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The generation of nonlinear modulated waves is investigated in the framework of hydrodynamics using a model of coupled oscillators. In this model, the separatrices between each pair of vortices may be viewed as individual oscillators and are described by a phenomenological one-dimensional discrete complex Ginzburg-Landau equation involving first- and second-nearest neighbor couplings. A theoretical approach based on the linear stability analysis predicts regions of modulational instability, governed by both the first and second-nearest neighbor couplings. From numerical investigations of different wave patterns that may be driven by the modulational instability, it appears that analytical predictions are correctly verified. For wave number in the unstable regions, an initial condition whose amplitude is slightly modulated breaks into a train of unstable patterns. This phenomenon agrees with the description of amplification of the spectral component of the perturbation and its harmonics, as well.

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The purpose of this paper is to extend the application of the modified, generalized, rational harmonic balance method to the determination of approximate analytical solutions to general single-degree-of-freedom oscillator equations. We propose a simple form of rational periodic function that depends on a parameter r such that 0<|r|<1 for the approximating solution. We show how one can determine this parameter analytically as a function of oscillation amplitude for a general nonlinear equation. Three examples are used to demonstrate the procedure and its accuracy.

We consider the stability of a continuous wave solution in line of vortices. The weakly nonlinear dynamics are governed by a one-dimensional (1D) discrete complex cubic Ginzburg–Landau (DCCGL) equation. In particular, within the framework of linear stability analysis, we show how the Lange and Newell criterion for the Benjamin–Feir instability is modified by the lattice effects as well as the second neighbour coupling. A noteworthy feature of the present study is that the initial condition is not disintegrated into trains of unstable patterns which have the shape of solitons in the absence of second neighbour couplings and quintic term. The simulations also show that the main observed physical signatures of this instability are a variety of propagative patterns for short-time scales and chaotic states for long-time scales where the linear perturbation analysis fails.