Stability of new exact solutions of the nonlinear Schrödinger equation in a Pöschl–Teller external potential
Abstract
Here, we discuss the stability properties of the solutions of the general nonlinear Schrödinger equation in 1 + 1 dimensions in an external potential derivable from a paritytime ($$ \newcommand{\PT}{\mathcal{PT}} \PT$$ ) symmetric superpotential $W(x)$ that we considered earlier in Kevrekidis et al (2015 Phys. Rev. E 92 042901). In particular we consider the nonlinear partial differential equation $$ \{i \, \partial_t + \partial_x^2  V(x) + g \vert \psi(x, t) \vert ^{2\kappa} \} \, \psi(x, t) = 0 \>, $$ for arbitrary nonlinearity parameter κ, where $$g= \pm1$$ and V is the well known Pöschl–Teller potential which we allow to be repulsive as well as attractive. Using energy landscape methods, linear stability analysis as well as a time dependent variational approximation, we derive consistent analytic results for the domains of instability of these new exact solutions as a function of the strength of the external potential and κ. For the repulsive potential we show that there is a translational instability which can be understood in terms of the energy landscape as a function of a stretching parameter and a translation parameter being a saddle near the exact solution. In this case, numerical simulations show that if we start with the exact solution, the initial wave function breaks into two pieces traveling in opposite directions. If we explore the slightly perturbed solution situations, a 1% change in initial conditions can change significantly the details of how the wave function breaks into two separate pieces. For the attractive potential, changing the initial conditions by 1% modifies the domain of stability only slightly. For the case of the attractive potential and negative g perturbed solutions merely oscillate with the oscillation frequencies predicted by the variational approximation.
 Authors:

 Univ. of New Hampshire, Durham, NH (United States)
 The Santa Fe Institute, Santa Fe, NM (United States); Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
 Savitribai Phule Pune Univ., Pune (India)
 National Science Foundation, Arlington, VA (United States); Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
 Pontifical Catholic Univ. of Chile, Santiago (Chile)
 Texas A & M Univ., College Station, TX (United States)
 Texas A & M Univ., College Station, TX (United States); St. Petersburg State Univ., St. Petersburg (Russia); Institute for Information Transmission Problems, Moscow (Russia)
 Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
 Publication Date:
 Research Org.:
 Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
 Sponsoring Org.:
 USDOE Laboratory Directed Research and Development (LDRD) Program
 OSTI Identifier:
 1514934
 Report Number(s):
 LAUR1723542
Journal ID: ISSN 17518113
 Grant/Contract Number:
 89233218CNA000001
 Resource Type:
 Accepted Manuscript
 Journal Name:
 Journal of Physics. A, Mathematical and Theoretical
 Additional Journal Information:
 Journal Volume: 50; Journal Issue: 50; Journal ID: ISSN 17518113
 Publisher:
 IOP Publishing
 Country of Publication:
 United States
 Language:
 English
 Subject:
 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Mathematics; PTsymmetric superpotential; variational approximation; translational instability; Derrick’s theorem; collective coordinate
Citation Formats
Dawson, John F., Cooper, Fred, Khare, Avinash, Mihaila, Bogdan, Arévalo, Edward, Lan, Ruomeng, Comech, Andrew, and Saxena, Avadh. Stability of new exact solutions of the nonlinear Schrödinger equation in a Pöschl–Teller external potential. United States: N. p., 2017.
Web. https://doi.org/10.1088/17518121/aa9006.
Dawson, John F., Cooper, Fred, Khare, Avinash, Mihaila, Bogdan, Arévalo, Edward, Lan, Ruomeng, Comech, Andrew, & Saxena, Avadh. Stability of new exact solutions of the nonlinear Schrödinger equation in a Pöschl–Teller external potential. United States. https://doi.org/10.1088/17518121/aa9006
Dawson, John F., Cooper, Fred, Khare, Avinash, Mihaila, Bogdan, Arévalo, Edward, Lan, Ruomeng, Comech, Andrew, and Saxena, Avadh. Fri .
"Stability of new exact solutions of the nonlinear Schrödinger equation in a Pöschl–Teller external potential". United States. https://doi.org/10.1088/17518121/aa9006. https://www.osti.gov/servlets/purl/1514934.
@article{osti_1514934,
title = {Stability of new exact solutions of the nonlinear Schrödinger equation in a Pöschl–Teller external potential},
author = {Dawson, John F. and Cooper, Fred and Khare, Avinash and Mihaila, Bogdan and Arévalo, Edward and Lan, Ruomeng and Comech, Andrew and Saxena, Avadh},
abstractNote = {Here, we discuss the stability properties of the solutions of the general nonlinear Schrödinger equation in 1 + 1 dimensions in an external potential derivable from a paritytime ($ \newcommand{\PT}{\mathcal{PT}} \PT$ ) symmetric superpotential $W(x)$ that we considered earlier in Kevrekidis et al (2015 Phys. Rev. E 92 042901). In particular we consider the nonlinear partial differential equation $ \{i \, \partial_t + \partial_x^2  V(x) + g \vert \psi(x, t) \vert ^{2\kappa} \} \, \psi(x, t) = 0 \>, $ for arbitrary nonlinearity parameter κ, where $g= \pm1$ and V is the well known Pöschl–Teller potential which we allow to be repulsive as well as attractive. Using energy landscape methods, linear stability analysis as well as a time dependent variational approximation, we derive consistent analytic results for the domains of instability of these new exact solutions as a function of the strength of the external potential and κ. For the repulsive potential we show that there is a translational instability which can be understood in terms of the energy landscape as a function of a stretching parameter and a translation parameter being a saddle near the exact solution. In this case, numerical simulations show that if we start with the exact solution, the initial wave function breaks into two pieces traveling in opposite directions. If we explore the slightly perturbed solution situations, a 1% change in initial conditions can change significantly the details of how the wave function breaks into two separate pieces. For the attractive potential, changing the initial conditions by 1% modifies the domain of stability only slightly. For the case of the attractive potential and negative g perturbed solutions merely oscillate with the oscillation frequencies predicted by the variational approximation.},
doi = {10.1088/17518121/aa9006},
journal = {Journal of Physics. A, Mathematical and Theoretical},
number = 50,
volume = 50,
place = {United States},
year = {2017},
month = {11}
}
Figures / Tables:
Works referenced in this record:
Interplay between paritytime symmetry, supersymmetry, and nonlinearity: An analytically tractable case example
journal, October 2015
 Kevrekidis, Panayotis G.; Cuevas–Maraver, Jesús; Saxena, Avadh
 Physical Review E, Vol. 92, Issue 4
Making sense of nonHermitian Hamiltonians
journal, May 2007
 Bender, Carl M.
 Reports on Progress in Physics, Vol. 70, Issue 6
The Physics of NonHermitian Operators
journal, July 2006
 Geyer, Hendrik; Heiss, Dieter; Znojil, Miloslav
 Journal of Physics A: Mathematical and General, Vol. 39, Issue 32
$\mathcal{PT}$ Symmetric Periodic Optical Potentials
journal, February 2011
 Makris, K. G.; ElGanainy, R.; Christodoulides, D. N.
 International Journal of Theoretical Physics, Vol. 50, Issue 4
Physical realization of symmetric potential scattering in a planar slab waveguide
journal, February 2005
 Ruschhaupt, A.; Delgado, F.; Muga, J. G.
 Journal of Physics A: Mathematical and General, Vol. 38, Issue 9
Beam Dynamics in $\mathcal{P}\mathcal{T}$ Symmetric Optical Lattices
journal, March 2008
 Makris, K. G.; ElGanainy, R.; Christodoulides, D. N.
 Physical Review Letters, Vol. 100, Issue 10
Visualization of Branch Points in $\mathcal{P}\mathcal{T}$ Symmetric Waveguides
journal, August 2008
 Klaiman, Shachar; Günther, Uwe; Moiseyev, Nimrod
 Physical Review Letters, Vol. 101, Issue 8
Bloch Oscillations in Complex Crystals with $\mathcal{P}\mathcal{T}$ Symmetry
journal, September 2009
 Longhi, S.
 Physical Review Letters, Vol. 103, Issue 12
Dynamic localization and transport in complex crystals
journal, December 2009
 Longhi, S.
 Physical Review B, Vol. 80, Issue 23
Spectral singularities and Bragg scattering in complex crystals
journal, February 2010
 Longhi, S.
 Physical Review A, Vol. 81, Issue 2
Observation of parity–time symmetry in optics
journal, January 2010
 Rüter, Christian E.; Makris, Konstantinos G.; ElGanainy, Ramy
 Nature Physics, Vol. 6, Issue 3
Observation of $\mathcal{P}\mathcal{T}$ Symmetry Breaking in Complex Optical Potentials
journal, August 2009
 Guo, A.; Salamo, G. J.; Duchesne, D.
 Physical Review Letters, Vol. 103, Issue 9
Parity–time synthetic photonic lattices
journal, August 2012
 Regensburger, Alois; Bersch, Christoph; Miri, MohammadAli
 Nature, Vol. 488, Issue 7410
Experimental study of active LRC circuits with $\mathcal{PT}$ symmetries
journal, October 2011
 Schindler, Joseph; Li, Ang; Zheng, Mei C.
 Physical Review A, Vol. 84, Issue 4
$\mathcal{PT}$symmetric electronics
journal, October 2012
 Schindler, J.; Lin, Z.; Lee, J. M.
 Journal of Physics A: Mathematical and Theoretical, Vol. 45, Issue 44
Observation of PT phase transition in a simple mechanical system
journal, March 2013
 Bender, Carl M.; Berntson, Bjorn K.; Parker, David
 American Journal of Physics, Vol. 81, Issue 3
Parity–timesymmetric whisperinggallery microcavities
journal, April 2014
 Peng, Bo; Özdemir, Şahin Kaya; Lei, Fuchuan
 Nature Physics, Vol. 10, Issue 5
Supersymmetric Optical Structures
journal, June 2013
 Miri, MohammadAli; Heinrich, Matthias; ElGanainy, Ramy
 Physical Review Letters, Vol. 110, Issue 23
Supersymmetric mode converters
journal, April 2014
 Heinrich, Matthias; Miri, MohammadAli; Stützer, Simon
 Nature Communications, Vol. 5, Issue 1
Supersymmetry in quantum mechanics
journal, August 1985
 Gendenshteĭn, L. É; Krive, I. V.
 Soviet Physics Uspekhi, Vol. 28, Issue 8
Supersymmetry and quantum mechanics
journal, January 1995
 Cooper, Fred; Khare, Avinash; Sukhatme, Uday
 Physics Reports, Vol. 251, Issue 56
A new PT symmetric complex Hamiltonian with a real spectrum
journal, December 1999
 Bagchi, B.; Roychoudhury, R.
 Journal of Physics A: Mathematical and General, Vol. 33, Issue 1
Generating Complex Potentials with real Eigenvalues in Supersymmetric Quantum Mechanics
journal, June 2001
 Bagchi, B.; Mallik, S.; Quesne, C.
 International Journal of Modern Physics A, Vol. 16, Issue 16
sl(2, ) as a complex Lie algebra and the associated nonHermitian Hamiltonians with real eigenvalues
journal, September 2000
 Bagchi, B.; Quesne, C.
 Physics Letters A, Vol. 273, Issue 56
Real and complex discrete eigenvalues in an exactly solvable onedimensional complex invariant potential
journal, April 2001
 Ahmed, Zafar
 Physics Letters A, Vol. 282, Issue 6
Supersymmetrygenerated onewayinvisible $\mathcal{PT}$ symmetric optical crystals
journal, March 2014
 Midya, Bikashkali
 Physical Review A, Vol. 89, Issue 3
Bemerkungen zur Quantenmechanik des anharmonischen Oszillators
journal, March 1933
 P�schl, G.; Teller, E.
 Zeitschrift f�r Physik, Vol. 83, Issue 34
Gray solitons on the surface of water
journal, January 2014
 Chabchoub, A.; Kimmoun, O.; Branger, H.
 Physical Review E, Vol. 89, Issue 1
Gaussian wavepacket dynamics: Semiquantal and semiclassical phasespace formalism
journal, November 1994
 Pattanayak, Arjendu K.; Schieve, William C.
 Physical Review E, Vol. 50, Issue 5
Note on Exchange Phenomena in the Thomas Atom
journal, July 1930
 Dirac, P. A. M.
 Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 26, Issue 3
Comments on Nonlinear Wave Equations as Models for Elementary Particles
journal, September 1964
 Derrick, G. H.
 Journal of Mathematical Physics, Vol. 5, Issue 9
Modulational Stability of Ground States of Nonlinear Schrödinger Equations
journal, May 1985
 Weinstein, Michael I.
 SIAM Journal on Mathematical Analysis, Vol. 16, Issue 3
Generalized travelingwave method, variational approach, and modified conserved quantities for the perturbed nonlinear Schrödinger equation
journal, July 2010
 Quintero, Niurka R.; Mertens, Franz G.; Bishop, A. R.
 Physical Review E, Vol. 82, Issue 1
Quantum dynamics in a timedependent variational approximation
journal, December 1986
 Cooper, Fred; Pi, SoYoung; Stancioff, Paul N.
 Physical Review D, Vol. 34, Issue 12
Universal Critical Power for Nonlinear Schrödinger Equations with a Symmetric Double Well Potential
journal, November 2009
 Sacchetti, Andrea
 Physical Review Letters, Vol. 103, Issue 19