Transmission of rogue wave signals through a modified Noguchi electrical transmission network.

A distributed electrical transmission network with dispersive elements that consists of a large number of identical unit cells is considered. Using the reductive perturbation method in the semidiscrete limit, we show that the voltage for the network is described by a nonlinear Schrödinger (NLS) equation with an external linear potential. Using such a NLS equation, the propagation of the first- and second-order rogue waves in the system is predicted and analyzed quantitatively and qualitatively. The rogue waves are expected to propagate for the bandwidth frequencies where the network may exhibit modulational instability. The effects of relevant network parameters on the characteristics of the rogue wave parameters are investigated. We show how to manipulate the relevant network parameters as well as the propagating frequency either to amplify the propagation of the rogue waves through the network, or to prevent rogue waves from being amplified in the network under consideration.

Management of matter-wave solitons in Bose-Einstein condensates with time-dependent atomic scattering length in a time-dependent parabolic complex potential.

In this paper, we consider a Gross-Pitaevskii (GP) equation with a time-dependent nonlinearity and a spatiotemporal complex linear term which describes the dynamics of matter-wave solitons in Bose-Einstein condensates (BECs) with time-dependent interatomic interactions in a parabolic potential in the presence of feeding or loss of atoms. We establish the integrability conditions under which analytical solutions describing the modulational instability and the propagation of both bright and dark solitary waves on a continuous wave background are obtained. The obtained integrability conditions also appear as the conditions under which the solitary waves of the BECs can be managed by controlling the functional gain or loss parameter. For specific BECs, the dynamics of bright and dark solitons are investigated analytically through the found exact solutions of the GP equation. Our results show that under the integrability conditions, the gain or loss parameter of the GP equation can be used to manage the motion of both bright and dark solitons. We show that for BECs with loss (gain) of atoms, the bright and dark solitons during their propagation have a compression (broadening) in their width. Furthermore, under a safe range of parameters and under the integrability conditions, it is possible to squeeze a bright soliton of BECs with loss of atoms into the assumed peak matter density, which can provide an experimental tool for investigating the range of validity of the 1D GP equation. Our results also reveal that under the conditions of the solitary wave management, neither the injection or the ejection of atoms from the condensate affects the soliton peak during its propagation.

Solitonlike pulses along a modified Noguchi nonlinear electrical network with second-neighbor interactions: Analytical studies.

A modified lossless nonlinear Noguchi transmission network with second-neighbor interactions is considered. In the semidiscrete limit, we apply the reductive perturbation method and show that the dynamics of modulated waves propagating through the network are governed by an NLS equation with linear external potential. Classes of exact solitonic solutions of this network equation are derived, proving possible transmission of both bright and dark solitonlike pulses through the network. The effects of both the coupling second-neighbor parameter L_{3} and the strength λ of the linear potential on the dynamics of modulated waves through the network are investigated. One of the main results of our work is that with the introduction of the second neighbors in the network, two solitary signals, either two bright solitary signals or one bright and one dark solitary signal, may simultaneously propagate at the same frequency through the network.

Modeling of matter-wave solitons in a nonlinear inductor-capacitor network through a Gross-Pitaevskii equation with time-dependent linear potential.

A lossless nonlinear LC transmission network is considered. With the use of the reductive perturbation method in the semidiscrete limit, we show that the dynamics of matter-wave solitons in the network can be modeled by a one-dimensional Gross-Pitaevskii (GP) equation with a time-dependent linear potential in the presence of a chemical potential. An explicit expression for the growth rate of a purely growing modulational instability (MI) is presented and analyzed. We find that the potential parameter of the GP equation of the system does not affect the different regions of the MI. Neglecting the chemical potential in the GP equation, we derive exact analytical solutions which describe the propagation of both bright and dark solitary waves on continuous-wave (cw) backgrounds. Using the found exact analytical solutions of the GP equation, we investigate numerically the transmission of both bright and dark solitary voltage signals in the network. Our numerical studies show that the amplitude of a bright solitary voltage signal and the depth of a dark solitary voltage signal as well as their width, their motion, and their behavior depend on (i) the propagation frequencies, (ii) the potential parameter, and (iii) the amplitude of the cw background. The GP equation derived in this paper with a time-dependent linear potential opens up different ideas that may be of considerable theoretical interest for the management of matter-wave solitons in nonlinear LC transmission networks.

Dynamics of modulated waves in a lossy modified Noguchi electrical transmission line.

We study analytically the dynamics of modulated waves in a dissipative modified Noguchi nonlinear electrical network. In the continuum limit, we use the reductive perturbation method in the semidiscrete limit to establish that the propagation of modulated waves in the network is governed by a dissipative nonlinear Schrödinger (NLS) equation. Motivated with a solitary wave type of solution to the NLS equation, we use both the direct method and the Weierstrass's elliptic function method to present classes of bright, kink, and dark solitary wavelike solutions to the dissipative NLS equation of the network. Through the exact solitary wavelike solutions to the dissipative NLS equation, we investigate the effects of the dissipative elements of the network on wave propagation. We show that the wave amplitude decreases and its width increases when the dissipative element of the network increases. It has been also found that the dissipative element of the network can be used to manipulate the motion of solitary waves through the network. This work presents a good analytical approach of investigating the propagation of solitary waves through discrete electrical transmission lines and is very important for studying modulational instability.

Analytical study of dynamics of matter-wave solitons in lossless nonlinear discrete bi-inductance transmission lines.

We consider a lossless one-dimensional nonlinear discrete bi-inductance electrical transmission line made of N identical unit cells. When lattice effects are considered, we use the reductive perturbation method in the semidiscrete limit to show that the dynamics of modulated waves can be modeled by the classical nonlinear Schrödinger (CNLS) equation, which describes the modulational instability and the propagation of bright and dark solitons on a continuous-wave background. Our theoretical analysis based on the CNLS equation predicts either two or four frequency regions with different behavior concerning the modulational instability of a plane wave. With the help of the analytical solutions of the CNLS equation, we investigate analytically the effects of the linear capacitance CS on the dynamics of matter-wave solitons in the network. Our results reveal that the linear parameter CS can be used to manipulate the motion of bright, dark, and kink soliton in the network.

Phase engineering, modulational instability, and solitons of Gross-Pitaevskii-type equations in 1+1 dimensions.

Motivated by recent proposals of "collisionally inhomogeneous" Bose-Einstein condensates (BECs), which have a spatially modulated scattering length, we introduce a phase imprint into the macroscopic order parameter governing the dynamics of BECs with spatiotemporal varying scattering length described by a cubic Gross-Pitaevskii (GP) equation and then suitably engineer the imprinted phase to generate the modified GP equation, also called the cubic derivative nonlinear Schrödinger (NLS) equation. This equation describes the dynamics of condensates with two-body (attractive and repulsive) interactions in a time-varying quadratic external potential. We then carry out a theoretical analysis which invokes a lens-type transformation that converts the cubic derivative NLS equation into a modified NLS equation with only explicit temporal dependence. Our analysis suggests a particular interest in a specific time-varying potential with the strength of the magnetic trap ~1/(t+t(*))(2). For a time-varying quadratic external potential of this kind, an explicit expression for the growth rate of a purely growing modulational instability is presented and analyzed. We point out the effect of the imprint parameter and the parameter t(*) on the instability growth rate, as well as on the solitary waves of the BECs.

Transverse stability of solitary waves propagating in coupled nonlinear dispersive transmission lines.

In the semidiscrete limit and in suitably scaled coordinates, the voltage of a system of coupled nonlinear dispersive transmission lines is described by a nonlinear Schrödinger equation. This equation is used to study the transverse stability of solitary waves of the system. Exact results for the growth rate and the corresponding perturbation function of linear transverse perturbations are obtained in terms of the network's and soliton's parameters.

Modulational instability criteria for coupled nonlinear transmission lines with dispersive elements.

We investigate the modulational instability of Stokes wave solutions on a system of coupled nonlinear electrical transmission lines with dispersive elements. In the continuum limit, and in suitable scaled coordinates, the voltage on the system is described by the two-dimensional coupled nonlinear Schrödinger equations. The set of coupled nonlinear Schrödinger equations obtained is analyzed via a perturbation approach. No assumption is made on the signs of the relevant coefficients such as the coefficients of nonlinearity and the coupling coefficients. A set of explicit criteria of modulational stability and modulational instability is derived and analyzed. It is numerically shown that the effect of the dispersive elements in the line is to decrease the instability region and the instability growth rate.

Exact solutions of the derivative nonlinear Schrödinger equation for a nonlinear transmission line.

We consider the derivative nonlinear Schrödinger equation with constant potential as a model for wave propagation on a discrete nonlinear transmission line. This equation can be derived in the small amplitude and long wavelength limit using the standard reductive perturbation method and complex expansion. We construct some exact soliton and elliptic solutions of the mentioned equation by perturbation of its Stokes wave solutions. We find that for some values of the coefficients of the equation and for some parameters of solutions, the graphical representations show some kinds of symmetries such as mirror symmetry and rotational symmetry.