Heteroclinic Structure of Parametric Resonance in the Nonlinear Schrödinger Equation

Abstract

We show that the nonlinear stage of modulational instability induced by parametric driving in the defocusing nonlinear Schrödinger equation can be accurately described by combining mode truncation and averaging methods, valid in the strong driving regime. The resulting integrable oscillator reveals a complex hidden heteroclinic structure of the instability. A remarkable consequence, validated by the numerical integration of the original model, is the existence of breather solutions separating different Fermi-Pasta-Ulam recurrent regimes. Our theory also shows that optimal parametric amplification unexpectedly occurs outside the bandwidth of the resonance (or Arnold tongues) arising from the linearised Floquet analysis. [12,13]. While the concept of PR originates in the linear world [14], PRs deeply impact also the behavior of nonlinear conservative systems. However, the full nonlinear dynamics of PRs is relatively well understood only for lowdimensional Hamiltonian systems [2,15,16]. Conversely, the analysis of extended systems described by PDEs with periodicity in the evolution variable [17] is essentially limited to determine the region of parametric instability (Arnold tongues) via Floquet analysis [18][19][20][21], while the nonlinear stage of PR past the linearized growth of the unstable modes remains mostly unexplored.In this letter, taking the periodic defocusing nonlinear Schrödinger equation (NLSE) as a widespread example describing, e.g. periodic management of atom condensates [12,19,20,22], optical beam propagation in layered media [21], or optical fibers with periodic dispersion [18,23,24], we show that the PR gives rise to quasi-periodic evolutions which exhibit on average FermiPasta-Ulam (FPU) recurrence [25] with a remarkably complex (but ordered) underlying phase-plane structure. Such structure describes the continuation into the nonlinear regime of the modulational instability (MI) of a background solution, uniquely due to the parametric forcing (with zero forcing the defocusing NLSE is stable). A byproduct of this structure is the existence of breatherlike solutions [26]. This fact suggests the intriguing possibility of observing rogue waves [27], which are commonly associated with breathers [28], in the defocusing NLSE [29]. On the other hand, the richness of such structure allows us to remarkably predict that optimal parametric amplification occurs at a critical frequency where the system lies off-resonance (outside the PR bandwidth). Our approach retains its validity in the most interesting regime of strong parametric driving, where the system is found to exhibit a remarkably ordered structure despite its broken translational symmetry and integrability. In this sense the physics differs from other integrable models exhibiting a complex nonlinear dynamics of MI already in the undriven regime (e.g. focusing NLSE [26,[30][31][32][33][34][35][36]), around which chaos can develop under weak periodic perturbations [31,37,38].We consider the following periodic NLSEreferring, without loss of genera...

Graded-index breathing solitons from Airy pulses in multimode fibers

“…1(d) we can see that the peak power evolution of the breathing soliton changes nearly periodically while the fast and irregular intensity oscillations can be identified for each breathing period. This behavior is similar to the pump power evolution for the parametric resonance breathers within the periodically modulated dispersion single-mode fiber [46]. These can be attributed to both the power decay in the form of the persistent dispersive wave emissions and the adiabatically reshaped pulse through the periodic perturbation given by ( ) g ξ during each breathing cycle.…”

Recurrence in the high-order nonlinear Schrödinger equation: A low-dimensional analysis

“…This approach proved effective in other nearlyintegrable systems [33], where exact solutions cannot be constructed. This is the case of HONLS: the Akhmediev breather [34,35], the Peregrine soliton [36,37], conventionally regarded as prototypes of rogue waves [38,39], only represent approximate solutions [40].…”

Stabilization of Unsteady Nonlinear Waves by Phase-Space Manipulation

“…The present mechanism can also occur where the NLSE provides a leading-order description of nonlinear MI, such as Bose-Einstein condensation [29] and optics, where quasi-stabilization has been interpreted in terms of solitons [30]. Indeed, this approach can be extended to other models with a homoclinic structure [31], and even to settings such as parametric resonance described by strongly non-integrable models [32].…”