Nodal Two-Dimensional Solitons in Nonlinear Parametric Resonance

The parametrically driven damped nonlinear Schrödinger equation serves as an amplitude equation for a variety of resonantly forced oscillatory systems on the plane. In this note, we consider its nodal soliton solutions. We show that although the nodal solitons are stable against radially-symmetric perturbations for sufficiently large damping coefficients, they are always unstable to azimuthal perturbations. The corresponding break-up scenarios are studied using direct numerical simulations. Typically, the nodal solutions break into symmetric “necklaces” of stable nodeless solitons. 1. Two-dimensional localised oscillating structures, commonly referred to as oscillons, have been detected in experiments on vertically vibrated layers of granular material [1], Newtonian fluids and suspensions [2,3]. Numerical simulations established the existence of stable oscillons in a variety of pattern-forming systems, including the Swift-Hohenberg and Ginsburg-Landau equations, perioddoubling maps with continuous spatial coupling, semicontinuum theories and hydrodynamic models [3,4]. These simulations provided a great deal of insight into the phenomenology of the oscillons; however, the mechanism by which they acquire or loose their stability remained poorly understood. In order to elucidate this mechanism, a simple model of a parametrically forced oscillatory medium was proposed recently [5]. The model comprises a two-dimensional lattice of diffusively coupled, vertically vibrated pendula. When driven at the frequency close to their double natural frequency, the pendula execute almost synchronous librations whose slowly varying amplitude satisfies the 2D parametrically driven, damped nonlinear Schrödinger (NLS) equation. The NLS equation was shown to support radially-symmetric, bell-shaped (i.e. nodeless) solitons which turned out to be stable for sufficiently large values of the damping coefficient. These stationary solitons of the amplitude equation correspond to the spatio-temporal envelopes of the oscillons in the original lattice system. By reducing the NLS to a finite-dimensional system in the vicinity of the soliton, its stabilisation mechanism (and hence, the oscillon’s stabilisation mechanism) was clarified [5]. In the present note we consider a more general class of radially-symmetric solitons of the parametrically driven, damped NLS on the plane, namely soli2 N.V. Alexeeva and E.V. Zemlyanaya 0 5 10 15 −1 0 1 2 3 Figure1. Solutions of eq.(3): R0(r) (thin continuous line), R1(r) (thick line), R2(r) (dashed). tons with nodes. We will demonstrate that these solitons are unstable against azimuthal modes, and analyse the evolution of this instability. 2. The parametrically driven, damped NLS equation has the form: iψt +∇ ψ + 2|ψ|ψ − ψ = hψ∗ − iγψ. (1) Here ∇ = ∂/∂x + ∂/∂y. Eq.(1) serves as an amplitude equation for a wide range of nearly-conservative two-dimensional oscillatory systems under parametric forcing. This equation was also used as a phenomenological model of nonlinear Faraday resonance in water [3]. The coefficient h > 0 plays the role of the driver’s strength and γ > 0 is the damping coefficient. We start with the discussion of its nodeless solitons and their stability. The exact (though not explicit) stationary radially-symmetric solution is given by ψ0 = Ae −iθ R0(Ar), (2) where r = x + y, A = 1 + √ h2 − γ2, θ = 1 2 arcsin (γ h )