC 27599, United States |
3. | Department of Applied Physics and Applied Mathematics, Department of Mathematics, Columbia University, New York City, NY 10024, United States |
References:
[1] |
M. Albiez, R. Gati, J. Fölling, S. Hunsmann, M. Cristiani and M. K. Oberthaler, Direct observation of tunneling and nonlinear self-trapping in a single Bosonic Josephson junction, Phys. Rev. Lett., 95 (2005), 010402.
doi: 10.1103/PhysRevLett.95.010402. |
[2] |
R. W. Boyd, Nonlinear Optics, 3rd edition, Academic Press, 2008. |
[3] |
P. Byrd and M. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists, 2nd edition, Springer-Verlag, 1971. |
[4] |
X. Chen, Collapsing estimates and the rigorous derivation of the 2d cubic nonlinear Schrödinger equation with anisotropic switchable quadratic traps, J. Math. Pures Appl., 98 (2012), 450-478.
doi: 10.1016/j.matpur.2012.02.003. |
[5] |
X. Chen, On the rigorous derivation of the 3D cubic nonlinear Schrödinger equation with a quadratic trap, Arch. Ration. Mech. Anal., 210 (2013), 365-408.
doi: 10.1007/s00205-013-0645-5. |
[6] |
X. Chen and J. Holmer, Focusing quantum many-body dynamics: The rigorous derivation of the 1D focusing cubic nonlinear Schrödinger equation, preprint, arXiv:1308.3895, (2013). |
[7] |
L. Erdős, B. Schlein and H.-T. Yau, Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems, Inventiones mathematicae, 167 (2007), 515-614.
doi: 10.1007/s00222-006-0022-1. |
[8] |
A. Giorgilli and L. Galgani, Rigorous estimates for the series expansions of Hamiltonian perturbation theory, Celest. Mech. Dyn. Astr., 37 (1985), 95-112.
doi: 10.1007/BF01230921. |
[9] |
R. H. Goodman, Hamiltonian Hopf bifurcations and dynamics of NLS/GP standing-wave modes, J. Phys. A: Math. Theor., 44 (2011), 425101, 28 pp.
doi: 10.1088/1751-8113/44/42/425101. |
[10] |
R. H. Goodman, Bifurcations of relative periodic orbits in a reduction of the nonlinear Schrödinger equation with a multiple-well potential, in preparation, (2014). |
[11] |
I. S. Gradshteyn and I. M Ryzhik, Table of Integrals, Series, and Products, 7th edition, Elsevier, 2007. |
[12] |
E. Harrell, Double wells, Comm. Math. Phys., 75 (1980), 239-261.
doi: 10.1007/BF01212711. |
[13] |
T. Kapitula, P. G. Kevrekidis and Z. Chen, Three is a crowd: Solitary waves in photorefractive media with three potential wells, SIAM J. Appl. Dyn. Syst., 5 (2006), 598-633.
doi: 10.1137/05064076X. |
[14] |
E.-W. Kirr, P. G. Kevrekidis and D. E. Pelinovsky, Symmetry breaking bifurcation in nonlinear Schrödinger equation with symmetric potentials, Comm. Math. Phys., 308 (2011), 795-844.
doi: 10.1007/s00220-011-1361-3. |
[15] |
E.-W. Kirr, P. G. Kevrekidis, E. Shlizerman and M. I. Weinstein, Symmetry breaking bifurcation in nonlinear Schrödinger/Gross-Pitaevskii equations, SIAM J. Math. Anal., 40 (2008), 566-604.
doi: 10.1137/060678427. |
[16] |
G. Kovačič and S. Wiggins, Orbits homoclinic to resonances, with an application to chaos in a model of the forced and damped sine-Gordon equation, Phys. D, 57 (1992), 185-225.
doi: 10.1016/0167-2789(92)90092-2. |
[17] |
J. L. Marzuola and M. I. Weinstein, Long time dynamics near the symmetry breaking bifurcation for nonlinear Schrödinger/Gross-Pitaevskii equations, DCDS-A, 28 (2010), 1505-1554.
doi: 10.3934/dcds.2010.28.1505. |
[18] |
K. Meyer, G. Hall and D. Offin, Introduction to Hamiltonian Dynamical Systems and the N-Body Problem, Applied Mathematical Sciences, 90, Springer, 2010. |
[19] |
K. R. Meyer, Jacobi elliptic functions from a dynamical systems point of view, Am. Math. Mon., 108 (2001), 729-737.
doi: 10.2307/2695616. |
[20] |
N. N. Nekhoroshev, Behavior of Hamiltonian systems close to integrable, Funct. Anal. Appl., 5 (1971), 338-339.
doi: 10.1007/BF01086753. |
[21] |
A. C. Newell and J. V. Moloney, Nonlinear Optics, Advanced Book Program, Westview Press, 2003.
doi: 10.1007/978-94-009-0591-7_4. |
[22] |
NIST Digital Library of Mathematical Functions, http://dlmf.nist.gov/,, Release 1.0.5 of 2012-10-01., (): 2012.
|
[23] |
D. Pelinovsky and T. Phan, Normal form for the symmetry-breaking bifurcation in the nonlinear Schrödinger equation, J. Diff. Eq., 253 (2012), 2796-2824.
doi: 10.1016/j.jde.2012.07.007. |
[24] |
L. Pitaevskii and S. Stringari, Bose-Einstein Condensation, International Series of Monographs on Physics, 116, The Clarendon Press, Oxford University Press, Oxford, 2003. |
[25] |
E. Shlizerman and V. Rom-Kedar, Hierarchy of bifurcations in the truncated and forced nonlinear Schrödinger model, Chaos, 15 (2005), 013107.
doi: 10.1063/1.1831591. |
[26] |
G. Theocharis, P. G. Kevrekidis, D. J. Frantzeskakis and P. Schmelcher, Symmetry breaking in symmetric and asymmetric double-well potentials, Phys. Rev. E, 74 (2006), 056608.
doi: 10.1103/PhysRevE.74.056608. |
[27] |
J. Yang, Classification of solitary wave bifurcations in generalized nonlinear media, Stud. Appl. Math., 129 (2012), 133-162.
doi: 10.1111/j.1467-9590.2012.00549.x. |
[28] |
J. Yang, Stability analysis for pitchfork bifurcations of solitary waves in generalized nonlinear Schrödinger equations, Phys. D., 244 (2012), 50-67.
doi: 10.1016/j.physd.2012.10.006. |
show all references
References:
[1] |
M. Albiez, R. Gati, J. Fölling, S. Hunsmann, M. Cristiani and M. K. Oberthaler, Direct observation of tunneling and nonlinear self-trapping in a single Bosonic Josephson junction, Phys. Rev. Lett., 95 (2005), 010402.
doi: 10.1103/PhysRevLett.95.010402. |
[2] |
R. W. Boyd, Nonlinear Optics, 3rd edition, Academic Press, 2008. |
[3] |
P. Byrd and M. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists, 2nd edition, Springer-Verlag, 1971. |
[4] |
X. Chen, Collapsing estimates and the rigorous derivation of the 2d cubic nonlinear Schrödinger equation with anisotropic switchable quadratic traps, J. Math. Pures Appl., 98 (2012), 450-478.
doi: 10.1016/j.matpur.2012.02.003. |
[5] |
X. Chen, On the rigorous derivation of the 3D cubic nonlinear Schrödinger equation with a quadratic trap, Arch. Ration. Mech. Anal., 210 (2013), 365-408.
doi: 10.1007/s00205-013-0645-5. |
[6] |
X. Chen and J. Holmer, Focusing quantum many-body dynamics: The rigorous derivation of the 1D focusing cubic nonlinear Schrödinger equation, preprint, arXiv:1308.3895, (2013). |
[7] |
L. Erdős, B. Schlein and H.-T. Yau, Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems, Inventiones mathematicae, 167 (2007), 515-614.
doi: 10.1007/s00222-006-0022-1. |
[8] |
A. Giorgilli and L. Galgani, Rigorous estimates for the series expansions of Hamiltonian perturbation theory, Celest. Mech. Dyn. Astr., 37 (1985), 95-112.
doi: 10.1007/BF01230921. |
[9] |
R. H. Goodman, Hamiltonian Hopf bifurcations and dynamics of NLS/GP standing-wave modes, J. Phys. A: Math. Theor., 44 (2011), 425101, 28 pp.
doi: 10.1088/1751-8113/44/42/425101. |
[10] |
R. H. Goodman, Bifurcations of relative periodic orbits in a reduction of the nonlinear Schrödinger equation with a multiple-well potential, in preparation, (2014). |
[11] |
I. S. Gradshteyn and I. M Ryzhik, Table of Integrals, Series, and Products, 7th edition, Elsevier, 2007. |
[12] |
E. Harrell, Double wells, Comm. Math. Phys., 75 (1980), 239-261.
doi: 10.1007/BF01212711. |
[13] |
T. Kapitula, P. G. Kevrekidis and Z. Chen, Three is a crowd: Solitary waves in photorefractive media with three potential wells, SIAM J. Appl. Dyn. Syst., 5 (2006), 598-633.
doi: 10.1137/05064076X. |
[14] |
E.-W. Kirr, P. G. Kevrekidis and D. E. Pelinovsky, Symmetry breaking bifurcation in nonlinear Schrödinger equation with symmetric potentials, Comm. Math. Phys., 308 (2011), 795-844.
doi: 10.1007/s00220-011-1361-3. |
[15] |
E.-W. Kirr, P. G. Kevrekidis, E. Shlizerman and M. I. Weinstein, Symmetry breaking bifurcation in nonlinear Schrödinger/Gross-Pitaevskii equations, SIAM J. Math. Anal., 40 (2008), 566-604.
doi: 10.1137/060678427. |
[16] |
G. Kovačič and S. Wiggins, Orbits homoclinic to resonances, with an application to chaos in a model of the forced and damped sine-Gordon equation, Phys. D, 57 (1992), 185-225.
doi: 10.1016/0167-2789(92)90092-2. |
[17] |
J. L. Marzuola and M. I. Weinstein, Long time dynamics near the symmetry breaking bifurcation for nonlinear Schrödinger/Gross-Pitaevskii equations, DCDS-A, 28 (2010), 1505-1554.
doi: 10.3934/dcds.2010.28.1505. |
[18] |
K. Meyer, G. Hall and D. Offin, Introduction to Hamiltonian Dynamical Systems and the N-Body Problem, Applied Mathematical Sciences, 90, Springer, 2010. |
[19] |
K. R. Meyer, Jacobi elliptic functions from a dynamical systems point of view, Am. Math. Mon., 108 (2001), 729-737.
doi: 10.2307/2695616. |
[20] |
N. N. Nekhoroshev, Behavior of Hamiltonian systems close to integrable, Funct. Anal. Appl., 5 (1971), 338-339.
doi: 10.1007/BF01086753. |
[21] |
A. C. Newell and J. V. Moloney, Nonlinear Optics, Advanced Book Program, Westview Press, 2003.
doi: 10.1007/978-94-009-0591-7_4. |
[22] |
NIST Digital Library of Mathematical Functions, http://dlmf.nist.gov/,, Release 1.0.5 of 2012-10-01., (): 2012.
|
[23] |
D. Pelinovsky and T. Phan, Normal form for the symmetry-breaking bifurcation in the nonlinear Schrödinger equation, J. Diff. Eq., 253 (2012), 2796-2824.
doi: 10.1016/j.jde.2012.07.007. |
[24] |
L. Pitaevskii and S. Stringari, Bose-Einstein Condensation, International Series of Monographs on Physics, 116, The Clarendon Press, Oxford University Press, Oxford, 2003. |
[25] |
E. Shlizerman and V. Rom-Kedar, Hierarchy of bifurcations in the truncated and forced nonlinear Schrödinger model, Chaos, 15 (2005), 013107.
doi: 10.1063/1.1831591. |
[26] |
G. Theocharis, P. G. Kevrekidis, D. J. Frantzeskakis and P. Schmelcher, Symmetry breaking in symmetric and asymmetric double-well potentials, Phys. Rev. E, 74 (2006), 056608.
doi: 10.1103/PhysRevE.74.056608. |
[27] |
J. Yang, Classification of solitary wave bifurcations in generalized nonlinear media, Stud. Appl. Math., 129 (2012), 133-162.
doi: 10.1111/j.1467-9590.2012.00549.x. |
[28] |
J. Yang, Stability analysis for pitchfork bifurcations of solitary waves in generalized nonlinear Schrödinger equations, Phys. D., 244 (2012), 50-67.
doi: 10.1016/j.physd.2012.10.006. |
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