arXiv subjects

Sergey V. Dmitriev

Publications and source records attributed to Sergey V. Dmitriev.

44 records · Page 3Linked to original sources

arXiv2011

Solitons in a chain of PT-invariant dimers

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.

physics.optics↗

arXiv2009

Kink ratchets in the Klein-Gordon lattice free of the Peierls-Nabarro potential

A discrete Klein-Gordon model with asymmetric potential that supports kinks free of the Peierls-Nabarro potential (PNp) is constructed. Undamped ratchet of kinks under harmonic AC driving force is investigated in this model numerically and contrasted with the kink ratchets in the conventional discrete model where kinks experience the PNp. We show that the PNp-free kinks exhibit ratchet dynamics very much different from that reported for the conventional lattice kinks which experience PNp [see, e.g., Phys. Rev. E {\bf 73}, 066621 (2006)]. Particularly, we could not observe any significant influence of the discreteness parameter on the acceleration of PNp-free kinks induced by the AC driving.

nlin.PS↗

arXiv2008

Radiationless energy exchange in three-soliton collisions

We revisit the problem of the three-soliton collisions in the weakly perturbed sine-Gordon equation and develop an effective three-particle model allowing to explain many interesting features observed in numerical simulations of the soliton collisions. In particular, we explain why collisions between two kinks and one antikink are observed to be practically elastic or strongly inelastic depending on relative initial positions of the kinks. The fact that the three-soliton collisions become more elastic with an increase in the collision velocity also becomes clear in the framework of the three-particle model. The three-particle model does not involve internal modes of the kinks, but it gives a qualitative description to all the effects observed in the three-soliton collisions, including the fractal scattering and the existence of short-lived three-soliton bound states. The radiationless energy exchange between the colliding solitons in weakly perturbed integrable systems takes place in the vicinity of the separatrix multi-soliton solutions of the corresponding integrable equations, where even small perturbations can result in a considerable change in the collision outcome. This conclusion is illustrated through the use of the reduced three-particle model.

nlin.PS↗

arXiv2008

High-speed kinks in a generalized discrete $ϕ^4$ model

We consider a generalized discrete $ϕ^4$ model and demonstrate that it can support exact moving kink solutions in the form of tanh with an arbitrarily large velocity. The constructed exact moving solutions are dependent on the specific value of the propagation velocity. We demonstrate that in this class of models, given a specific velocity, the problem of finding the exact moving solution is integrable. Namely, this problem originally expressed as a three-point map can be reduced to a two-point map, from which the exact moving solutions can be derived iteratively. It was also found that these high-speed kinks can be stable and robust against perturbations introduced in the initial conditions.

nlin.SI↗

arXiv2007

Exact Static Solutions of a Generalized Discrete ϕ^4 Model Including Short-Periodic Solutions

For a five-parameter discrete $ϕ^4$ model, we derive various exact static solutions, including the staggered ones, in the form of the basic Jacobi elliptic functions $\sn$, $\cn$, and $\dn$, and also in the form of their hyperbolic function limits such as kink ($\tanh$) and single-humped pulse ($\sech$) solutions. Such solutions are admitted by the considered model in seven cases, two of which have been discussed in the literature, and the remaining five cases are addressed here. We also obtain $\sin$e, staggered $\sin$e as well as a large number of short-periodic static solutions of the generalized 5-parameter model. All the Jacobi elliptic, hyperbolic and trigonometric function solutions (including the staggered ones) are translationally invariant (TI), i.e., they can be shifted along the lattice by an arbitrary $x_0$, but among the short-periodic solutions there are both TI and non-TI solutions. The stability of these solutions is also investigated. Finally, the constructed Jacobi elliptic function solutions reveal four new types of cubic nonlinearity with the TI property.

nlin.SI↗

arXiv2007

Exact Moving and Stationary Solutions of a Generalized Discrete Nonlinear Schrodinger Equation

We obtain exact moving and stationary, spatially periodic and localized solutions of a generalized discrete nonlinear Schrödinger equation. More specifically, we find two different moving periodic wave solutions and a localized moving pulse solution. We also address the problem of finding exact stationary solutions and, for a particular case of the model when stationary solutions can be expressed through the Jacobi elliptic functions, we present a two-point map from which all possible stationary solutions can be found. Numerically we demonstrate the generic stability of the stationary pulse solutions and also the robustness of moving pulses in long-term dynamics.

nlin.PS↗

arXiv2007

Exact static solutions to a translationally invariant discrete $ϕ^4$ model

For a discrete, translationally-invariant $ϕ^4$ model introduced by Barashenkov {\it et al.} [Phys. Rev. E {\bf 72}, 35602R (2005)], we provide the momentum conservation law and demonstrate how the first integral of the static version of the discrete model can be constructed from a Jacobi elliptic function (JEF) solution. The first integral can be written in the form of a nonlinear map from which {\it any} static solution supported by the model can be constructed. A set of JEF solutions, including the staggered ones, is derived. We also report on the stability analysis for the static bounded solutions and exemplify the dynamical behavior of the unstable solutions. This work provides a road-map, through this illustrative example, on how to fully analyze translationally-invariant models in terms of their static problem, its first integral, their full-set of static solutions and associated conservation laws.

nlin.SI↗

arXiv2006

Comparative Study of Different Discretizations of the $ϕ^4$ Model

We examine various recently proposed discretizations of the well-known $ϕ^4$ field theory. We compare and contrast the properties of their fundamental solutions including the nature of their kink-type solitary waves and the spectral properties of the linearization around such waves. We study these features as a function of the lattice spacing $h$, as one deviates from the continuum limit of $h \to 0$. We then proceed to a more ``stringent'' comparison of the models, by discussing the scattering properties of a kink-antikink pair for the different discretizations. These collisions are well-known to possess properties that quite sensitively depend on the initial speed even at the continuum limit. We examine how typical model behaviors are modified in the presence (and as a function) of discreteness and attempt to extract qualitative trends and issue pertinent warnings about some of the surprising resulting properties.

nlin.PS↗