Category Indian Institute of Science Education and Research

PT-Symmetric dimer in a generalized model of coupled nonlinear oscillators

J. Cuevas-Maraver, A. Khare, P.G. Kevrekidis, H. Xu, A. Saxena

In the present work, we explore the case of a general PT-symmetric dimer in the context of two both linearly and nonlinearly coupled cubic oscillators. To obtain an analytical handle on the system, we first explore the rotating wave approximation converting it into a discrete nonlinear Schrodinger type dimer. In the latter context, the stationary solutions and their stability are identified numerically but also wherever possible analytically. Solutions stemming from both symmetric and anti-symmetric special limits are identified. A number of special cases are explored regarding the ratio of coefficients of nonlinearity between oscillators over the intrinsic one of each oscillator. Finally, the considerations are extended to the original oscillator model, where periodic orbits and their stability are obtained. When the solutions are found to be unstable their dynamics is monitored by means of direct numerical simulations.
Pattern Formation and Solitons (nlin.PS); Chaotic Dynamics (nlin.CD); Exactly Solvable and Integrable Systems (nlin.SI)

Periodic and Hyperbolic Soliton Solutions of a Number of Nonlocal PT-Symmetric Nonlinear Equations

Avinash Khare, Avadh Saxena

For a number of nonlocal nonlinear equations such as nonlocal, nonlinear Schrodinger equation (NLSE), nonlocal Ablowitz-Ladik (AL), nonlocal, saturable discrete NLSE (DNLSE), coupled nonlocal NLSE, coupled nonlocal AL and coupled nonlocal, saturable DNLSE, we obtain periodic solutions in terms of Jacobi elliptic functions as well as the corresponding hyperbolic soliton solutions. Remarkably, in all the six cases, we find that unlike the corresponding local cases, all the nonlocal models simultaneously admit both the bright and the dark soliton solutions. Further, in all the six cases, not only \(\rm{Dn}(x,m)\) and \(\rm{Cn}(x,m)\) but even their linear superposition is shown to be an exact solution. Finally, we show that the coupled nonlocal NLSE not only admits solutions in terms of Lame polynomials of order 1, but it also admits solutions in terms of Lame polynomials of order 2, even though they are not the solutions of the uncoupled nonlocal problem. We also remark on the possible integrability in certain cases.
Pattern Formation and Solitons (nlin.PS)

A study of PT-symmetric Non-linear Schroedinger Equation

K. Nireekshan Reddy, Subhrajit Modak, Kumar Abhinav, Prasanta K. Panigrahi

Systems governed by the Non-linear Schroedinger Equation (NLSE) with various external PT-symmetric potentials are considered. Exact solutions have been obtained for the same through the method of ansatz, some of them being solitonic in nature. It is found that only the unbroken PT-symmetric phase is realized in these systems, characterized by real energies.
Quantum Physics (quant-ph)

PT-Symmetric Dimer of Coupled Nonlinear Oscillators

J. Cuevas, P.G. Kevrekidis, A. Saxena, A. Khare

We provide a systematic analysis of a prototypical nonlinear oscillator system respecting PT-symmetry i.e., one of them has gain and the other an equal and opposite amount of loss. Starting from the linear limit of the system, we extend considerations to the nonlinear case for both soft and hard cubic nonlinearities identifying symmetric and anti-symmetric breather solutions, as well as symmetry breaking variants thereof. We propose a reduction of the system to a Schr\”odinger type PT-symmetric dimer, whose detailed earlier understanding can explain many of the phenomena observed herein, including the PT phase transition. Nevertheless, there are also significant parametric as well as phenomenological potential differences between the two models and we discuss where these arise and where they are most pronounced. Finally, we also provide examples of the evolution dynamics of the different states in their regimes of instability.

Pattern Formation and Solitons (nlin.PS)

Nonlinear localized modes in PT-symmetric optical media with competing gain and loss

Bikashkali Midya, Rajkumar Roychoudhury

The existence and stability of the nonlinear spatial localized modes are investigated in parity-time symmetric optical media characterized by a generic complex hyperbolic refractive index distribution with competing gain and loss profile. The exact analytical expressions of the localized modes are found for all values of the competing parameter and in the presence of both the self-focusing and self-defocusing Kerr nonlinearity. The effect of competing gain/loss profile on the stability structure of these localized modes are discussed with the help of linear stability analysis followed by the direct numerical simulation of the governing equation. The spatial localized modes in two-dimensional geometry as well as the transverse power-flow density associated with these localized modes are also examined.

Optics (physics.optics)

Non-Hermitian Oscillator and R-deformed Heisenberg Algebra

Rajkumar Roychoudhury, Barnana Roy, Partha Pratim Dube

A non-Hermitian generalized oscillator model, generally known as the Swanson model, has been studied in the framework of R-deformed Heisenberg algebra. The non-Hermitian Hamiltonian is diagonalized by generalized Bogoliubov transformation. A set of deformed creation annihilation operators is introduced whose algebra shows that the transformed Hamiltonian has conformal symmetry. The spectrum is obtained using algebraic technique. The superconformal structure of the system is also worked out in detail. An anomaly related to the spectrum of the Hermitian counterpart of the non-Hermitian Hamiltonian with generalized ladder operators is shown to occur and is discussed in position dependent mass scenario.
Mathematical Physics (math-ph); Quantum Physics (quant-ph)

From particle in a box to PT -symmetric systems via isospectral deformation

Philip Cherian, Kumar Abhinav, P. K. Panigrahi

A family of PT -symmetric complex potentials are obtained which is isospectral to free particle in an infinite complex box in one dimension (1-D). These are generalizations to the cosec2(x) potential, isospectral to particle in a real infinite box. In the complex plane, the infinite box is extended parallel to the real axis having a real width, which is found to be an integral multiple of a constant quantum factor, arising due to boundary conditions necessary for maintaining the PT -symmetry of the superpartner. As the spectra of the particle in a box is still real, it necessarily picks out the unbroken PT -sector of its superpartner, thereby invoking a close relation between PT -symmetry and SUSY for this case.
Mathematical Physics (math-ph); Quantum Physics (quant-ph)

Conserved Correlation in PT -symmetric Systems: Scattering and Bound States

Kumar Abhinav, Arun Jayannavar, P. K. Panigrahi

For one-dimensional PT -symmetric systems, it is observed that the non-local product obtained from the continuity equation can be interpreted as a conserved corre- lation function. This leads to physical conclusions, regarding both discrete and continuum states of such systems. Asymptotic states are shown to have necessarily broken PT -symmetry, leading to modified scattering and transfer matrices. This yields restricted boundary conditions, e.g., in- cidence from both sides, analogous to that of the proposed PT CPA laser. The interpretation of left and right states leads to a Hermitian S-matrix, resulting in the non-conservation of the flux. This further satisfies a duality condition, identical to the optical analogues. However, the non-local conserved scalar implements alternate boundary conditions in terms of in and out states, leading to the pseudo-Hermiticity condition in terms of the scattering matrix. Interestingly, when PT -symmetry is preserved, it leads to stationary states with real energy, naturally inter- pretable as bound states. The broken PT -symmetric phase is also captured by this correlation, with complex-conjugate pair of energies, interpreted as resonances.
Quantum Physics (quant-ph)