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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 - A, 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).  doi: 10.1103/PhysRevLett.95.010402.  Google Scholar

[2]

R. W. Boyd, Nonlinear Optics,, 3rd edition, (2008).   Google Scholar

[3]

P. Byrd and M. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists,, 2nd edition, (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.  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.  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, (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.  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.  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).  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, (2007).   Google Scholar

[12]

E. Harrell, Double wells,, Comm. Math. Phys., 75 (1980), 239.  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.  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.  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.  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.  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.  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, (2010).   Google Scholar

[19]

K. R. Meyer, Jacobi elliptic functions from a dynamical systems point of view,, Am. Math. Mon., 108 (2001), 729.  doi: 10.2307/2695616.  Google Scholar

[20]

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

[21]

A. C. Newell and J. V. Moloney, Nonlinear Optics,, Advanced Book Program, (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.   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.  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, (2003).   Google Scholar

[25]

E. Shlizerman and V. Rom-Kedar, Hierarchy of bifurcations in the truncated and forced nonlinear Schrödinger model,, Chaos, 15 (2005).  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).  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.  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.  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).  doi: 10.1103/PhysRevLett.95.010402.  Google Scholar

[2]

R. W. Boyd, Nonlinear Optics,, 3rd edition, (2008).   Google Scholar

[3]

P. Byrd and M. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists,, 2nd edition, (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.  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.  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, (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.  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.  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).  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, (2007).   Google Scholar

[12]

E. Harrell, Double wells,, Comm. Math. Phys., 75 (1980), 239.  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.  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.  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.  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.  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.  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, (2010).   Google Scholar

[19]

K. R. Meyer, Jacobi elliptic functions from a dynamical systems point of view,, Am. Math. Mon., 108 (2001), 729.  doi: 10.2307/2695616.  Google Scholar

[20]

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

[21]

A. C. Newell and J. V. Moloney, Nonlinear Optics,, Advanced Book Program, (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.   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.  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, (2003).   Google Scholar

[25]

E. Shlizerman and V. Rom-Kedar, Hierarchy of bifurcations in the truncated and forced nonlinear Schrödinger model,, Chaos, 15 (2005).  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).  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.  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.  doi: 10.1016/j.physd.2012.10.006.  Google Scholar

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