Stability of exact solutions of the nonlinear Schrödinger equation in an external potential having supersymmetry and parity-time symmetry
- Santa Fe Inst. (SFI), Santa Fe, NM (United States); Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
- Savitribai Phule Pune Univ., Pune (India). Physics Dept.
- Texas A & M Univ., College Station, TX (United States). Dept. of Mathematics; Inst. for Information Transmission Problems, Moscow (Russia)
- National Science Foundation, Arlington, VA (United States); Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
- Univ. of New Hampshire, Durham, NH (United States). Dept. of Physics
- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
We discuss the stability properties of the solutions of the general nonlinear Schrödinger equation (NLSE) in 1+1 dimensions in an external potential derivable from a parity-time ($${ \mathcal P }{ \mathcal T }$$) symmetric superpotential $W(x)$ that we considered earlier, Kevrekidis et al (2015 Phys. Rev. E 92 042901). In particular we consider the nonlinear partial differential equation $$\{{\rm{i}}\,{\partial }_{t}+{\partial }_{x}^{2}-{V}^{-}(x)+| \psi (x,t){| }^{2\kappa }\}\,\psi (x,t)=0,$$ for arbitrary nonlinearity parameter κ. We study the bound state solutions when $${V}^{-}(x)\,=(1/4-{b}^{2}){\text{sech}}^{2}(x)$$, which can be derived from two different superpotentials $W(x)$, one of which is complex and $${ \mathcal P }{ \mathcal T }$$ symmetric. Using Derrick's theorem, as well as a time dependent variational approximation, we derive exact analytic results for the domain of stability of the trapped solution as a function of the depth b 2 of the external potential. We compare the regime of stability found from these analytic approaches with a numerical linear stability analysis using a variant of the Vakhitov–Kolokolov (V–K) stability criterion. The numerical results of applying the V–K condition give the same answer for the domain of stability as the analytic result obtained from applying Derrick's theorem. Our main result is that for $$\kappa \gt 2$$ a new regime of stability for the exact solutions appears as long as $$b\gt {b}_{{\rm{crit}}}$$, where $${b}_{{\rm{crit}}}$$ is a function of the nonlinearity parameter κ. In the absence of the potential the related solitary wave solutions of the NLSE are unstable for $$\kappa \gt 2$$.
- Research Organization:
- Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)
- Sponsoring Organization:
- USDOE Laboratory Directed Research and Development (LDRD) Program
- Grant/Contract Number:
- 89233218CNA000001
- OSTI ID:
- 1499332
- Report Number(s):
- LA-UR-16-22361
- Journal Information:
- Journal of Physics. A, Mathematical and Theoretical, Vol. 50, Issue 1; ISSN 1751-8113
- Publisher:
- IOP PublishingCopyright Statement
- Country of Publication:
- United States
- Language:
- English
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