# 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 parity-time ($$ \newcommand{\PT}{\mathcal{PT}} \PT$$ ) 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):
- LA-UR-17-23542

Journal ID: ISSN 1751-8113

- 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 1751-8113

- Publisher:
- IOP Publishing

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Mathematics; PT-symmetric 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:10.1088/1751-8121/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:10.1088/1751-8121/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:10.1088/1751-8121/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 parity-time ($ \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 = {10.1088/1751-8121/aa9006},

journal = {Journal of Physics. A, Mathematical and Theoretical},

number = 50,

volume = 50,

place = {United States},

year = {2017},

month = {11}

}

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