Stable Complexes of Parametrically Driven, Damped Nonlinear Schrödinger Solitons

Abstract

Since solitons of the parametrically driven damped nonlinear Schrödinger equation do not have oscillatory tails, it was suggested that they cannot form bound states. We show that this equation does support solitonic complexes, with the mechanism of their formation being different from the standard tail-overlap mechanism. One of the arising stationary complexes is found to be stable in a wide range of parameters, others unstable. PACS numbers: 42.65.Tg, 05.45.Yv Motivation.-Bound states of solitons and solitary pulses are attracting increasing attention in nonlinear optics [1-5], dynamics of fluids [6-9], and excitable media [10]. Stable bound states can compete with free solitons as alternative attractors. This is detrimental in nonlinear optics, for example, where the interaction between adjacent pulses poses limitations to the stable operation of transmission lines and information storage elements. Unstable solitonic complexes are not meaningless either; they serve as intermediate states visited by the system when in the spatiotemporal chaotic regime.Here, we consider solitonic complexes in the parametrically driven damped nonlinear Schrödinger equation,

Dynamics of counterpropagating waves in parametrically driven systems: dispersion vs. advection

“…It is important to note that the number of Fourier modes required for the numerical integration of Eqs. (7)- (9) should not be too large since otherwise the long wavelength assumption (2) and (3) can be violated. This is because the parabolic approximation produced by the amplitude equations (dotted line in Fig.…”

“…When ν > √ 3 and γ − ≤ γ ≤ γ + , where γ ∓ = R ± 1 + (R 2 ± − ν) 2 and R 2 ± = (2ν ± √ ν 2 − 3)/3, there are three such states, say, A(x, t), t > 0, necessarily develops a front for large t along the characteristic of each intersection point. The presence of weak dispersion thus smooths the solution of the hyperbolic system [17] preventing the formation of infinite gradients and the resulting solutions (Figs.…”

“…This equation has been extensively studied in the literature (see, e.g. [1][2][3] and references therein). As the forcing amplitude is increased, the trivial solution (corresponding to a spatially homogeneous state) loses stability to frequency-locked standing waves (SW).…”

Low-Dimensional Description of Pulses under the Action of Global Feedback Control

“…Thus the two-pulse solution is unstable for 1−p C < 0 and stable for 1−p C > 0 and C = C − , and this is opposite to the stability of the single-pulse solution, at least if we do not consider the possibility of the oscillatory instability. Note that multipulse solitons were studied formerly for the parametrically driven [5,6] and ac-driven [4] damped nonlinear Schrödinger equation.…”

Soliton complexity in the damped-driven nonlinear Schrödinger equation: Stationary to periodic to quasiperiodic complexes

“…Stable solitonic complexes, or bound states, were detected in a variety of soliton-bearing partial differential equations [6][7][8][9][10][11][12][13][14][15]. One mechanism of complex formation is the trapping of the soliton in a potential well formed by the undulating tail of its partner [7,8].…”