We study propagation and on-off switching of two colliding soliton sequences in the presence of second-order dispersion, Kerr nonlinearity, linear loss, cubic gain, and quintic loss. Employing a Lotka-Volterra (LV) model for dynamics of soliton amplitudes along with simulations with two perturbed coupled nonlinear Schrödinger (NLS) equations, we show that stable long-distance propagation can be achieved for a wide range of the gain-loss coefficients, including values that are outside of the perturbative regime. Furthermore, we demonstrate robust on-off and off-on switching of one of the sequences by an abrupt change in the ratio of cubic gain and quintic loss coefficients, and extend the results to pulse sequences with periodically alternating phases. Our study significantly strengthens the recently found relation between collision dynamics of sequences of NLS solitons and population dynamics in LV models, and indicates that the relation might be further extended to solitary waves of the cubic-quintic Ginzburg-Landau equation.

We report several major theoretical steps towards realizing stable long-distance multichannel soliton transmission in Kerr nonlinear waveguide loops. We find that transmission destabilization in a single waveguide is caused by resonant formation of radiative sidebands and investigate the possibility to increase transmission stability by optimization with respect to the Kerr nonlinearity coefficient γ. Moreover, we develop a general method for transmission stabilization, based on frequency dependent linear gain-loss in Kerr nonlinear waveguide couplers, and implement it in twochannel and three-channel transmission. We show that the introduction of frequency dependent loss leads to significant enhancement of transmission stability even for non-optimal γ values via decay of radiative sidebands, which takes place as a dynamic phase transition. For waveguide couplers with frequency dependent linear gain-loss, we observe stable oscillations of soliton amplitudes due to decay and regeneration of the radiative sidebands. PACS numbers: 42.65.Tg, 42.81.Dp, 42.81.Qb 1 I. INTRODUCTIONThe rates of transmission of information in broadband optical waveguide systems can be significantly increased by transmitting many pulse sequences through the same waveguide [1][2][3]. This is achieved by the wavelength-division-multiplexing (WDM) method, where each pulse sequence is characterized by the central frequency of its pulses, and is therefore called a frequency channel. Applications of these WDM or multichannel systems include fiber optics communication lines [1][2][3], data transfer between computer processors through silicon waveguides [4,5], and multiwavelength lasers [6,7]. Since pulses from different frequency channels propagate with different group velocities, interchannel pulse collisions are very frequent, and can therefore lead to severe transmission degradation [1]. Soliton-based transmission is considered to be advantageous compared with other transmission formats, due to the stability and shape-preserving properties of the solitons, and as a result, has been the focus of many studies [1-3]. These studies have shown that effects of Kerr nonlinearity on interchannel collisions, such as cross-phase modulation and four-wave-mixing, are among the main impairments in soliton-based WDM fiber optics transmission. Furthermore, various methods for mitigation of Kerr-induced effects, such as filtering and dispersion-management, have been developed [2,3]. However, the problem of achieving stable long-distance propagation of optical solitons in multichannel Kerr nonlinear waveguide loops remains unresolved. The challenge in this case stems from two factors. First, any radiation emitted by the solitons stays in the waveguide loop, and therefore, the radiation accumulates. Second, the radiation emitted by solitons from a given channel at frequencies of the solitons in the other channels undergoes unstable growth and develops into radiative sidebands. Due to radiation accumulation and to the fact that the sidebands form at the frequencie...

We study the dynamics of emission of radiation (small-amplitude waves) in fast collisions between two solitons of the nonlinear Schrödinger (NLS) equation in the presence of weak cubic loss. We calculate the radiation dynamics by a perturbation technique with two small parameters: the cubic loss coefficient 3 and the reciprocal of the group velocity difference 1/β. The agreement between the perturbation theory predictions and the results of numerical simulations with the full coupled-NLS propagation model is very good for large β values, and is good for intermediate β values. Additional numerical simulations with four simplified NLS models show that the differences between perturbation theory and simulations for intermediate β values are due to the effects of Kerr nonlinearity on interpulse interaction in the collision. Thus, our study demonstrates that the perturbation technique that was originally developed to study radiation dynamics in fast soliton collisions in the presence of conservative perturbations can also be employed for soliton collisions in the presence of dissipative perturbations.

A hybrid method is developed based on the spectral and finite-difference methods for solving the inhomogeneous Zerilli equation in time-domain. The developed hybrid method decomposes the domain into the spectral and finite-difference domains. The singular source term is located in the spectral domain while the solution in the region without the singular term is approximated by the higher-order finite-difference method.The spectral domain is also split into multi-domains and the finite-difference domain is placed as the boundary domain. Due to the global nature of the spectral method, a multi-domain method composed of the spectral domains only does not yield the proper power-law decay unless the range of the computational domain is large. The finitedifference domain helps reduce boundary effects due to the truncation of the computational domain. The multi-domain approach with the finite-difference boundary domain method reduces the computational costs significantly and also yields the proper powerlaw decay.Stable and accurate interface conditions between the finite-difference and spectral domains and the spectral and spectral domains are derived. For the singular source term, we use both the Gaussian model with various values of full width at half maximum and a localized discrete δ-function. The discrete δ-function was generalized to adopt the Gauss-Lobatto collocation points of the spectral domain.The gravitational waveforms are measured. Numerical results show that the developed hybrid method accurately yields the quasi-normal modes and the power-law decay profile. The numerical results also show that the power-law decay profile is less sensitive to the shape of the regularized δ-function for the Gaussian model than expected. The Gaussian model also yields better results than the localized discrete δ-function.

We demonstrate that the amplitudes of optical solitons in nonlinear multisequence optical waveguide coupler systems with weak linear and cubic gain-loss exhibit large stable oscillations along ultra-long distances. The large stable oscillations are caused by supercritical Hopf bifurcations of the equilibrium states of the Lotka-Volterra (LV) models for dynamics of soliton amplitudes. The predictions of the LV models are confirmed by numerical simulations with the coupled cubic nonlinear Schrödinger (NLS) propagation models with 2 ≤ N ≤ 4 pulse sequences. Thus, we provide the first demonstration of intermediate nonlinear amplitude dynamics in multisequence soliton systems, described by the cubic NLS equation. Our findings are also an important step towards realization of spatio-temporal chaos with multiple periodic sequences of colliding NLS solitons.

We study transmission stabilization against radiation emission in soliton-based nonlinear optical waveguides with weak linear gain-loss, cubic loss, and delayed Raman response. We show by numerical simulations with perturbed nonlinear Schrödinger propagation models that transmission quality in waveguides with frequency independent linear gain and cubic loss is not improved by the presence of delayed Raman response due to the lack of an efficient mechanism for suppression of radiation emission. In contrast, we find that the presence of delayed Raman response leads to significant enhancement of transmission quality in waveguides with frequency dependent linear gain-loss and cubic loss. Enhancement of transmission quality in the latter waveguides is enabled by the separation of the soliton's spectrum from the radiation's spectrum due to the Raman-induced selffrequency shift and by efficient suppression of radiation emission due to the frequency dependent linear gain-loss. Further numerical simulations demonstrate that the enhancement of transmission quality in waveguides with frequency dependent linear gain-loss, cubic loss, and delayed Raman response is similar to transmission quality enhancement in waveguides with linear gain, cubic loss, and guiding filters with a varying central frequency.

We consider the Klein-Gordon and sine-Gordon type equations with a pointlike potential, which describes the wave phenomenon in disordered media with a defect. The singular potential term yields a critical phenomenon-that is, the solution behavior around the critical parameter value bifurcates into two extreme cases. Finding such critical parameter values and the associated statistical quantities demands a large number of individual simulations with different parameter values. Pinpointing the critical value with arbitrary accuracy is even more challenging. In this work, we adopt the generalized polynomial chaos (gPC) method to determine the critical values and the mean solutions around such values.First, we consider the critical value associated with the strength of the singular potential for the Klein-Gordon equation. We show the existence of a critical behavior with certain boundary conditions. Then we expand the solution in the random variable associated with the parameter. The obtained partial differential equations are solved using the Chebyshev collocation method. Due to the existence of the singularity, the Gibbs phenomenon appears in the solution, yielding a slow convergence of the numerically computed critical value. To deal with the singularity, we adopt the consistent spectral collocation method. The gPC method, along with the consistent Chebyshev method, determines the critical value and the mean solution highly efficiently.We then consider the sine-Gordon equation, for which the critical value is associated with the initial velocity of the kink soliton solution. The critical behavior in this case is that the solution passes through (particle-pass), is trapped by (particle-capture), or is reflected by (particle-reflection) the singular potential if the initial velocity of the soliton solution is greater than, equal to, or less than * Corresponding author the critical value, respectively. Due to the nonlinearity of the equation, we use the gPC mean value rather than reconstructing the solution to find the critical parameter. Numerical results show that the critical value can be determined efficiently and accurately by using the proposed method. The results are also compared with the results using the Monte-Carlo method.

We consider the nonlinear Schrödinger equation with a point-like source term. The soliton interaction with such a singular potential yields a critical solution behavior. That is, for the given value of the potential strength and the soliton amplitude, there exists a critical velocity of the initial soliton solution, around which the solution is either trapped by or transmitted through the potential. In this paper, we propose an efficient method for finding such a critical velocity by using the generalized polynomial chaos method. For the proposed method, we assume that the soliton velocity is a random variable and expand the solution in the random space using the orthogonal polynomials. The proposed method finds the critical velocity accurately with spectral convergence. Thus the computational complexity is much reduced. Numerical results for the smaller and higher values of the potential strength confirm the spectral convergence of the proposed method.