American Institute of Mathematical Sciences

Advanced
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

Roy H. Goodman 1, , Jeremy L. Marzuola 2, and  Michael I. Weinstein 3,

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].
Keywords: shadowing., tunneling, Hamiltonian systems, Nonlinear Schrödinger equation, effective dynamics.
Mathematics Subject Classification: Primary: 35Q55, 35Q6.
Citation: Roy H. Goodman, Jeremy L. Marzuola, Michael I. Weinstein. Self-trapping and Josephson tunneling solutions to the nonlinear Schrödinger / Gross-Pitaevskii equation. Discrete & 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.  Google Scholar

[2]

R. W. Boyd, Nonlinear Optics, 3rd edition, Academic Press, 2008.  Google Scholar

[3]

P. Byrd and M. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists, 2nd edition, Springer-Verlag, 1971.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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). Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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). Google Scholar

[11]

I. S. Gradshteyn and I. M Ryzhik, Table of Integrals, Series, and Products, 7th edition, Elsevier, 2007.  Google Scholar

[12]

E. Harrell, Double wells, Comm. Math. Phys., 75 (1980), 239-261. doi: 10.1007/BF01212711.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

K. Meyer, G. Hall and D. Offin, Introduction to Hamiltonian Dynamical Systems and the N-Body Problem, Applied Mathematical Sciences, 90, Springer, 2010.  Google Scholar

[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.  Google Scholar

[20]

N. N. Nekhoroshev, Behavior of Hamiltonian systems close to integrable, Funct. Anal. Appl., 5 (1971), 338-339. doi: 10.1007/BF01086753.  Google Scholar

[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.  Google Scholar

[22]

NIST Digital Library of Mathematical Functions, http://dlmf.nist.gov/,, Release 1.0.5 of 2012-10-01., (): 2012.  http://dlmf.nist.gov/,+" target="_new" title="Go to article in Google Scholar"> Google Scholar

[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.  Google Scholar

[24]

L. Pitaevskii and S. Stringari, Bose-Einstein Condensation, International Series of Monographs on Physics, 116, The Clarendon Press, Oxford University Press, Oxford, 2003.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

R. W. Boyd, Nonlinear Optics, 3rd edition, Academic Press, 2008.  Google Scholar

[3]

P. Byrd and M. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists, 2nd edition, Springer-Verlag, 1971.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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). Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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). Google Scholar

[11]

I. S. Gradshteyn and I. M Ryzhik, Table of Integrals, Series, and Products, 7th edition, Elsevier, 2007.  Google Scholar

[12]

E. Harrell, Double wells, Comm. Math. Phys., 75 (1980), 239-261. doi: 10.1007/BF01212711.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

K. Meyer, G. Hall and D. Offin, Introduction to Hamiltonian Dynamical Systems and the N-Body Problem, Applied Mathematical Sciences, 90, Springer, 2010.  Google Scholar

[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.  Google Scholar

[20]

N. N. Nekhoroshev, Behavior of Hamiltonian systems close to integrable, Funct. Anal. Appl., 5 (1971), 338-339. doi: 10.1007/BF01086753.  Google Scholar

[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.  Google Scholar

[22]

NIST Digital Library of Mathematical Functions, http://dlmf.nist.gov/,, Release 1.0.5 of 2012-10-01., (): 2012.  http://dlmf.nist.gov/,+" target="_new" title="Go to article in Google Scholar"> Google Scholar

[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.  Google Scholar

[24]

L. Pitaevskii and S. Stringari, Bose-Einstein Condensation, International Series of Monographs on Physics, 116, The Clarendon Press, Oxford University Press, Oxford, 2003.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

Walid K. Abou Salem, Xiao Liu, Catherine Sulem. Numerical simulation of resonant tunneling of fast solitons for the nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems, 2011, 29 (4) : 1637-1649. doi: 10.3934/dcds.2011.29.1637
[2]

Takahisa Inui. Global dynamics of solutions with group invariance for the nonlinear schrödinger equation. Communications on Pure & Applied Analysis, 2017, 16 (2) : 557-590. doi: 10.3934/cpaa.2017028
[3]

Sandra Lucente, Eugenio Montefusco. Non-hamiltonian Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 761-770. doi: 10.3934/dcdss.2013.6.761
[4]

D.G. deFigueiredo, Yanheng Ding. Solutions of a nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems, 2002, 8 (3) : 563-584. doi: 10.3934/dcds.2002.8.563
[5]

Hartmut Schwetlick, Daniel C. Sutton, Johannes Zimmer. Effective Hamiltonian dynamics via the Maupertuis principle. Discrete & Continuous Dynamical Systems - S, 2020, 13 (4) : 1395-1410. doi: 10.3934/dcdss.2020078
[6]

José Luis López. A quantum approach to Keller-Segel dynamics via a dissipative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2601-2617. doi: 10.3934/dcds.2020376
[7]

Fengshuang Gao, Yuxia Guo. Multiple solutions for a nonlinear Schrödinger systems. Communications on Pure & Applied Analysis, 2020, 19 (2) : 1181-1204. doi: 10.3934/cpaa.2020055
[8]

Pavel I. Naumkin, Isahi Sánchez-Suárez. On the critical nongauge invariant nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems, 2011, 30 (3) : 807-834. doi: 10.3934/dcds.2011.30.807
[9]

Tarek Saanouni. Remarks on the damped nonlinear Schrödinger equation. Evolution Equations & Control Theory, 2020, 9 (3) : 721-732. doi: 10.3934/eect.2020030
[10]

Younghun Hong. Scattering for a nonlinear Schrödinger equation with a potential. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1571-1601. doi: 10.3934/cpaa.2016003
[11]

Alexander Komech, Elena Kopylova, David Stuart. On asymptotic stability of solitons in a nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1063-1079. doi: 10.3934/cpaa.2012.11.1063
[12]

Dario Bambusi, A. Carati, A. Ponno. The nonlinear Schrödinger equation as a resonant normal form. Discrete & Continuous Dynamical Systems - B, 2002, 2 (1) : 109-128. doi: 10.3934/dcdsb.2002.2.109
[13]

Hans Zwart, Yann Le Gorrec, Bernhard Maschke. Relating systems properties of the wave and the Schrödinger equation. Evolution Equations & Control Theory, 2015, 4 (2) : 233-240. doi: 10.3934/eect.2015.4.233
[14]

Mohamad Darwich. On the $L^2$-critical nonlinear Schrödinger Equation with a nonlinear damping. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2377-2394. doi: 10.3934/cpaa.2014.13.2377
[15]

Fuzhong Cong, Jialin Hong, Hongtian Li. Quasi-effective stability for nearly integrable Hamiltonian systems. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 67-80. doi: 10.3934/dcdsb.2016.21.67
[16]

Partha Guha, Indranil Mukherjee. Hierarchies and Hamiltonian structures of the Nonlinear Schrödinger family using geometric and spectral techniques. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1677-1695. doi: 10.3934/dcdsb.2018287
[17]

Alessio Pomponio, Simone Secchi. A note on coupled nonlinear Schrödinger systems under the effect of general nonlinearities. Communications on Pure & Applied Analysis, 2010, 9 (3) : 741-750. doi: 10.3934/cpaa.2010.9.741
[18]

Jiabao Su, Rushun Tian, Zhi-Qiang Wang. Positive solutions of doubly coupled multicomponent nonlinear Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 2143-2161. doi: 10.3934/dcdss.2019138
[19]

David Usero. Dark solitary waves in nonlocal nonlinear Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2011, 4 (5) : 1327-1340. doi: 10.3934/dcdss.2011.4.1327
[20]

Huifang Jia, Gongbao Li, Xiao Luo. Stable standing waves for cubic nonlinear Schrödinger systems with partial confinement. Discrete & Continuous Dynamical Systems, 2020, 40 (5) : 2739-2766. doi: 10.3934/dcds.2020148

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (46)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy