Related papers: Stabilization of unsteady nonlinear waves by phase…
The two-dimensional cubic nonlinear Schrodinger equation admits a large family of one-dimensional bounded traveling-wave solutions. All such solutions may be written in terms of an amplitude and a phase. Solutions with piecewise constant…
Accurate simulations of ice sheet dynamics, mantle convection, lava flow, and other highly viscous free-surface flows involve solving the coupled Stokes/free-surface equations. In this paper, we theoretically analyze the stability and…
In this paper, we study local well-posedness and orbital stability of standing waves for a singularly perturbed one-dimensional nonlinear Klein-Gordon equation. We first establish local well-posedness of the Cauchy problem by a fixed point…
In this paper, we determine the spectral instability of periodic odd waves for the defocusing fractional cubic nonlinear Schr\"odinger equation. Our approach is based on periodic perturbations that have the same period as the standing wave…
We study stationary capillary-gravity waves in a two-dimensional body of water that rests above a flat ocean bed and below vacuum. This system is described by the Euler equations with a free surface. Our main result states that there exist…
In this paper, we determine the transverse instability of periodic standing wave solutions for the generalized Schr\"odinger equation with fractional power nonlinearity. The existence of periodic waves is determined by using a constrained…
We study the strong instability of standing waves $e^{i\omega t}\phi_\omega(x)$ for nonlinear Schr\"{o}dinger equations with an $L^2$-supercritical nonlinearity and an attractive inverse power potential, where $\omega\in\mathbb{R}$ is a…
The modulation instability (MI) is responsible for the disintegration of a regular nonlinear wave train and can lead to strong localizations in a from of rogue waves. This mechanism has been studied in a variety of nonlinear dispersive…
We prove an asymptotic stability result for the water wave equations linearized around small solitary waves. The equations we consider govern irrotational flow of a fluid with constant density bounded below by a rigid horizontal bottom and…
This thesis is concerned with dynamics of conservative nonlinear waves on bounded domains. In general, there are two scenarios of evolution. Either the solution behaves in an oscillatory, quasiperiodic manner or the nonlinear effects cause…
Using a standing light wave trap, a stable quasi-one-dimensional attractive dilute-gas Bose-Einstein condensate can be realized. In a mean-field approximation, this phenomenon is modeled by the cubic nonlinear Schr\"odinger equation with…
We demonstrate the first successful non-invasive stabilisation of nonlinear travelling waves in a straight cylindrical pipe using time-delayed feedback control (TDF) working in various symmetry subspaces. By using an approximate linear…
In the framework of the ordinary non-relativistic quantum mechanics, it is known that a quantum particle in a rapidly-oscillating bound potential with vanishing time average can be scattered off or even trapped owing to the phenomenon of…
The dynamical stability of solitary lattice waves in non-integrable FPUT chains is a longstanding open problem and has been solved so far only in a certain asymptotic regime, namely by Friesecke and Pego for the KdV limit, in which the…
A nonlinear Schrodinger equation arising from light propagation down an inhomogeneous medium is considered. The inhomogeneity is reflected through a non-uniform coefficient of the non-linear term in the equation. In particular, a…
This paper deals with the existence of travelling wave solutions for a general one-dimensional nonlinear Schr\"odinger equation. We construct these solutions by minimizing the energy under the constraint of fixed momentum. We also prove…
Hydrodynamic instability of a gravity-driven flow down an inclined plane is investigated in the presence of a floating elastic plate which rests on the top surface of the flow. Linear instability of the system with respect to infinitesimal…
We study the structure and stability of discrete breathers (both pinned and mobile) in two-dimensional nonlinear anisotropic Schrodinger lattices. Starting from a set of identical one-dimensional systems we develop the continuation of the…
In this paper, we prove the nonlinear stability under localized perturbations of spectrally stable time-periodic source defects of reaction-diffusion systems. Consisting of a core that emits periodic wave trains to each side, source defects…
In this topical review we explore the dynamics of nonlinear lattices with a particular focus to Fermi-Pasta-Ulam-Tsingou type models that arise in the study of elastic media and, more specifically, granular crystals. We first revisit the…