violation of the fluctuation dissipation theorem in glassy systems basic notions and the numerical evidence this review reports on the research done during past years on violations of the fluctuation dissipation theorem fdt in glassy systems it is focused on the existence of a quasi fluctuation dissipation theorem qfdt in glassy systems and the current supporting knowledge gained from numerical simulation studies it covers a broad range of non stationary aging and stationary driven systems such as structural glasses spin glasses coarsening systems ferromagnetic models at criticality trap models models with entropy barriers kinetically constrained models sheared systems and granular media the review is divided into four main parts 1 an introductory section explaining basic notions related to the existence of the fdt in equilibrium and its possible extension to the glassy regime qfdt 2 a description of the basic analytical tools and results derived in the framework of some exactly solvable models 3 a detailed report of the current evidence in favour of the qfdt and 4 a brief digression on the experimental evidence in its favour this review is intended for inexpert readers who want to learn about the basic notions and concepts related to the existence of the qfdt as well as for the more expert readers who may be interested in more specific results

conditional stability of solitary waves propagating in elastic rods abstract the problem of dynamic stability of solitary wave solutions of the equations describing bending oscillations of inextensible elastic rods is treated the governing equations describe sufficiently large displacements though we are restricted to small strains we show that under the condition of well posedness of the initial value problem in some specific sense the family of solitary wave solutions of the mentioned equations is stable for the planar elastic rod the framework of the analysis is largely based on the spectral properties of the linearized hamiltonian h we show that for planar oscillations h is positively semidefinite subject to a certain constraint which implies the stability

stability and instability of periodic travelling wave solutions for the critical korteweg de vries and nonlinear schrodinger equations abstract in this paper we establish new results about the existence stability and instability of periodic travelling wave solutions related to the critical korteweg de vries equation u t 5 u 4 u x u x x x 0 and the critical nonlinear schrodinger equation i v t v x x v 4 v 0 the periodic travelling wave solutions obtained in our study tend to the classical solitary wave solutions in the infinite wavelength scenario the stability approach is based on the theory developed by angulo natali in j angulo f natali positivity properties of the fourier transform and the stability of periodic travelling wave solutions siam j math anal 40 2008 1123 1151 for positive periodic travelling wave solutions associated to dispersive evolution equations of korteweg de vries type the instability approach is based on an extension to the periodic setting of arguments found in bona souganidis strauss j l bona p e souganidis w a strauss stability and instability of solitary waves of korteweg de vries type proc roy soc london ser a 411 1987 395 412 regarding the critical schrodinger equation stability instability theories similar to the critical korteweg de vries equation are obtained by using the classical grillakis shatah strauss theory in m grillakis j shatah w strauss stability theory of solitary waves in the presence of symmetry ii j funct anal 94 1990 308 348 m grillakis j shatah w strauss stability theory of solitary waves in the presence of symmetry i j funct anal 74 1987 160 197 the arguments presented in this investigation have prospects for the study of the stability of periodic travelling wave solutions of other nonlinear evolution equations

simulations of singularity dynamics in liquid crystal flows a c0 finite element approach in this paper we present a c 0 finite element method for a 2d hydrodynamic liquid crystal model which is simpler than existing c 1 element methods and mixed element formulation the energy law is formally justified and the energy decay is used as a validation tool for our numerical computation a splitting method combined with only a few fixed point iteration for the penalty term of the director field is applied to reduce the size of the stiffness matrix and to keep the stiffness matrix time independent the latter avoids solving a linear system at every time step and largely reduces the computational time especially when direct linear system solvers are used our approach is verified by comparing its computational results with those obtained by c 1 elements and by mixed formulation through numerical experiments of a few other splittings and explicit implicit strategies we recommend a fast and reliable algorithm for this model a number of examples are computed to demonstrate the algorithm

blow up and instability of solitary wave solutions to a generalized kadomtsev petviashvili equation in this paper we consider a generalized kadomtsev petviashvili equation in the form ut uxxx upux uyy x y e r2 t 0 it is shown that the solutions blow up in finite time for the supercritical power of nonlinearity p 4 3 with p the ratio of an even to an odd integer moreover it is shown that the solitary waves are strongly unstable if 2 p 4 that is the solutions blow up in finite time provided they start near an unstable solitary wave

dispersion of low energy waves for the generalized benjamin bona mahony equation on considere les equations suivantes u t u x f u x u xxx 0 1 et u t u x f u x u xxt 0 2 avec f r r est c on donne le comportement des solutions ondes a basse energie pour certains f

count of eigenvalues in the generalized eigenvalue problem we study isolated and embedded eigenvalues in the generalized eigenvalue problem defined by two self adjoint operators with a positive essential spectrum and a finite number of isolated eigenvalues the generalized eigenvalue problem determines the spectral stability of nonlinear waves in infinite dimensional hamiltonian systems the theory is based on pontryagin s invariant subspace theorem and extends beyond the scope of earlier papers of pontryagin krein grillakis and others our main results are i the number of unstable and potentially unstable eigenvalues equals the number of negative eigenvalues of the self adjoint operators ii the total number of isolated eigenvalues of the generalized eigenvalue problem is bounded from above by the total number of isolated eigenvalues of the self adjoint operators and iii the quadratic forms defined by the two self adjoint operators are strictly positive on the subspace related to the continuous spectrum of the generalized eigenvalue problem applicatio

stability and instability of solitary waves of korteweg de vries type considered herein are the stability and instability properties of solitary wave solutions of a general class of equations that arise as mathematical models for the unidirectional propagation of weakly nonlinear dispersive long waves special cases for which our analysis is decisive include equations of the korteweg de vries and benjamin ono type necessary and sufficient conditions are formulated in terms of the linearized dispersion relation and the nonlinearity for the solitary waves to be stable

phase field simulations of interfacial dynamics in viscoelastic fluids using finite elements with adaptive meshing this paper describes a novel numerical algorithm for simulating interfacial dynamics of non newtonian fluids the interface between two immiscible fluids is treated as a thin mixing layer across which physical properties vary steeply but continuously the property and evolution of the interfacial layer is governed by a phase field variable that obeys a cahn hilliard type of convection diffusion equation this circumvents the task of directly tracking the interface and produces the correct interfacial tension from the free energy stored in the mixing layer viscoelasticity and other types of constitutive equations can be incorporated easily into the variational phase field framework the greatest challenge of this approach is in resolving the sharp gradients at the interface this is achieved by using an efficient adaptive meshing scheme governed by the phase field variable the finite element scheme easily accommodates complex flow geometry and the adaptive meshing makes it possible to simulate large scale two phase systems of complex fluids in two dimensional and axisymmetric three dimensional implementations the numerical toolkit is applied here to drop deformation in shear and elongational flows rise of drops and retraction of drops and torii some of these solutions serve as validation of the method and illustrate its key features while others explore novel physics of viscoelasticity in the deformation and evolution of interfaces

fluctuation response relations for multitime correlations we show that time correlation functions of arbitrary order for any random variable in a statistical dynamical system can be calculated as higher order response functions of the mean history of the variable the response is to a control term added as a modification to the master equation for statistical distributions the proof of the relations is based upon a variational characterization of the generating functional of the time correlations the same fluctuation response relations are preserved within moment closures for the statistical dynamical system when these are constructed via the variational rayleigh ritz procedure for the two time correlations of the moment variables themselves the fluctuation response relation is equivalent to an onsager regression hypothesis for the small fluctuations for correlations of higher order there is a further effect in addition to such linear propagation of fluctuations present instantaneously the dynamical generation of correlations by nonlinear interaction of fluctuations in general we discuss some physical and mathematical aspects of the ans atze required for an accurate calculation of the time correlations we also comment briefly upon the computational use of these relations which is well suited for automatic differentiation tools an example will be given of a simple closure for turbulent energy decay which illustrates the numerical application of the relations

stability of periodic traveling waves for complex modified korteweg de vries equation abstract we study the existence and stability of periodic traveling wave solutions for complex modified korteweg de vries equation we also discuss the problem of uniform continuity of the data solution mapping

an isochoric domain deformation method for computing steady free surface flows with conserved volumes abstract the domain deformation method has been applied successfully to steady state free surface flows where the volume of the flow domain is unknown v f de almeida gas liquid counterflow through constricted passages ph d thesis university of minnesota minneapolis mn 1995 p a sackinger p r schunk r r rao a newton raphson pseudo solid domain mapping technique for free and moving boundary problems a finite element implementation j comput phys 125 1996 83 103 l c musson two layer slot coating ph d thesis university of minnesota minneapolis mn 2001 however this method does not handle effectively problems where the volume of the flow domain is known a priori this work extends the original domain deformation method to a new isochoric domain deformation method to account for the volume conservation like in the original domain deformation method the unknown shape of the flow domain is mapped onto a reference domain by using the equations of an elastic pseudo solid the difference with the original method is that this pseudo solid is considered incompressible because of the incompressibility the pseudo pressure of the mapping appears as a lagrange multiplier in the equations and it is determined only up to an arbitrary uniform datum by analyzing the coupled fluid flow mapping problem we show that in the finite element setting such pressure datum can be specified by replacing one continuity equation in the fluid domain

spectral stability and time evolution of n solitons in the kdv hierarchy this paper concerns spectral stability and time evolution of n solitons in the korteweg de vries kdv hierarchy with mixed commuting time flows spectral stability problem is analysed by using a pair of self adjoint operators with finite numbers of negative eigenvalues we show that the absence of unstable eigenvalues in the stability problem is related to the absence of negative eigenvalues of these operators in the constrained function spaces time evolution of n solitons is uniquely characterized from the inverse scattering transform technique

guaranteed quality triangular mesh generation for domains with curved boundaries guaranteed quality unstructured meshing algorithms facilitate the development of automatic meshing tools however these algorithms require domains discretized using a set of linear segments leading to numerical errors in domains with curved boundaries we introduce an extension of ruppert s delaunay refinement algorithm to two dimensional domains with curved boundaries and prove that the same quality bounds apply with curved boundaries as with straight boundaries we provide implementation details for two dimensional boundary patches such as lines circular arcs cubic parametric curves and interpolated splines we present guaranteed quality triangular meshes generated with curved boundaries and propose solutions to some problems associated with the use of curved boundaries copyright 2002 john wiley sons ltd

on the stability of internal waves the extended kdv equation ut uux u2ux uxxx 0 is widely used as a model describing internal waves in ideal fluids the equation admits a family of negative and positive solitary waves c these solitary waves exhibit the typical broadening effect seen in internal waves it is shown here that all solitary wave solutions of the extended kdv equation are orbitally stable the proof of stability is based on the general theory of grillakis et al 1987 j funct anal 74 160 for equations of the form ut je u which have two conserved integrals e u and v u a spectral analysis of the linear operator reduces the question of orbital stability to the question of whether the scalar function d c e c cv c is convex to prove the stability an explicit calculation showing the convexity is performed

on the decay of solutions of the generalized benjamin bona mahony equation on etudie le comportement des solutions du probleme aux valeurs initiales u t u x u p u x u xxt 0 x r t o u x 0 x x r p 0 entier on suppose p 4 alors il existe 0 tel que si 1 1 5 0 et x r ou a ne depend pas de x ou t

an adaptive meshfree method for phase field models of biomembranes part i we present an adaptive meshfree method for phase field models of biomembranes max ent smooth approximants can deal with the high order phase field equations the method resolves adaptively the models at an affordable computational cost convergence to sharp interface limit is shown with small regularization parameters we present an adaptive meshfree method to approximate phase field models of biomembranes in such models the helfrich curvature elastic energy the surface area and the enclosed volume of a vesicle are written as functionals of a continuous phase field which describes the interface in a smeared manner such functionals involve up to second order spatial derivatives of the phase field leading to fourth order euler lagrange partial differential equations pde the solutions develop sharp internal layers in the vicinity of the putative interface and are nearly constant elsewhere thanks to the smoothness of the local maximum entropy max ent meshfree basis functions we approximate numerically this high order phase field model with a direct ritz galerkin method the flexibility of the meshfree method allows us to easily adapt the grid to resolve the sharp features of the solutions thus the proposed approach is more efficient than common tensor product methods e g finite differences or spectral methods and simpler than unstructured c 0 finite element methods applicable by reformulating the model as a system of second order pde the proposed method implemented here under the assumption of axisymmetry allows us to show numerical evidence of convergence of the phase field solutions to the sharp interface limit as the regularization parameter approaches zero in a companion paper we present a lagrangian method based on the approximants analyzed here to study the dynamics of vesicles embedded in a viscous fluid

an energy law preserving c0 finite element scheme for simulating the kinematic effects in liquid crystal dynamics in this paper we use finite element methods to simulate the hydrodynamical systems governing the motions of nematic liquid crystals in a bounded domain w we reformulate the original model in the weak form which is consistent with the continuous dissipative energy law for the flow and director fields in w 1 2 s w s 0 is an arbitrarily small number this enables us to use convenient conformal c 0 finite elements in solving the problem moreover a discrete energy law is derived for a modified midpoint time discretization scheme a fixed iterative method is used to solve the resulted nonlinear system so that a matrix free time evolution may be achieved and velocity and director variables may be solved separately a number of hydrodynamical liquid crystal examples are computed to demonstrate the effects of the parameters and the performance of the method

final state problem for class of nonlinear nonlocal dispersive equation abstract we study the large time asymptotics of the solutions to nonlinear dispersive equations of the form 0 1 t u p i x x u x u 3 t 0 x r where p i x is a pseudo differential operator defined by the symbol p in this paper we consider the case that p is nonlocal and nonhomogeneous and show the existence of the solution to 0 1 around a given modified free solution

blow up of solutions for the dissipative dullin gottwald holm equation with arbitrary coefficients abstract we study the cauchy problem for the weakly dissipative dullin gottwald holm equation describing unidirectional propagation of surface waves in a shallow water regime u t 2 u x x t c 0 u x 3 u u x u x x x u 2 u x x 2 2 u x u x x u u x x x in the present paper we demonstrate the simple conditions on the initial data that lead to blow up of the solution in finite time

convergence of petviashvili s iteration method for numerical approximation of stationary solutions of nonlinear wave equations we analyze a heuristic numerical method suggested by v i petviashvili in 1976 for approximation of stationary solutions of nonlinear wave equations the method is used to construct numerically the solitary wave solutions such as solitons lumps and vortices in a space of one and higher dimensions assuming that the stationary solution exists we find conditions when the iteration method converges to the stationary solution and when the rate of convergence is the fastest the theory is illustrated with examples of physical interest such as generalized korteweg de vries benjamin ono zakharov kuznetsov kadomtsev petviashvili and klein gordon equations

the ivp for the benjamin ono zakharov kuznetsov equation in low regularity sobolev spaces abstract in this paper we study the initial value problem associated with the benjamin ono zakharov kuznetsov equation such equation appears as a two dimensional generalization of the benjamin ono equation when transverse effects are included via weak dispersion of zakharov kuznetsov type we prove that the initial value problem is locally well posed in the usual l 2 r 2 based sobolev spaces h s r 2 s 11 8 and in some weighted sobolev spaces to obtain our results most of the arguments are accomplished taking into account the ones for the benjamin ono equation

numerical simulation of moving contact line problems using a volume of fluid method abstract moving contact lines are implemented in a volume of fluid scheme with piecewise linear interface construction interfacial tension is treated as a continuous body force computed from numerical derivatives of a smoothed volume of fluid function two methods for implementing the contact angle condition are investigated the first extrapolates the volume of fluid function beyond the flow domain on the basis of the condition that its gradient is perpendicular to the interface and that the normal to the interface at the wall is determined by the contact angle the second method treats the problem as a three phase situation and mimics the classical argument of young it is found that the latter approach introduces an artificial localized flow and the extrapolation method is preferable slip is a crucial factor in the spreading of contact lines the numerical method introduces slip at the discrete level effectively introducing a slip length on the order of the mesh size

decay of solutions of some nonlinear wave equations abstract we study the large time behaviour of solutions to the initial value problem for the korteweg de vries equation and for the regularized long wave equation with a dissipative term appended using energy estimates a maximum principle and a transformation of cole hopf type sharp rates of temporal decay of certain norms of the solution are obtained

numerical study of blow up and dispersive shocks in solutions to generalized korteweg de vries equations abstract we present a detailed numerical study of solutions to general korteweg de vries equations with critical and supercritical nonlinearity both in the context of dispersive shocks and blow up we study the stability of solitons and show that they are unstable against being radiated away and blow up in the l 2 critical case the blow up mechanism by martel merle and raphael can be numerically identified in the limit of small dispersion it is shown that a dispersive shock always appears before an eventual blow up in the latter case always the first soliton to appear will blow up it is shown that the same type of blow up as for the perturbations of the soliton can be observed which indicates that the theory by martel merle and raphael is also applicable to initial data with a mass much larger than the soliton mass we study the scaling of the blow up time t in dependence of the small dispersion parameter and find an exponential dependence t and that there is a minimal blow up time t 0 greater than the critical time of the corresponding hopf solution for 0 to study the cases with blow up in detail we apply the first dynamic rescaling for generalized korteweg de vries equations this allows to identify the type of the singularity

an index theorem for the stability of periodic travelling waves of korteweg de vries type we consider the stability of periodic travelling wave solutions to a generalized korteweg de vries gkdv equation and prove an index theorem relating the number of unstable and potentially unstable eigenvalues to geometric information on the classical mechanics of the travelling wave ordinary differential equation we illustrate this result with several examples including the integrable kdv and modified kdv equations the l 2 critical kdv 4 equation that arises in the study of blow up and the kdv equation which is an idealized model for plasmas

existence and asymptotic decay for a boussinesq type model abstract in this article we study the initial value problem associated with a five parameter boussinesq type system we prove local existence and uniqueness of the solution and that the supremum norm of the solution decays algebraically to zero as 1 t 1 3 when t approaches to infinity provided the initial data are sufficiently small and regular we further present a high accurate spectral numerical method to approximate the solutions and validate the theoretical results

stability of solitary waves for a nonlinear dispersive system in a critical case we study a nonlinear dispersive system with critical power nonlinearities building on the works of weinstein and angulo bona linares and scialom a conditional stability modulo the group of translations in space and dilations is proved for a solitary wave solution

existence and stability of traveling waves for a class of nonlocal nonlinear equations in this article we are concerned with the existence and orbital stability of traveling wave solutions of a general class of nonlocal wave equations utt luxx b u p 1 u xx p 1 the main characteristic of this class of equations is the existence of two sources of dispersion characterized by two coercive pseudo differential operatorsl and b members of the class arise as mathematical models for the propagation of dispersive waves in a wide variety of situations for instance all boussinesq type equations and the so called double dispersion equation are members of the class we first establish the existence of traveling wave solutions to the nonlocal wave equations considered we then obtain results on the orbital stability or instability of traveling waves for the case l i corresponding to a class of klein gordon type equations we give an almost complete characterization of the values of the wave velocity for which the traveling waves are orbitally stable or unstable by blow up

the cauchy problem for dissipative benjamin ono equation in weighted sobolev spaces abstract we study the well posedness in weighted sobolev spaces for the initial value problem ivp associated with the dissipative benjamin ono dbo equation we establish persistence properties of the solution flow in the weighted sobolev spaces z s r h s r l 2 x 2 r d x s r 0 we also prove some unique continuation properties in these spaces in particular such results of unique continuation show that our results of well posedness are sharp

stability of solitary waves in nonlinear composite media the system of equations for planar waves in elastic composite media in the presence of anisotropy is considered in anisotropic case two two parametric families of solitary waves are found in an explicit form in case of the absence of anisotropy these two families coalesce into the unique three parametric family the solitary wave solutions are found to be orbitally stable in a certain range of their phase speeds range of stability both in an anisotropic as well as in an isotropic materials it is also shown that the initial value problem for the governing equations is locally well posed which is needed to prove the stability result the local well posedness of the initial value problem along with stability of solitary waves implies global existence result provided the initial data lie in a neighbourhood of a stable solitary wave this complements the previous results of blow up for this type of equations

computation of multiphase systems with phase field models phase field models offer a systematic physical approach for investigating complex multiphase systems behaviors such as near critical interfacial phenomena phase separation under shear and microstructure evolution during solidification however because interfaces are replaced by thin transition regions diffuse interfaces phase field simulations require resolution of very thin layers to capture the physics of the problems studied this demands robust numerical methods that can efficiently achieve high resolution and accuracy especially in three dimensions we present here an accurate and efficient numerical method to solve the coupled cahn hilliard navier stokes system known as model h that constitutes a phase field model for density matched binary fluids with variable mobility and viscosity the numerical method is a time split scheme that combines a novel semi implicit discretization for the convective cahn hilliard equation with an innovative application of high resolution schemes employed for direct numerical simulations of turbulence this new semi implicit discretization is simple but effective since it removes the stability constraint due to the nonlinearity of the cahn hilliard equation at the same cost as that of an explicit scheme it is derived from a discretization used for diffusive problems that we further enhance to efficiently solve flow problems with variable mobility and viscosity moreover we solve the navier stokes equations with a robust time discretization of the projection method that guarantees better stability properties than those for crank nicolson based projection methods for channel geometries the method uses a spectral discretization in the streamwise and spanwise directions and a combination of spectral and high order compact finite difference discretizations in the wall normal direction the capabilities of the method are demonstrated with several examples including phase separation with and without shear in two and three dimensions the method effectively resolves interfacial layers of as few as three mesh points the numerical examples show agreement with analytical solutions and scaling laws where available and the 3d simulations in the presence of shear reveal rich and complex structures including strings

quasi incompressible cahn hilliard fluids and topological transitions one of the fundamental problems in simulating the motion of sharp interfaces between immiscible fluids is a description of the transition that occurs when the interfaces merge and reconnect it is well known that classical methods involving sharp interfaces fail to describe this type of phenomena following some previous work in this area we suggest a physically motivated regularization of the euler equations which allows topological transitions to occur smoothly in this model the sharp interface is replaced by a narrow transition layer across which the fluids may mix the model describes a flow of a binary mixture and the internal structure of the interface is determined by both diffusion and motion an advantage of our regularization is that it automatically yields a continuous description of surface tension which can play an important role in topological transitions an additional scalar field is introduced to describe the concentration of one of the fluid components and the resulting system of equations couples the euler or navier stokes and the cahn hilliard equations the model takes into account weak non locality dispersion associated with an internal length scale and localized dissipation due to mixing the non locality introduces a dimensional surface energy dissipation is added to handle the loss of regularity of solutions to the sharp interface equations and to provide a mechanism for topological changes in particular we study a non trivial limit when both components are incompressible the pressure is kinematic but the velocity field is non solenoidal quasi incompressibility to demonstrate the effects of quasi incompressibility we analyse the linear stage of spinodal decomposition in one dimension we show that when the densities of the fluids are not perfectly matched the evolution of the concentration field causes fluid motion even if the fluids are inviscid in the limit of infinitely thin and well separated interfacial layers an appropriately scaled quasi incompressible euler cahn hilliard system converges to the classical sharp interface model in order to investigate the behaviour of the model outside the range of parameters where the sharp interface approximation is sufficient we consider a simple example of a change of topology and show that the model permits the transition to occur without an associated singularity

instability of nonlinear bound states we establish a sharp instability theorem for the bound states of lowest energy of the nonlinear klein gordon equation utt u f u 0 and the nonlinear schrodinger equation iut u f u 0

the wahlquist estabrook method for evolution equations with a small parameter a technique of approximating pseudopotentials an approach is suggested to construct approximate solutions for nonlinear partial differential equations non integrable by the inverse scattering transform the method is applied to find an approximate lax pair and conservation laws for the kawahara equation

orbital stability of double solitons for the benjamin ono equation in this paper we prove the orbital stability of double solitons for the benjamin ono equation in the case of the kdv equation this stability has been proved in 17 parts of the proof given there rely on the fact that the euler lagrange equations for the conserved quantities of the kdv equation are ordinary differential equations since this is not the case for the benjamin ono equation new methods are required our approach consists in using a new invariant for multi solitons and certain new identities motivated by the sylvester law of inertia

stability of the solitary wave solutions to a coupled bbm system abstract in this work we present a stability criteria for the solitary wave solutions to a bbm system that contains coupled nonlinear terms using the idea by bona chen and karakashian 5 and exploiting the accurate point spectrum information of the associated schrodinger operator we improve the stability results previously gotten by pereira 15

remarks on solitary waves of the generalized two dimensional benjamin ono equation abstract in this paper we study the generalized 2d bo equation in two dimensions u t h u xx u p u x x u yy 0 x y r 2 t 0 we classify the existence and non existence of solitary waves depending on the sign of and on the nonlinearity we also prove nonlinear stability and some regularity properties of such waves

stability and instability of the kdv solitary wave under the kp i flow we consider the kp i and gkp i equations in mathbb r times mathbb r 2 pi mathbb z we prove that the kdv soliton with subcritical speed 0 c in the spirit of the work by duyckaerts and merle gafa 18 1787 1840 2009 we sharpen our previous instability result and construct a global solution which is different from the solitary wave and its translates and which converges to the solitary wave as time goes to infinity this last result also holds for the gkp i equation

solitary waves of nonlinear dispersive evolution equations with critical power nonlinearities we study the nonlinear schrodinger nls generalized korteweg de vries gkdv and generalized benjamin ono equations gbo with critical power nonlinearities the ground state solitary wave is conditionally stable modulo the group of translations in space and phase and dilations this is a natural extension of the result for subcritical nonlinearities where solitary waves are stable modulo translations alone to a case where solutions may blow up we also present results on global existence for gbo and the structure of singularities for certain solutions of gbo which are analogues of previous results by the author for nls and gkdv

on imposing dynamic contact angle boundary conditions for wall bounded liquid gas flows we present an efficient scheme within the phase field framework for imposing dynamic contact angle boundary conditions for wall bounded flows of two immiscible incompressible fluids with large density ratios first we develop an algorithm for imposing the dynamic contact angle boundary conditions to the cahn hilliard equation our algorithm consists of two components i we ignore the boundary conditions and transform the cahn hilliard equation into two nominally de coupled helmholtz type equations ii we treat the dynamic contact angle boundary conditions in such a manner that the two helmholtz type equations are truly de coupled then we combine this algorithm together with a scheme for variable density navier stokes equations we developed recently to form an efficient method for the coupled system of navier stokes and cahn hilliard equations for contact line problems involving large density ratios the overall method can deal with moving contact lines under dynamic and also static contact angle boundary conditions it is endowed with several attractive features that make the method very efficient in particular computations for all flow variables are completely decoupled the resultant linear algebraic systems after discretization for all flow variables involve only constant and time independent coefficient matrices which can be pre computed during pre processing even though the coupled navier stokes cahn hilliard system involves variable density and variable viscosity ample numerical simulations of wall bounded air water two phase flows have been presented to demonstrate the capability of the method for dealing with contact line problems under dynamic and static contact angle conditions involving large density ratios

the ivp for the benjamin ono zakharov kuznetsov equation in weighted sobolev spaces abstract in this paper we study the initial value problem associated with the benjamin ono zakharov kuznetsov equation we prove that the ivp for such equation is locally well posed in the usual sobolev spaces h s r 2 s 2 and in the anisotropic spaces h s 1 s 2 r 2 s 2 2 s 1 s 2 we also study the persistence properties of the solution and local well posedness in the weighted sobolev class z s r h s r 2 l 2 1 x 2 y 2 r d x d y where s 2 r 0 and s 2 r unique continuation properties of the solution are also established these continuation principles show that our persistence properties are sharp

stability theory of solitary waves in the presence of symmetry ii abstract consider an abstract hamiltonian system which is invariant under a group of operators we continue to study the effect of the group invariance on the stability of solitary waves applications are given to bound states and traveling wave solutions of nonlinear wave equations

nonlinear response theory for markov processes simple models for glassy relaxation the theory of nonlinear response for markov processes obeying a master equation is formulated in terms of time dependent perturbation theory for the green s functions and general expressions for the response functions up to third order in the external field are given the nonlinear response is calculated for a model of dipole reorientations in an asymmetric double well potential a standard model in the field of dielectric spectroscopy the static nonlinear response is finite with the exception of a certain temperature t 0 determined by the value of the asymmetry in a narrow temperature range around t 0 the modulus of the frequency dependent cubic response shows a peak at a frequency on the order of the relaxation rate and it vanishes for both low frequencies and high frequencies at temperatures at which the static response is finite lower and higher than t 0 the modulus is found to decay monotonously from the static limit to zero at high frequencies in addition results of calculations for a trap model with a gaussian density of states are presented in this case the cubic response depends on the specific dynamical variable considered and also on the way the external field is coupled to the kinetics of the model in particular a set of different dynamical variables that gives rise to identical shapes of the linear susceptibility and only to different temperature dependencies of the relaxation times is considered it is found that the frequency dependence of the nonlinear response functions however strongly depends on the particular choice of the variables the results are discussed in the context of recent theoretical and experimental findings regarding the nonlinear response of supercooled liquids and glasses

simulation of moving contact lines in two phase polymeric fluids we numerically study the influence of polymer additives on contact line dynamics using a sharp interface model an additional term which accounts for the polymer stress is added to the two phase navier stokes equation the polymer stress is modeled by the fene p model the coupled dynamic equations are solved using the finite difference method the interface is tracked using marker particles and the interface conditions are imposed using the immersed boundary method simulations in a couette geometry are carried out to study the effect of the polymer stress on the moving contact line it is found that the large velocity gradient near the moving contact line results in large polymer stress which retards the contact line motion the polymer stress increases with the polymer viscosity but is insensitive to the change of the weissenberg number the strength of the polymer stress is also found to increase with the capillary number especially when the capillary number is large we also study the relation between the apparent contact angle and the capillary number for newtonian fluids the relation fits well with cox s analytic result however deviation occurs for polymeric fluids the critical capillary number at which the transition to an unsteady state happens decreases as the polymer viscosity increases

the symmetric regularized long wave equation well posedness and nonlinear stability abstract the focus of the present work is the symmetric regularized long wave equation we prove that the initial value problem for this equation is locally and globally well posed in h per s h per s 1 and h s r h s 1 r if s 0 we also prove the existence and nonlinear stability of periodic travelling wave solutions of cnoidal type for the equation mentioned above

b spline quasi interpolation based numerical methods for some sobolev type equations in this article we intend to use quadratic and cubic b spline quasi interpolants to develop higher order numerical methods for some sobolev type equations in one space dimension our aim is also to compare the performance of the proposed methods in terms of the accuracy and the rate of convergence we also discuss another approach to the cubic b spline quasi interpolation based method where we achieve fourth order of accuracy in space we theoretically establish the order of accuracy for the three proposed methods and also establish the l 2 stability in the linear case using von neumann analysis as a particular case of the sobolev type equations we take the equal width and the benjamin bona mahony burgers equations and perform several numerical experiments to support our theoretical results

long time dynamics of variable coefficient modified korteweg de vries solitary waves we study the long time behavior of solutions to the korteweg de vries type equation partial derivative t u partial derivative x partial derivative 2 x u f u b t x u with initial conditions cl

instability of a class of dispersive solitary waves this paper studies the stability and instability properties of solitary wave solutions x ct of a general class of evolution equations of the form mu t f u x 0 which support weakly nonlinear dispersive waves it turns out that depending on their speed c and the relation between the dispersion i e the order of the pseudodifferential operator and the nonlinearity travelling waves may be stable or unstable sharp conditions to that effect are given

a sharp interface isogeometric solution to the stefan problem abstract in the present paper the stefan problem is solved by enriching an underlying nurbs based isogeometric approximation with an explicitly defined sharp interface on which a hybrid function derivative condition is isoparametrically described since the geometry of the enrichment is explicitly defined normals and curvatures are explicitly computed at any point on the interface thus the enriched approximation naturally captures the interfacial discontinuity in temperature gradient and naturally enables the imposition of gibbs thomson condition the blending of the enrichment with the underlying approximation requires an estimate of distance to the enriching geometry from a quadrature point and the parametric value of the footpoint on the enriching geometry these quantities are computed efficiently in the present paper using an algebraic estimate of distance coupled with an algebraic point projection method these algebraic schemes rely on implicitization of the parametric curve and are shown to be more efficient and robust than newton raphson iterations procedures for adaptive time stepping refinement and coarsening of geometry are developed to increase the stability and efficiency of the developed methodology several numerical examples of classical and dendritic stefan problem are presented to demonstrate the methodology

solitary wave solutions of the benjamin equation considered here is a model equation put forward by benjamin that governs approximately the evolution of waves on the interface of a two fluid system in which surface tension effects cannot be ignored our principal focus is the traveling wave solutions called solitary waves and three aspects will be investigated a constructive proof of the existence of these waves together with a proof of their stability is developed continuation methods are used to generate a scheme capable of numerically approximating these solitary waves the computer generated approximations reveal detailed aspects of the structure of these waves they are symmetric about their crests but unlike the classical korteweg de vries solitary waves they feature a finite number of oscillations the derivation of the equation is also revisited to get an idea of whether or not these oscillatory waves might actually occur in a natural setting

instability of solitary wave solutions for the generalized bo zk equation in this paper we study the generalized bo zk equation in two space dimensions ut upux huxx euxyy 0 we review the existence theory for solitary waves and prove that they are nonlinearly unstable if p belongs to the range 4 3 p 4 we also establish strichartz type estimates

scalable information inequalities for uncertainty quantification abstract in this paper we demonstrate the only available scalable information bounds for quantities of interest of high dimensional probabilistic models scalability of inequalities allows us to a obtain uncertainty quantification bounds for quantities of interest in the large degree of freedom limit and or at long time regimes b assess the impact of large model perturbations as in nonlinear response regimes in statistical mechanics c address model form uncertainty i e compare different extended models and corresponding quantities of interest we demonstrate some of these properties by deriving robust uncertainty quantification bounds for phase diagrams in statistical mechanics models

3d phase field simulations of interfacial dynamics in newtonian and viscoelastic fluids this work presents a three dimensional finite element algorithm based on the phase field model for computing interfacial flows of newtonian and complex fluids a 3d adaptive meshing scheme produces fine grid covering the interface and coarse mesh in the bulk it is key to accurate resolution of the interface at manageable computational costs the coupled navier stokes and cahn hilliard equations plus the constitutive equation for non newtonian fluids are solved using second order implicit time stepping within each time step newton iteration is used to handle the nonlinearity and the linear algebraic system is solved by preconditioned krylov methods the phase field model with a physically diffuse interface affords the method several advantages in computing interfacial dynamics one is the ease in simulating topological changes such as interfacial rupture and coalescence another is the capability of computing contact line motion without invoking ad hoc slip conditions as validation of the 3d numerical scheme we have computed drop deformation in an elongational flow relaxation of a deformed drop to the spherical shape and drop spreading on a partially wetting substrate the results are compared with numerical and experimental results in the literature as well as our own axisymmetric computations where appropriate excellent agreement is achieved provided that the 3d interface is adequately resolved by using a sufficiently thin diffuse interface and refined grid since our model involves several coupled partial differential equations and we use a fully implicit scheme the matrix inversion requires a large memory this puts a limit on the scale of problems that can be simulated in 3d especially for viscoelastic fluids

sufficient conditions for stability of solitary wave solutions of model equations for long waves abstract we consider solitary wave solutions of model equations for long waves that feature a general form of linear dispersion sufficient conditions for the non linear stability of such solutions are derived these conditions are shown to obtain for the korteweg de vries equation and certain of its generalizations such as the benjamin ono equation and the intermediate long wave equation

three dimensional fully adaptive simulations of phase field fluid models we present an efficient numerical methodology for the 3d computation of incompressible multi phase flows described by conservative phase field models we focus here on the case of density matched fluids with different viscosity model h the numerical method employs adaptive mesh refinements amr in concert with an efficient semi implicit time discretization strategy and a linear multi level multigrid to relax high order stability constraints and to capture the flow s disparate scales at optimal cost only five linear solvers are needed per time step moreover all the adaptive methodology is constructed from scratch to allow a systematic investigation of the key aspects of amr in a conservative phase field setting we validate the method and demonstrate its capabilities and efficacy with important examples of drop deformation kelvin helmholtz instability and flow induced drop coalescence

nonlocal models for nonlinear dispersive waves abstract considered herein are model equations for the unidirectional propagation of small amplitude nonlinear dispersive long waves such as those governed by the classical korteweg de vries equation of special interest are physical situations in which the linear dispersion relation is not appropriately approximated by a polynomial so that the operator modelling dispersive effect is nonlocal particular cases in view here are the benjamin ono equation and the intermediate long wave equation which arise in internal wave theory and smith s equation which governs certain types of continental shelf waves the initial value problem for these equations is shown to be globally well posed in the classical sense including continuous dependence upon the initial data and in certain cases upon the modelling of nonlinear and dispersive effects whilst the results are stated for the specific equations listed above the techniques utilized are seen to have a considerable range of generality as regards application to nonlinear dispersive evolution equations particularly worthy of note is our theorem implying that solutions of the intermediate long wave equation converge strongly to solutions of the korteweg de vries equation or to solutions of the benjamin ono equation in appropriate asymptotic limits

the regularized benjamin ono and bbm equations well posedness and nonlinear stability abstract nonlinear stability of nonlinear periodic solutions of the regularized benjamin ono equation and the benjamin bona mahony equation with respect to perturbations of the same wavelength is analytically studied these perturbations are shown to be stable we also develop a global well posedness theory for the regularized benjamin ono equation in the periodic and in the line setting in particular we show that the cauchy problem in both periodic and nonperiodic case cannot be solved by an iteration scheme based on the duhamel formula for negative sobolev indices

lyapunov stability of ground states of nonlinear dispersive evolution equations on presente une nouvelle demonstration de la stabilite orbitale des solitons d etat fondamental de l equation de schrodinger non lineaire pour une grande classe de non linearites on demontre la stabilite de l onde solitaire pour l equation de korteweg de vries generalisee

instability of travelling waves for weakly coupled kdv systems abstract in this paper we consider the weakly coupled kdv system 0 1 u t u x x f u f u u v x v t v x x g v f v u v x where is a small parameter assuming that for 0 the decoupled system has a travelling wave we first give a sufficient condition for the persistence of such a wave secondly assuming that each wave for the unperturbed system is stable and under a generic condition we show that at least for a one sided interval for the parameter the persistent wave is unstable

efficient spectral galerkin method ii direct solvers of second and fourth order equations using chebyshev polynomials efficient direct solvers based on the chebyshev galerkin methods for second and fourth order equations are presented they are based on appropriate base functions for the galerkin formulation that lead to discrete systems with special structured matrices that can be efficiently inverted numerical results indicate that the direct solvers presented in this paper are significantly more accurate and efficient than those based on the chebyshev tau method

fluctuation dissipation relations for markov processes the fluctuation dissipation relation is calculated for stochastic models obeying a master equation with continuous time in the general case of a nonstationary process there appears to be no simple relation between the response and the correlation also if one considers stationary processes the linear response cannot be expressed via time derivatives of the correlation function alone in this case an additional function which has rarely been discussed previously is required this so called asymmetry depends on the two times also relevant for the response and the correlation and it vanishes under equilibrium conditions the asymmetry can be expressed in terms of the propagators and the transition rates of the master equation but it is not related to any physical observable in an obvious way it is found that the behavior of the asymmetry strongly depends on the nature of the dynamical variable considered in the calculation of the correlation and the response if one is concerned with a variable which randomizes with any transition among the states of the system the asymmetry vanishes in most cases this is in contrast to the situation for other classes of variables in particular for trap models of glassy relaxation the fluctuation dissipation ratio strongly depends on the observable and the asymmetry plays a dominant role in the determination of this ratio also if only neutral variables are considered some implications of a nonvanishing asymmetry with regard to the definition of an effective temperature are discussed

wave collapse and instability of solitary waves of a generalized kadomtsev petviashvili equation abstract the solutions of a family of generalized kadomtsev petviasvili equation in two diemnsions u t u m u x u xxx x 2 u yy are studied with appropriate boundary conditions and constraints on the initial data for m in a dense set m 0 and for 2 1 if m 4 then the solution can blow up in finite time we show that if 1 m 4 a solitary wave so exists which is unstable if m 4 3 in this and related problems th e instability of solitary waves is associated with onset of wave collapse we give numerical evidence to support this

nonlinear response and fluctuation dissipation relations a unified derivation of the off equilibrium fluctuation dissipation relations fdrs is given for ising and continuous spins to arbitrary order within the framework of markovian stochastic dynamics knowledge of the fdrs allows one to develop zero field algorithms for the efficient numerical computation of the response functions two applications are presented in the first one the problem of probing for the existence of a growing cooperative length scale is considered in those cases like in glassy systems where the linear fdr is of no use the effectiveness of an appropriate second order fdr is illustrated in the test case of the edwards anderson spin glass in one and two dimensions in the second application the important problem of the definition of an off equilibrium effective temperature through the nonlinear fdr is considered it is shown that in the case of coarsening systems the effective temperature derived from the second order fdr is consistent with the one obtained from the linear fdr

nonlinear stability of gardner breathers abstract we show that breather solutions of the gardner equation a natural generalization of the kdv and mkdv equations are h 2 r stable through a variational approach we characterize gardner breathers as minimizers of a new lyapunov functional and we study the associated spectral problem through i the analysis of the spectrum of explicit linear systems spectral stability and i i controlling degenerated directions by using low regularity conservation laws

the gardner equation and the l 2 stability of the n soliton solution of the korteweg de vries equation multi soliton solutions of the korteweg de vries equation kdv are shown to be globally l2 stable and asymptotically stable in the sense of martel and merle the proof is surprisingly simple and combines the gardner transform which links the gardner and kdv equations together with the martel merle tsai and martel merle recent results on stability and asymptotic stability in the energy space applied this time to the gardner equation as a by product the results of maddocks sachs and merle vega are improved in several directions

regular article calculation of two phase navier stokes flows using phase field modeling phase field models provide a way to model fluid interfaces as having finite thickness this can allow the computation of interface movement and deformation on fixed grids this paper applies phase field modeling to the computation of two phase incompressible navier stokes flows the navier stokes equations are modified by the addition of the continuum forcing email protected f where c is the composition variable and f is c s chemical potential the equation for interface advection is replaced by a continuum advective diffusion equation with diffusion driven by c s chemical potential gradients the paper discusses how solutions to these equations approach those of the original sharp interface navier stokes equations as the interface thickness e and the diffusivity both go to zero the basic flow physics of phase field interfaces is discussed straining flows can thin or thicken an interface and this must be resisted by a high enough diffusion on the other hand too large a diffusion will overly damp the flow these two constraints result in an upper bound for the diffusivity of o e and a lower bound of o e 2 within these two bounds the phase field navier stokes equations appear to generate an o e error relative to the exact sharp interface equations an o h 2 e 2 numerical method is introduced that is energy conserving in the sense that creation of interface energy by convection is always balanced by an equal decrease in kinetic energy caused by surface tension forcing an o h 4 e 4 compact scheme is introduced that takes advantage of the asymptotic comparatively smooth behavior of the chemical potential for o e accurate phase field models the optimum path to convergence for this scheme appears to be email protected 4 5 the asymptotic rate of convergence corresponding to this is o h 4 5 but results at practical resolutions show that the practical convergence of the method is generally considerably faster than linear extensive analysis and computations show that this scheme is very effective and accurate it allows the accurate calculation of two phase flows with interfaces only two cells wide computational results are given for linear capillary waves and for rayleigh taylor instabilities the first set of computations is compared to exact solutions of the diffuse interface equations and of the original sharp interface equations the rayleigh taylor computations test the ability of the method to compute highly deforming flows these flows include near singular phenomena such as interface coalescences and breakups contact line movement and the formation and breakup of thin wall films grid refinement studies are made and rapid convergence is found for macroscopic flow features such as instability growth rate and propagation speed wavelength and the general physical characteristics of the instability and mass transfer rates

fully discrete galerkin methods for the korteweg de vries equation abstract fully discrete schemes for the numerical simulation of solutions of the periodic initial value problem for the korteweg de vries equation are introduced implemented and tested of special interest are stable schemes featuring rates of convergence of order higher than two in both the spatial and temporal variable a careful analysis of the relative and absolute efficiency of these schemes is carried out and one of the schemes is applied to demonstrate that solutions of certain generalized korteweg de vries equations apparently may develop singularities in finite time

instability of nonlinear dispersive solitary waves we consider linear instability of solitary waves of several classes of dispersive long wave models they include generalizations of kdv bbm regularized boussinesq equations with general dispersive operators and nonlinear terms we obtain criteria for the existence of exponentially growing solutions to the linearized problem the novelty is that we dealt with models with nonlocal dispersive terms for which the spectra problem is out of reach by the evans function technique for the proof we reduce the linearized problem to study a family of nonlocal operators which are closely related to properties of solitary waves a continuation argument with a moving kernel formula is used to find the instability criteria these techniques have also been extended to study instability of periodic waves and of the full water wave problem

efficient spectral galerkin method i direct solvers of second and fourth order equations using legendre polynomials this paper presents some efficient algorithms based on the legendre galerkin approximations for the direct solution of the second and fourth order elliptic equations the key to the efficiency of these algorithms is to construct appropriate base functions which lead to systems with sparse matrices for the discrete variational formulations the complexities of the algorithms are a small multiple of n d 1 operations for a d dimensional domain with n 1 d unknowns while the convergence rates of the algorithms are exponential for problems with smooth solutions in addition the algorithms can be effectively parallelized since the bottlenecks of the algorithms are matrix matrix multiplications

traveling waves to a burgers korteweg de vries type equation with higher order nonlinearities abstract in this paper first we survey some recent advances in the study of traveling wave solutions to the burgers korteweg de vries equation and some comments are given then we study a burgers korteweg de vries type equation with higher order nonlinearities a qualitative analysis to a two dimensional autonomous system which is equivalent to the burgers kdv type equation is presented and indicates that under certain conditions the burgers korteweg de vries type equation has neither nontrivial bell profile solitary waves nor periodic waves finally a solitary wave solution is obtained by means of the first integral method which is based on the ring theory of commutative algebra

global existence of smooth solutions and stability of solitary waves for a generalized boussinesq equation certain generalizations of one of the classical boussinesq type equations u tt u xx u 2 u xx xx are considered it is shown that the initial value problem for this type of equation is always locally well posed it is also determined that the special solitary wave solutions of these equations are nonlinearly stable for a range of their phase speeds these two facts lead to the conclusion that initial data lying relatively close to a stable solitary wave evolves into a global solution of these equations this contrasts with the results of blow up obtained recently by kalantarov and ladyzhenskaya for the same type of equation and casts additional light upon the results for the original version of this class of equations obtained via inverse scattering theory by deift tomei and trubowitz

on well posedness for the benjamin ono equation we prove existence and uniqueness of solutions for the benjamin ono equation with data in h s mathbb r s 1 4 moreover the flow is holder continuous in weaker topologies