Stable Periodic Waves in Coupled Kuramoto–Sivashinsky–Korteweg–de Vries Equations

Abstract

Periodic waves are investigated in a system composed of a Kuramoto -Sivashinsky -Korteweg -de Vries (KS-KdV) equation linearly coupled to an extra linear dissipative one. The model describes, e.g., a two-layer liquid film flowing down an inclined plane. It has been recently shown that the system supports stable solitary pulses. We demonstrate that a perturbation analysis, based on the balance equation for the net field momentum, predicts the existence of stable cnoidal waves (CnWs) in the same system. It is found that the mean value u 0 of the wave field u in the main subsystem, but not the mean value of the extra field, affects the stability of the periodic waves. Three different areas can be distinguished inside the stability region in the parameter plane (L,u 0 ), where L is the wave's period. In these areas, stable are, respectively, CnWs with positive velocity, constant solutions, and CnWs with negative velocity. Multistability, i.e., the coexistence of several attractors, including the waves with several maxima per period, appears at large value of L. The analytical predictions are completely confirmed by direct simulations. Stable waves are also found numerically in the limit of vanishing dispersion, when the KS-KdV equation goes over into the KS one.

Interaction of solitary pulses in active dispersive–dissipative media

“…In the present paper, we consider the generalized Kuramoto-Sivashinsky (gKS) equation, also known as the Kuramoto-Sivashinsky-Korteweg-de Vries equation and the Benney equation (Feng et al, 2002),…”

The Coupled Kuramoto-Sivashinsky-KdV Equations for Surface Wave in Multilayered Liquid Films

“…Inequalities (9) and (10) in Theorem 1 are obtained for the Cauchy initial data. Obviously, it is also valid for the initialboundary value problems with periodic boundary conditions studied for the periodic waves of KS-KdV system in [11].…”

Global solutions for coupled Kuramoto-Sivashinsky-KdV system

“…Previous research by the authors of [3,4,5,6,9] on system (1.1) mainly studied the stability of the harmonic wave mode, for example, the stability of steady-state soliton solutions is analyzed by perturbation theory and wave mode in [6]. In [2], linear stability is analyzed in the context of the energy estimate and the local solution is established for the Cauchy problem to (1.1).…”