renormalization group and asymptotics of solutions of nonlinear parabolic equations we present a general method for studying long time asymptotics of nonlinear parabolic partial differential equations the method does not rely on a priori estimates such as the maximum principle it applies to systems of coupled equations to boundary conditions at infinity creating a front and to higher possibly fractional differential linear terms we present in detail the analysis for nonlinear diffusion type equations with initial data falling off at infinity and also for data interpolating between two different stationary solutions at infinity in an accompanying paper 5 the method is applied to systems of equations where some variables are slaved such as the complex ginzburg landau equation c 1994 john wiley sons inc

existence and stability of propagating fronts for an autocatalytic reaction diffusion system we study a one dimensional reaction diffusion system which describes an isothermal autocatalytic chemical reaction involving both a quadratic a b 2b and a cubic a 2b 3b autocatalysis the parameters of this system are in the ratio d d b d a of the diffusion constants of the reactant a and the autocatalyst b and the relative activity ii of the cubic reaction first for all values of d 0 and k greater than or equal to 0 we prove the existence of a family of propagating fronts or travelling waves describing the advance of the reaction in particular in the quadratic case k 0 we recover the results of billingham and needham phil trans r sec london a 334 1991 1 24 then if d is close to 1 and k is sufficiently small we prove using energy functionals that these propagating fronts are stable against small perturbations in exponentially weighted sobolev spaces this extends part of the results that are known for the scalar equation to which our system reduces when d 1 c 1998 elsevier science b v

spectral theory for the periodic sine gordon equation a concrete viewpoint a summary of the spectral theory for quasiperiodic sine and sinh gordon equations is given analogies with whole line solitons and scattering theory motivates the discussion the relation between the ingredients in the inverse spectral solution of the periodic sine gordon equation and physical characteristics of sine gordon waves is emphasized the explicit topics covered are summarized in the table of contents in the introduction

stability of strong travelling waves for a non local time delayed reaction diffusion equation the paper is concerned with a non local time delayed reaction diffusion equation we prove the time asymptotic stability of a travelling wavefront without a smallness assumption on its wavelength i e the so called strong wavefront by means of the technical weighted energy method when the initial perturbation around the wave is small the exponential convergent rate is also given selection of a suitable weight plays a key role in the proof

analytic calculation of phase and group velocities of p waves in orthorhombic media abstractwe have developed an approximate method to calculate the p wave phase and group velocities for orthorhombic media two forms of analytic approximations for p wave velocities in orthorhombic media were built by analogy with the five parameter moveout approximation and the four parameter velocity approximation for transversely isotropic media respectively they are called the generalized moveout approximation gma type approximation and the fomel approximation respectively we have developed approximations for elastic and acoustic orthorhombic media we have characterized the elastic orthorhombic media in voigt notation and we can describe the acoustic orthorhombic media by introducing the modified alkhalifah s notation our numerical evaluations indicate that the gma type and fomel approximations are accurate for elastic and acoustic orthorhombic media with strong anisotropy and the gma type approximation is comparable with the approximation recently proposed by sripanich and fomel potential

generalized restricted lie algebras and representations of the zassenhaus algebra abstract letfbe a field of characteristicp 0 la generalized restricted lie algebra overf andp l the primitivep envelope ofl a close relation betweenl representations andp l representations is established in particular the irreducible reduced modules oflfor any l coincide with the irreducible 0 reduced modules ofp l where 0 p l is a trivial extension of from this result the determination of all irreducible representations of the zassenhaus algebra is completed and the dimensions of the corresponding modules are also given

construction of breather solutions for nonlinear klein gordon equations on periodic metric graphs abstract the purpose of this paper is to construct small amplitude breather solutions for a nonlinear klein gordon equation posed on a periodic metric graph via spatial dynamics and center manifold reduction the major difficulty occurs from the irregularity of the solutions the persistence of the approximately constructed pulse solutions under higher order perturbations is obtained by symmetry and reversibility arguments

normal forms and quadratic nonlinear klein gordon equations on construit une transformation qui change l equation a non linearite cubique que l on sait resoudre a 3 1 dimensions on demontre que la solution du probleme original a le meme comportement asymptotique que l equation de klein gordon non lineaire

stability of moving fronts in the ginzburg landau equation we use renormalization group ideas to study stability of moving fronts in the ginzburg landau equation in one spatial dimension in particular we prove stability of the real fronts under complex perturbations this extends the results of aronson and weinberger to situations where the maximum principle is inapplicable and constitutes a step in proving the general marginal stability hypothesis for the ginzburg landau equation

estimating anisotropy parameters and traveltimes in the τ p domain the presence of anisotropy influences many aspects of seismic wave propagation and has therefore implications for conventional processing schemes to estimate the anisotropy we need both forward modelling and inversion tools exact forward modelling in anisotropic media is generally done by raytracing however we present a new and fast method using the p transform to calculate exact p and sv reflection moveout curves in stratified laterally homogeneous anisotropic media which requires no ray tracing results are exact even if the sv waves display cusps in addition we show how the same method can be used for parameter estimation since inversion for anisotropic parameters is very nonunique we develop expressions requiring only a reduced number of parameters nevertheless predictions using these expressions are more accurate than taylor series expansions and are also able to handle cusps in the sv traveltime curves in addition layer stripping is a linear process therefore both effective aver

wavepacket preservation under nonlinear evolution we study nonlinear systems of hyperbolic pde s in mathbb r d the hyperbolicity is understood in a wider sense namely multiple roots of the characteristic equation are allowed and dispersive equations are permitted they describe wave propagation in dispersive nonlinear media such as for example electromagnetic waves in nonlinear photonic crystals the initial data is assumed to be a finite sum of wavepackets referred to as a multi wavepacket the wavepackets and the medium nonlinearity are characterized by two principal small parameters and varrho where i frac 1 beta is a factor describing spatial extension of involved wavepackets ii frac 1 varrho is a factor describing the relative magnitude of the linear part of the evolution equation compared to its nonlinearity a key element in our approach is a proper definition of a wavepacket remarkably the introduced definition has a flexibility sufficient for a wavepacket to preserve its defining properties under a general nonlinear evolution for long times in particular the corresponding wave vectors and the band numbers of involved wavepackets are conserved quantities we also prove that the evolution of a multi wavepacket is described with high accuracy by a properly constructed system of envelope equations with a universal nonlinearity the universal nonlinearity is obtained by a time averaging applied to the original nonlinearity in simpler cases the averaged system turns into a system of nonlinear schrodinger equations

propagation of ultra short optical pulses in cubic nonlinear media abstract we derive a partial differential equation that approximates solutions of maxwell s equations describing the propagation of ultra short optical pulses in nonlinear media and which extends the prior analysis of alterman and rauch phys lett a 264 2000 390 diffractive nonlinear geometric optics for short pulses preprint 2002 we discuss non rigorously conditions under which this approximation should be valid but the main contributions of this paper are 1 an emphasis on the fact that the model equation for short pulse propagation may depend on the details of the optical susceptibility in the wavelength regime under consideration 2 a numerical comparison of solutions of this model equation with solutions of the full nonlinear partial differential equation 3 a local well posedness result for the model equation and 4 a proof that in contrast to the nonlinear schrodinger equation which models slowly varying wavetrains this equation has no smooth pulse solutions which propagate with fixed shape and speed

non linear stability of modulated fronts for the swift hohenberg equation we consider front solutions of the swift hohenberg equation t u 1 x 2 2 u 2 u u 3 these are traveling waves which leave in their wake a periodic pattern in the laboratory frame using renormalization techniques and a decomposition into bloch waves we show the non linear stability of these solutions it turns out that this problem is closely related to the question of stability of the trivial solution for the model problem t u x t x 2 u x t 1 tanh x ct u x t u x t p with p 3 in particular we show that the instability of the perturbation ahead of the front is entirely compensated by a diffusive stabilization which sets in once the perturbation has hit the bulk behind the front

newton s method and periodic solutions of nonlinear wave equations we prove the existence of periodic solutions of the nonlinear wave equation satisfying either dirichlet or periodic boundary conditions on the interval o the coefficients of the eigenfunction expansion of this equation satisfy a nonlinear functional equation using a version of newton s method we show that this equation has solutions provided the nonlinearity g x u satisfies certain generic conditions of nonresonance and genuine nonlinearity 1993 john wiley sons inc

generalized nonelliptic moveout approximation in τ p domain we have derived a generalized nonelliptic moveout approximation in the p domain that is a companion to the generalized nonhyperbolic moveout approximation in the t x domain approximation requires five parameters three parameters that can be determined from the zero slowness ray and two parameters that can be determined from one additional ray we convert the generalized approximation from the p domain to the t x domain and vice versa the generalized nonelliptic moveout approximation in the p domain admits other moveout approximations as special cases several analytical and numerical tests showed a superior accuracy of the generalized nonelliptic moveout approximation

stability of fronts for a kpp system ii the critical case abstract we study a system of n 1 coupled semilinear parabolic equations on the real line which depends on a small parameter and reduces to the scalar kolmogorov petrovsky piscounov kpp equation when 0 under appropriate scaling the system has a family of traveling fronts parametrized by their speed when 2 as in the scalar kpp case the case of critical speed 2 say is investigated and it is shown that the system inherits some crucial properties of the kpp equation when is small in particular the asymptotic stability of the front in a local and semiglobal sense first we describe the properties of the front and then apply functional arguments to prove its local stability in an adequate weighted sobolev space moreover the decay rate of the perturbations is shown to be polynomial in time finally we show a semiglobal stability property of the front which also is inherited from the scalar kpp equation

theory of interval traveltime parameter estimation in layered anisotropic media abstractmoveout approximations for reflection traveltimes are typically based on a truncated taylor expansion of traveltime squared around the zero offset the fourth order taylor expansion involves normal moveout velocities and quartic coefficients we have derived general expressions for layer stripping second and fourth order parameters in horizontally layered anisotropic strata and specified them for two important cases horizontally stacked aligned orthorhombic layers and azimuthally rotated orthorhombic layers in the first of these cases the formula involving the out of symmetry plane quartic coefficients has a simple functional form and possesses some similarity to the previously known formulas corresponding to the 2d in symmetry plane counterparts in vertically transversely isotropic vti media the error of approximating effective parameters by using approximate vti formulas can be significant in comparison with the exact formulas that we have derived we have proposed a framework for derivin

front propagation into unstable states universal algebraic convergence towards uniformly translating pulled fronts fronts that start from a local perturbation and propagate into a linearly unstable state come in two classes pulled fronts and pushed fronts the term pulled front expresses that these fronts are pulled along by the spreading of linear perturbations about the unstable state accordingly their asymptotic speedv equals the spreading speed of perturbations whose dynamics is governed by the equations linearized about the unstable state the central result of this paper is the analysis of the convergence of asymptotically uniformly traveling pulled fronts towardsv we show that when such fronts evolve from sufficiently steep initial conditions which initially decay faster than e x forx 1 t hey have auniversal relaxation behavioras timet 1 the velocity of a pulled front always relaxes algebraically like v t d v 3 2t c 3 p d d 2 t 3 2 c o 1 t 2 the parametersv and d are determined through a saddle point analysis from the equation of motion linearized about the unstable invaded state this front velocity is independent of the precise value of the front amplitude which one tracks to measure the front position the interior of the front is essentially slaved to the leading edge and develops universally as x t d8v t x rt dt 0 v t 0 c o 1 t 2 where8v x vt is a uniformly translating front solution with velocity v v our result which can be viewed as a general center manifold result for pulled front propagation is derived in detail for the well known nonlinear diffusion equation of type t d 2 x c 3 where the invaded unstable state is d 0 even for this simple case the subdominant t 3 2 term extends an earlier result of bramson our analysis is then generalized to more general sets of partial differential equations with higher spatial or temporal derivatives to pdes with memory kernels and also to difference equations such as those that occur in numerical finite difference codes our universal result for pulled fronts thus implies independence i of the level curve which is used to track the front position ii of the precise nonlinearities iii of the precise form of the linear operators in the dynamical equation and iv of the precise initial conditions as long as they are sufficiently steep the only remainders of the explicit form of the dynamical equation are the nonlinear solutions 8v and the three saddle point parameters v and d as our simulations confirm all our analytical predictions in every detail it can be concluded that we have a complete analytical understanding of the propagation mechanism and relaxation behavior of pulled fronts if they are uniformly translating for t 1 an immediate consequence of the slow algebraic relaxation is that the standard moving boundary approximation breaks down for weakly curved pulled fronts in two or three dimensions in addition to our main result for pulled fronts we also discuss the propagation and convergence of fronts emerging from

velocity independent time domain seismic imaging using local event slopes by estimating local event slopes in prestack seismic reflection data it is possible to accomplish all common time domain imaging tasks from normal moveout to prestack time migration without the need to estimate seismic velocities or any other attributes local slopes contain complete information about the reflection geometry once they are estimated seismic velocities and all other moveout parameters turn into data attributes and are directly mappable from the prestack data domain into the time migrated image domain i develop an analytical theory which defines the transformation from data and local slopes to the image space for different time domain imaging operators computational experiments with synthetic and field data examples confirm theoretical expectations and demonstrate practical effectiveness of the proposed method

normal moveout revisited inhomogeneous media and curved interfaces the equation of normal moveout t2 t02 x2 va2 is valid for a reflection from the base of a single homogeneous and isotropic bed but is only an approximation in the real world of multilayered inhomogeneous media and curved interfaces using the theory of geometrical optics we can find another second order equation which represents hyperbolas that are also symmetrical about the time axis however the centers of these hyperbolas do not coincide with the center of coordinates but are shifted along the time axis the equation describing this second type of hyperbola is t tp t0 2 tp2 x2 v12 where tp is the time of focusing depth and v1 the velocity of the input medium this equation is not only more accurate than the usual normal moveout but its use is more economical on a vector computer because the traditional dynamic correction is a static correction in the tp analysis this procedure makes it possible to compute velocities for all the samples of all the stacked traces and produces a velocity secti

generalized moveout approximation for qp and qsv waves in a homogeneous transversely isotropic medium the moveout approximations can be used in kinematic modeling velocity analysis and time migration the generalized moveout approximation involves five approximation parameters and has several known approximations as special cases a method is demonstrated for determining parameters of the generalized nonhyperbolic moveout approximation for qp and qsv waves in a homogeneous transversely isotropic medium with vertical symmetry axis vti medium the additional parameters for the generalized approximation are computed from the hyperbolic asymptote at infinite offset comparison with a few well known moveout approximations for higher order terms in the taylor series and asymptotic behavior shows that the generalized moveout approximation is superior to other nonhyperbolic approximations a few numerical examples for qp and qsv waves in a vti medium also indicate that the generalized approximation performs the best

generalized nonhyperbolic moveout approximation reflection moveout approximations are commonly used for velocity analysis stacking and time migration a novel functional form for approximating the moveout of reflection traveltimes at large offsets is introduced in comparison with the classic hyperbolic approximation which uses only two parameters zero offset time and moveout velocity this form involves five parameters that can be determined in a known medium from zero offset computations and from tracing one nonzero offset ray it is called a generalized approximation because it reduces to some known three parameter forms with a particular choice of coefficients by testing the accuracy of the proposed approximation with analytical and numerical examples the new approximation is shown to bring an improvement in accuracy of several orders of magnitude compared to known analytical approximations which makes it as good as exact for many practical purposes

breather solutions in periodic media for nonlinear wave equations existence proofs for breathers are very rare in the spatially homogeneous case up to rescaling the sine gordon equation partial 2 t u partial 2 x u sin u is the only nonlinear wave equation which is known to possess breather solutions for nonlinear wave equations in periodic media no examples of breather solutions have been known so far using spatial dynamics center manifold theory and bifurcation theory for periodic systems we construct for the first time such time periodic solutions of finite energy for a nonlinear wave equation s x partial 2 t u x t partial 2 x u x t q x u x t r x u x t 3 with spatially periodic coefficients s q and r on the real axis such breather solutions play an important role in theoretical scenarios where photonic crystals are used as optical storage

approximate invariant manifolds up to exponentially small terms abstract this paper is devoted to analytic vector fields near an equilibrium for which the linearized system is split in two invariant subspaces e 0 dim m 0 e 1 dim m 1 under light diophantine conditions on the linear part we prove that there is a polynomial change of coordinate in e 1 allowing to eliminate in the e 1 component of the vector field all terms depending only on the coordinate u 0 e 0 up to an exponentially small remainder this main result enables to prove the existence of analytic center manifolds up to exponentially small terms and extends to infinite dimensional vector fields in the elliptic case our results also proves with very light assumptions on the linear part in e 1 that for initial data very close to a certain analytic manifold the solution stays very close to this manifold for a very long time which means that the modes in e 1 stay very small

asymptotic stability of the critical pulled front in a lotka volterra competition model abstract we prove that the critical pulled front of lotka volterra competition systems is nonlinearly asymptotically stable more precisely we show that perturbations of the critical front decay algebraically with rate t 3 2 in a weighted l space our proof relies on pointwise semigroup methods and utilizes in a crucial way that the faster decay rate t 3 2 is a consequence of the lack of an embedded zero of the evans function at the origin for the linearized problem around the critical front

nonlinear stability of taylor vortices in infinite cylinders we consider the taylor couette problem in an infinitely extended cylindrical domain in the case when couette flow is weakly unstable and a family of spatially periodic equilibria called the taylor vortices has bifurcated from this trivial ground state we show that those taylor vortices which are not linearly unstable in the sense of eckhaus are in fact nonlinearly stable with respect to small spatially localized perturbations the main difficulty in showing this result stems from the fact that on unbounded cylindrical domains the taylor vortices are only linearly marginally stable with continuous spectrum up to the imaginary axis bloch wave representations of the solutions and renormalization theory allow us to show that the nonlinear problem behaves asymptotically like the linearized one which is under a diffusive regime

pattern forming pulled fronts bounds and universal convergence we analyze the dynamics of pattern forming fronts which propagate into an unstable state and whose dynamics is of the pulled type so that their asymptotic speed is equal to the linear spreading speed v we discuss a method that allows to derive bounds on the front velocity and which hence can be used to prove for among others the swift hohenberg equation the extended fisher kolmogorov equation and the cubic complex ginzburg landau equation that the dynamically relevant fronts are of the pulled type in addition we generalize the derivation of the universal power law convergence of the dynamics of uniformly translating pulled fronts to both coherent and incoherent pattern forming fronts the analysis is based on a matching analysis of the dynamics in the leading edge of the front to the behavior imposed by the nonlinear region behind it numerical simulations of fronts in the swift hohenberg equation are in full accord with our analytical predictions

nonpersistence of breather families for the perturbed sine gordon equation we show that up to one exception and as a consequence of first order perturbation theory only it is impossible that a large portion of the well known family of breather solutions to the sine gordon equation could persist under any nontrivial perturbation of the form u tt u xx sin u varepsilon delta left u right o left varepsilon 2 right where is an analytic function in anarbitrarily small neighbourhood ofu 0 improving known results we analyze and overcome the particular difficulties that arise when one allows the domain of analyticity of to be small the single exception is a one dimensional linear space of perturbation functions under which the full family of breathers does persist up to first order in e

propagating fronts and the center manifold theorem we prove the existence of propagating front solutions for the swift hohenberg equation sh using the center manifold theorem we reduce the problem to a two dimensional system of ordinary differential equations they describe stationary solutions and front solutions of the partial differential equation sh we identify the well known amplitude equation as the lowest order approximation to the equation of motion on the center manifold

traveltime approximations for a layered transversely isotropic medium we consider multiple transmitted reflected and converted qp qsv waves or multiple transmitted and reflected sh waves in a horizontally layered medium that is transversely isotropic with a vertical symmetry axis vti traveltime and offset horizontal distance between a source and receiver not necessarily in the same layer are expressed as functions of horizontal slowness these functions are given in terms of a taylor series in slowness in exactly the same form as for a layered isotropic medium the coefficients depend on the parameters of the anisotropic layers through which the wave has passed and there is no weak anisotropy assumption using classical formulas the traveltime or traveltime squared can then be expressed as a taylor series in even powers of offset these taylor series give rise to a shifted hyperbola traveltime approximation and a new continued fraction approximation described by four parameters that match the taylor series up to the sixth power in offset further approximations give several simplified continued fraction approximations all of which depend on three parameters zero offset traveltime nmo velocity and a heterogeneity coefficient the approximations break down when there is a cusp in the group velocity for the qsv wave numerical studies indicate that approximations of traveltime squared are generally better than those for traveltime a new continued fraction approximation that depends on three parameters is more accurate than the commonly used continued fraction approximation and the shifted hyperbola

renormalization group and the ginzburg landau equation we use renormalization group methods to prove detailed long time asymptotics for the solutions of the ginzburg landau equations with initial data approaching asx different spiraling stationary solutions a universal pattern is formed depending only on this asymptotics at spatial infinity

the generalized restricted representations of graded lie algebras of cartan type abstract let f be an algebraically closed field of characteristic p 2 in this paper the concepts of generalized restricted lie algebras and their generalized restricted representations over f are introduced any graded lie algebra of cartan type l x m n 2 x w s h or k is a generalized restricted lie algebra associated with a standard basis e of l and s a positive integer max 1 i m n i the irreducible generalized restricted representations of l are determined which are in a one to one correspondence with p sd weight functions here d is the dimension of the standard torus in l 1 x k e m 0 the descriptions of the blocks and a brauer type reciprocity are obtained

algebraic group actions in the cohomology theory of lie algebras of cartan type abstract given a lie algebra g of cartan type we construct an infinite dimensional cocommutative hopf algebrad g u g which is the analog of the distribution algebra of a connected reductive algebraic group we show that the simpleu g modules lift to ad g u g structure this additional structure is used to formulate and prove relative projectivity theorems for lie algebras of cartan type the support varieties of certaind g u g modules are computed by using these algebraic group techniques

a centre manifold theorem for hyperbolic pdes a version of the centre manifold theorem is established which is suitable for quasilinear hyperbolic equations as an application the benard problem for a viscoelastic fluid is discussed

solutions without phase slip for the ginsburg landau equation we consider the ginsburg landau equation for a complex scalar field in one dimension and consider initial data which have two different stationary solutions as their limits in space asx if these solutions are not very different then we show that the initial data will evolve to a stationary solution by a phase melting process which avoids phase slips i e which does not go through zero amplitude

the non linear stability of front solutions for parabolic partial differential equations for the ginzburg landau equation and similar reaction diffusion equations on the line we show convergence of complex perturbations of front solutions towards the front solutions by exhibiting a coercive functional

dimensions of the simple restricted modules for the restricted contact lie algebra according to 2 the finite dimensional restricted simple lie algebras over an algebraically closed field of characteristic p 7 are either classical or of cartan type lusztig has a conjecture see 8 p 294 which provides a formula for the dimensions of the simple modules for the classical algebras in this paper modules are assumed to be left finitedimensional and restricted anderson jantzen and soergel 1 have recently shown that for p a 0 the lusztig conjecture follows from an analogous conjecture also due to lusztig for certain quantum groups they have also pointed out that this latter conjecture is implied by recent work of kazhdan and lusztig and a paper by casian although casian s paper has not yet convinced all of its readers the simple modules for the cartan type algebras fall into two classes there is a large class see 11 and 5 consisting of modules that are induced from simple modules for the homogeneous component of degree zero extended trivially to positive components constructed as mixed products in 11 the degree zero homogeneous components of cartan type algebras are either classical or nearly so therefore the lusztig conjecture can be used to get a formula for the dimensions of the modules in this class the small class consists of modules corresponding to exceptional weights in 11 the dimensions of these modules were determined for three of the four cartan type algebras namely the witt special and hamiltonian algebras there the exceptional weights turned out to be fundamental dominant weights in this paper we obtain the dimensions assuming p is not too small in the remaining case of the contact algebra in the process we determine the contragredients of the simple modules

power gradient velocity model a proposed power gradient velocity model incorporates several well known velocity models as special cases the model covers a wide range of possible velocity distributions and has four parameters giving more flexibility in velocity model manipulation for this nonlinear model the kinematic characteristics offset traveltime parameteric equations traveltime parameters relative geometric spreading and propagator phase are computed the characteristics are investigated with respect to a parameter responsible for nonlinearity of velocity distribution the inversion of travel time parameters was studied in three and four parameter frameworks

3d generalized nonhyperboloidal moveout approximation abstractmoveout approximations are commonly used in velocity analysis and time domain seismic imaging we revisit the previously proposed generalized nonhyperbolic moveout approximation and develop its extension to the 3d multiazimuth case the advantages of the generalized moveout approximation are its high accuracy and its ability to reduce to several other known approximations with particular choices of parameters the proposed 3d functional form involves 17 independent parameters instead of five as in the 2d case these parameters can be defined by zero offset traveltime attributes and four additional far offset rays in our tests the proposed approximation achieves significantly higher accuracy than previously proposed 3d approximations

an existence theory for three dimensional periodic travelling gravity capillary water waves with bounded transverse profiles this article presents a rigorous existence theory for three dimensional gravity capillary water waves which are uniformly translating and periodic in one horizontal spatial direction x and have a nontrivial transverse profile in the other z the hydrodynamic equations are formulated as an infinite dimensional hamiltonian system in which z is the time like variable and a centre manifold reduction technique is applied to demonstrate that the problem is locally equivalent to a finite dimensional hamiltonian system of ordinary differential equations a family of straight lines c1 c2 in bond number froude number parameter space is identified which are associated with codimension one bifurcation phenomena at each point on one of these lines two real eigen values of the linear problem become purely imaginary by passing through zero there are also codimension two points the line ck intersects each of the lines ck 1 ck 2 in precisely one point general statements concerning the existence of waves which are periodic or quasiperiodic in z are made by applying standard tools in hamiltonian systems theory to the reduced equations moreover a critical curve in parameter space is found at which a wave with a spatially localised and exponentially decaying transverse profile bifurcates from zero this curve is piecewise linear it contains one line segment from each of c1 c2

an asymptotic expression for the splitting of separatrices of the rapidly forced pendulum the measure of the splitting of the separatrices of the rapidly forced pendulum ddot x sin x mu sin frac t varepsilon is considered as a model problem that has been studied by different authors here e are small parameters e 0 but otherwise independent the following formula for the angle between separatrices is established alpha frac pi 2 varepsilon frac mu cosh frac pi 2 varepsilon left 1 o mu varepsilon 2 right

splitting of separatrices for standard and semistandard mappings abstract we investigate chirikov s standard mapping which is a model for the onset of nonintegrability in nonlinear hamiltonian systems the asymptotic formula when the deviation from an integrable system is small for the angle of intersection of separatrices at a homoclinic point is derived on the basis of approximation by the green percival semistandard mapping in the complex domain computed data on the splitting of separatrices for the semi standard map are presented from which we derive the pre exponential coefficient in the formula for splitting of separatrices in the standard mapping

normal hyperbolicity of center manifolds and saint venant s principle the concept of normal hyperbolicity of center manifolds is generalized to infinite dimensional differential equations in particular to elliptic problems in cylindrical domains it is shown that all solutions u staying close to the center manifold for t l l satisfy an estimate of the form left u t tilde u t right leqslant ce alpha l t where c and are independent of l and is a solution on the center manifold these results are applied to saint venant s principle for the static deformation of nonlinearly elastic prismatic bodies the use of the center manifold permits the effective treatment of the general case of non zero resultant forces and moments acting on each cross section

modulating pulse solutions for a class of nonlinear wave equations we consider modulating pulse solutions for a nonlinear wave equation on the infinite line such a solution consists of a permanent pulse like envelope steadily advancing in the laboratory frame and modulating an underlying wave train the problem is formulated as an infinite dimensional dynamical system with one stable one unstable and infinitely many neutral directions using a partial normal form and invariant manifold theory we establish the existence of modulating pulse solutions which decay to small amplitude disturbances at large distances

modulating pulse solutions for quasilinear wave equations abstract this paper presents an existence proof for symmetric modulating pulse solutions of a quasilinear wave equation modulating pulse solutions consist of a pulse like envelope advancing in the laboratory frame and modulating an underlying wave train they are also referred to as moving breathers since they are time periodic in a moving frame of reference the problem is formulated as an infinite dimensional dynamical system with two stable two unstable and infinitely many neutral directions using a partial normal form and a generalisation of local invariant manifold theory to the quasilinear setting we prove the existence of modulating pulses on arbitrarily large but finite domains in space and time

breathers on diatomic fermi pasta ulam lattices we prove the existence of breathers spatially localized and time periodic oscillations in diatomic fermi pasta ulam fpu chains with arbitrary mass ratio this completes an existence result by livi spicci and mackay valid for large mass ratio the problem is formulated as a mapping in a loop space and analyzed via a discrete spatial centre manifold reduction plane wave solutions of the linearized system have frequencies in a higher optic band or a lower acoustic band for frequencies close to band edges all small amplitude solutions of the nonlinear system lie on a finite dimensional centre manifold which reduces the problem locally to the study of a finite dimensional mapping for good parameter values the map admits homoclinic orbits to 0 corresponding to discrete breathers when the fpu interaction potential satisfies a hardening condition we find breathers with frequencies slightly above the optic band or in the gap slightly above the acoustic band for a potential satisfying the opposite softening condition we obtain breathers with frequencies in the gap slightly below the optic band 2004 elsevier b v all rights reserved

periodic nonlinear waves on a half line nontrivial solutions of the equationutt uxx g u which are 2 periodic int and which decay asx are shown to exist ifg a 0 andg 0 1 breather like solutions which also decay asx can be interpreted as homoclinic solutions in thex dynamics their existence is still in question for generalg

polynomial normal forms with exponentially small remainder for analytic vector fields a key tool in the study of the dynamics of vector fields near an equilibrium point is the theory of normal forms invented by poincare which gives simple forms to which a vector field can be reduced close to the equilibrium in the class of formal vector valued vector fields the problem can be easily solved whereas in the class of analytic vector fields divergence of the power series giving the normalizing transformation generally occurs nevertheless the study of the dynamics in a neighborhood of the origin can very often be carried out via a normalization up to finite order this paper is devoted to the problem of optimal truncation of normal forms for analytic vector fields in r m more precisely we prove that for any vector field in r m admitting the origin as a fixed point with a semi simple linearization the order of the normal form can be optimized so that the remainder is exponentially small we also give several examples of non semi simple linearization for which this result is still true

modulating pulse solutions to quadratic quasilinear wave equations over exponentially long length scales this paper presents an existence proof for modulating pulse solutions to a wide class of quadratic quasilinear klein gordon equations of the form partial t 2 u partial x 2 u u f 1 u partial x u partial t u partial x 2 u f 2 u partial x u partial t u modulating pulse solutions consist of a pulse like envelope advancing in the laboratory frame and modulating an underlying wave train they are also referred to as moving breathers since they are time periodic in a moving frame of reference the problem is formulated as an infinite dimensional dynamical system with three stable three unstable and infinitely many neutral directions by transforming part of the equation into a normal form with an exponentially small remainder term and using a generalisation of local invariant manifold theory to the quasilinear setting we prove the existence of small amplitude modulating pulses on domains in space whose length is exponentially large compared to the magnitude of the pulse

small generalized breathers with exponentially small tails for klein gordon equations abstract we consider a class of nonlinear klein gordon equation u t t u x x u f u and show that generically there exist small breathers with exponentially small tails

the time dependent amplitude equation for the swift hohenberg problem precise estimates for the validity of the amplitude approximation for the swift hohenberg equation are given in a fully time dependent framework it is shown that small solutions of ordero which are modulated like stationary solutions have an evolution which is well described in the amplitude approximation for a time of ordero 2 for the proofs we use techniques for nonlinear semigroups and oscillatory integrals

inverse spectral theory for uniformly open gaps in a weighted sturm liouville problem abstract motivated by a pde existence problem we study the inverse problem for a weighted sturm liouville operator l s associated with the eigenvalue problem y s x y 0 where s is a real valued periodic even function that is bounded from below by a positive constant and belongs to the l 2 based sobolev space h r 0 1 r 1 choosing gap lengths and gap midpoints as coordinates we define a spectral map g that assigns to a coefficient s the structure of the spectrum of l s we find that g is a real analytic isomorphism locally around s 1 which in particular implies the existence of coefficients s h r 0 1 r 3 2 whose spectrum features band structure with all gaps uniformly open around the gap midpoints this result paves the way for the construction of so called breathers in nonlinear wave equations with such coefficients s apart from the novelty of treating the inverse spectral problem for the full banach scale h r 0 1 r 1 the local nature of our result allows more concise and transparent proofs in particular instead of using any preliminary transformations we treat the weighted problem directly by adapting techniques used for schrodinger operators with distribution potentials

representations of cartan type lie algebras in characteristic p in this paper we discuss the representations of cartan type lie algebras in characteristic p 2 from the viewpoint of reducing rank when the character is regular semisimple for generalized witt algebras we can essentially reduce higher rank representations to lower rank representations

diffusive stability of spatial periodic solutions of the swift hohenberg equation we are interested in the nonlinear stability of the eckhaus stable equilibria of the swift hohenberg equation on the infinite line with respect to small integrable perturbations the difficulty in showing stability for these stationary solutions comes from the fact that their linearizations possess continuous spectrum up to zero the nonlinear stability problem is treated with renormalization theory which allows us to show that the nonlinear terms are irrelevant i e that the fully nonlinear problem behaves asymptotically as the linearized one which is under a diffusive regime

generalized restricted representations of the zassenhaus superalgebras abstract let f be an algebraically closed field of prime characteristic p 2 and n n let z n be the zassenhaus superalgebra defined over f which as the simplest non restricted simple lie superalgebra is the superversion of the zassenhaus algebra more precisely z n is the lie superalgebra of the special super derivations of the superalgebra n here n is the tensor product of the divided power algebra of one variable and the grassmann superalgebra of one variable in this paper we study generalized restricted simple modules over the zassenhaus superalgebra z n classification of isomorphism classes of generalized restricted simple modules and their dimensions are precisely determined a sufficient and necessary condition for irreducibility of generalized restricted kac modules is provided

a theory of normal moveout three geophysical principles are shown to be sufficient to determine the most general practical normal moveout nmo equation the principles are reciprocity in a common midpoint cmp gather finite slowness and exact constant velocity limit the resulting equation is the shifted hyperbola nmo equation that has three parameters comparisons at both near and far offsets between the shifted hyperbola nmo equation and the results for layered media assign geophysical meaning to the parameters two of the parameters zero offset time and nmo velocity are constants and control the very near offset behavior the third parameter is dimensionless and controls the far offset behavior of the nmo curve but it may be a function of offset so as to exactly fit any traveltime curve the parameters may be found by a linear least squares fit to data the theory applies to all offsets for nonturning wave reflections in an isotropic earth for both p waves and converted p sv waves

non linear wave and schrodinger equations i instability of periodic and quasiperiodic solutions we investigate stability of periodic and quasiperiodic solutions of linear wave and schrodinger equations under non linear perturbations we show in the case of the wave equations that such solutions are unstable for generic perturbations for the schrodinger equations periodic solutions are stable while the quasiperiodic ones are not we extend these results to periodic solutions of non linear equations

a simple global characterization for normal forms of singular vector fields we derive a new global characterization of the normal forms of amplitude equations describing the dynamics of competing order parameters in degenerate bifurcation problems using an appropriate scalar product in the space of homogeneous vector polynomials we show that the resonant terms commute with the group generated by the adjoint of the original critical linear operator this leads to a very efficient constructive method to compute both the nonlinear coefficients and the unfolding of the normal form explicit examples and results obtained when there are additional symmetries are also presented

the validity of modulation equations for extended systems with cubic nonlinearities modulation equations play an essential role in the understanding of complicated systems near the threshold of instability here we show that the modulation equation dominates the dynamics of the full problem locally at least over a long time scale for systems with no quadratic interaction term we develop a method which is much simpler than previous ones it involves a careful bookkeeping of errors and an estimate of gronwall type as an example for the dissipative case we find that the ginzburg landau equation is the modulation equation for the swift hohenberg problem moreover the method also enables us to handle hyperbolic problems the nonlinear schrodinger equation is shown to describe the modulation of wave packets in the sine gordon equation

existence and stability of traveling fronts in the extended fisher kolmogorov equation abstract we study traveling wave solutions to a general fourth order differential equation that is a singular perturbation of the fisher kolmogorov equation we apply the geometric method for singularly perturbed systems to show that for every positive wavespeed there exists a traveling wave also we find that there exists a critical wavespeed c which divides these solutions into monotonic c c and oscillatory c c solutions we show that the monotonic fronts are locally stable under perturbations in appropriate weighted sobolev spaces by using various energy functionals

persistence and stability of discrete vortices in nonlinear schrodinger lattices abstract we study discrete vortices in the two dimensional nonlinear schrodinger lattice with small coupling between lattice nodes the discrete vortices in the anti continuum limit of zero coupling represent a finite set of excited nodes on a closed discrete contour with a non zero charge using the lyapunov schmidt reductions we analyze continuation and termination of the discrete vortices for small coupling between lattice nodes an example of a square discrete contour is considered that includes the vortex cell also known as the off site vortex we classify families of symmetric and asymmetric discrete vortices that bifurcate from the anti continuum limit we predict analytically and confirm numerically the number of unstable eigenvalues associated with each family of such discrete vortices