computation of homoclinic solutions to periodic orbits in a reduced water wave problem abstract this paper concerns homoclinic solutions to periodic orbits in a fourth order hamiltonian system arising from a reduction of the classical water wave problem in the presence of surface tension these solutions correspond to travelling solitary waves which converge to non decaying ripples at infinity an analytical result of amick and toland 1992 showing the existence of such homoclinic orbits to small amplitude periodic orbits in a singular limit is extended numerically also a related result by amick and mcleod 1991 showing the non existence of homoclinic solutions to zero is motivated geometrically a general boundary value method is constructed for continuation of homoclinic orbits to periodic orbits in hamiltonian and reversible systems numerical results are presented using the path following software auto showing that the amick toland solutions persist well away from the singular limit and for large amplitude periodic orbits special account is taken of the phase shift between the two periodic solutions in the asymptotic limits furthermore new multi modal homoclinic solutions to periodic orbits are shown to exist under a transversality hypothesis which is verified a posteriori by explicit computation continuation of these new solutions reveals limit points with respect to the singular parameter

a weak formulation of roe s approximate riemann solver abstract this paper presents a weak formulation of roe s approximate riemann solver based on a notion of weak solution to nonlinear hyperbolic systems in nonconservative form this weak formulation is applied to the euler equations for real gases and the homogeneous equilibrium two phase flow model numerical results for a two phase flow heat addition problem are given

numerical investigation of a modified family of centered schemes applied to multiphase equations with nonconservative sources systems of hyperbolic partial differential equations are frequently used to model the flow of multiphase mixtures these equations often contain sources referred to as nozzling terms that cannot be posed in divergence form and have proven to be particularly challenging in the development of finite volume methods upwind schemes have recently shown promise in properly resolving the steady wave solution of the associated multiphase riemann problem however these methods require a full characteristic decomposition of the system eigenstructure which may be either unavailable or computationally expensive central schemes such as the kurganov tadmor kt family of methods require minimal characteristic information which makes them easily applicable to systems with an arbitrary number of phases however the proper implementation of nozzling terms in these schemes has been mathematically ambiguous the primary objectives of this work are twofold first an extension of the kt family of schemes is proposed that formally accounts for the nonconservative nozzling sources this modification results in a semidiscrete form that retains the simplicity of its predecessor and introduces little additional computational expense second this modified method is applied to multiple but equivalent forms of the multiphase equations to perform a numerical study by solving several one dimensional test problems both ideal and mie gruneisen equations of state are used with the results compared to an analytical solution this study demonstrates that the magnitudes of the resulting numerical errors are sensitive to the form of the equations considered and suggests an optimal form to minimize these errors finally a separate modification of the wave propagation speeds used in the kt family is also suggested that can reduce the extent of numerical diffusion in multiphase flows

strange attractors of infinitesimal widths in the bifurcation diagram with an unusual mechanism of onset nonlinear dynamics in coupled fuzzy control systems ii abstract a new type of onset of chaos in a multidimensional dissipative system is reported that has been observed under a circumstance in which a delicate balance of order formation and chaos is materialized two limit cycles are connected directly to a sequence of strange attractors the widths in the bifurcation feigenbaum diagram of which converge to zero in a range close to the onset the measures of the extent of chaos are also zero there and the trajectories behave regularly in a macroscopic scale the chaos zones appear alternately with the windows of resonance frequency locking in the bifurcation diagram whose number of periodicity decreases as a system parameter is increased from an infinity in the manner of arithmetical progression 28 n 2 with n being integers down to unity concomitantly the extent of chaos develops and thereby grows measurable also the ranges of both the chaos and resonance zones in the bifurcation diagram become wider two lorenz plots of quantum numbers n and n 1 are always necessary to characterize the n th chaos zone for a chaotic state they appear in such a manner as to sandwich the identity map y x this chaos is readily interrupted by a limit cycle since one of the lorenz maps is crossed with y x by a small change of the control parameter moreover the tangential parts of the individual lorenz plots to y x have as many folds as their quantum numbers they form clusters each of which consists of many stable points being located densely at the limit of onset n both the intervals between these stable points and the folds lying between two neighboring stable points converge to zero this is the origin of the unmeasurable degree of chaos

resonance regions for mathieu type dynamical systems on a torus abstract mathieu type dynamical systems on a torus are torus maps or differential equations whose right hand side components are trigonometric polynomials the asymptotic behaviour of resonance regions for mathieu type dynamical systems close to the integrable ones is drastically different from the behaviour of generic systems because the scaling law of resonance regions depends on the degree of the trigonometric polynomial or its newton polyhedron we prove this scaling law and compare our results with numerical experiments

on low regularity of the ostrovsky stepanyams and tsimring equation abstract in this paper we prove by using the fourier restriction norm method that the initial value problem of the ostrovsky stepanyams and tsimring equation u t u x x x h u x h u x x x u u x 0 x r t 0 where 0 and h denotes the usual hilbert transformation is locally well posed in the sobolev space h s r for any s 5 4 which improve our former result in zhao and cui 2009 5

solving shallow water systems in 2d domains using finite volume methods and multimedia sse instructions the goal of this paper is to construct efficient parallel solvers for 2d hyperbolic systems of conservation laws with source terms and nonconservative products the method of lines is applied at every intercell a projected riemann problem along the normal direction is considered which is discretized by means of well balanced roe methods the resulting 2d numerical scheme is explicit and first order accurate in m j castro j a garcia j m gonzalez c pares a parallel 2d finite volume scheme for solving systems of balance laws with nonconservative products application to shallow flows comput methods appl mech engrg 196 2006 2788 2815 a domain decomposition method was used to parallelize the resulting numerical scheme which was implemented in a pc cluster by means of mpi techniques in this paper in order to optimize the computations a new parallelization of simd type is performed at each mpi thread by means of sse streaming simd extensions which are present in common processors more specifically as the most costly part of the calculations performed at each processor consists of a huge number of small matrix and vector computations we use the intel c integrated performance primitives small matrix library to make easy the use of this library which is implemented using assembler and sse instructions we have developed a c wrapper of this library in an efficient way some numerical tests were carried out to validate the performance of the c small matrix wrapper the specific application of the scheme to one layer shallow water systems has been implemented on a pc s cluster the correct behavior of the one layer model is assessed using laboratory data

hllc type riemann solver for the baer nunziato equations of compressible two phase flow we first construct an approximate riemann solver of the hllc type for the baer nunziato equations of compressible two phase flow for the subsonic wave configuration the solver is fully nonlinear it is also complete that is it contains all the characteristic fields present in the exact solution of the riemann problem in particular stationary contact waves are resolved exactly we then implement and test a new upwind variant of the path conservative approach such schemes are suitable for solving numerically nonconservative systems finally we use locally the new hllc solver for the baer nunziato equations in the framework of finite volume discontinuous galerkin finite element and path conservative schemes we systematically assess the solver on a series of carefully chosen test problems

spatio temporal dimensionality in the overall complex dynamics of an experimental cable mass system an experimental model of an elastic cable mass hanging at in phase or out of phase vertically moving supports is considered system parameters are adjusted to produce two different conditions of multiple internal resonance nonregular dynamics are analyzed in various frequency ranges including meaningful external resonance conditions attention is devoted to characterization of system dimensionality in terms of both time and spatial complexity the aims of this paper are i to give a general overview of the richness and robustness of different quasiperiodic and homoclinic bifurcation scenarios to chaos in various regions of the control parameter space ii to characterize steady nonregular response through delay embedding technique for attractor reconstruction and proper orthogonal decomposition of spatio temporal flow and iii to identify spatial configuration variables experimental eigenfunctions contributing mostly to nonregular dynamics thus obtaining hints about possible reduced models for reproducing complex regimes system dimensionality will be evaluated both by relating the dimension of attractors to the dimension of the linear phase space and from the dominating proper orthogonal modes

implicit spectral methods for wave propagation problems abstract the numerical solution of a non linear wave equation can be obtained by using spectral methods to resolve the unknown in space and the standard crank nicolson differencing scheme to advance the solution in time we have analyzed iterative techniques for solving the non linear equations that arise from such implicit time stepping schemes for the k dv and the k p equations we derived predictor corrector method that retain the full accuracy of the implicit method with minimal stability restrictions on the size of the time step some numerical examples show the propagation of interacting solitons

behavior of a three torus in truncated navier stokes equations abstract the presence of a three torus in a seven mode truncation of the three dimensional navier stokes equations is investigated numerically by means of cross section and power spectra furthermore by taking advantage of particular features of the model rotation vectors circle maps and torus maps can be computed with high accuracy and used to study the dynamics in particular some interesting phenomena of partial phase locking are described in deep detail the three torus which arises via a hopf bifurcation and persists in a wide parameter range is found to break and originate a strange attractor the onset of chaos and the associated bifurcation point can be defined quite precisely

local well posedness of the ostrovsky stepanyams and tsimring equation in sobolev spaces of negative indices abstract in this paper we prove by using the method of bilinear estimate in the bourgain type spaces that the initial value problem of the ost equation u t u x x x h u x h u x x x u u x 0 x r t 0 where 0 and h denotes the usual hilbert transformation is locally well posed in the sobolev space h s r when s 1

on the low regularity of the benney lin equation abstract we consider the low regularity of the benney lin equation u t u u x u x x x u x x u x x x x u x x x x x 0 we established the global well posedness for the initial value problem of benney lin equation in the sobolev spaces h s r for 0 s 2 improving the well posedness result of biagioni and linares h a biaginoi f linares on the benney lin and kawahara equation j math anal appl 211 1997 131 152 for s 2 we also prove some ill posedness issues

a well balanced flux vector splitting scheme designed for hyperbolic systems of conservation laws with source terms abstract we propose a way to construct robust numerical schemes for the computations of numerical solutions of one and two dimensional hyperbolic systems of balance laws in order to reduce the computational cost we selected the family of flux vector splitting schemes we reformulate the source terms as nonconservative products and treat them directly in the definition of the numerical fluxes by means of generalized jump relations this is applied to a 1d shallow water system with topography and to a 2d simplified model of two phase flows with damping effects numerical results and comparisons with a classical centered discretizations scheme are supplied

an efficient riemann solver for unsteady flows with non ideal gases a riemann solver is presented for the euler equations of gas dynamics with real gases this represents a more efficient version of an algorithm originally presented by the author

well balanced high order extensions of godunov s method for semilinear balance laws this paper is concerned with the development of well balanced high order numerical schemes for systems of balance laws with a linear flux function whose coefficients may be variable first well balanced first order numerical schemes are obtained based on the use of exact solvers of riemann problems that include both the flux and the source terms godunov s methods so obtained are extended to higher order schemes by using a technique of reconstruction of states the main contribution of this paper is to introduce a reconstruction technique that preserves the well balanced property of godunov s methods some numerical experiments are presented to verify in practice the properties of the developed numerical schemes

a new inexact alternating directions method for monotone variational inequalities the alternating directions method adm is an effective method for solving a class of variational inequalities vi when the proximal and penalty parameters in sub vi problems are properly selected in this paper we propose a new adm method which needs to solve two strongly monotone sub vi problems in each iteration approximately and allows the parameters to vary from iteration to iteration the convergence of the proposed adm method is proved under quite mild assumptions and flexible parameter conditions

the relationship between the direct and weak formulations of a linearised riemann solver for systems of conservation laws abstract the direct and weak formulations of a linearised riemann solver attributable to roe and toumi are shown to be equivalent for certain systems of conservation laws and particular parameter vectors it is demonstrated however that in other cases this is not necessarily true

renormalization in a circle map with two inflection points abstract an extended version of the sine circle map becomes non invertible by developing simultaneously two critical points in part of parameter space universality in the corresponding transition from quasiperiodicity to chaos manifests itself in the organisation of doubly superstable orbits four operators are needed to generate all such orbits the existence of non trivial equivalence relations for operation sequences makes the present theory different from other multi operator renormalisation approaches possible implications of the theory in higher dimensional systems are discussed

well posedness of the cauchy problem for ostrovsky stepanyams and tsimring equation with low regularity data abstract in this paper we prove that the initial value problem of the ost equation u t u x x x h u x h u x x x u u x 0 x r t 0 where 0 and h denotes the usual hilbert transformation is locally well posed in the sobolev space h s r when s 3 4 and globally well posed in h s r when s 0

discrete equations for physical and numerical compressible multiphase mixtures we have recently proposed in 21 a compressible two phase unconditionally hyperbolic model able to deal with a wide range of applications interfaces between compressible materials shock waves in condensed multiphase mixtures homogeneous two phase flows bubbly and droplet flows and cavitation in liquids one of the difficulties of the model as always in this type of physical problems was the occurrence of non conservative products in 21 we have proposed a discretisation technique that was without any ambiguity only in the case of material interfaces not in the case of shock waves this model was extended to several space dimensions in 24 in this paper thanks to a deeper analysis of the model we propose a class of schemes that are able to converge to the correct solution even when shock waves interact with volume fraction discontinuities this analysis provides a more accurate estimate of closure terms but also an accurate resolution method for the conservative fluxes as well as non conservative terms even for situations involving discontinuous solutions the accuracy of the model and method is clearly demonstrated on a sequence of difficult test problems

on a perturbation of the benjamin ono equation abstract we prove using the fourier restriction norm method that the initial value problem associated to a perturbation of the benjamin ono equation u t u u x h u x x h u x u x x 0 where x r t 0 0 and h denotes the usual hilbert transform is locally well posed in the sobolev space h s r for any s 1 2 and globally well posed in h s r when s 0 for s 1 we also prove some ill posedness issues

shock capturing for supercritical free surface flows in curvilinear channel geometries abstract a shock capturing scheme is developed for the shallow water equations governing steady supercritical flow in two dimensions in channels with a curvilinear coordinate geometry the scheme incorporates a special averaging procedure of the flow variables as well as details of the transformation that maps a nonrectangular physical domain for a general channel transition

dissipative toda rayleigh lattice and its oscillatory modes a detailed theoretical and experimental analysis of the possible oscillatory regimes of the dissipative toda rayleigh lattice system is provided it is shown that the system has n 1 oscillatory modes with different space time scales and two rotatory modes using its analog electronic circuit implementation we also show with a simple and robust method how switching between modes occurs

multi hump stationary waves for a korteweg de vries equation with nonlocal perturbations abstract periodic and solitary wave solutions are investigated numerically for a perturbed korteweg de vries equation with unstable and dissipation terms in hilbert transform u t uu x u xxx h u x h u xxx 0 a family of solitary wave solutions s 1 s 2 s n whose members are distinguished by the number of humps is numerically identified the tails of these waves decay as o 1 x 2 when x irrespective of the magnitude of it is also found that for a given there exist families of periodic wave solutions p 1 p 2 p n which originate from one near sinusoidal wave and end up in the infinite periodicity to the corresponding solitary waves the numerical results are consistent with the theoretical estimates based on the conservation properties

characterizing bifurcations and classes of motion in the transition to chaos through 3d tori of a continuous experimental system in solid mechanics abstract the dynamics of a continuous experimental system exhibiting quasiperiodic transition to chaos has been investigated an elastic cable mass system hanging at in phase vertically moving supports is considered with vibrational parameters in the neighbourhood of external and internal resonance conditions adjusted to be able to introduce three torus dynamics measurements of the nonregular response are made in frequency ranges including primary resonance of the first symmetric in plane mode of the cable quantitative characterisation of regular and nonregular attractors and of the configuration variables involved in the motion is made by means of delay embedding technique and proper orthogonal decomposition of spatio temporal flow attention is devoted to analyse in depth the scenario of transition to chaos exhibited by the model steady quasiperiodic motions on three tori partial and full phase locking of the flow and chaotic attractors are observed in spite of the dynamics variedness the systematic characterisation of the response allows us to set the bifurcation behaviour of the experimental system in the framework of theoretic numeric transition to chaos through three dimensional tori

a comparison of the different extensions of a weak formulation of an approximate riemann solver for supercritical flows and their relationship to existing schemes abstract a weak formulation of the roe linearised riemann solver proposed recently is examined in the context of the equations governing steady supercritical flows two possible alternatives present themselves and it is shown that these are equivalent and moreover are equivalent to existing schemes

a finite difference scheme for steady supersonic two dimensional compressible flow of real gases a finite difference scheme is presented for the solution of the two dimensional equations of steady supersonic compressible flow of real gases the scheme incorparates numerical characteristic decomposition is shock capturing by design and incorporates space marching as a result of the assumption that the flow is wholly supersonic in at least one space dimension results are shown for problems involving oblique hydraulic jumps and reflection from a wall

a turbulence model with stochastic soliton motion a dissipative benjamin ono equation is used to study fluid and plasma turbulence the system is studied by an exact nonlinear mode truncation method in which a finite number of poles are used to present the solution the justification of the pole expansion approach is discussed with the proof of a completeness theorem the stability and spectrum analysis show that asymptotic behavior of the system is completely represented by a finite number of nonlinear modes the behavior of those nonlinear modes resembles solitons and exhibits a wide range of bifurcation phenomena and routes to turbulence

on extending some primal dual interior point algorithms from linear programming to semidefinite programming this work concerns primal dual interior point methods for semidefinite programming sdp that use a search direction originally proposed by helmberg et al siam j optim 6 1996 pp 342 361 and kojima shindoh and hara siam j optim 7 1997 pp 86 125 and recently rediscovered by monteiro siam j optim 7 1997 pp 663 678 in a more explicit form in analyzing these methods a number of basic equalities and inequalities were developed in kojima shindoh and hara and also in monteiro through different means and in different forms in this paper we give a concise derivation of the key equalities and inequalities for complexity analysis along the exact line used in linear programming lp producing basic relationships that have compact forms almost identical to their counterparts in lp we also introduce a new formulation of the central path and variable metric measures of centrality these results provide convenient tools for deriving polynomiality results for primal dual algorithms extended from lp to sdp using the aforementioned and related search directions we present examples of such extensions including the long step infeasible interior point algorithm of zhang siam j optim 4 1994 pp 208 227

torus maps and the problem of a one dimensional optical resonator with a quasiperiodically moving wall abstract we study the problem of the asymptotic behavior of the electromagnetic field in an optical resonator one of whose walls is at rest and the other is moving quasiperiodically with d 2 incommensurate frequencies we show that this problem can be reduced to a problem about the behavior of the iterates of a map of the d dimensional torus that preserves a foliation by irrational straight lines in particular the jacobian of this map has d 1 eigenvalues equal to 1 we present rigorous and numerical results about several dynamical features of such maps we also show how these dynamical features translate into properties for the field in the cavity in particular we show that when the torus map satisfies a kam theorem which happens for a cantor set of positive measure of parameters the energy of the electromagnetic field remains bounded when the torus map is in a resonant region which happens in open sets of parameters inside the gaps of the previous cantor set the energy grows exponentially

chaotic motion on hamiltonian tori kam tori are well known to be that finite part of phase space in which the motion of a weakly perturbed classical hamiltonian system remains integrable moreover they are barriers in the phase space of systems with two degrees of freedom the authors show that certain hamiltonian systems contain invariant non regular tori with compressible flow using as an example an electron moving in electromagnetic fields which are periodic in space they demonstrate i that strange attractors and repellers of well known autonomous or quasi periodically time driven systems may occur on such strange tori ii that one may find barriers consisting of non regular hamiltonian tori in systems with any number of degrees of freedom the flow on such barriers is non chaotic though and iii that certain kam tori transform into non regular tori rather than breaking up when the perturbation becomes strong

the benjamin ono equation in weighted sobolev spaces abstract in this paper we continue the study of the initial value problem t u x u 2 2 x u u 0 in the spaces y s h s r l 2 r where 0 s 3 2 and l r 2 r denotes the set of all complex valued measurable functions such that f r 2 r 1 x 2 r f x 2 dx our main result is the proof of well posedness of the aforementioned problem in y 2 0 1 the proof involves parabolic regularization the riesz thorin interpolation theorem and kato s theory of linear evolution equations of hyperbolic type

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

strange attractors that are not chaotic abstract it is shown that in certain types of dynamical systems it is possible to have attractors which are strange but not chaotic here we use the word strange to refer to the geometry or shape of the attracting set while the word chaotic refers to the dynamics of orbits on the attractor in particular the exponential divergence of nearby trajectories we first give examples for which it can be demonstrated that there is a strange nonchaotic attractor these examples apply to a class of maps which model nonlinear oscillators continuous time which are externally driven at two incommensurate frequencies it is then shown that such attractore are persistent under perturbations which preserve the original system type i e there are two incommensurate external driving frequencies this suggests that for systems of the typw which we have considered nonchaotic strange attractors may be expected to occur for a finite interval of parameter values on the other hand when small perturbations which do not preserve the system type are numerically introduced the strange nonchaotic attractor is observed to be converted to a periodic or chaotic orbit thus we conjecture that in general continuous time systems flows which are not externally driven at two incommensurate frequencies should not be expected to have strange nonchaotic attractors except possibly on a set of measure zero in the parameter space

chaotic attractors on a 3 torus and torus break up abstract two coupled driven van der pol oscillators can have three frequency quasiperiodic attractors which lie on a 3 torus the evidence presented in this paper indicates that the torus is destroyed when the stable and unstable manifolds of an unstable orbit become tangent furthermore no chaotic orbits lying on a torus were observed suggesting that in most cases at least in the case of this system orbits do not become chaotic before their tori are destroyed to expedite the calculations a method was developed which can be used to determine if an orbit is on a torus without actually displaying that orbit the method also described in this paper was designed specifically for our system the basic idea however could be used for studying attractors of other systems very few modifications of the method if any would be necessary when studying systems with the number of degrees of freedom equal to that of our van der pol system

homoclinic orbits in reversible systems and their applications in mechanics fluids and optics abstract this survey article reviews the theory and application of homoclinic orbits to equilibria in reversible continuous time dynamical systems where the homoclinic orbit and the equilibrium are both reversible the focus is on even order reversible systems in four or more dimensions local theory generic argument and global existence theories are examined for each qualitatively distinct linearisation several recent results such as coalescence caused by non transversality and the reversible orbit flip bifurcation are covered a number of open problems are highlighted applications are reviewed to systems arising from a variety of disciplines with the aid of numerical methods three examples are presented in detail one of which is infinite dimensional

an approximate linearised riemann solver for the three dimensional euler equations for real gases using operator splitting abstract an approximate linearised riemann solver is presented for the solution of the euler equations of gas dynamics in three dimensions with a general equation of state the scheme incorporates operator splitting and is applied to the problem of mach 3 flow past a forward facing step for some specimen equations of state

three coupled oscillators mode locking global bifurcations and toroidal chaos abstract many systems exhibit behaviour which can be described by three coupled oscillators provided the coupling is not too strong such systems can be modelled by maps of the two torus to itself in this paper we describe and explain those aspects of the bifurcation diagram for two parameter families of torus maps that involve change of mode locking type we introduce the concepts of partial and full mode locking for the coupled oscillators these notions correspond to the presence of one or two rational relations between the frequencies respectively numerical investigation of a particular family of torus maps reveals an intricate web of global bifurcations in order to explain the results we first show that we can approximate maps in the neighbourhood of a rational translation to arbitrary order by the time 1 map of a flow on the torus then we analyse those condimension 1 and 2 bifurcations for flows on the torus which change the set of frequency ratios we find a large variety of bifurcation diagrams in particular many involving homotopically non trivial saddle connections next we ask how the picture changes when the time 1 maps of a family of flows are perturbed to a general family of diffeomorphisms we find toroidal chaos in the neighbourhood of all these bifurcations meaning that there exist orbits which perform a pseudo random sequence of rotations in different directions around the torus the corresponding behaviour for the coupled oscillators is that the frequency ratios perform a random walk finally we show how all these ingredients can be put together to give global scenarios for bifurcation for families of torus maps which we believe to be of general applicability to physical systems with three weakly coupled modes of oscillation

temporal evolutions and stationary waves for dissipative benjamin ono equation abstract initial value problems as well as stationary solitary and periodic waves are investigated for dissipative benjamin ono dbo equation multi hump stationary waves and their structures are identified numerically and the stability regions of stationary periodic waves are also examined numerically these results elucidate a close relation between irregular behaviours in the initial value problem and the multiplicity of stationary waves

stationary travelling wave solutions of an unstable kdv burgers equation abstract both solitary and periodic stationary travelling wave solutions are investigated numerically for an unstable korteweg de vries burgers equation u t uu x u xxx u u xx 0 0 a family of stationary solitary wave solutions whose members are distinguished by the number of humps is found for a given corresponding to each solitary wave thus found a family of stationary periodic waves with the same number of humps exists under periodic condition and ends up in the infinite periodicity to the corresponding solitary wave the numerical results are consistent with the theoretical estimates based on the conservation properties

attractors on an n torus quasiperiodicity versus chaos the occurrence of quasiperiodic motions in nonconservative dynamical systems is of great fundamental importance however current understanding concerning the question of how prevalent such motions should be is incomplete with this in mind the types of attractors which can exist for flows on an n torus are studied numerically for n 3 and 4 specifically nonlinear perturbations are applied to maps representing n frequency quasiperiodic attractors these perturbations can cause the original n frequency quasiperiodic attractors to bifurcate to other types of attractors our results show that for small and moderate nonlinearity the frequency of occurrence of quasiperiodic motions is as follows n frequency quasiperiodic attractors are the most common followed by n 1 frequency quasiperiodic attractors followed by periodic attractors however as the nonlinearity is further increased n frequency quasiperiodicity becomes less common ceasing to occur when the map becomes noninvertible chaotic attractors are very rare for n 3 for small to moderate nonlinearity but are somewhat more common for n 4 examination of the types of chaotic attractors that occur for n 3 reveals a rich variety of structure and dynamics in particular we see that there are chaotic attractors which apparently fill the entire n torus i e limit sets of orbits on these attractors are the entire torus furthermore these are the most common types of chaotic attractors at moderate nonlinearities

an improved prediction correction method for monotone variational inequalities with separable operators in this paper we study the variational inequality problems with a particular splitting structure in which the mapping f does not have an explicit form and only its function values can be employed in the numerical methods for solving such problems studies and applications of such problems can be found in fukushima 1992 3 glowinski 1984 4 glowinski 1989 5 and xu 2007 15 the paper he et al 2006 6 presents an efficient prediction correction method for such problems based on the predictor method in the latter this paper presents two classes of correction methods which are more convenient to be carried out than that in this reference numerical experiments show that the new method is effective

the nonlinear dynamics of the damped and driven toda chain ii fourier and lyapunov analysis of tori abstract the dynamics of a periodically damped and driven toda chain with periodic boundary conditions is investigated with the help of spectral energy bifurcation diagrams and lyapunov diagrams it can be shown that quasiperiodic motions on tori tk of dimension k 3 exist the technical problems concerning the fourier and lyapunov analysis of the motions on tori tk k 2 are addressed

on the well posedness for the generalized ostrovsky stepanyams and tsimring equation abstract in this work we prove that the initial value problem associated with equation u t u xxx h u x h u xxx u k u x 0 x r t 0 where 0 and h denotes the usual hilbert transform is locally well posed in sobolev spaces h s r for s 0 and k 1 2 3 and globally well posed in l 2 r

a new comment on the computation of non conservative products using roe type path conservative schemes abstract we are interested in the numerical approximation of the discontinuous solutions of non conservative hyperbolic systems we more precisely consider a non conservative formulation of the usual gas dynamics equations and show how to slightly modify the so called roe type path conservative schemes to properly capture the underlying shock discontinuities numerical evidences are proposed the present note follows a first comment on the computation of non conservative products in 1

discontinuous galerkin finite element methods for hyperbolic nonconservative partial differential equations we present space and space time discontinuous galerkin finite element dgfem formulations for systems containing nonconservative products such as occur in dispersed multiphase flow equations the main criterium we pose on the weak formulation is that if the system of nonconservative partial differential equations can be transformed into conservative form then the formulation must reduce to that for conservative systems standard dgfem formulations cannot be applied to nonconservative systems of partial differential equations we therefore introduce the theory of weak solutions for nonconservative products into the dgfem formulation leading to the new question how to define the path connecting left and right states across a discontinuity the effect of different paths on the numerical solution is investigated and found to be small we also introduce a new numerical flux that is able to deal with nonconservative products our scheme is applied to two different systems of partial differential equations first we consider the shallow water equations where topography leads to nonconservative products in which the known possibly discontinuous topography is formally taken as an unknown in the system second we consider a simplification of a depth averaged two phase flow model which contains more intrinsic nonconservative products

a weak formulation of roe s scheme for two dimensional unsteady compressible flows and steady supersonic flows abstract a weak formulation of roe s approximate riemann solver developed recently for one dimensional unsteady compressible flows is extended to two dimensions using operator splitting and to steady supersonic flows and it is shown that the resulting schemes are analogous to other methods for such flows the technique for steady flows draws on recent work in this area

alternating projection proximal methods for convex programming and variational inequalities we consider a mixed problem composed in part of finding a zero of a maximal monotone operator and in part of solving a monotone variational inequality problem we propose a solution method for this problem that alternates between a proximal step for the maximal monotone operator part and a projection type step for the monotone variational inequality part and analyze its convergence and rate of convergence this method extends a decomposition method of chen and teboulle math programming 64 1994 pp 81 101 for convex programming and yields as a by product new decomposition methods

a variable penalty alternating directions method for convex optimization we study a generalized version of the method of alternating directions as applied to the minimization of the sum of two convex functions subject to linear constraints the method consists of solving consecutively in each iteration two optimization problems which contain in the objective function both lagrangian and proximal terms the minimizers determine the new proximal terms and a simple update of the lagrangian terms follows we prove a convergence theorem which extends existing results by relaxing the assumption of uniqueness of minimizers another novelty is that we allow penalty matrices and these may vary per iteration this can be beneficial in applications since it allows additional tuning of the method to the problem and can lead to faster convergence relative to fixed penalties as an application we derive a decomposition scheme for block angular optimization and present computational results on a class of dual block angular problems

laminarizing effects of dispersion in an active dissipative nonlinear medium abstract the kuramoto sivashinsky equation has become a popular prototype for systems which exhibit spatial temporal chaos we show here that a linear dispersion term h xxx tends to arrest such irregular behavior in favor of spatially periodic cellular structures as is consistent with prior numerical and experimental observations the study includes a normal form analysis and a numerical investigation of periodic and solitary stationary wave solutions of the equation it is shown that by 1 1 the infinite families of stationary periodic waves of the kuramoto sivashinsky equation each ending in a solitary wave have been annihilated successively such that only a lone family of periodic waves consisting of one hump kdv pulses for 3 7 remains as the only periodic stationary wave attractors of the system these periodic waves have much larger domains of attraction than the strange attractors and hence tend to dominate spatial temporal chaos in an extended domain with significant dispersion

global bifurcation destroying the experimental torus t 2 we show experimentally the scenario of a two frequency torus t 2 breakdown in which a global bifurcation occurs due to the collision of a torus with an unstable periodic orbit creating a heteroclinic saddle connection followed by an intermittent behavior

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