Milena Stanislavova

On the vortices for the nonlinear Schrödinger equation in higher dimensions

We consider the nonlinear Schrödinger equation in n space dimensions and study the existence and stability of standing wave solutions of the form and For n=2k, (rj,θj) are polar coordinates in , j=1,2,…,k; for n=2k+1, (rj,θj) are polar coordinates in , (rk,θk,z) are cylindrical coordinates in , j=1,2,…,k−1. We show the existence of functions φw, which are constructed variationally as minimizers of appropriate constrained functionals. These waves are shown to be spectrally stable (with respect to perturbations of the same type), if 1<p<1+4/n. This article is part of the theme issue ‘Stability of nonlinear waves and patterns and related topics’.

Conditional stability theorem for the one dimensional Klein-Gordon equation

The paper addresses the conditional non-linear stability of the steady state solutions of the one-dimensional Klein-Gordon equation for large time. We explicitly construct the center-stable manifold for the steady state solutions using the modulation method of Soffer and Weinstein and Strichartz type estimates. The main difficulty in the one-dimensional case is that the required decay of the Klein-Gordon semigroup does not follow from Strichartz estimates alone. We resolve this issue by proving an additional weighted decay estimate and further refinement of the function spaces, which allows us to close the argument in spaces with very little time decay.

Periodic Traveling Waves of the Regularized Short Pulse and Ostrovsky Equations: Existence and Stability

We construct various periodic traveling wave solutions of the Ostrovsky/Hunter--Saxton/short pulse equation and its KdV regularized version. For the regularized short pulse model with small Coriolis parameter, we describe a family of periodic traveling waves which are a perturbation of appropriate KdV solitary waves. We show that these waves are spectrally stable. For the short pulse model, we construct a family of traveling peakons with corner crests. We show that the peakons are spectrally stable as well.

Asymptotic stability for spectrally stable Lugiato-Lefever solitons in periodic waveguides

We consider the Lugiato-Lefever model of optical fibers in the periodic context. Spectrally stable periodic steady states were constructed recently in the studies of Delcey and Haragus [Philos. Trans. R. Soc., A 376, 20170188 (2018)]; [Rev. Roumaine Math. Pures Appl. (to be published)]; and Hakkaev et al. (e-print arXiv:1806.04821). The spectrum of the linearization around such solitons consists of simple eigenvalues 0, −2α < 0, while the rest of it is a subset of the vertical line { μ : R μ = − α } . Assuming such a property abstractly, we show that the linearized operator generates a C0 semigroup and, more importantly, the semigroup obeys (optimal) exponential decay estimates. Our approach is based on the Gearhart-Pruss theorem, where the required resolvent estimates may be of independent interest. These results are applied to the proof of asymptotic stability with phase of the steady states.We consider the Lugiato-Lefever model of optical fibers in the periodic context. Spectrally stable periodic steady states were constructed recently in the studies of Delcey and Haragus [Philos. Trans. R. Soc., A 376, 20170188 (2018)]; [Rev. Roumaine Math. Pures Appl. (to be published)]; and Hakkaev et al. (e-print arXiv:1806.04821). The spectrum of the linearization around such solitons consists of simple eigenvalues 0, −2α < 0, while the rest of it is a subset of the vertical line { μ : R μ = − α } . Assuming such a property abstractly, we show that the linearized operator generates a C0 semigroup and, more importantly, the semigroup obeys (optimal) exponential decay estimates. Our approach is based on the Gearhart-Pruss theorem, where the required resolvent estimates may be of independent interest. These results are applied to the proof of asymptotic stability with phase of the steady states.

On the spectral stability of ground states of semi-linear Schrödinger and Klein-Gordon equations with fractional dispersion

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.

On the dynamics of the four‐dimensional rigid body in a quadratic potential field

We study the nondegenerated solutions of the rotation of a four‐dimensional rigid body in a quadratic potential field. This problem has 6 degrees of freedom. We obtain 143 topologically different solutions and explicit formulas in Prym theta‐functions.