Effect of the Random Field on the Dynamics of Pulsating, Erupting, and Creeping Solitons in the Cubic-Quintic Complex Ginzburg-Landau Equation

It is shown that the dynamics of pulsating, erupting, and creeping (PEC) solitons obtained from the one-dimensional cubic-quintic complex Ginzburg-Landau equation can be drastically modified in the presence of a random background field. It is found that, when the random field is applied to a pulselike initial profile, multiple soliton trains are formed for the parameters of the pulsating and erupting solitons. Furthermore, as the strength of the gain term increases, the multiple pulsating or erupting solitons transform into fixed-shape stable solitons. This may be important for a practical use such as to generate stable femtosecond pulses. For the case of creeping soliton parameters, the presence of the random field does not generate multiple solitons, however, it induces a rapidly twisting or traveling soliton with a fixed-shape, of which stability can be also controlled by the gain term. - PACS numbers: 42.65.Tg, 03.40.Kf, 05.70.Ln, 47.20.Ky.

Dynamics of Pulsating, Erupting, and Creeping Solitons in the Cubic- Quintic Complex Ginzburg-Landau Equation under the Modulated Field

It is shown that the dynamics of the pulsating, erupting, and creeping (PEC) solitons in the one-dimensional cubic-quintic complex Ginzburg-Landau equation can be drastically modified in the presence of a modulated field. We first perform the linear instability analysis of continuous-wave (CW) and obtain the gain by the modulational instability (MI). It is found that the CW states applied by the weakly modulated field always transform into fronts for the parameters of the PEC solitons. We then show that, when the modulated field is applied to the pulse-like initial profile, multiple solitons are formed for the parameters of the pulsating and erupting solitons. Furthermore, as the strength of the gain term increases, the multiple pulsating or erupting solitons transform into fixedshape stable solitons. This may be important for a practical use such as to generate multiple stable femtosecond pulses. For the case of creeping soliton parameters, the presence of a modulated field does not generate multiple solitons, however, the initial profile transforms into an irregularly pulsating soliton or evolves into a fixed-shape soliton as the strength of the gain term is increased. - PACS numbers: 42.65.Tg, 03.40.Kf, 05.70.Ln, 47.20.Ky

Factorization of the constants of motion

A complete set of first integrals, or constants of motion, for a model system is constructed using “factorization”, as described below. The system is described by the effective Feynman Lagrangian L = [Formula: see text], with one of the simplest, nontrivial, potentials V(x) = (1/2)m ω2x2 selected for study. Four new, explicitly time-dependent, constants of the motion ci±, i = 1, 2 are defined for this system. While [Formula: see text]ci± ≠ 0, [Formula: see text]ci± = [Formula: see text]ci± + [Formula: see text]ci± + [Formula: see text]ci± + · · · = 0 along an extremal of L. The Hamiltonian H is shown to equal a sum of products of the ci±, and verifies [Formula: see text] = 0. A second, functionally independent constant of motion is also constructed as a sum of the quadratic products of ci±. It is shown that these derived constants of motion are in involution.PACS Nos.: 02.30.Jr, 02.30.Ik, 02.60.Cb, 02.30.Hq, 05.70.Ln, 02.50.–r