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:
 Journal Article: 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. doi: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. doi: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. doi: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},
issn = {17518113},
number = 50,
volume = 50,
place = {United States},
year = {2017},
month = {11}
}
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