Solitons in ${\mathscr{P}}{\mathscr{T}}$-symmetric ladders of optical waveguides

Alexeeva, N. V.; Barashenkov, I. V.; Kivshar, Yuri S.

doi:10.1088/1367-2630/aa8fdd

Cited by 24 publications

(15 citation statements)

References 92 publications

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“…It can be checked that, as well as it is obvious in Eqs. (37) and (38), the N (σ, k) dependence corresponding to the defocusing nonlinearity satisfies the above-mentioned anti-VK criterion, ∂N/∂k < 0 [79], at all values of σ. Accordingly, all the pinned modes are stable, both at γ = 0 and γ > 0, similar to the case of the self-defocusing cubic nonlinearity [63].…”

Section: Exact Solutions With the Self-defocusing Nonlinearitymentioning

confidence: 82%

Attraction centers and parity-time-symmetric delta-functional dipoles in critical and supercritical self-focusing media

2019

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We introduce a model based on the one-dimensional nonlinear Schrödinger equation (NLSE) with the critical (quintic) or supercritical self-focusing nonlinearity. We demonstrate that a family of solitons, which are unstable in this setting against the critical or supercritical collapse, is stabilized by pinning to an attractive defect, that may also include a parity-time (PT )-symmetric gain-loss component. The model can be realized as a planar waveguide in nonlinear optics, and in a super-Tonks-Girardeau bosonic gas. For the attractive defect with the delta-functional profile, a full family of the pinned solitons is found in an exact analytical form. In the absence of the gain-loss term, the solitons' stability is investigated in an analytical form too, by means of the Vakhitov-Kolokolov criterion; in the presence of the PT -balanced gain and loss, the stability is explored by means of numerical methods. In particular, the entire family of pinned solitons is stable in the quintic (critical) medium if the gain-loss term is absent. A stability region for the pinned solitons persists in the model with an arbitrarily high power of the self-focusing nonlinearity. A weak gain-loss component gives rise to intricate alternations of stability and instability in the system's parameter plane. Those solitons which are unstable under the action of the supercritical self-attraction are destroyed by the collapse. On the other hand, if the self-attraction-driven instability is weak and the gain-loss term is present, unstable solitons spontaneously transform into localized breathers, while the collapse does not occur. The same outcome may be caused by a combination of the critical nonlinearity with the gain and loss. Instability of the solitons is also possible when the PT -symmetric gain-loss term is added to the subcritical nonlinearity. The system with self-repulsive nonlinearity is briefly considered too, producing completely stable families of pinned localized states.

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Section: Exact Solutions With the Self-defocusing Nonlinearitymentioning

confidence: 82%

Attraction centers and parity-time-symmetric delta-functional dipoles in critical and supercritical self-focusing media

2019

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“…For this reason, antisymmetric solitons are not considered in detail below below (they may be made stable in a discrete version of Eqs. (2) [59]). As mentioned above, an essential difference of the PT -symmetric system (2) from its conservative counterpart, with γ 0 = 0, is that, at A 2 > A 2 crit , unstable PT -symmetric solitons are not replaced by stable asymmetric ones (cf.…”

Section: The Modelmentioning

confidence: 99%

“…The PT symmetry in an optical waveguide (as well as its CP counterpart) may naturally combine with the material Kerr nonlinearity, giving rise to propagation models based on cubic nonlinear Schrödinger equations (NLSEs) with the complex potentials subject to condition (1). These models may generate PT -symmetric solitons, which were addressed in many theoretical works [18], [23]- [59], [15] (see also reviews [41,42]), and experimentally demonstrated too [34]. Although the presence of the gain and loss makes PT -symmetric media dissipative, solitons exist in them in continuous families, similar to the commonly known situation in conservative models [43], while usual dissipative solitons exist as isolated solutions (attractors, if they are stable) [44].…”

Section: Introductionmentioning

confidence: 99%

Dynamical control of solitons in a parity-time-symmetric coupler by periodic management

Fan

Malomed

2019

Communications in Nonlinear Science and Numerical Simulation

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We consider a dual-core nonlinear waveguide with the parity-time (PT ) symmetry, realized in the form of equal gain and loss terms carried by the coupled cores. To expand a previously found stability region for solitons in this system, and explore possibilities for the development of dynamical control of the solitons, we introduce "management" in the form of periodic sinusoidal variation of the loss-gain (LG) coefficients, along with synchronous variation of the inter-core coupling (ICC) constant. This system, which can be realized in optics (in the temporal and spatial domains alike), features strong robustness when amplitudes of the variation of the LG and ICC coefficients keep a ratio equal to that of their constant counterparts, allowing one to find exact solutions for PTsymmetric solitons. A stability region for the solitons is identified in terms of the management amplitude and period, as well as the soliton's amplitude. In the long-period regime, the solitons evolve adiabatically, making it possible to predict their stability boundaries in an analytical form. The system keeping the Galilean invariance, collisions between moving solitons are considered too. Slowly moving solitons undergo multiple collisions, but eventually separate.

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“…The eigenvalues are symmetric relative to the imaginary axis. To be precise, if λ 0 is an eigenvalue with the eigenvector v 0 = (a, b) T , then −λ 0 is another eigenvalue with the eigenvector σ 3v0 = (a, −b) T by the Hamiltonian symmetry σ 3 L =Lσ 3 .…”

Section: Krein Signature For the Nls Equationmentioning

confidence: 99%

“…The linear Schrödinger equation with a complex-valued PT -symmetric potential was considered in [35], where the indefinite PT -inner product with the induced PT -Krein signature was introduced in the exact correspondence with the Krein signature for the Hamiltonian spectral problem (1.9). Coupled non-Hamiltonian PT -symmetric systems with constant coefficients were considered in [2,3] (see also [48]), where the linearized problem was block-diagonalized to the form for which the Krein signature of eigenvalues can be introduced. A Hamiltonian version of the PT -symmetric system of coupled oscillators was considered in [11,12], where the Krein signature of eigenvalues was introduced by using the corresponding Hamiltonian.…”

Section: Introductionmentioning

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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Solitons in ${\mathscr{P}}{\mathscr{T}}$-symmetric ladders of optical waveguides

Cited by 24 publications

References 92 publications

Attraction centers and parity-time-symmetric delta-functional dipoles in critical and supercritical self-focusing media

Attraction centers and parity-time-symmetric delta-functional dipoles in critical and supercritical self-focusing media

Dynamical control of solitons in a parity-time-symmetric coupler by periodic management

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

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