Multisoliton Solutions of the Complex Ginzburg-Landau Equation

Abstract

We present novel stable solutions which are soliton pairs and trains of the 1D complex GinzburgLandau equation (CGLE), and analyze them. We propose that the distance between the pulses and the phase difference between them is defined by energy and momentum balance equations. We present a two-dimensional phase plane ("interaction plane") for analyzing the stability properties and general dynamics of two-soliton solutions of the CGLE. [S0031-9007(97) [3], and spatially extended nonequilibrium systems [4,5]. In optics, it is useful in analyzing optical transmission lines [6,7], passively mode-locked fiber lasers [8,9], and spatial optical solitons [10]. In each case, the problem of the interaction of two, individually stable, juxtaposed elementary coherent structures (i.e., solitons) is crucial for understanding the general behavior of the system [11,12].Stable pulse-like solutions of the quintic CGLE have been found by Thual and Fauve [13]. Minimal requirements for their stability have been obtained in [14]. In the conservative limit, these solutions can be considered as perturbations of the nonlinear Schrödinger equation (NLSE) solitons [15]. The continuous transition of these solutions from the conservative limit to the gradient limit in the parameter space of the CGLE has been studied in Ref. [16]. Although the dynamical properties of these pulselike solutions, their collisions and interactions are different from those of solitons of integrable systems, they have been called "solitons" in a number of works. We follow this tradition, and also call them "solitons" or "soliton solutions."For the nonlinear Schrödinger equation, two solitons have zero binding energy. Hence, any nonlinear superposition of two solitons is neutrally stable, and can be made unstable with a very small perturbation. On the other hand, for the NLSE, there is no stationary solution in the form of two solitons with equal amplitudes and velocities and with a fixed separation. Frequently, real systems are not described by integrable equations (e.g., the NLSE), but by Hamiltonian generalizations of the NLSE. For these systems, the interaction between the pulses becomes inelastic, so that two-soliton solutions of the perturbed NLSE (when they exist) are unstable due to the energy exchange between the pulses [17]. The situation changes completely for nonconservative systems. Each soliton then has its own internal balance of energy which maintains its constant amplitude. Fixing the amplitudes effectively reduces the number of degrees of freedom in the system of two solitons and can make it stable. Bound states of two solitons in these systems were first analyzed by Malomed [18]. Using standard perturbation analysis for soliton interaction, he showed that stationary solutions in the form of bound states of two solitons, which are in-phase or out-of-phase, may exist. We also confirm that they do exist. However, careful numerical 0031-9007͞97͞79(21)͞4047(5)$10.00

Discrete family of dissipative soliton pairs in mode-locked fiber lasers

“…In the lumped model no stable or even stationary BS solutions with that phase difference could be identified. For the CQGLE model, but just for anomalous dispersion, this case has been studied earlier [21], and it was shown, that stable BS solutions can exist.…”

“…From previous investigations [13,14,15,16,17,18,19,20,21] it is known that BS formation is caused by linear and nonlinear dissipative effects (right hand side of Eq. (7)).…”

“…In previous papers [17,18,21] it was shown that inphase and out-of-phase first level BS are unstable. From our propagation simulations we anticipate that the corresponding higher level BSs are unstable as well.…”

“…For certain parameter sets this equation was shown to support solitary wave solutions, called dissipative solitons, as well as bound states (BS) of pairs of SPs [13,14,15,16,17,18,19,20,21]. Careful numerical investigations [16] have demonstrated BS formation from a pair of initially resting SPs.…”

“…Furthermore it was shown [17,18] that out-of-phase and in-phase SPs can form unstable stationary BS solutions, because they represent saddle points in the phase plane. In the framework of the CQGLE stable BSs were discovered by Akhmediev et al [21] as stable two-soliton solutions with a π/2 phase difference of the peaks. They were observed for anomalous dispersion only.…”