an energy conserving parabolic equation for elastic media the accuracy of the parabolic equation pe method is improved for range dependent elastic media the energy conserving elastic pe is a generalization of the energy conserving acoustic pe a range dependent waveguide is approximated by a sequence of range independent regions a linear approximation is derived for the nonlinear condition corresponding to conservation of compressional energy flux at the vertical interfaces between range independent regions the rotated elastic pe is applied to generate reference solutions for demonstrating accuracy the energy conserving elastic pe is applied to solve problems involving the propagation of an interface wave along a sloping ocean bottom and a problem involving mode cutoff and coupling into shear wave beams in the ocean bottom the latter problem is analogous to a cutoff problem that was first solved with the acoustic pe f b jensen and w a kuperman j acoust soc am 67 1564 1566 1980

the construction of one unstable manifold for the dissipative henon mapping the unstable manifold of a saddle point of the henon mapping is constructed analytically via a contraction mapping for a range of parameter values where the second fixed point is a stable node one invariant piece of this manifold connects the saddle with the second fixed point rigorous error bounds are derived for the each step of the iterative procedure it is demonstrated that an algebraic approximation with known accuracy can be given of the unstable manifold

a model for the unstable manifold of the bursting behavior in the 2d navier stokes flow quasi periodic and bursting behaviors of the two dimensional 2d navier stokes flow are analyzed the tools used are the proper orthogonal decomposition pod method and the artificial neural network ann method the pod is used to extract coherent structures and prominent features from pde simulations of a quasi periodic regime and a bursting regime eigenfunctions of the two regimes were related by the symmetries of the 2d navier stokes equations three eigenfunctions that represent the dynamics of the quasi periodic regime and two eigenfunctions associated with the unstable manifold of the bursting regime were derived calculations of the pod eigenfunctions are performed on the fourier amplitudes in a comoving frame inverse fourier transform is applied to represent the pod eigenfunctions in both streamfunction and vorticity formulations so that the number of relevant eigenfunctions for streamfunction and vorticity data is the same projection onto the two eigenfunctions associated with the unstable manifold reduces the data to two time series processing these time series through an ann results in a low dimensional model describing the unstable manifold of the bursting regime that can be used to predict the onset of a burst

on equations of kp type we discuss the local cauchy problem for the generalised kadomtsev petviashvili equation namely in the periodic and nonperiodic settings

stable and unstable manifolds of the henon mapping by using a parametric representation of the stable and unstable manifolds we prove that for some given values of the parameter in particular in the case first investigated by henon the henon mapping has a transversal homoclinic orbit

crises sudden changes in chaotic attractors and transient chaos abstract the occurrence of sudden qualitative changes of chaotic dynamics as a parameter is varied is discussed and illustrated it is shown that such changes may result from the collision of an unstable periodic orbit and a coexisting chaotic attractor we call such collisions crises phenomena associated with crises include sudden changes in the size of chaotic attractors sudden appearances of chaotic attractors a possible route to chaos and sudden destructions of chaotic attractors and their basins this paper presents examples illustrating that crisis events are prevalent in many circumstances and systems and that just past a crisis certain characteristic statistical behavior whose type depends on the type of crisis occurs in particular the phenomenon of chaotic transients is investigated the examples discussed illustrate crises in progressively higher dimension and include the one dimensional quadratic map the two dimensional henon map systems of ordinary differential equations in three dimensions and a three dimensional map in the case of our study of the three dimensional map a new route to chaos is proposed which is possible only in invertible maps or flows of dimension at least three or four respectively based on the examples presented the following conjecture is proposed almost all sudden changes in the size of chaotic attractors and almost all sudden destruction or creations of chaotic attractors and their basins are due to crises

nonlinear wide angle paraxial acoustic propagation in shallow water channels a time domain model that describes wide angle paraxial propagation of acoustic pulses in shallow water is developed this model incorporates weak nonlinear effects and depth variability in both ambient density and sound speed derivations of paraxial approximations are based on an iterative approach in which the wide angle approximation is obtained by using a narrow angle equation to approximate the second range derivative in the two way equation scaling arguments are used to obtain a more tractable simplification of the equation the wide angle equation is solved numerically by splitting into components representing distinct physical processes and using a modified version of the time domain parabolic equation tdpe code m d collins j acoust soc am 84 2114 2125 1988 a high order upwind flux correction method is modified to handle the nonlinear component numerical results are presented for adiabatic propagation in a shallow isospeed channel it is demonstrated that nonlinear effects are significant even at small ranges if the peak source pressure is high enough both nonlinear and wide angle effects are illustrated and their differences are discussed

exploring partial control of chaotic systems in this paper we make a thorough exploration of the technique of partial control of chaotic systems this control technique allows one to keep the trajectories of a dynamical system close to a chaotic saddle even if the control applied is smaller than the effects of environmental noise in the system provided that the chaotic saddle is due to the existence of a horseshoelike mapping in phase space we state this here in a mathematically precise way using the conley moser conditions and we prove that they imply that our partial control strategy can be applied we also give an upper bound of the control noise ratio needed to achieve this goal and we describe how this technique can be applied for large noise values finally we study in detail the effect of imperfect targeting in our control technique all these results are illustrated numerically with the paradigmatic henon map

well posedness for the kadomtsev petviashvili ii equation and generalisations we show the local in time well posedness of the cauchy problem for the kadomtsev petviashvili ii equation for initial data in the non isotropic sobolev space h s 1 s 2 r 2 with s 1 j and s 2 0 on the h s 1 0 r 2 scale this result includes the full subcritical range without any additional low frequency assumption on the initial data more generally we prove the local in time well posedness of the cauchy problem for the following generalisation of the kp ii equation u t d x u x u 2 x x u yy 0 u 0 u 0 for 4 3 max 1 3 4 1 4 3 8 s 2 0 and u 0 h s 1 s 2 r 2 we deduce global well posedness for s 1 0 s 2 0 and real valued initial data

ergodic properties of the lozi mappings in this paper we construct the bowen ruelle measure for the lozi mapping an almost everywhere hyperbolic diffeomorphism of the plane we also derive some of its properties which are similar to those of an axiom a system

global well posedness of the kp i initial value problem in the energy space we prove that the kp i initial value problem begin cases partial tu partial x 3u partial x 1 partial y 2u partial x u 2 2 0 text on mathbb r 2 x y times mathbb r t u 0 phi end cases is globally well posed in the energy space mathbf e 1 mathbb r 2 big phi mathbb r 2 to mathbb r phi mathbf e 1 mathbb r 2 approx phi l 2 partial x phi l 2 big partial x 1 partial y phi big l 2 infty big

growing 1d and quasi 2d unstable manifolds of maps we present a new 1d algorithm for computing the global one dimensional unstable manifold of a saddle point of a map our method can be generalized to compute two dimensional unstable manifolds of maps with three dimensional state spaces this is shown here with a quasi 2d q2d algorithm for the special case of a quasiperiodically forced map which allows for a substantial simplification of the general case described in krauskopf and osinga 1998 the key idea is to grow the manifold in steps which consist of finding a new point on the manifold at a prescribed distance from the last point the speed of growth is determined only by the curvature of the manifold and not by the dynamics the performance of the 1d algorithm is demonstrated with a constructed test example and it is then used to compute one dimensional manifolds of a map modeling mixing in a stirring tank with the q2d algorithm we compute two dimensional unstable manifolds in the quasiperiodically forced henon map

invariant curves attractors and phase diagram of a piecewise linear map with chaos we introduce equations describing the invariant curves associated with periodic points in a wide class of two dimensional invertible maps which in the special case of the mapt x z 1 a x bz x can be solved by analytical methods in the dissipative case several branches of the separatrices of the fixed points as well as of the period 2 and 4 points are constructed the regions of the parameter space where a given type of strange attractor exists are located we point out that the disappearance of homoclinic intersections between the separatrices of the fixed point and that of heteroclinic intersections between the unstable manifolds of the period 2 points and the stable manifold of the fixed point may occur separately and the latter leads already to the appearance of a two piece strange attractor this phenomenon may happen at weak dissipation in other maps too in the conservative caseb 1 separatrices and certain invariant tori are calculated

time domain formulation for pulse propagation including nonlinear behavior at a caustic this paper derives a simple three dimensional time domain formulation for small angle propagation of finite amplitude acoustic pulses and weak shocks in a medium that can be range dependent the formulation is closely related to a full wave model and does not contain the artificial singularities of ray based models at caustics the simplicity of the formulation suggests that it may be of value for broadband linear as well as nonlinear propagation two derivations are given one heuristic and the other a formal series expansion from the fluid equations the result is a first order nonlinear progressive wave equation npe cast in a wave following frame of reference the npe is shown to be the nonlinear time domain counterpart of the frequency domain parabolic wave equation pe the npe gives a natural separation of terms governing refraction spreading and nonlinear steepening numerical calculations not involving fourier synthesis are presented using the npe to simulate the following cases a broadband linear pulse propagation in a waveguide b the development of an initially smooth nonlinear pulse into an n wave and c the behavior of a weak shock at a caustic cases a and b are compared with analytic solutions

constructing constrained invariant sets in multiscale continuum systems we present a method that we name the constrained invariant manifold method a visualization tool to construct stable and unstable invariant sets of a map or flow where the invariant sets are constrained to lie on a slow invariant manifold the construction of stable and unstable sets constrained to an unstable slow manifold is exemplified in a singularly perturbed model arising from a structural mechanical system consisting of a pendulum coupled to a viscoelastic rod additionally we extend the step and stagger method d sweet h nusse and j yorke phys rev lett 86 2261 2001 to calculate a delta pseudoorbit on a chaotic saddle constrained to the slow manifold in order to be able to compute the lyapunov exponents of the saddle

on the propagation of regularities in solutions of the fifth order kadomtsev petviashvili ii equation abstract in this paper we study special properties of solutions of the initial value problem ivp associated to the fifth order kadomtsev petviashvili ii equation mainly for initial data h 1 r 2 whose restriction belongs to h m x 0 r for some m z m 2 and x 0 r we prove that the restriction of the corresponding solution u t belongs to h m b r for any b r and any t 0 in other words the extra regularity on the data propagates in the solutions in the direction of the dispersion

three dimensional time domain paraxial approximations for ocean acoustic wave propagation one way narrow and wide angle three dimensional time domain paraxial approximations to the wave equation are developed to model acoustic propagation the approximate equations are designed to be appropriate for ocean applications including pulse propagation with volume attenuation and variable density first differential equations for acoustic pressure are obtained from a thermodynamic model which incorporates attenuation due to the presence of acoustically absorbing chemical species the method of multiple scales is then used to generate partial differential equations with paraxial characteristics comparisons are made to corresponding previous results results from an operator formalism using pade approximants are subsequently compared with the corresponding multiscaling results appropriate boundary initial and interface conditions are described for the model equations stability of the three dimensional narrow angle approximation is demonstrated by means of an energy integral analytical expressi

a two way parabolic equation for acoustic backscattering in the ocean the parabolic equation pe method is generalized to handle backscattered acoustic energy in the ocean the two way pe is based on the single scattering approximation and the approach of two way coupled modes in which range dependent environments are approximated by a sequence of range independent regions at the vertical boundaries between regions the solution of the two way pe is required to satisfy two continuity conditions the range derivative in one of the conditions is replaced by a higher order pe depth operator the reflected and transmitted fields that satisfy these conditions are computed with an efficient iteration scheme the outgoing and incoming fields are propagated by two way range marching the two way pe which is presently implemented for two dimensional problems is a practical method for solving large scale reverberation problems the accuracy of the two way pe is demonstrated by comparisons with reference solutions the two way pe is applied to simulate the localization of a source of backscattering using the method of back propagation

how often do simple dynamical processes have infinitely many coexisting sinks this work concerns the nature of chaotic dynamical processes sheldon newhouse wrote on dynamical processes depending on a parameter xx 1 t xn wherex is in the plane such as might arise when studying poincare return maps for autonomous differential equations in ir3 he proved that if the system is chaotic there will very often be existing parameter values for which there are infinitely many periodic attractors coexisting in a bounded region of the plane and that such parameter values would be dense in some interval the fact that infinitely many coexisting sinks can occur brings into question the very nature of the foundations of chaotic dynamical processes we prove for an apparently typical situation that newhouse s construction yields only a set of parameter values of measure zero

applications and time domain solution of higher order parabolic equations in underwater acoustics a higher order parabolic equation and the corresponding higher order time domain parabolic equation are derived from a pade series and solved numerically these models provide accurate solutions for problems involving very wide angle propagation e g propagation in the nearfield or over a hard ocean bottom propagation in domains in which sound speed variations are large e g propagation in deep water deep within the ocean bottom in high speed ocean bottoms and possibly of different wave types and propagation out to very long ranges the possibility of modeling elastic wave propagation with a similar approach is considered

a higher order parabolic equation for wave propagation in an ocean overlying an elastic bottom a higher order elastic parabolic equation hepe is derived for wave propagation in depth dependent and weakly range dependent fluid solid media galerkin s method is used to discretize the depth operators in the hepe within layers in which depth variations in the lame constants and density are continuous discontinuities in material properties are handled with centered differences for the interface conditions between layers the numerical solution of the hepe also involves the method of alternating directions and crank nicolson integration the hepe is applied to underwater wave propagation problems involving a water column over an elastic bottom including a weakly range dependent problem the accuracy of the hepe is demonstrated with benchmark calculations

a procedure for finding numerical trajectories on chaotic saddles abstract examples are common in dynamical systems in which there are regions containing chaotic sets that are not attractors if almost every trajectory eventually leaves some region but the region contains a chaotic set then typical trajectories will behave chaotically for a while and then will leave the region and so we will observe chaotic transients the main objective that will be addressed is the dynamic restraint problem given a region that contains a chaotic set but does not contain an attractor find a chaotic trajectory numerically that remains in the regionfor an arbituarily long period of time systems with horseshoes have such regions as do systems with fractal basin boundaries as does the henon map for suitably chosen parameters we present a numerical technique for finding trajectories which will stay in such chaotic sets for arbitrarily long periods of time and it leads to a saddle straddle trajectory the method is called the pim triple procedure since it is based on so called pim triples a pim proper interior maximum triple is three points a c b in a straight line segment such that the interior point c i e c is between a and b has the maximum escape time that is its escape time from the region is greater than the escape time of both a and b proper means the segment from a to b is smaller than a previously obtained segment we show rigorously that the pim triple procedure works in ideal situations we find it works well even in less than ideal cases this procedure can also be used for the computation of lyapunov exponents furthermore the accessible pim triple procedure a refined pim triple procedure for finding accessible trajectories on the chaotic saddle will also be discussed

global well posedness for the kp i equation we prove the global well posedness of the kp i equation in bbb r 2 for data of arbitrary size in a suitable sobolev class we use a compactness method to get local solutions we extend the solutions thanks to some conservation laws an almost conserved quantity and strichartz estimates

a multiscale derivation of a new parabolic equation which includes density variations abstract a new parabolic equation is obtained from the acoustic equation by the multiscale method the new equation incorporates the effects of a variable ocean density the density can be smooth or piecewise smooth thus the new formulation alleviates the need for interfacial conditions when the density is stratified in a piecewise constant fashion it also reduces to the standard p e when the density is constant the new equation has the same conservation law as the p e a difference equation is presented which has a discrete version of the same law

a generalized westervelt equation for nonlinear medical ultrasound a model equation is derived for nonlinear medical ultrasound unlike the existing models which use spatial coordinates material coordinates are used and hence a model for a heterogeneous medium is able to be derived the equation is a generalization of the westervelt equation and includes the nonlinearity relaxation and heterogeneity of soft tissue the validity of the generalized westervelt equation as a model equation for a piola kirchoff acoustic pressure and as an equation for the acoustic pressure is discussed in the second case it turns out that the model follows from two geometric approximations which are valid when the radius of curvature of the phase fronts is much larger than the particle displacements the model is exact for plane waves and includes arbitrary nonlinearity in the stress strain relation

on computing the entropy of the henon attractor in a recent article d ruelle inlecture notes in physics no 80 springer berlin 1978 has conjectured that for the henon attractor its measure theoretic entropy should be equal to its characteristic exponent this result is known to be true for systems which satisfy smale s axiom a in this article we report the results of our computations which suggest that ruelle s conjecture may be true for the henon attractor further in our study we are confronted with fundamental questions which suggest that certain existence theorems from ergodic theory are not sufficient from a computational point of view

on constructing markov partitions by computer two methods to construct markov partitions for two dimensional systems are proposed one is based on the existence of a known or easily accessible by numerical analysis hyperbolic fixed point the other one which is more general is derived from bowen s proof of the existence theorem of markov partitions for hyperbolic systems the methods are successfully implemented in two cases of hyperbolic systems the linear automorphism left begin array 20 c 1 1 1 2 end array right of the 2 torus and a nonlinear perturbation of it the methods are applied also to the henon mapping in such nonhyperbolic case however they produce partitions of the henon attractor which lack some essential properties

smoothness of solutions of nonlinear ode smoothness of a c infty function f is measured by carleman sequence m m k 0 infty we say f in c infty m 0 1 if f k t leq cr k m k with c r 0 as an extension of our results in 2 we prove the following type statements let u be c infty function on 0 1 such that u prime phi u t where phi z t is an entire function of order 1 finite type on z and of a nonanalytic carleman class on t then u is in the same carleman class

multilinear weighted convolution of l 2 functions and applications to nonlinear dispersive equations the x s b spaces as used by beals bourgain kenig ponce vega klainerman machedon and others are fundamental tools to study the low regularity behavior of nonlinear dispersive equations it is of particular interest to obtain bilinear or multilinear estimates involving these spaces by plancherel s theorem and duality these estimates reduce to estimating a weighted convolution integral in terms of the l 2 norms of the component functions in this paper we systematically study weighted convolution estimates on l 2 as a consequence we obtain sharp bilinear estimates for the kdv wave and schrodinger x s b spaces

extracting qualitative dynamics from experimental data abstract we consider the notion of qualitative information and the practicalities of extracting it from experimental data our approach based on a theorem of takens draws on ideas from the generalized theory of information known as singular system analysis due to bertero pike and co workers we illustrate our technique with numerical data from the chaotic regime of the lorenz model

plotting stable manifolds error estimates and noninvertible maps a numerical procedure is described that can accurately compute the stable manifold of a saddle fixed point for a map of r2 even if the map has no inverse conventional algorithms use the inverse map to compute an approximation of the unstable manifold of the fixed point we rigorously analyze the errors that arise in the computation and guarantee that they are small we also argue that a simpler nonrigorous algorithm nevertheless produces highly accurate representations of the stable manifold

partial control of chaotic systems in a region in phase space where there is a chaotic saddle all initial conditions will escape from it after a transient with the exception of a set of points of zero lebesgue measure the action of an external noise makes all trajectories escape faster attempting to avoid those escapes by applying a control smaller than noise seems to be an impossible task here we show however that this goal is indeed possible based on a geometrical property found typically in this situation the existence of a horseshoe the horseshoe implies that there exist what we call safe sets which assures that there is a general strategy that allows one to keep trajectories inside that region with control smaller than noise we call this type of control partial control of chaos

simple models for bifurcations creating horseshoes we present a class of simple models for global bifurcations creating horseshoes some properties known for henon mappings are easily obtained for these models such as e g the existence of nontrivial hyperbolic sets kneading sequences techniques allow us to exhibit explicit differences with the global bifurcation diagram for maps of the interval explicit examples displaying wild hyperbolic sets and infinitely many sinks are also given as an illustration of the simplicity of these models

on the cauchy problem for the zakharov system abstract we study the local cauchy problem in time for the zakharov system 1 1 and 1 2 governing langmuir turbulence with initial data u 0 n 0 t n 0 h k h lscr h l 1 in arbitrary space dimension we define a natural notion of criticality according to which the critical values of k l are 2 3 2 2 2 using a method recently developed by bourgain we prove that the zakharov system is locally well posed for a variety of values of k l the results cover the whole subcritical range for 4 for 3 they cover only part of it and the lowest admissible values are k l 1 2 0 for 2 3 and k l 0 1 2 for 1 as a by product of the one dimensional result we prove well posedness of the benney system 1 14 and 1 15 governing the interaction of short and long waves for the same values of k l

on finite energy solutions of the kp i equation we prove that the flow map of the kadomtsev petviashvili i kp i equation is not uniformly continuous on bounded sets of the natural energy space

noise reduction finding the simplest dynamical system consistent with the data abstract a novel method is described for noise reduction in chaotic experimental data whose dynamics are low dimensional in addition we show how the approach allows experimentalists to use many of the same techniques that have been essential for the analysis of nonlinear systems of ordinary differential equations and difference equations

on the low regularity of the fifth order kadomtsev petviashvili i equation we consider the fifth order kadomtsev petviashvili i kp i equation as tu 3 x u 5 x u 1 x 2 y u uux 0 while r we introduce an interpolated energy space es to consider the well posedeness of the initial value problem ivp of the fifth order kp i equation we obtain the local well posedness of ivp of the fifth order kp i equation in es for 0 s 1 to obtain the local well posedness we present a bilinear estimate in the bourgain space in the framework of the interpolated energy space it crucially depends on the dyadic decomposed strichartz estimate the fifth order dispersive smoothing effect and maximal estimate

the parameterization method for invariant manifolds iii overview and applications abstract we describe a method to establish existence and regularity of invariant manifolds and at the same time to find simple maps which are conjugated to the dynamics on them the method establishes several invariant manifold theorems for instance it reduces the proof of the usual stable manifold theorem near hyperbolic points to an application of the implicit function theorem in banach spaces we also present several other applications of the method

stochastic bifurcation in a driven laser system experiment and theory we analyze the effects of stochastic perturbations in a physical example occurring as a higher dimensional dynamical system the physical model is that of a class b laser which is perturbed stochastically with finite noise the effect of the noise perturbations on the dynamics is shown to change the qualitative nature of the dynamics experimentally from a stochastic periodic attractor to one of chaoslike behavior or noise induced chaos to analyze the qualitative change we apply the technique of the stochastic frobenius perron operator l billings et al phys rev lett 88 234101 2002 to a model of the experimental system our main result is the identification of a global mechanism to induce chaoslike behavior by adding stochastic perturbations in a realistic model system of an optics experiment in quantifying the stochastic bifurcation we have computed a transition matrix describing the probability of transport from one region of phase space to another which approximates the stochastic frobenius perron operator this mechanism depends on both the standard deviation of the noise and the global topology of the system our result pinpoints regions of stochastic transport whereby topological deterministic dynamics subjected to sufficient noise results in noise induced chaos in both theory and experiment

the time domain solution of the wide angle parabolic equation including the effects of sediment dispersion the wide angle time domain parabolic equation tdpe which is the inverse fourier transform of the wide angle parabolic equation pe is derived a numerical solution for the model is described and a benchmark calculation is presented the narrow angle tdpe is also considered and its error is analyzed and compared with the error of the narrow angle pe the tdpe is compared with the progressive wave equation which is shown to be restricted to narrow angle propagation for practical purposes in the sediment attenuation is assumed to depend linearly on frequency and the corresponding causal dispersion law is assumed the model is used to show that the effect of sediment dispersion on pulse propagation in the ocean can be significant

wada basins and chaotic invariant sets in the henon heiles system the h enon heiles hamiltonian is investigated in the context of chaotic scattering in the range of energies where escaping from the scattering region is possible special attention is paid to the analysis of the different nature of the orbits and the the invariant sets such as the stable and unstable manifolds and the chaotic saddle furthermore a discussion on the average decay time associated to the typical chaotic transients which are present in this problem is presented the main goal of this paper is to show by using various computational methods that the corresponding exit basins of this open hamiltonian are not only fractal but they also verify the more restrictive property of wada we argue that this property is verified by typical open hamiltonian systems with three or more escapes

chaotic principle some applications to developed turbulence some models for developed turbulence are considered they are shown to obey a large fluctuations theorem and one of them also obeys a response reciprocity relation of onsager type this illustrates and extends ideas and techniques developed in earlier works mainly for nonequilibrium problems in statistical mechanics

approximating stable and unstable manifolds in experiments we introduce a procedure to reveal invariant stable and unstable manifolds given only experimental data we assume a model is not available and show how coordinate delay embedding coupled with invariant phase space regions can be used to construct stable and unstable manifolds of an embedded saddle we show that the method is able to capture the fine structure of the manifold is independent of dimension and is efficient relative to previous techniques

on the low regularity of the modified korteweg de vries equation with a dissipative term abstract we consider a dissipative version of the modified korteweg de vries equation u t u x x x u x x u 3 x 0 we prove global well posedness results on the associated cauchy problem in the sobolev spaces h s r for s 1 4 while for s 1 2 we prove some ill posedness issues

1 soliton solution of the generalized kp equation with generalized evolution this paper obtains the 1 soliton solution of three variants of the generalized kp equation with generalized evolution the solitary wave ansatz is used to carry out the integration of such equations the parameter domain is also identified in the process the numerical simulations are also obtained in this paper

the cauchy problem for higher order kp equations abstract we study the local well posedness of higher order kp equations our well posedness results make an essential use of a global smoothing effect for the linearized equation established in ben artzi and saut preprint 1997 injected into the framework of fourier transform restriction spaces introduced by bourgain our ill posedness results rely on the existence of solitary wave solutions and on scaling arguments the method was first applied in the context of the kdv and schrodinger equations cf birnir et al ann inst h poincare anal non lineaire 13 1996 529 535 birnir et al j london math soc 53 1996 551 559

a higher order energy conserving parabolic equqation for range dependent ocean depth sound speed and density outgoing solutions of the wave equation including parabolic equation pe and normal mode solutions are usually formulated so that pressure is continuous with range for range dependent problems the accuracy of normal mode solutions has been improved by conserving energy rather than maintaining continuity of pressure porter et al the problem of energy conservation in one way equations j acoust soc am 89 1058 1067 1991 this approach is applied to derive a higher order energy conserving pe that provides improved accuracy for problems involving large ocean bottom slopes and large range and depth variations in sound speed and density a special numerical approach and complex pade coefficients are applied to suppress gibbs oscillations the back propagated half space field an improved pe starter is applied to handle wide propagation angles reference solutions generated with a complex ray model and with the rotated pe are used for comparison

a bilinear estimate with applications to the kdv equation u x 0 u0 x where u0 h r our principal aim here is to lower the best index s for which one has local well posedness in h r i e existence uniqueness persistence and continuous dependence on the data for a finite time interval whose size depends on u0 hs equation in 1 1 was derived by korteweg and de vries 21 as a model for long wave propagating in a channel a large amount of work has been devoted to the existence problem for the ivp 1 1 for instance see 9 10 the inverse scattering method applies to this problem and under appropriate decay assumptions on the data several existence results have been established see 5 6 14 28 33 another approach inherited from hyperbolic problems relies on the energy estimates and in particular shows that 1 1 is locally well posed in h r for s 3 2 see 2 3 12 29 30 31 using these results and conservation laws global in time well posedness in h r s 2 was established see 3 12 30 also global in time weak solutions in the energy space h r were constructed in 34 in 13 and 22 a local smoothing effect for solutions of 1 1 was discovered this combined with the conservation laws was used in 13 and 22 to construct global in time weak solutions with data in h r and even in l r in 16 we introduced oscillatory integral techniques to establish local well posedness of 1 1 in h r s 3 4 and hence global in time well posedness in h r s 1 in 16 we showed how to obtain the above mentioned result by picard iteration in an appropriate function space in 4 j bourgain introduced new function spaces adapted to the linear operator t 3 x for which there are good bilinear estimates for the nonlinear term x u 2 using these spaces bourgain was able to establish local well posedness of 1 1 in h r l r and hence by a conservation

bifurcation to infinitely many sinks this paper considers one parameter families of diffeomorphisms ft in two dimensions which have a curve of dissipative saddle periodic pointspt i e ftn pt pt and detdftn pt 1 the family is also assumed to create new homoclinic intersections of the stable and unstable manifolds ofpt as the parameter varies throught0 gavirlov and silnikov proved that if the new homoclinic intersections are created nondegenerately att0 then there is an infinite cascade of periodic sinks i e there are parameter valuestn accumulating att0 for which there is a sink of periodn gs2 sect 4 we show that this result is true for real analytic diffeomorphisms even if the homoclinic intersection is created degenerately we give computer evidence to show that this latter result is probably applicable to the henon map fora near 1 392 andb equal 0 3