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We consider PT-symmetric, nonlocal nonlinear Schrodinger equation on metric graphs. Vertex boundary conditions are derived from the conservation laws. Soliton solutions are obtained for simplest graph topologies, such as star and tree graphs. Integrability of the problem is shown by proving existence of infinite number of conservation laws. more...

We consider branched Josephson junction created by planar superconductors connected to each other through the Y-junction insulator. Assuming that the structure interacts with the external constant magnetic field, we study static sine-Gordon solitons in such system by modeling them in terms of the stationary sine-Gordon equation on metric graph. Exact analytical solutions of the problem are obtained and their stability is analyzed. more...

We consider a damped, parametrically driven discrete nonlinear Klein-Gordon equation, that models coupled pendula and micromechanical arrays, among others. To study the equation, one usually uses a small-amplitude wave ansatz, that reduces the equation into a discrete nonlinear Schr��dinger equation with damping and parametric drive. Here, we justify the approximation by looking for the error bound with the method of energy estimates. Furthermore, we prove the local and global existence of {solutions to the discrete nonlinear} Schr��dinger equation. To illustrate the main results, we consider numerical simulations showing the dynamics of errors made by the discrete nonlinear equation. We consider two types of initial conditions, with one of them being a discrete soliton of the nonlinear Schr��dinger equation, that is expectedly approximate discrete breathers of the nonlinear Klein-Gordon equation., arXiv admin note: text overlap with arXiv:1911.01631 more...

In this work we investigate a one-dimensional parity-time (PT)-symmetric magnetic metamaterial consisting of split-ring dimers having gain or loss. Employing a Melnikov analysis we study the existence of localized travelling waves, i.e. homoclinic or heteroclinic solutions. We find conditions under which the homoclinic or heteroclinic orbits persist. Our analytical results are found to be in good agreement with direct numerical computations. For the particular nonlinearity admitting travelling kinks, numerically we observe homoclinic snaking in the bifurcation diagram. The Melnikov analysis yields a good approximation to one of the boundaries of the snaking profile., 15 pages, 4 figures, Wave Motion 76 (2018) 9-18. arXiv admin note: text overlap with arXiv:1304.0556 by other authors more...

Two effectively one-dimensional parallel coupled Bose-Einstein condensates in the presence of external potentials are studied. The system is modelled by linearly coupled Gross-Pitaevskii equations. In particular, the interactions of grey-soliton-like solutions representing analogues of superconducting Josephson fluxons as well as coupled dark solitons are discussed. A theoretical approximation based on variational formulations to calculate the oscillation frequency of the grey-soliton-like solution is derived and a qualitatively good agreement is obtained., 26 pages, 17 figures. Appeared in "Spontaneous Symmetry Breaking, Self-Trapping, and Josephson Oscillations", B.A. Malomed, ed. (Springer, Berlin, 2013) more...

Travelling solitary waves in the one-dimensional discrete nonlinear Schr��dinger equation (DNLSE) with saturable onsite nonlinearity are studied. A variational approximation (VA) for the solitary waves is derived in an analytical form. The stability is also studied by means of the VA, demonstrating that the solitons are stable, which is consistent with previously published results. Then, the VA is applied to predict parameters of travelling solitons with non-oscillatory tails (\textit{embedded solitons}, ESs). Two-soliton bound states are considered too. The separation distance between the solitons forming the bound state is derived by means of the VA. A numerical scheme based on the discretization of the equation in the moving coordinate frame is derived and implemented using the Newton--Raphson method. In general, a good agreement between the analytical and numerical results is obtained. In particular, we demonstrate the relevance of the analytical prediction of characteristics of the embedded solitons., This paper is to appear in Journal of Physics A more...

We consider a parametrically driven damped discrete nonlinear Schr��dinger (PDDNLS) equation. Analytical and numerical calculations are performed to determine the existence and stability of fundamental discrete bright solitons. We show that there are two types of onsite discrete soliton, namely onsite type I and II. We also show that there are four types of intersite discrete soliton, called intersite type I, II, III, and IV, where the last two types are essentially the same, due to symmetry. Onsite and intersite type I solitons, which can be unstable in the case of no dissipation, are found to be stabilized by the damping, whereas the other types are always unstable. Our further analysis demonstrates that saddle-node and pitchfork (symmetry-breaking) bifurcations can occur. More interestingly, the onsite type I, intersite type I, and intersite type III-IV admit Hopf bifurcations from which emerge periodic solitons (limit cycles). The continuation of the limit cycles as well as the stability of the periodic solitons are computed through the numerical continuation software Matcont. We observe subcritical Hopf bifurcations along the existence curve of the onsite type I and intersite type III-IV. Along the existence curve of the intersite type I we observe both supercritical and subcritical Hopf bifurcations., to appear in "Spontaneous Symmetry Breaking, Self-Trapping, and Josephson Oscillations in Nonlinear Systems", B.A. Malomed, ed. (Springer, Berlin, 2012) more...

A periodically inhomogeneous Schrodinger equation is considered. The inhomogeneity is reflected through a non-uniform coefficient of the linear and non-linear term in the equation. Due to the periodic inhomogeneity of the linear term, the system may admit spectral bands. When the oscillation frequency of a localized solution resides in one of the finite band gaps, the solution is a gap soliton, characterized by the presence of infinitely many zeros in the spatial profile of the soliton. Recently, how to construct such gap solitons through a composite phase portrait is shown. By exploiting the phase-space method and combining it with the application of a topological argument, it is shown that the instability of a gap soliton can be described by the phase portrait of the solution. Surface gap solitons at the interface between a periodic inhomogeneous and a homogeneous medium are also discussed. Numerical calculations are presented accompanying the analytical results., Comment: Comments and suggestions are welcome more...

We introduce a system of two linearly coupled discrete nonlinear Schr\"{o}dinger equations (DNLSEs), with the coupling constant subject to a rapid temporal modulation. The model can be realized in bimodal Bose-Einstein condensates (BEC). Using an averaging procedure based on the multiscale method, we derive a system of averaged (autonomous) equations, which take the form of coupled DNLSEs with additional nonlinear coupling terms of the four-wave-mixing type. We identify stability regions for fundamental onsite discrete symmetric solitons (single-site modes with equal norms in both components), as well as for two-site in-phase and twisted modes, the in-phase ones being completely unstable. The symmetry-breaking bifurcation, which destabilizes the fundamental symmetric solitons and gives rise to their asymmetric counterparts, is investigated too. It is demonstrated that the averaged equations provide a good approximation in all the cases. In particular, the symmetry-breaking bifurcation, which is of the pitchfork type in the framework of the averaged equations, corresponds to a Hopf bifurcation in terms of the original system., Comment: 6 pages, 3 figures more...

We consider a spatially non-autonomous discrete sine-Gordon equation with constant forcing and its continuum limit(s) to model a 0-$��$ Josephson junction with an applied bias current. The continuum limits correspond to the strong coupling limit of the discrete system. The non-autonomous character is due to the presence of a discontinuity point, namely a jump of $��$ in the sine-Gordon phase. The continuum models admits static solitary waves which are called $��$-kinks and are attached to the discontinuity point. For small forcing, there are three types of $��$-kinks. We show that one of the kinks is stable and the others are unstable. There is a critical value of the forcing beyond all static $��$-kinks fail to exist. Up to this value, the (in)stability of the $��$-kinks can be established analytically in the strong coupling limits. Applying a forcing above the critical value causes the nucleation of $2��$-kinks and -antikinks. Besides a $��$-kink, the unforced system also admits a static $3��$-kink. This state is unstable in the continuum models. By combining analytical and numerical methods in the discrete model, it is shown that the stable $��$-kink remains stable, and that the unstable $��$-kinks cannot be stabilized by decreasing the coupling. The $3��$-kink does become stable in the discrete model when the coupling is sufficiently weak., figures are not included due to file size limit more...

We consider the dynamics of two coupled miscible Bose-Einstein condensates, when an obstacle is dragged through them. The existence of two different speeds of sound provides the possibility for three dynamical regimes: when both components are subcritical, we do not observe nucleation of coherent structures; when both components are supercritical they both form dark solitons in one dimension (1D) and vortices or rotating vortex dipoles in two dimensions (2D); in the intermediate regime, we observe the nucleation of a structure in the form of a dark-antidark soliton in 1D; subcritical component; the 2D analog of such a structure, a vortex-lump, is also observed., Comment: 4 pages, 4 figures, submitted to Phys Rev E more...

We consider a spatially nonautonomous discrete sine-Gordon equation with constant forcing and its continuum limit(s) to model a 0-pi Josephson junction with an applied bias current. The continuum limits correspond to the strong coupling limit of the discrete system. The nonautonomous character is due to the presence of a discontinuity point, namely, a jump of pi in the sine-Gordon phase. The continuum model admits static solitary waves which are called pi-kinks and are attached to the discontinuity point. For small forcing, there are three types of pi-kinks. We show that one of the kinks is stable and the others are unstable. There is a critical value of the forcing beyond which all static pi- kinks fail to exist. Up to this value, the (in) stability of the pi-kinks can be established analytically in the strong coupling limits. Applying a forcing above the critical value causes the nucleation of 2 pi-kinks and -antikinks. Besides a pi- kink, the unforced system also admits a static 3 pi-kink. This state is unstable in the continuum models. By combining analytical and numerical methods in the discrete model, it is shown that the stable pi-kink remains stable and that the unstable pi-kinks cannot be stabilized by decreasing the coupling. The 3 pi- kink does become stable in the discrete model when the coupling is sufficiently weak. more...

We consider the dynamics of two coupled miscible Bose-Einstein condensates, when an obstacle is dragged through them. The existence of two different speeds of sound provides the possibility for three dynamical regimes: when both components are subcritical, we do not observe nucleation of coherent structures; when both components are supercritical they both form dark solitons in one dimension (1D) and vortices or rotating vortex dipoles in two dimensions; in the intermediate regime, we observe the nucleation of a structure in the form of a dark-antidark soliton in 1D; the 2D analog of such a structure, a vortex-lump, is also observed. more...

We consider the dynamics of two coupled miscible Bose-Einstein condensates, when an obstacle is dragged through them. The existence of two different speeds of sound provides the possibility for three dynamical regimes: when both components are subcritical, we do not observe nucleation of coherent structures; when both components are supercritical they both form dark solitons in one dimension (1D) and vortices or rotating vortex dipoles in two dimensions (2D); in the intermediate regime, we observe the nucleation of a structure in the form of a dark-antidark soliton in 1D; subcritical component; the 2D analog of such a structure, a vortex-lump, is also observed., 4 pages, 4 figures, submitted to Phys Rev E more...

We study the mobility of solitons in lattices with quadratic (χ(2), alias second-harmonic-generating) nonlinearity. Using the notion of the Peierls-Nabarro potential and systematic numerical simulations, we demonstrate that, in contrast with their cubic (χ(3)) counterparts, the discrete quadratic solitons are mobile not only in the one-dimensional (1D) setting, but also in two dimensions (2D), in any direction. We identify parametric regions where an initial kick applied to a soliton leads to three possible outcomes: staying put, persistent motion, or destruction. On the 2D lattice, the solitons survive the largest kick and attain the largest speed along the diagonal direction. © 2007 The American Physical Society. more...

In the present work, we study dark solitons in dynamical lattices with the saturable nonlinearity and compare them to those in lattices with the cubic nonlinearity. This comparison has become especially relevant in light of recent experimental developments in the former context. The stability properties of the fundamental waves, for both onsite and intersite modes, are examined analytically and corroborated by numerical results. Our findings indicate that for both models onsite solutions are stable for sufficiently small values of the coupling between adjacent nodes, while intersite solutions are always unstable. The nature of the instability (which is oscillatory for onsite solutions at large coupling and exponential for inter-site solutions) is probed via the dynamical evolution of unstable solitary waves through appropriately crafted numerical experiments; typically, these computations result in dynamic motion of the originally stationary solitary waves. Another key finding, consistent with recent experimental results, is that the instability growth rate for the saturable nonlinearity is found to be smaller than that of the cubic case. © 2007 The American Physical Society. more...