Stability of new exact solutions of the nonlinear Schrödinger equation in a Pöschl–Teller external potential
Autor: | Avinash Khare, Edward Arévalo, Bogdan Mihaila, Fred Cooper, Ruomeng Lan, Avadh Saxena, John F. Dawson, Andrew Comech |
---|---|
Rok vydání: | 2017 |
Předmět: |
Statistics and Probability
Physics Oscillation Superpotential FOS: Physical sciences General Physics and Astronomy Statistical and Nonlinear Physics Pattern Formation and Solitons (nlin.PS) Function (mathematics) Nonlinear Sciences - Pattern Formation and Solitons 01 natural sciences Domain (mathematical analysis) 010305 fluids & plasmas Nonlinear system symbols.namesake Exact solutions in general relativity Modeling and Simulation 0103 physical sciences symbols 010306 general physics Wave function Nonlinear Schrödinger equation Mathematical Physics Mathematical physics |
Zdroj: | Journal of Physics A: Mathematical and Theoretical. 50:505202 |
ISSN: | 1751-8121 1751-8113 |
DOI: | 10.1088/1751-8121/aa9006 |
Popis: | We discuss the stability properties of the solutions of the general nonlinear \Schrodinger\ equation (NLSE) in 1+1 dimensions in an external potential derivable from a parity-time ($\PT$) symmetric superpotential $W(x)$ that we considered earlier \cite{PhysRevE.92.042901}. In particular we consider the nonlinear partial differential equation $ \{ i \, \partial_t + \partial_x^2 - V(x) + g | \psi(x,t) |^{2\kappa} \} \, \psi(x,t) = 0 \>, $ for arbitrary nonlinearity parameter $\kappa$, where $g= \pm1$ and $V$ is the well known P{\"o}schl-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 $\kappa$. For the repulsive potential (and $g=+1$) 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 (and $g=+1$), 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. Comment: 16 pages, 10 figures |
Databáze: | OpenAIRE |
Externí odkaz: |