Category Tomsk State University

Kink scattering from a parity-time-symmetric defect in the \(\phi^4\) model

Danial Saadatmand, Sergey V. Dmitriev, Denis I. Borisov, Panayotis G. Kevrekidis, Minnekhan A. Fatykhov, Kurosh Javidan

In this paper, we study the \(\phi^4\) kink scattering from a spatially localized PT-symmetric defect and the effect of the kink’s internal mode (IM) is discussed. It is demonstrated that if a kink hits the defect from the gain side, a noticeable IM is excited, while for the kink coming from the opposite direction the mode excitation is much weaker. This asymmetry is a principal finding of the present work. Similar to the case of the sine-Gordon kink studied earlier, it is found that the \(\\phi^4\) kink approaching the defect from the gain side always passes through the defect, while in the opposite case it must have sufficiently large initial velocity, otherwise it is trapped by the loss region. It is found that for the kink with IM the critical velocity is smaller, meaning that the kink bearing IM can pass more easily through the loss region. This feature, namely the “increased transparency” of the defect as regards the motion of the kink in the presence of IM is the second key finding of the present work. A two degree of freedom collective variable model offered recently by one of the co-authors is shown to be capable of reproducing both principal findings of the present work. A simpler, analytically tractable single degree of freedom collective variable method is used to calculate analytically the kink phase shift and the kink critical velocity sufficient to pass through the defect. Comparison with the numerical results suggests that the collective variable method is able to predict these parameters with a high accuracy.

http://arxiv.org/abs/1411.5857
Pattern Formation and Solitons (nlin.PS)

Hermitian Hamiltonian equivalent to a given non-Hermitian one. Manifestation of spectral singularity

Boris F. Samsonov

One of the simplest non-Hermitian Hamiltonians first proposed by Schwartz (1960 Commun. Pure Appl. Math. 13 609) which may possess a spectral singularity is analyzed from the point of view of non-Hermitian generalization of quantum mechanics. It is shown that \(\eta\) operator, being a second order differential operator, has supersymmetric structure. Asymptotic behavior of eigenfunctions of a Hermitian Hamiltonian equivalent to the given non-Hermitian one is found. As a result the corresponding scattering matrix and cross section are given explicitly. It is demonstrated that the possible presence of the spectral singularity in the spectrum of the non-Hermitian Hamiltonian may be detected as a resonance in the scattering cross section of its Hermitian counterpart. Nevertheless, just at the singular point the equivalent Hermitian Hamiltonian becomes undetermined.

http://arxiv.org/abs/1207.2525
Mathematical Physics (math-ph); Quantum Physics (quant-ph)

Supersymmetric η operators

Boris F. Samsonov

Being chosen as a differential operator of a special form, metric \(\eta\) operator becomes unitary equivalent to a one-dimensional Hermitian Hamiltonian with a natural supersymmetric structure. We show that fixing the superpartner of this Hamiltonian permits to determine both the metric operator and corresponding non-Hermitian Hamiltonian. Moreover, under an additional restriction on the non-Hermitian Hamiltonian, it becomes a superpartner of another Hermitian Hamiltonian.

http://arxiv.org/abs/1207.2522

Mathematical Physics (math-ph); Quantum Physics (quant-ph)