We derive a priori estimates on the absorbing ball in L2 for the stabilized and destabilized Kuramoto-Sivashinsky (KS) equations, and for a sixth-order analog, the Nikolaevskiy equation, and in each case obtain bounds whose parameter dependence is demonstrably optimal. This is done by extending a Lyapunov function construction developed by Bronski and Gambill (Nonlinearity 19, 2023–2039 (2006))...

We report a laboratory investigation of weak turbulence of water surface waves in the gravity-capillary crossover. By using time-space-resolved profilometry and a bicoherence analysis, we observe that the nonlinear processes involve three-wave resonant interactions. By studying the solutions of the resonance conditions, we show that the nonlinear interaction is dominantly one dimensional and in...

The dynamic characteristics of a controller designed by the gain-scheduling approach can be strongly dependent on the realisation adopted; that is, the manner in which the local linear controller designs are combined to obtain a nonlocal controller. The purpose of the present paper is to investigate the choice of appropriate realisations for general MIMO gain-scheduled controllers. An extended ...

The modulational instability (MI) of plane waves in nonlocal Kerr media is studied for a general response function. Several generic properties are proven mathematically, with emphasis on how new gain bands are formed through a bifurcation process when the degree of nonlocality, sigma, passes certain bifurcation values and how the bandwidth and maximum of each individual gain band depends on sig...

this paper is concerned with the study of the existence of positive solutions for a navier boundaryvalue problem involving the p-biharmonic operator; the right hand side of problem is a nonsmoothfunctional with variable parameters. the existence of at least three positive solutions is establishedby using nonsmooth version of a three critical points theorem for discontinuous functions. our resul...

in this paper, the stability characteristics of single-walled carbon nanotubes (swcnts) under the action of axial load are investigated. to this end, a nonlocal flügge shell model is developed to accommodate the small length scale effects. the analytical rayleigh-ritz method with beam functions is applied to the variational statement derived from the flügge-type buckling equations. molecular dy...

We give a brief overview of the recent advances in nonlinear singular optics that studies the propagation and stability of optical vortices in nonlinear media, with the emphasis on the properties of vortex solitons and rotating azimuthons. In general, self-focusing nonlinearity generates the azimuthal instability of vortex beams, but it can support novel types of stable (or metastable) self-tra...

On the basis of the competing cubic-quintic nonlinearity model, stability (instability) of continuous waves in nonlocal random non-Kerr nonlinear media is studied analytically and numerically. Fluctuating media parameters are modeled by the Gaussian white noise. It is shown that for different response functions of a medium nonlocality suppresses, as a rule, both the growth rate peak and bandwid...

All-optical signal processing enables modulation and transmission speeds not achievable using electronics alone. However, its practical applications are limited by the inherently weak nonlinear effects that govern photon-photon interactions in conventional materials, particularly at high switching rates. Here, we show that the recently discovered nonlocal optical behaviour of plasmonic nanorod ...

Solitary acoustic pulses can propagate along the surface of a coated homogeneous and inhomogeneous medium. It is shown how these nonlinear surface acoustic waves evolve out of initial pulselike conditions generated by pulsed laser excitation and how they can be monitored by optical detection. The solitary pulse shapes at the surface are computed on the basis of an evolution equation with nonloc...