Solitons in a nonlinear Schrödinger equation withPT-symmetric potentials and inhomogeneous nonlinearity: Stability and excitation of nonlinear modes

Abstract

We report branches of explicit expressions for nonlinear modes in parity-time (PT ) symmetric potentials of several types. For the single-well and double-well potentials the found solutions are two-parametric and appear to be stable even when the PT -symmetry of respective underlying linear models is broken. Based on the examples of these solutions we describe an algorithm of excitation of a stable nonlinear mode in a model, whose linear limit is unstable. The method is based on the adiabatic change of the control parameter driving the mode along a branch bifurcating from a stable linear mode. The suggested algorithm is confirmed by extensive numerical simulations.

Stability of matter-wave solitons in a density-dependent gauge theory

“…We return to the zero-eigenvalue problem described by Eqs. (23) and (24), which contain a homogeneous and an inhomogeneous eigenvalue problem for the eigenvector components u(x) and v(x) respectively. The general solution…”

“…which are a standard reflectionless potential [70,71] and a gain-loss distribution for the excitations respectively [21][22][23][24]. Together, they form a modified (hyperbolic) Scarf-II potential with the following properties [21]:…”

“…Most notable is the Vakhitov-Kolokolov criterion [15,16], which connects the linear stability of bright solitons to two key properties: the number of negative eigenvalues in the spectral problem, and the behaviour of the power integral with respect to the propagation constant. In addition to this pioneering work, several studies have addressed the addition of perturbative terms in the model, such as but not limited to, the excitation of internal modes in non-Kerr media [17][18][19], the effects of third-order dispersion and self-steepening [10,20], and more recently for PTsymmetric potentials [21][22][23][24], which describe media with complex refractive indices.…”

Perturbative dynamics of stationary states in nonlinear parity-time symmetric coupler

“…Since then there has been numerous works on PT symmetry in optics, both in experimental as well as theoretical settings. PT Symmetry is studied in various contexts such as: Bragg solitons in nonlinear PT-symmetric periodic potential [9], continuous and discrete Schrodinger systems with PTsymmetric nonlinearities [10][11][12], bright and dark solitons and existence of optical rogue waves [13][14][15][16][17][18][19], modulation instability in nonlinear PT-symmetric structures [20][21], optical oligomers [22][23][24][25][26][27][28][29], optical mesh lattices [30][31][32][33], unidirectional invisibility [34], non-reciprocity and power oscillations [35,36], field propagation in linear and nonlinear stochastic PT coupler [37], optical mode conversion and transmission on photonic circuits [38] and so on. In coupled waveguide systems, the PT phase transition is characterized by exponential growth and decay of optical power.…”

Rational Solutions of a Weakly Coupled Nonlocal Nonlinear Schrödinger Equation

“…The concept of PT-symmetry, based on the non-Hermitian Hamiltonians [2,[19][20][21][22], has recently attracted much attention [23], in particular in the fields of optics and photonics [9,24,25]. This concept offers a fertile ground for PT-related notions and experiments.…”