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)

I. V. Barashenkov, Sergey V. Suchkov, Andrey A. Sukhorukov, Sergey V. Dmitriev, Yuri S. Kivshar

We show that the parity-time (PT) symmetric coupled optical waveguides with gain and loss support localised oscillatory structures similar to the breathers of the classical \(\phi^4\) model. The power carried by the PT-breather oscillates periodically, switching back and forth between the waveguides, so that the gain and loss are compensated on the average. The breathers are found to coexist with solitons and be prevalent in the products of the soliton collisions. We demonstrate that the evolution of the small-amplitude breather’s envelope is governed by a system of two coupled nonlinear Schrodinger equations, and employ this Hamiltonian system to show that the small-amplitude PT-breathers are stable.

http://arxiv.org/abs/1211.1835

Pattern Formation and Solitons (nlin.PS); Optics (physics.optics)

Sergey V. Suchkov, Boris A. Malomed, Sergey V. Dmitriev, Yuri S. Kivshar

Dynamics of a chain of interacting parity-time invariant nonlinear dimers is investigated. A dimer is built as a pair of coupled elements with equal gain and loss. A relation between stationary soliton solutions of the model and solitons of the discrete nonlinear Schrodinger (DNLS) equation is demonstrated. Approximate solutions for solitons whose width is large in comparison to the lattice spacing are derived, using a continuum counterpart of the discrete equations. These solitons are mobile, featuring nearly elastic collisions. Stationary solutions for narrow solitons, which are immobile due to the pinning by the effective Peierls-Nabarro potential, are constructed numerically, starting from the anti-continuum limit. The solitons with the amplitude exceeding a certain critical value suffer an instability leading to blowup, which is a specific feature of the nonlinear PT-symmetric chain, making it dynamically different from DNLS lattices. A qualitative explanation of this feature is proposed. The instability threshold drops with the increase of the gain-loss coefficient, but it does not depend on the lattice coupling constant, nor on the soliton’s velocity.

http://arxiv.org/abs/1110.1501

Optics (physics.optics)