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
Effective estimates of the higher Sobolev norms for the Kuramoto-Sivashinsky equation
Milena Stanislavova Atanas Stefanov
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
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.
On the Lipschitzness of the solution map for the 2 D Navier-Stokes system
Atanas Stefanov
We consider the Navier-Stokes system on R2. It is well-known that solutions with $L^2$ data become instantly smooth and persist globally. In this note, we show that the solution map is Lipschitz, when acting in $L^\infty $Hσ (R2) and $L^2_t$Hσ+1 (R2), when $0\leq $ σ<1. This generalizes an earlier result of Gallagher and Planchon [7], who showed the Lipschitzness in $L^2$(R2). The question for the Lipschitzness of the map in Hσ (R2), σ$\geq 1$ remains an interesting open problem, which hinges upon the validity of an endpoint estimate for the trilinear form $(\phi, v, w)\to \int$R2(∂Φ/∂x ∂v/∂y - ∂Φ/∂y ∂v/∂x)wdx.
keywords: $L^2$ solution maps 2 D Navier-Stokes equation stability of solutions.
Smoothing-Strichartz estimates for the Schrodinger equation with small magnetic potential
Vladimir Georgiev Atanas Stefanov Mirko Tarulli
The work treats smoothing and dispersive properties of solutions to the Schrödinger equation with magnetic potential. Under suitable smallness assumption on the potential involving scale invariant norms we prove smoothing - Strichartz estimate for the corresponding Cauchy problem. An application that guarantees absence of pure point spectrum of the corresponding perturbed Laplace operator is discussed too.
keywords: Strichartz estimates smoothing properties. Schrödinger Equation
Dmitry Pelinovsky Milena Stanislavova Atanas Stefanov
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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