# American Institute of Mathematical Sciences

January  2015, 35(1): 225-246. doi: 10.3934/dcds.2015.35.225

## 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

Received  November 2013 Revised  April 2014 Published  August 2014

We study the long-time behavior of solutions to the nonlinear Schrödinger / Gross-Pitaevskii equation (NLS/GP) with a symmetric double-well potential. NLS/GP governs nearly-monochromatic guided optical beams in weakly coupled waveguides with both linear and nonlinear (Kerr) refractive indices and zero absorption, as well as the behavior of Bose-Einstein condensates. For small $L^2$ norm (low power), the solution executes beating oscillations between the two wells. There is a power threshold at which a symmetry breaking bifurcation occurs. The set of guided mode solutions splits into two families of solutions. One type of solution is concentrated in either well of the potential, but not both. Solutions in the second family undergo tunneling oscillations between the two wells. A finite dimensional reduction (system of ODEs) derived in [17] is expected to well-approximate the PDE dynamics on long time scales. In particular, we revisit this reduction, find a class of exact solutions and shadow them in the (NLS/GP) system by applying the approach of [17].
Citation: Roy H. Goodman, Jeremy L. Marzuola, Michael I. Weinstein. Self-trapping and Josephson tunneling solutions to the nonlinear Schrödinger / Gross-Pitaevskii equation. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 225-246. doi: 10.3934/dcds.2015.35.225
##### 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. [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. [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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