arXiv subjects

Milena Stanislavova

Publications and source records attributed to Milena Stanislavova.

17 recordsLinked to original sources

arXiv2025

Asymptotic attraction with algebraic rates toward fronts of dispersive-diffusive Burgers equations

Burgers equation is a classic model, which arises in numerous applications. At its very core it is a simple conservation law, which serves as a toy model for various dynamics phenomena. In particular, it supports explicit heteroclinic solutions, both fronts and backs. Their stability has been studied in details. There has been substantial interest in considering dispersive and/or diffusive modifications, which present novel dynamical paradigms in such simple setting. More specificaly, the KdV-Burgers model has been showed to support unique fronts (not all of them monotone!) with fixed values at $\pm \infty$. Many articles, among which \cite{Pego}, \cite{NS1}, \cite{NS2}, have studied the question of stability of monotone (or close to monotone) fronts. In a breakthrough paper, \cite{BBHY}, the authors have extended these results in several different directions. They have considered a wider range of models. The fronts do not need to be monotone, but are subject of a spectral condition instead. Most importantly the method allows for large perturbations, as long as the heteroclinic conditions at $\pm \infty$ are met. That is, there is asymptotic attraction to the said fronts or equivalently the limit set consist of one point. The purpose of this paper is to extend the results of \cite{BBHY} by providing explicit algebraic rates of convergence as $t\to \infty$. We bootstrap these results from the results in \cite{BBHY} using additional energy estimates for two important examples namely KdV-Burgers and the fractional Burgers problem. These rates are likely not optimal, but we conjecture that they are algebraic nonetheless.

math.AP↗

arXiv2024

Existence and stability for the travelling waves of the Benjamin equation

In the seminal work of Benjamin,\cite{Ben} in the late 70's, he has derived the ubiquitous Benjamin model, which is a reduced model in the theory of water waves. Notably, it contains two parameters in its dispersion part and under some special circumstances, it turns into the celebrated KdV or the Benjamin-Ono equation, During the90's, there was renewed interest in it. Benjamin, \cite{Ben1}, \cite{Ben2} studied the problem for existence of solitary waves, followed by works of Bona-Chen, \cite{BC}, Albert-Bona-Restrepo, \cite{ABR}, Pava, \cite{Pava1}, who have showed the existence of travelling waves, mostly by variational, but also bifurcation methods. Some results about the stability became available, but unfortunately, those were restricted to either small waves or Benjamin model, close to a distinguished (i.e. KdV or BO) limit. Quite recently, in \cite{ADM}, Abdallah, Darwich and Molinet, proved existence, orbital stability and uniqueness results for these waves, but only for large values of $\f{c}{\ga^2}>>1$. In this article, we present an alternative constrained maximization procedure for the construction of these waves, for the full range of the parameters, which allows us to ascertain their spectral stability. Moreover, we extend this construction to all $L^2$ subcritical cases (i.e. power nonlinearities $(|u|^{p-2}u)_x$, $2<p\leq 6$). Finally, we propose a different procedure, based on a specific form of the Sobolev embedding inequality, which works for all powers $2<p<\infty$, but produces some unstable waves, for large $p$. Some open questions and a conjecture regarding this last result are proposed for further investigation.

math.AP↗

arXiv2023

Spectral stability of periodic waves for the Zakharov system

The paper concerns with the stability of periodic travelling waves of dnoidal type of the Zakharov system. This problem was considered in Angulo-Brango, Nonlinearity'11, where it was shown that subject to a technical condition on the perturbation, such waves are orbitally stable, with respect to perturbations of the same period. Our main result fills up the gap created by the aforementioned technical condition. More precisely, we show that for all natural values of the parameters, the periodic dnoidal waves are spectrally stable.

math.AP↗

arXiv2022

On the stability of the periodic waves for the Benney system

We analyze the Benney model for interaction of short and long waves in resonant water wave interactions. Our particular interest is in the periodic traveling waves, which we construct and study in detail. The main results are that, for all natural values of the parameters, the periodic dnoidal waves are spectrally stable with respect to perturbations of the same period. For another natural set of parameters, we construct the snoidal waves, which exhibit instabilities, in the same setup. Our results are the first instability results in this context. On the other hand, the spectral stability established herein improves significantly upon the work Angulo-Corcho-Hakkaev, which established stability of the dnoidal waves, on a subset of parameter space, by relying on the Grillakis-Shatah theory. Our approach, which turns out to give definite answer for the entire domain of parameters, relies on the instability index theory. Interestingly, end even though the linearized operators are explicit, our spectral analysis requires subtle and detailed analysis of matrix Schrödinger operators in the periodic context, which support some interesting features.

math.AP↗

arXiv2021

On the Barashenkov-Bogdan-Zhanlav solitons and their stability

The Barashenkov-Bogdan-Zhanlav solitons $u_\pm$ for the forced NLS/Lugiato-Lefever model on the line are considered. While the instability of $u_+$ was established in the original paper, \cite{B1}, the analogous question for $u_-$ was only considered heuristically and numerically. We rigorously analyze the stability of $u_-$ in the various regime of the parameters. In particular, we show that $u_-$ is spectrally stable for small pump strength $h$. Moreover, $u_-$ remains spectrally stable until a pair of neutral eigenvalues of negative Krein signature hits another pair of eigenvalues, which has emanated from the edge of the continuous spectrum, \cite{B1, BBK, ABP}. After the collision, an instability is conjectured and numerically observed in previous works, \cite{B1}.

math.AP↗

arXiv2021

On the stability of periodic waves for the cubic derivative NLS and the quintic NLS

We study the periodic cubic derivative non-linear Schrödinger equation (dNLS) and the (focussing) quintic non-linear Schrödinger equation (NLS). These are both $L^2$ critical dispersive models, which exhibit threshold type behavior, when posed on the line ${\mathbb R}$. We describe the (three parameter) family of non-vanishing bell-shaped solutions for the periodic problem, in closed form. The main objective of the paper is to study their stability with respect to co-periodic perturbations. We analyze these waves for stability in the framework of the cubic DNLS. We provide a criteria for stability, depending on the sign of a scalar quantity. The proof relies on an instability index count, which in turn critically depends on a detailed spectral analysis of a self-adjoint matrix Hill operator. We exhibit a region in parameter space, which produces spectrally stable waves. We also provide an explicit description of the stability of all bell-shaped traveling waves for the quintic NLS, which turns out to be a two parameter subfamily of the one exhibited for DNLS. We give a complete description of their stability - as it turns out some are spectrally stable, while other are spectrally unstable, with respect to co-periodic perturbations.

math.AP↗

arXiv2020

Ground states for the nonlinear Schrödinger equation under a general trapping potential

The classical Schrödinger equation with a harmonic trap potential $V(x)=|x|^2$, describing the quantum harmonic oscillator, has been studied quite extensively in the last twenty years. Its ground states are bell-shaped and unique, among localized positive solutions. In addition, they have been shown to be non-degenerate and (strongly) orbitally stable. All of these results, produced over the course of many publications and multiple authors, rely on ODE methods specifically designed for the Laplacian and the power function potential. In this article, we provide a wide generalization of these results. More specifically, we assume sub-Laplacian fractional dispersion and a very general form of the trapping potential $V$, with the driving linear operator in the form $H=(-Δ)^s+V, 0<s\leq 1$. We show that the normalized waves of such semi-linear fractional Schrödinger equation exist, they are bell-shaped, provided that the non-linearity is of the form $|u|^{p-1} u, p<1+\frac{4 s}{n}$. In addition, we show that such waves are non-degenerate, and strongly orbitally stable. Most of these results are new even in the classical case $H=-Δ+V$, where $V$ is a general trapping potential considered herein.

math.AP↗

arXiv2018

On the energy decay rates for the 1D damped fractional Klein-Gordon equation

We consider the fractional Klein-Gordon equation in one spatial dimension, subjected to a damping coefficient, which is non-trivial and periodic, or more generally strictly positive on a periodic set. We show that the energy of the solution decays at the polynomial rate $O(t^{-\frac{s}{4-2s}})$ for $0< s<2 $ and at some exponential rate when $s\geq 2$. Our approach is based on the asymptotic theory of $C_0$ semigroups in which one can relate the decay rate of the energy in terms of the resolvent growth of the semigroup generator. The main technical result is a new observability estimate for the fractional Laplacian, which may be of independent interest.

math.AP↗

arXiv2018

On the generation of stable Kerr frequency combs in the Lugiato-Lefever model of periodic optical waveguides

We consider the Lugiato-Lefever (LL) model of optical fibers. We construct a two parameter family of steady state solutions, i.e. Kerr frequency combs, for small pumping parameter $h>0$ and the correspondingly (and necessarily) small detuning parameter, $α>0$. These are $O(1)$ waves, as they are constructed as bifurcation from the standard cnoidal solutions of the cubic NLS. We identify the spectrally stable ones, and more precisely, we show that the spectrum of the linearized operator contains the eigenvalues $0, -2α$, while the rest of it is a subset of $ \{μ: \Reμ=-α\}$. This is in line with the expectations for effectively damped Hamiltonian systems, such as the LL model.

math.AP↗

arXiv2016

Spectral stability for classical periodic waves of the Ostrovsky and short pulse models

We consider the Ostrovsky and short pulse models in a symmetric spatial interval, subject to periodic boundary conditions. For the Ostrovsky case, we revisit the classical periodic traveling waves and for the short pulse model, we explicitly construct traveling waves in terms of Jacobi elliptic functions. For both examples, we show spectral stability, for all values of the parameters. This is achieved by studying the non-standard eigenvalue problems in the form $L u=\la u'$, where $L$ is a Hill operator.

math.AP↗

arXiv2014

Spectral stability analysis for standing waves of a perturbed Klein-Gordon equation

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.

nlin.PS↗

arXiv2012

Spectral stability for subsonic traveling pulses of the Boussinesq `abc' system

We consider the spectral stability of certain traveling wave solutions of the Boussinesq `abc' system. More precisely, we consider the explicit $sech^2(x)$ like solutions of the form $(\vp(x-w t), ψ(x- w t)=(\vp, const. \vp)$, exhibited by M. Chen (1998) and we provide a complete rigorous characterization of the spectral stability in all cases for which $a=c<0, b>0$.

math.AP↗

arXiv2012

Linear stability analysis for periodic traveling waves of the Boussinesq equation and the KGZ system

The question for linear stability of spatially periodic waves for the Boussinesq equation (the cases $p=2,3$) and the Klein-Gordon-Zakharov system is considered. For a wide class of solutions, we completely and explicitly characterize their linear stability (instability respectively), when the perturbations are taken with the same period $T$. In particular, our results allow us to completely recover the linear stability results, in the limit $T\to \infty$, for the whole line case.

math.AP↗

arXiv2011

Linear stability analysis for traveling waves of second order in time PDE's

We develop a general theory for linear stability of traveling waves of second order in time PDE's. More precisely, we introduce an explicitly computable index $\om^*\in (0, \infty]$ (depending on the self-adjoint part of the linearized operator) so that the wave is stable if and only if $|c|\geq \om^*$. The results are applicable both in the periodic case and in the whole line case. As an application, we consider three classical models - the Boussinesq equation, the Klein-Gordon-Zakharov (KGZ) system and the fourth order beam equation. For the Boussinesq model and the KGZ system (and as a direct application of the main results), we compute explicitly the set of speeds which give rise to linearly stable traveling waves (and for all powers of $p$ in the case of Boussinesq). This result is new for the KGZ system, while it generalizes the results of Alexander-Sachs, which apply to the case $p=2$. For the beam equation, we provide an explicit formula (depending of the function $\|\vp_c'\|_{L^2}$), which works for all $p$ and for both the periodic and the whole line cases. Our results complement (and exactly match, whenever they exist) the results of a long line of investigation regarding the related notion of orbital stability of the same waves.

math.AP↗

arXiv2010

Transverse instability for periodic waves of KP-I and Schrödinger equations

We consider the quadratic and cubic KP - I and NLS models in $1+2$ dimensions with periodic boundary conditions. We show that the spatially periodic travelling waves (with period $K$) in the form $u(t,x,y)=\vp(x-c t)$ are spectrally and linearly unstable, when the perturbations are taken to be with the same period. This strong instability implies other instabilities considered recently - for example with respect to perturbations with periods $nK, n=2, 3, ...$ or bounded perturbations.

math.AP↗

arXiv2007

The Kuramoto-Sivashinsky equation in R^1 and R^2: effective estimates of the high-frequency tails and higher Sobolev norms

We consider the Kuramoto-Sivashinsky (KS) equation in finite domains of the form $[-L,L]^d$. Our main result provides refined Gevrey estimates for the solutions of the one dimensional differentiated KS, which in turn imply effective new estimates for higher Sobolev norms of the solutions in terms of powers of $L$. We illustrate our method on a simpler model, namely the regularized Burger's equation. We also show local well-posedness for the two dimensional KS equation and provide an explicit criteria for (eventual) blow-up in terms of its $L^2$ norm. The common underlying idea in both results is that {\it a priori} control of the $L^2$ norm is enough in order to conclude higher order regularity and allows one to get good estimates on the high-frequency tails of the solutions.

math.DS↗

arXiv2006

Attractors for the viscous Camassa-Holm equation

We show that the viscous Camassa-Holm equation subject to an external force, and where the viscosity term is given by second order differential operator in divergence form has a global attractors in the energy space $H^1$. Moreover, we establish an asymptotic smoothing effect.

math.DS↗