Interaction of solitary pulses in active dispersive–dissipative media

Abstract

We examine weak interaction and formation of bound states of pulses for the generalized Kuramoto-Sivashinsky (gKS) equation, which is one of the simplest prototypes describing active media with energy supply, dissipation, dispersion, and nonlinearity. We derive a system of ordinary differential equations describing the leading-order dynamics of the pulses of the gKS equation and prove a criterion for the existence of a countable infinite or finite number of bound states. Our theory is corroborated by computations of the full equation.

Wavy regimes of film flow down a fiber

“…The equation is also often referred to as the "generalized KS" equation [21,37,38], and it is the KS equation appropriately extended to include dispersion (∂ xxx H ). As before, we look for TW solutions in their moving frame, ξ = x − ct. Stretching the moving coordinate as ξ = √ ηX, the amplitude as H = η −3/2 A/2, and the speed as (40) where δ K = 3η 3/2 log(α) is a parameter that measures the relative importance of dispersion.…”

Rigorous coherent-structure theory for falling liquid films: Viscous dispersion effects on bound-state formation and self-organization

“…As it has been emphasized in the Introduction, at sufficiently large distances from the inlet of the film, the dynamics of the free surface is dominated by the presence of localized coherent structures, each of which resembling (infinite-domain) solitary pulses, which continuously interact with each other as quasi particles through attractions and repulsions. The objective of this section is to appropriately extend the recently developed coherent-structure interaction theory for the solitary pulses of the gKS equation 16,20,21 to the two-field model given by (3). This is by far a non-trivial task as we shall demonstrate, e.g.…”

“…However, for the present problem, as well as for the gKS equation analyzed recently in Refs. 16,20, and 21, the pulses are inherently unstable with the zero eigenvalue of the linearized interaction operator embedded into the essential spectrum. This in turn suggests that the usual projection procedure used in previous studies, such as those on weak-interaction approaches for the gKS equation (e.g.…”

Pulse dynamics in a power-law falling film

“…It is important to remark, however, that the fact that the discrete part of the spectrum is embedded into the continuous part makes it difficult to elucidate the true mechanisms for the instability of the pulse, not to mention that a rigorous analysis of the relevant discrete eigenfunctions would require one to study the infinite-domain limit of the system (Pradas et al 2011), something which is also crucial for an accurate description of pulse interactions (see § 4 below). Following the works by Pego & Weinstein (1994) (see also Sandstede & Scheel 2000) one can make use of exponentially weighted functional spaces to guarantee that any disturbance originated in the flat-film area of the solution will be filtered away, allowing then to study the infinite-domain properties of a single pulse (see also the works by Tseluiko et al (2010b) and Tseluiko & Kalliadasis (2014) for the gKS equation and Pradas et al (2011) for the Newtonian case). We shall therefore proceed by formulating the above equations in a weighted space.…”

“…Sandstede & Scheel 2000). Due to the convectively unstable character of film, a rigorous study of the spectral properties of the system requires a careful and rather technical analysis (see Pradas et al (2011) for the Newtonian second-order model or Chang & Demekhin (2002), Tseluiko et al (2010a), Tseluiko, Saprykin & Kalliadasis (2010b) and Tseluiko & Kalliadasis (2014) for an analysis of the generalised Kuramoto-Sivashinsky (gKS) equation). In particular, it was found in these studies that the spectrum of L consists of two parts, continuous and discrete, which are responsible for the flat-film solution stability and the stability of localised solitary waves, respectively.…”