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A circular array of optical waveguides collectively coupled with a central core is investigated. Nonlinear losses, both linear and nonlinear coupling as well as energy transfer between neighboring array elements and between the array and the core are allowed. The concept is ideal for the design of high power stable amplifiers as well as of all-optical data processing devices in optical communications. The existence of stable steady-state continuous wave modes as well as of localized solitary and breathing type modes is demonstrated. These properties render the proposed system functionally rich, far more controllable than a planar one and easier to stabilize.

We study the stability of breather solutions of a dissipative cubic discrete NLS with localized forcing. The breathers are similar to the ones found for the Hamiltonian limit of the system. In the case of linearly stable multi-peak breathers the combination of dissipation and localized forcing also leads to stability, and the apparent damping of internal modes that make the energy around multi-peak breathers nondefinite. This stabilizing effect is however accompanied by overdamping for relatively small values of the dissipation parameter, and the appearance of near-zero stable eigenvalues.

We illustrate the analytical control of the Akhmediev breather frequency in Bose–Einstein condensate in a time varying parabolic trap. This is achieved by varying Feshbach managed scattering length R(t) within the integrability framework. Recognizing the mathematical relation between R(t) and parabolic trap with the linear Schrödinger equation, we use isospectral Hamiltonian approach, which leads to a one parameter family of nonlinearity control functions, directly affecting the characteristic scales of the solution space, for the same trapping potential. The chirp equation and the trap equation being same, the phase of the wave for the obtained class of nonlinearity function remains identical. For explicating the general applications of the present approach, we exhibit breather frequency control in a number of cases, with asymptotically vanishing nonlinearity, for expulsive parabolic trap. Interestingly, the same procedure is found to provide a free parameter, which controls the Rogue wave amplitude.

We study shelf-like breathers and dispersive shock phenomena in a discrete nonlinear Schrödinger (DNLS) equation with a nonlocal nonlinearity. The system models laser light propagation in waveguide arrays made from a nematic liquid crystal substratum. Shelf-like breathers are studied in the regime of small linear intersite coupling, and we report some new theoretical existence and stability results. We also study numerically the evolution from nearby dam-break and more general jump initial conditions for stronger linear intersite coupling. In the defocusing case, we see rarefaction and shock wave profiles, superposed with oscillations. Some of the hyperbolic features of the observed profiles are described approximately by continuous NLS hydrodynamics. Nonlocality is seen to lead to some smoothing of the rapid oscillations seen in the local DNLS.

We study the evolution of a quantum discrete nonlinear Schrödinger (DNLS) system using as initial conditions coherent states corresponding to points in the vicinity of breather solutions of the classical system. We consider various examples of stable and unstable breathers and examine the distance between exactly evolved states and coherent states with parameters that evolve according to classical dynamics. Initial conditions near stable breathers and their vicinity are seen to lead to recurrences to small distances between the two evolving states. Similar recurrences are not observed for initial conditions near unstable breathers.

In this paper, the optical soliton and solitary wave solutions of the $(2+1)$-dimensional Mel’nikov equation are investigated using the Kudryashov $R$ function technique. The Kudryashov $R$ function approach has various features that significantly facilitate symbolic computing, particularly for highly dispersive nonlinear equations. In computations, this approach has the benefit of not requiring the use of a certain function form. This approach gives an algorithm that is straightforward, efficient, and simple for finding solitary wave solutions. In addition, this approach is very influential and reliable when it comes to discovering hyperbolic function solutions of nonlinear equations. Many new hyperbolic function solutions have been obtained from the governing equation by using this technique. In addition, numerous types of soliton solutions describing various structures of optical solitons are retrieved. Using this method, breather, W-shaped, bell shaped, and bright soliton solutions have been generated from the governing equation. From the obtained results, it can be asserted that the applied approach may be a useful tool for addressing more highly nonlinear problems in various fields. By choosing particular values for the relevant parameters, the dynamic features of some breather, W-shaped, bell shaped and bright soliton solutions to the $(2+1)$-dimensional Mel’nikov equation have been displayed in 3D, 2D and contour graphs.

In this paper we discuss some ideas on how to define the concept of quasi-integrability. Our ideas stem from the observation that many field theory models are "almost" integrable; i.e. they possess a large number of "almost" conserved quantities. Most of our discussion will involve a certain class of models which generalize the sine-Gordon model in (1 + 1) dimensions. As will be mentioned many field configurations of these models look like those of the integrable systems and so appear to be close to those in integrable model. We will then attempt to quantify these claims looking in particular, both analytically and numerically, at field configurations with scattering solitons. We will also discuss some preliminary results obtained in other models.

This paper retrieves the investigation of rational solitons via symbolic computation with logarithmic transformation and ansatz functions approach for the $(3+1)$-dimensional generalized Konopelchenko–Dubrovsky–Kaup-Kupershmidt (GKDKK) equation in fluid mechanics, ocean dynamics and plasma physics. We find two categories of M-shaped rational solitons and their dynamics will be revealed through graphs by choosing the suitable values of involved parameters. In addition, two categories of M-shaped rational solitons and their interactions with kink waves will be analyzed. Furthermore, homoclinic breathers, multi-wave and kink cross rational solitons will be investigated. The periodic, rational, dark, bright, Weierstrass elliptic function and positive soliton solutions will also be retrieved with the aid of Sub-ODE approach. Moreover, stability characteristics of solutions will be evaluated.

This template retrieves M-shaped rational solitons and their interactions with kink waves, homoclinic breathers, multiwave, Grey-black optical solitons, periodic cross-rational solitons and kink cross-rational solitons propagation in optical fibers where self-phase modulation (SPM) is negligibly minor and therefore removed. The proposed equation contains spatio-temporal dispersions (STD), of second and third orders, to recompense for small group velocity dispersion (GVD). All new analytical solutions are found by utilizing the symbolic computation with logarithmic transformation and ansatz functions approach. Moreover, stability characteristics of all solutions are found.