Exactly Solvable Wadati Potentials in the PT-Symmetric Gross-Pitaevskii Equation

Barashenkov, I. V.; Zezyulin, Dmitry A.

doi:10.1007/978-3-319-31356-6_9

Cited by 18 publications

(22 citation statements)

References 28 publications

(52 reference statements)

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“…Note that there's a small region of stability for the nonlinear modes Φ of small amplitudes, as it was shown in [27]. Figure 2 (a)-(f) shows the instability bifurcation for the Scarf II potential (4) studied in [6] in the focusing case with g = 1. Here V 0 = 3, γ = −3.7, and the second branch of the nonlinear modes Φ is considered.…”

Section: Numerical Examplesmentioning

confidence: 71%

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Krein signature for instability of PT-symmetric states

Chernyavsky

Pelinovsky

2018

Physica D: Nonlinear Phenomena

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Krein quantity is introduced for isolated neutrally stable eigenvalues associated with the stationary states in the PT -symmetric nonlinear Schrödinger equation. Krein quantity is real and nonzero for simple eigenvalues but it vanishes if two simple eigenvalues coalesce into a defective eigenvalue. A necessary condition for bifurcation of unstable eigenvalues from the defective eigenvalue is proved. This condition requires the two simple eigenvalues before the coalescence point to have opposite Krein signatures. The theory is illustrated with several numerical examples motivated by recent publications in physics literature.

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Section: Numerical Examplesmentioning

confidence: 71%

“…This choice ensures that Φ remains nonzero up to 16 decimals on the interior grid points {x j } j=N −1 j=1 . The algorithm was tested on the exact solution derived in [6] for the Scarf II potential (4) with V 0 = 1 and µ = γ = −1:…”

Section: Numerical Approximationsmentioning

confidence: 99%

Krein signature for instability of PT-symmetric states

Chernyavsky

Pelinovsky

2018

Physica D: Nonlinear Phenomena

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“…A class of potentials introduced by Wadati [100] of the special form W 2 (x) − idW (x)/dx, where W (x) is a real-valued function, is attracting increasing attention in context of PT-symmetric nonlinear systems [101][102][103]. The PT-symmetric Wadati potentials have a unique feature of supporting continuous families of asymmetric solitons, in contrast to other PT-symmetric potentials [81,104].…”

Section: Solitons In Localized Potentialsmentioning

confidence: 99%

“…The PT-symmetric Wadati potentials have a unique feature of supporting continuous families of asymmetric solitons, in contrast to other PT-symmetric potentials [81,104]. A method of generation of Wadati potentials supporting exact localised solutions of the associated Gross-Pitaevskii equations has been reported in [103]. Generalized Wadati-like potentials of the form W 2 (x) + aW (x) + idW (x)/dx were considered in Refs.…”

Section: Solitons In Localized Potentialsmentioning

confidence: 99%

Nonlinear switching and solitons in PT‐symmetric photonic systems

Suchkov

Sukhorukov

Huang

et al. 2016

Laser & Photonics Reviews

347

263

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One of the challenges of the modern photonics is to develop all-optical devices enabling increased speed and energy efficiency for transmitting and processing information on an optical chip. It is believed that the recently suggested ParityTime (PT) symmetric photonic systems with alternating regions of gain and loss can bring novel functionalities. In such systems, losses are as important as gain and, depending on the structural parameters, gain compensates losses. Generally, PT systems demonstrate nontrivial non-conservative wave interactions and phase transitions, which can be employed for signal filtering and switching, opening new prospects for active control of light. In this review, we discuss a broad range of problems involving nonlinear PT-symmetric photonic systems with an intensitydependent refractive index. Nonlinearity in such PT symmetric systems provides a basis for many effects such as the formation of localized modes, nonlinearly-induced PT-symmetry breaking, and all-optical switching. Nonlinear PT-symmetric systems can serve as powerful building blocks for the development of novel photonic devices targeting an active light control.

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“…Most of them are related to the vector NLS, i.e. k = 1 and u is an n-component vector; for n = 2 this is the famous Manakov model [28], see also [5,8].…”

Section: Introductionmentioning

confidence: 99%

On Nonlocal Models of Kulish-Sklyanin Type and Generalized Fourier Transforms

Gerdjikov

2017

Advanced Computing in Industrial Mathematics

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A special class of multicomponent NLS equations, generalizing the vector NLS and related to the BD.I-type symmetric are shown to be integrable through the inverse scattering method (ISM). The corresponding fundamental analytic solutions are constructing thus reducing the inverse scattering problem to a RiemannHilbert problem. We introduce the minimal sets of scattering data T which determines uniquely the scattering matrix and the potential Q of the Lax operator. The elements of T can be viewed as the expansion coefficients of Q over the 'squared solutions' that are natural generalizations of the standard exponentials. Thus we demonstrate that the mapping T → Q is a generalized Fourier transform. Special attention is paid to two special representatives of this MNLS with three-component and five components which describe spinor (F = 1 and F = 2, respectively) BoseEinstein condensates.

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Exactly Solvable Wadati Potentials in the PT-Symmetric Gross-Pitaevskii Equation

Cited by 18 publications

References 28 publications

Krein signature for instability of PT-symmetric states

Krein signature for instability of PT-symmetric states

Nonlinear switching and solitons in PT‐symmetric photonic systems

On Nonlocal Models of Kulish-Sklyanin Type and Generalized Fourier Transforms

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