Viscous Shocks in the Destabilized Kuramoto-Sivashinsky Equation

Abstract

We study stationary periodic solutions of the Kuramoto-Sivashinsky (KS) model for complex spatio-temporal dynamics in the presence of an additional linear destabilizing term. In particular, we show the phase space origins of the previously observed stationary “viscous shocks” and related solutions. These arise in a reversible four-dimensional dynamical system as perturbed heteroclinic connections whose tails are joined through a reinjection mechanism due to the linear term. We present numerical evidence that the transition to the KS limit contains a rich bifurcation structure even within the class of stationary reversible solutions.

Computing absolute and essential spectra using continuation

“…The equation L(λ, ν)w = 0 is obtained upon substituting the spatio-temporal Floquet-Bloch ansatz u(t, x) = exp(λt + νx)w(t, x) into the original linearized problem, where the temporal frequency ω = −iλ and the spatial wavenumber k = −iν are typically complex. The continuation procedure outlined below can be readily adapted to this type of problem, e.g., the Kuramoto-Sivashinsky equation [11]. We give details for the implementation for one-dimensional boundary-value problems using the continuation software AUTO.…”

Fixed points of a destabilized Kuramoto–Sivashinsky equation

“…We obtain from (9), (10), and by introducing ρ k = −η 4 k 4 + βη 2 k 2 + α that a k = ρ k a k + γ η k 2 k−1 n=1 a n a k−n − γ ηk ∞ n=1 a n a k+n = ρ k a k + A k N(a) (11) for all k ∈ N, where a ∈ l 2 (N) in accordance with Section 1.1 (where we considered the equivalent l 2 (Z)) and…”

“…By introducing a destabilizing term αu, shock-like stationary solutions were observed by Wittenberg [10], and Rademacher and Wittenberg [11]. Motivated by this, we consider the general form u t + νu xxxx + βu xx + γ uu x = αu, (1) where α ≥ 0 and β, γ ∈ R. Without the destabilizing term αu, this is the same form that has been subjected to rigorous, computer-assisted studies by Zgliczyński and Mischaikow [12] and Zgliczyński [13][14][15], validating stationary and periodic solutions, and providing means for time integration.…”

Consistent Riccati expansion and exact solutions of the Kuramoto–Sivashinsky equation

“…(1) describes the fluctuation of the position of a flame front, the motion of a fluid going down a vertical wall, or a spatially uniform oscillating chemical reaction in a homogeneous medium. The KS equation also arises from the minimal ingredients necessary to observe interesting bifurcations in a simplified equation for a complex amplitude in fluid dynamics [8,9]. The solitary wave solutions of KS equation (1) was given in [10] by the extended tanh expansion and its Painlevé integrability and Bäcklund transformation have been studied in [11].…”