Coherent structure interactions driven by excited hidden modes

Understanding the dynamics and interaction of coherent structures, such as solitary pulses, is an active topic of fundamental research in nonlinear science. It is well known that a group of interacting pulses may lead to a variety of dynamical regimes, from well-organized steady states to spatio-temporal chaos. An example is a liquid film flowing down a vertical/inclined substrate, a relatively simple open-flow hydrodynamic system, ubiquitous in many engineering applications, such as heat exchangers and chemical reactor columns. This system exhibits a rich variety of spatio-temporal structures that are generic to a large class of hydrodynamic and other nonlinear systems.

Several studies have looked into the dynamics and interactions of pulses. The basic idea is to assume a superposition of interacting pulses, allowing for a "multiparticle description" of the system. However, previous analyses, including our own, focused on weakly interacting pulses, and little is known about dynamic states characterized by strongly interacting pulses, a challenging problem. The purpose of this work is precisely to analyze and quantify strong interactions between coherent structures in falling films. In this direction, we put forward a new coherent structure interaction framework encompassing both strong and weak interactions, underpinning also the previous theories on weak interaction. Specifically, we show that under some conditions a system of two pulses undergoes a transition from a regime of decaying oscillatory dynamics to self-sustained oscillations. Detailed examination of this transition shows that it is governed by a novel mechanism---a peculiar and unusual Hopf bifurcation in which a hidden complex conjugate resonance pair crosses the imaginary axis in the complex plane. We show that such a resonance pair results from the splitting of the resonance pole of the single-pulse system. This is a hidden object that is not part of the spectrum of the linearised operator, and only becomes visible when using appropriate weighted functional spaces. Spectral properties similar to those of the falling film problem, responsible for the existence of resonance poles, are prevalent in a wide class of active-dissipative systems, ranging from fluid dynamics to reaction-diffusion systems.