Existence, Stability and Dynamics of Nonlinear Modes in a 2D PartiallyPT Symmetric Potential

D'Ambroise, Jennie; Kevrekidis, P. G.

doi:10.3390/app7030223

Cited by 2 publications

(3 citation statements)

References 24 publications

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“…(21) In such cases of double symmetries, there is no need for the potential to have a special form as in Eq. (18). In addition, the soliton branch that emerges from the spectrum edge possesses both symmetries whereas the bifurcating branch loses one of the symmetries although it retains the other.…”

Section: Symmetry Breaking In Two-dimensional Potentialsmentioning

confidence: 98%

“…On the one hand, work on complex, asymmetric so-called Wadati potentials has produced mono-parametric continuous families of stationary solutions [15,16]. On the other hand, the notion of partial PT -symmetry has been explored, e.g., with models that possess the symmetry in one of the directions but not in another [17,18]. In fact, in the recent work of [19,20] that motivated the present study, it was shown that to identify critical points one can localize a soliton 1 in a way such that its intensity has a vanishing total overlap with the imaginary part of the potential, assuming that the real part of the potential is proportional to the anti-derivative of the imaginary part (but without making any assumptions on the parity of either).…”

Section: Introductionmentioning

confidence: 99%

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Nonlinear Beam Propagation in a Class of Complex Non- P T $$\mathcal {PT}$$ -Symmetric Potentials

Cuevas–Maraver

Kevrekidis

Frantzeskakis

et al. 2018

Springer Tracts in Modern Physics

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The subject of PT -symmetry and its areas of application have been blossoming over the past decade. Here, we consider a nonlinear Schrödinger model with a complex potential that can be tuned controllably away from being PTsymmetric, as it might be the case in realistic applications. We utilize two parameters: the first one breaks PT -symmetry but retains a proportionality between the imaginary and the derivative of the real part of the potential; the second one, detunes from this latter proportionality. It is shown that the departure of the potential from the PT -symmetric form does not allow for the numerical identification of exact stationary solutions. Nevertheless, it is of crucial importance to consider the dynamical evolution of initial beam profiles. In that light, we define a suitable notion of optimization and find that even for non PT -symmetric cases, the beam dynamics, both in 1D and 2D -although prone to weak growth or decay-suggests that the optimized profiles do not change significantly under propagation for specific parameter regimes.

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Section: Symmetry Breaking In Two-dimensional Potentialsmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Nonlinear Beam Propagation in a Class of Complex Non- P T $$\mathcal {PT}$$ -Symmetric Potentials

Cuevas–Maraver

Kevrekidis

Frantzeskakis

et al. 2018

Springer Tracts in Modern Physics

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show abstract

“…On the practical side of specific applications, it would be especially relevant to consider, e.g., two-dimensional problems involving vorticity in settings such as the one of [1]. Also, more recently partially PT -symmetric settings have been introduced in [17,53] where one dimension retains the symmetry and the other dimension does not. Considering the applicability of the ideas herein in such systems or in systems with complex, yet non-PT -symmetric potentials with families of solutions [33,36] would also be of interest.…”

Section: Summary and Further Directionsmentioning

confidence: 99%

Krein Signature in Hamiltonian and P T $$\mathbb {PT}$$ -Symmetric Systems

Chernyavsky

Pelinovsky

2018

Springer Tracts in Modern Physics

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We explain the concept of Krein signature in Hamiltonian and PT -symmetric systems on the case study of the one-dimensional Gross-Pitaevskii equation with a real harmonic potential and an imaginary linear potential. These potentials correspond to the magnetic trap, and a linear gain/loss in the mean-field model of cigar-shaped Bose-Einstein condensates. For the linearized Gross-Pitaevskii equation, we introduce the real-valued Krein quantity, which is nonzero if the eigenvalue is neutrally stable and simple and zero if the eigenvalue is unstable. If the neutrally stable eigenvalue is simple, it persists with respect to perturbations. However, if it is multiple, it may split into unstable eigenvalues under perturbations. A necessary condition for the onset of instability past the bifurcation point requires existence of two simple neutrally stable eigenvalues of opposite Krein signatures before the bifurcation point. This property is useful in the parameter continuations of neutrally stable eigenvalues of the linearized Gross-Pitaevskii equation.

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Existence, Stability and Dynamics of Nonlinear Modes in a 2D PartiallyPT Symmetric Potential

Cited by 2 publications

References 24 publications

Nonlinear Beam Propagation in a Class of Complex Non- P T $$\mathcal {PT}$$ -Symmetric Potentials

Nonlinear Beam Propagation in a Class of Complex Non- P T $$\mathcal {PT}$$ -Symmetric Potentials

Krein Signature in Hamiltonian and P T $$\mathbb {PT}$$ -Symmetric Systems

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