scholarly journals Coding of nonlinear states for the Gross–Pitaevskii equation with periodic potential

2013 ◽  
Vol 254 ◽  
pp. 29-45 ◽  
Author(s):  
G.L. Alfimov ◽  
A.I. Avramenko
Keyword(s):  
Periodic Potential ◽  
Pitaevskii Equation

scholarly journals Gap solitons for the repulsive Gross-Pitaevskii equation with periodic potential: Coding and method for computation

2017 ◽  
Vol 22 (4) ◽  
pp. 1207-1229 ◽  
Author(s):  
Georgy L. Alfimov ◽  
◽  
Pavel P. Kizin ◽  
Dmitry A. Zezyulin ◽  
Keyword(s):  
Periodic Potential ◽  
Pitaevskii Equation ◽  
Gap Solitons

Bifurcation of nonlinear bound states in the periodic Gross-Pitaevskii equation with 𝒫𝒯-symmetry

2019 ◽  
Vol 150 (1) ◽  
pp. 171-204
Author(s):  
Tomáš Dohnal ◽  
Dmitry Pelinovsky
Keyword(s):  
Bound States ◽  
Periodic Potential ◽  
Sufficient Conditions ◽  
Amplitude Equation ◽  
One Dimension ◽  
Real Interval ◽  
Pitaevskii Equation ◽  
Spectral Bands ◽  
Cubic Nonlinear Schrödinger Equation ◽  
Small Imaginary Part

AbstractThe stationary Gross–Pitaevskii equation in one dimension is considered with a complex periodic potential satisfying the conditions of the 𝒫𝒯 (parity-time reversal) symmetry. Under rather general assumptions on the potentials, we prove bifurcations of 𝒫𝒯-symmetric nonlinear bound states from the end points of a real interval in the spectrum of the non-selfadjoint linear Schrödinger operator with a complex 𝒫𝒯-symmetric periodic potential. The nonlinear bound states are approximated by the effective amplitude equation, which bears the form of the cubic nonlinear Schrödinger equation. In addition, we provide sufficient conditions for the appearance of complex spectral bands when the complex 𝒫𝒯-symmetric potential has an asymptotically small imaginary part.

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