generalized neighbor interaction models induced by nonlinear lattices it is shown that the tight binding approximation of the nonlinear schrodinger equation with a periodic linear potential and periodic in space nonlinearity coefficient gives rise to a number of nonlinear lattices with complex both linear and nonlinear neighbor interactions the obtained lattices present nonstandard possibilities among which we mention a quasilinear regime where the pulse dynamics obeys essentially the linear schrodinger equation we analyze the properties of such models both in connection to their modulational stability as well as in regard to the existence and stability of their localized solitary wave solutions

discrete vector on site vortices we study discrete vortices in coupled discrete nonlinear schrodinger equations we focus on the vortex cross configuration that has been experimentally observed in photorefractive crystals stability of the single component vortex cross in the anti continuum limit of small coupling between lattice nodes is proved in the vector case we consider two coupled configurations of vortex crosses namely the charge one vortex in one component coupled in the other component to either the charge one vortex forming a double charge vortex or the charge negative one vortex forming a so called hidden charge vortex we show that both vortex configurations are stable in the anti continuum limit if the parameter for the inter component coupling is small and both of them are unstable when the coupling parameter is large in the marginal case of the discrete two dimensional manakov system the double charge vortex is stable while the hidden charge vortex is linearly unstable analytical predictions are corroborated with numerical observations that show good agreement near the anti continuum limit but gradually deviate for larger couplings between the lattice nodes

the canonical connection of a bi lagrangian manifold we prove that the canonical connection of a bi lagrangian manifold introduced by hess is a levi civita connection showing that a bi lagrangian manifold i e a symplectic manifold endowed with two transversal lagrangian foliations is endowed with a canonical semi riemannian metric

computing absolute and essential spectra using continuation a continuation approach to the computation of essential and absolute spectra of dierential operators on the real line is presented the advantage of this approach compared with direct eigenvalue computations for the discretized operator are the ecient and accurate computation of selected parts of the spectrum typically those near the imaginary axis and the option to compute nonlinear travelling waves and selected eigenvalues or other stability indicators simultaneously in order to locate accurately the onset to instability we also discuss the implementation and usage of this approach with the software package auto and provide example computations for the fitzhugh nagumo and the complex ginzburg landau equation

dark bright discrete solitons a numerical study of existence stability and dynamics in the present work we numerically explore the existence and stability properties of dierent types of congurations of dark bright solitons dark bright soliton pairs and pairs of dark bright and dark solitons in discrete settings starting from the anti continuum limit we nd that while single discrete dark bright solitons have similar stability properties to discrete dark solitons their pairs may only be stable if the bright components are in phase and are always unstable if the bright components are out of phase pairs of dark bright solitons with dark ones have similar stability properties as individual dark or dark bright ones lastly we consider collisions between dark bright solitons and between a dark bright one and a dark one especially in the latter and in the regime where the underlying lattice structure matters we nd a wide range of potential dynamical outcomes depending on the

linear theory of faraday instability in viscous liquids the linear stability of the plane free surface of a viscous liquid on a horizontal plate under vertical sinusoidal oscillation is analysed theoretically the free surface of a laterally unbounded liquid of any depth h may always be excited to standing waves if the external acceleration is raised above a critical value a text c for a fixed external frequency omega solutions are possible only within certain bands of wave numbers k for a given forcing amplitude above a text c that is within tongue like stability zones in the a k plane the analysis for a shallow layer of viscous fluids shows new qualitative behaviours compared to the nearly inviscid theory it predicts a series of bicritical points where both harmonic and subharmonic solutions exist for the same forcing amplitude and forcing frequency this makes harmonic solutions possible at the onset in a laterally large container which is qualitatively different from the results of nearly inviscid theory for a low viscosity fluid of small depths the damping coefficient may be considered proportional to nu omega 1 2 h in contrast to nu kappa 2 predicted by the nearly inviscid theory an approximate analytic expression is derived for the lower part of the lowest marginal curve in cases when the depth of the liquid is much larger than the thickness of the viscous boundary layer formed at the bottom plate this approximate threshold agrees well with that of recent experiments with viscous liquids

intrinsic noise induced phase transitions beyond the noise interpretation we discuss intrinsic noise effects in stochastic multiplicative noise partial differential equations which are qualitatively independent of the noise interpretation ito vs stratonovich in particular in the context of noise induced ordering phase transitions we study a model which contrary to all cases known so far exhibits such ordering transitions when the noise is interpreted not only according to stratonovich but also to ito the main feature of this model is the absence of a linear instability at the transition point the dynamical properties of the resulting noise induced growth processes are studied and compared in the two interpretations and with a reference ginzburg landau type model a detailed discussion of a different numerical algorithm valid for both interpretations is also presented

pattern selection in a phase field model for directional solidification abstract a symmetric phase field model is used to study wavelength selection in two dimensions we study the problem in a finite system using a two pronged approach first we construct an action and minimizing this we obtain the most probable configuration of the system which we identify with the selected stationary state the minimization is constrained by the stationary solutions of stochastic evolution equations and is done numerically secondly additional support for this selected state is obtained from straightforward simulations of the dynamics from a variety of initial states

almost tangent and cotangent structures in the large we examine some global properties of integrable almost tangent and cotangent manifolds in particular we extend several results which essentially characterize tangent and cotangent bundles as respectively regular almost tangent and cotangent structures

state selection in the noisy stabilized kuramoto sivashinsky equation in this work we study the one dimensional stabilized kuramoto sivashinsky equation with additive uncorrelated stochastic noise the eckhaus stable band of the deterministic equation collapses to a narrow region near the center of the band this is consistent with the behavior of the phase diffusion constants of these states some connections to the phenomenon of state selection in driven out of equilibrium systems are made

vortex solutions of the discrete gross pitaevskii equation starting from the anti continuum limit in this paper we consider the existence stability and dynamical evolution of dark vortex states in the two dimensional defocusing discrete nonlinear schrodinger model a model of interest both to atomic physics and to nonlinear optics our considerations are chiefly based on initializing such vortex configurations at the anti continuum limit of zero coupling between adjacent sites and continuing them to finite values of the coupling systematic tools are developed for such continuations based on amplitude phase decompositions and explicit solvability conditions enforcing the vortex phase structure regarding the linear stability of such nonlinear waves we find that in a way reminiscent of their 1d analogs i e of discrete dark solitons the discrete defocusing vortices become unstable past a critical coupling strength and subsequently feature a cascade of alternating stabilization destabilization windows for any finite lattice although the results are mainly geared towards the uniform case we also consider the effect of harmonic trapping potentials often present in experimental atomic physics settings

electrohydrodynamic convection in liquid crystals driven by multiplicative noise sample stability we study the stochastic stability of a system described by two coupled ordinary differential equations parameterically driven by dichotomous noise with finite correlation time for a given realization of the driving noise a sample the long time behavior is described by an infinite product of random matrices the transfer matrix formalism leads to a frobenius perron equation which seems not solvable we use an alternative method to calculate the largest lyapunov exponent in terms of generalized hypergeometric functions at the threshold where the largest lyapunov exponent is zero we have an exact analytical expression also for the second lyapunov exponent the characteristic times of the system correspond to the inverse of the lyapunov exponents at the threshold the first characteristic time diverges and is thus well separated from the correlation time of the noise the second time however depending on control parameters may reach the order of the correlation time we compare the corresponding threshold with a threshold from a simple mean field decoupling and with the threshold describing stability of moments the different stability criteria give similar results if the characteristic times of the system and the noise are well separated the results may differ drastically if these times become of similar order digital simulation strongly confirms the criterion of sample stability the stochastic differential equations describe in the frame of a simple one dimensional model and a more realistic two dimensional model the appearance of normal rolls in nematic liquid crystals the superposition of a deterministic field with a fast stochastic field may lead to stable region that extends beyond the threshold values for deterministic or stochastic excitation alone forming thus a stable tongue in the space of control parameters for a certain measuring procedure the threshold curve may appear discontinuous as observed previously in experiment for a different set of material parameters the stable tongue is absent s1063 651x 98 14508 7

pade approximations of solitary wave solutions of the gross pitaevskii equation pade approximants are used to find approximate vortex solutions of any wind ing number in the context of gross pitaevskii equation for a uniform condensate and condensates with axisymmetric trapping potentials rational function and generalised rational function approximations of axisymmetric solitary waves of the gross pitaevskii equation are obtained in two and three dimensions these approximations are used to establish a new mechanism of vortex nucleation as a result of solitary wave interactions

finite size effects on instabilities of discrete breathers abstract when the two arcs of the continuous phonon spectrum of the floquet matrix of a discrete breather overlap on the unit circle the breather solution in the infinite lattice might be stable while the corresponding solutions in finite systems appear to be unstable more precisely when the model parameters vary the breather in the finite system exhibits a large number of collisions between the floquet eigenvalues belonging to the phonon spectrum these collisions correspond to complex cascades of instability thresholds followed near after by re entrant stability thresholds we interpret this complex structure on the basis of the band analysis of the matrix of the second variation of the action then we can predict that in the limit of an infinite system the number of instability and stability thresholds in the cascade diverges but simultaneously the maximum amplitude of the instabilities vanishes so that the breather in the infinite system recovers its linear stability as long as all its other localized modes remain stable this is the situation which is required in cretegny et al physica d 119 1998 73 87 for having inelastic phonon scattering with two channels we also analyze the size effects when a floquet eigenvalue associated with a localized mode collides with the floquet continuous phonon spectrum with different krein signature in contrast to the previous case the infinite system is unstable after the collision

on the moment stability of stochastic parametrically forced equations with rank one forcing we derive simplified formulas for analyzing the moment stability of stochastic parametrically forced linear systems this analysis extends the results in t blass and l a romero siam j control optim 51 2013 pp 1099 1127 where under the assumption that the stochastic excitation is small the stability of such systems was computed using a weighted sum of the extended power spectral density over the eigenvalues of the unperturbed operator in this paper we show how to convert this sum to a sum over the residues of the extended power spectral density for systems where the parametric forcing term is a rank one matrix this approach leads to an enormous simplification we give two examples of systems with rank one forcing including the problem of stochastically forced faraday waves

effect of multiplicative noise on parametric instabilities we report a study on the effect of external multiplicative noise on parametric instabilities using two different experimental systems an electronic rlc circuit parametrically pumped with a voltage variable capacitor and surface waves generated by vertically vibrating a layer of fluid the faraday instability both systems are forced by the superposition of a sinusoidal and a noisy component we study the statistical properties of the response of both systems to noisy parametric forcing and compare them with theoretical predictions when the detuning from parametric resonance is such that the bifurcation in the absence of noise is supercritical both systems behave in the same way under the influence of noise we find that the effect of noise is twofold on one hand it triggers the instability before its deterministic onset under the form of oscillatory bursts on the other hand it inhibits the nonlinearly saturated oscillatory response above the deterministic onset when the detuning is such that the bifurcation is subcritical we find that the two systems behave differently in the case of the electronic oscillator noise mostly triggers random transitions between the two states of the bistable region that exists in the absence of noise whereas in the surface wave experiment new states are created by noise and the bistable region is strongly enlarged

some geometric structures associated with a k symplectic manifold a canonical connection is attached to any k symplectic manifold we study the properties of this connection and its geometric applications to k symplectic manifolds in particular we prove that under some natural assumptions any k symplectic manifold admits an ehresmann connection discuss some corollaries of this result and find vanishing theorems for characteristic classes on a k symplectic manifold

additive global noise delays turing bifurcations we apply a stochastic center manifold method to the calculation of noise induced phase transitions in the stochastic swift hohenberg equation this analysis is applied to the reduced mode equations that result from fourier decomposition of the field variable and of the temporal noise the method shows a pitchfork bifurcation at lower perturbation order but reveals a novel additive noise induced postponement of the turing bifurcation at higher order good agreement is found between the theory and the numerics for both the reduced and the full system the results are generalizable to a broad class of nonlinear spatial systems