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Register a new userContinuous families of solitary waves in non-symmetric complex potentials: A Melnikov theory approach
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The existence of stationary solitary waves in symmetric and non-symmetric complex potentials is studied by means of Melnikovs perturbation method. The latter provides analytical conditions for the existence of such waves that bifurcate from the homogeneous nonlinear modes of the system and are located at specific positions with respect to the underlying potential. It is shown that the necessary conditions for the existence of continuous families of stationary solitary waves, as they arise from Melnikov theory, provide general constraints for the real and imaginary part of the potential, that are not restricted to symmetry conditions or specific types of potentials. Direct simulations are used to compare numerical results with the analytical predictions, as well as to investigate the propagation dynamics of the solitary waves.
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We consider the interplay between nonlocal nonlinearity and randomness for two different nonlinear Schrodinger models. We show that stability of bright solitons in presence of random perturbations increases dramatically with the nonlocality-induced finite correlation length of the noise in the transverse plane, by means of both numerical simulations and analytical estimates. In fact, solitons are practically insensitive to noise when the correlation length of the noise becomes comparable to the extent of the wave packet. We characterize soliton stability using two different criteria based on the evolution of the Hamiltonian of the soliton and its power. The first criterion allows us to estimate a time (or distance) over which the soliton preserves its form. The second criterion gives the life-time of the solitary wave packet in terms of its radiative power losses. We derive a simplified mean field approach which allows us to calculate the power loss analytically in the physically relevant case of weakly correlated noise, which in turn serves as a lower estimate of the life-time for correlated noise in general case.
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It is known that after a particular distance of evolution in fiber lasers, two (input) asymmetric soliton like pulses emerge as two (output) symmetric pulses having same and constant energy. We report such a compensation technique in dispersion managed fiber lasers by means of a semi-analytical method known as collective variable approach (CVA) with including third-order dispersion (TOD). The minimum length of fiber laser, at which the output symmetric pulses are obtained from the input asymmetric ones, is calculated for each and every pulse parameters numerically by employing Runge-Kutta fourth order method. The impacts of intercore linear coupling, asymmetric nature of initial parameters and TOD on the evolution of pulse parameters and on the minimum length are also investigated. It is found that strong intercore linear coupling and asymmetric nature of input pulse parameters result in the reduction of fiber laser length. Also, the role of TOD tends to increase the width of the pulses as well as their energies. Besides, chaotic patterns and bifurcation points on the minimum length of the fiber owing to the impact of TOD are also reported in a nutshell.
We investigate the generation and propagation of solitary waves in the context of the Hertz chain and Toda lattice, with the aim to highlight the similarities, as well as differences between these systems. We begin by discussing the kinetic and potential energy of a solitary wave in these systems, and show that under certain circumstances the kinetic and potential energy profiles in these systems (i.e., their spatial distribution) look reasonably close to each other. While this and other features, such as the connection between the amplitude and the total energy of the wave bear similarities between the two models, there are also notable differences, such as the width of the wave. We then study the dynamical behavior of these systems in response to an initial velocity impulse. For the Toda lattice, we do so by employing the inverse scattering transform, and we obtain analytically the ratio between the energy of the resulting solitary wave and the energy of the impulse, as a function of the impulse velocity; we then compare the dynamics of the Toda system to that of the Hertz system, for which the corresponding quantities are obtained through numerical simulations. In the latter system, we obtain a universality in the fraction of the energy stored in the resulting solitary traveling wave irrespectively of the size of the impulse. This fraction turns out to only depend on the nonlinear exponent. Finally, we investigate the relation between the velocity of the resulting solitary wave and the velocity of the impulse. In particular, we provide an alternative proof for the numerical scaling rule of Hertz type systems.
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