On the global attractor for the damped Benjamin-Bona-Mahony equation
Milena Stanislavova
Conference Publications 2005, 2005(Special): 824-832 doi: 10.3934/proc.2005.2005.824
We present a new necessary and sufficient condition to verify the asymptotic compactness of an evolution equation defined in an unbounded domain, which involves the Littlewood-Paley projection operators. We then use this condition to prove the existence of an attractor for the damped \bbme in the phase space $H^1({\bf R})$ by showing the solutions are point dissipative and asymptotically compact. Moreover the attractor is in fact smoother and it belongs to $H^{3/2-\ve}$ for every $\ve>0$.
keywords: Keywords and Phrases
Spectral stability analysis for standing waves of a perturbed Klein-Gordon equation
Aslihan Demirkaya Panayotis G. Kevrekidis Milena Stanislavova Atanas Stefanov
Conference Publications 2015, 2015(special): 359-368 doi: 10.3934/proc.2015.0359
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
Effective estimates of the higher Sobolev norms for the Kuramoto-Sivashinsky equation
Milena Stanislavova Atanas Stefanov
Conference Publications 2009, 2009(Special): 729-738 doi: 10.3934/proc.2009.2009.729
We consider the Kuramoto-Sivashinsky (KS) equation in finite domains of the form $[-L,L]$. Our main result provides effective new estimates for higher Sobolev norms of the solutions in terms of powers of $L$ for the one-dimentional differentiated KS. We illustrate our method on a simpler model, namely the regularized Burger's equation. The underlying idea in this result is that a priori control of the $L^2$ norm is enough in order to conclude higher order regularity and in fact, it allows one to get good estimates on the high-frequency tails of the solution.
keywords: Gevrey regularity regularized Burger's equation Kuramoto-Sivashinsky equation
Attractors for the viscous Camassa-Holm equation
Milena Stanislavova Atanas Stefanov
Discrete & Continuous Dynamical Systems - A 2007, 18(1): 159-186 doi: 10.3934/dcds.2007.18.159
We consider the viscous Camassa-Holm equation subject to an external force, where the viscosity term is given by second order differential operator in divergence form. We show that under some mild assumptions on the viscosity term, one has global well-posedness both in the periodic case and the case of the whole line. In the periodic case, we show the existence of global attractors in the energy space $H^1$, provided the external force is in the class $L^2(I)$. Moreover, we establish an asymptotic smoothing effect, which states that the elements of the attractor are in fact in the smoother Besov space B2 2, ∞$(I)$. Identical results (after adding an appropriate linear damping term) are obtained in the case of the whole line.
keywords: Viscous Camassa-Holm equation global solutions attractors.
Numerical results on existence and stability of standing and traveling waves for the fourth order beam equation
Aslihan Demirkaya Milena Stanislavova
Discrete & Continuous Dynamical Systems - B 2019, 24(1): 197-209 doi: 10.3934/dcdsb.2018097

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
Dmitry Pelinovsky Milena Stanislavova Atanas Stefanov
Discrete & Continuous Dynamical Systems - S 2012, 5(5): i-iii doi: 10.3934/dcdss.2012.5.5i
Partial differential equations viewed as dynamical systems on an infinite-dimensional space describe many important physical phenomena. Lately, an unprecedented expansion of this field of mathematics has found applications in areas as diverse as fluid dynamics, nonlinear optics and network communications, combustion and flame propagation, to mention just a few. In addition, there have been many recent advances in the mathematical analysis of differential difference equations with applications to the physics of Bose-Einstein condensates, DNA modeling, and other physical contexts. Many of these models support coherent structures such as solitary waves (traveling or standing), as well as periodic wave solutions. These coherent structures are very important objects when modeling physical processes and their stability is essential in practical applications. Stable states of the system attract dynamics from all nearby configurations, while the ability to control coherent structures is of practical importance as well. This special issue of Discrete and Continuous Dynamical Systems is devoted to the analysis of nonlinear equations of mathematical physics with a particular emphasis on existence and dynamics of localized modes. The unifying idea is to predict the long time behavior of these solutions. Three of the papers deal with continuous models, while the other three describe discrete lattice equations.

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On the spectral stability of ground states of semi-linear Schrödinger and Klein-Gordon equations with fractional dispersion
Wen Feng Milena Stanislavova Atanas Stefanov
Communications on Pure & Applied Analysis 2018, 17(4): 1371-1385 doi: 10.3934/cpaa.2018067

We consider standing wave solutions of various dispersive models with non-standard form of the dispersion terms. Using index count calculations, together with the information from a variational construction, we develop sharp conditions for spectral stability of these waves.

keywords: Spectral stability standing waves fractional NLS fractional KleinGordon equation

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