The existence of a global attractor for a Kuramoto-Sivashinsky type equation in 2D
Aslihan Demirkaya
We consider a variation of the Kuramoto-Sivashinsky Equation in space dimension two. We show that under some assumptions the equation is globally well-posed and posseses a global attractor in the periodic case. The analysis is based on the Lyapunov function approach, point dissipativeness and asymptotic compactness.
keywords: Attractors Kuramoto- Sivashinsky Equation
Numerical results on existence and stability of standing and traveling waves for the fourth order beam equation
Aslihan Demirkaya Milena Stanislavova

In this paper, we study numerically the existence and stability of some special solutions of the nonlinear beam equation: $u_{tt}+u_{xxxx}+u-|u|^{p-1} u = 0$ when $p = 3$ and $p = 5$. For the standing wave solutions $u(x, t) = e^{iω t}\varphi_{ω}(x)$ we numerically illustrate their existence using variational approach. Our numerics illustrate the existence of both ground states and excited states. We also compute numerically the threshold value $ω^*$ which separates stable and unstable ground states. Next, we study the existence and linear stability of periodic traveling wave solutions $u(x, t) = φ_c(x+ct)$. We present numerical illustration of the theoretically predicted threshold value of the speed $c$ which separates the stable and unstable waves.

keywords: Nonlinear beam equation standing waves traveling waves spectral stability orbital stability
Spectral stability analysis for standing waves of a perturbed Klein-Gordon equation
Aslihan Demirkaya Panayotis G. Kevrekidis Milena Stanislavova Atanas Stefanov
In the present work, we introduce a new $\mathcal{PT}$-symmetric variant of the Klein-Gordon field theoretic problem. We identify the standing wave solutions of the proposed class of equations and analyze their stability. In particular, we obtain an explicit frequency condition, somewhat reminiscent of the classical Vakhitov-Kolokolov criterion, which sharply separates the regimes of spectral stability and instability. Our numerical computations corroborate the relevant theoretical result.
keywords: standing waves. linear stability

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