spectra of sparse random matrices we compute the spectral density for ensembles of sparse symmetric random matrices using replica our formulation of the replica symmetric ansatz shares the symmetries of that suggested in a seminal paper by rodgers and bray symmetry with respect to permutation of replica and rotation symmetry in the space of replica but uses a different representation in terms of superpositions of gaussians it gives rise to a pair of integral equations which can be solved by a stochastic population dynamics algorithm remarkably our representation allows us to identify pure point contributions to the spectral density related to the existence of normalizable eigenstates our approach is not restricted to matrices defined on graphs with poissonian degree distribution matrices defined on regular random graphs or on scale free graphs are easily handled we also look at matrices with row constraints such as discrete graph laplacians our approach naturally allows us to unfold the total density of states into contributions coming from vertices of different local coordinations and an example of such an unfolding is presented our results are well corroborated by numerical diagonalization studies of large finite random matrices

characteristic polynomials of random matrices at edge singularities we have discussed earlier the correlation functions of the random variables mathrm det ensuremath lambda ensuremath x in which x is a random matrix in particular the moments of the distribution of these random variables are universal functions when measured in the appropriate units of the level spacing when the ensuremath lambda mathrm s instead of belonging to the bulk of the spectrum approach the edge a crossover takes place to an airy or to a bessel problem and we consider here these modified classes of universality furthermore when an external matrix source is added to the probability distribution of x various different phenomenona may occur and one can tune the spectrum of this source matrix to other critical points again there are remarkably simple formulas for arbitrary source matrices which allow us to compute the moments of the characteristic polynomials in these cases as well

index distribution of cauchy random matrices using a coulomb gas technique we compute analytically the probability that a large n n cauchy random matrix has n positive eigenvalues where n is called the index of the ensemble we show that this probability scales for large n as where is the dyson index of the ensemble the rate function c is computed in terms of single integrals that are easily evaluated numerically and amenable to an asymptotic analysis we find that the rate function around its minimum at 1 2 has a quadratic behavior modulated by a logarithmic singularity as a consequence the variance of the index scales for large n as var n cln n where c 2 2 is twice as large as the corresponding prefactor in the gaussian and wishart cases the analytical results are checked by numerical simulations and against an exact finite n formula which for 2 can be derived using orthogonal polynomials

averages of characteristic polynomials in random matrix theory we compute averages of products and ratios of characteristic polynomials associated with orthogonal unitary and symplectic ensembles of random matrix theory the pfaffian determinantal formulae for these averages are obtained and the bulk scaling asymptotic limits are found for ensembles with gaussian weights classical results for the correlation functions of the random matrix ensembles and their bulk scaling limits are deduced from these formulae by a simple computation we employ a discrete approximation method the problem is solved for discrete analogues of random matrix ensembles originating from representation theory and then a limit transition is performed exact pfaffian determinantal formulae for the discrete averages are proven using standard tools of linear algebra no application of orthogonal or skew orthogonal polynomials is needed 2005 wiley periodicals inc

conservative numerical simulation of multi component transport in two dimensional unsteady shallow water flow an explicit finite volume model to simulate two dimensional shallow water flow with multi component transport is presented the governing system of coupled conservation laws demands numerical techniques to avoid unrealistic values of the transported scalars that cannot be avoided by decreasing the size of the time step the presence of non conservative products such as bed slope and friction terms and other source terms like diffusion and reaction can make necessary the reduction of the time step given by the courant number a suitable flux difference redistribution that prevents instability and ensures conservation at all times is used to deal with the non conservative terms and becomes necessary in cases of transient boundaries over dry bed the resulting method belongs to the category of well balanced roe schemes and is able to handle steady cases with flow in motion test cases with exact solution including transient boundaries bed slope friction and reaction terms are used to validate the numerical scheme laboratory experiments are used to validate the techniques when dealing with complex systems as the k e model the results of the proposed numerical schemes are compared with the ones obtained when using uncoupled formulations

on the well balanced numerical discretization of shallow water equations on unstructured meshes we consider in this work a finite volume numerical approximation of weak solutions of the shallow water equations with varying topography on unstructured meshes relying on an alternative formulation of the shallow water equations that involves the free surface as a conservative variable instead of the water height we introduce a simple discretization of the bed slope source term together with some suitable conservative variables reconstructions the resulting scheme is automatically consistent and well balanced for any given consistent numerical flux for the homogeneous system we obtain a very simple formulation which do not need to be modified when second order accuracy muscl reconstructions are adopted additionally the positivity of the water height is preserved under a relevant stability condition as soon as the numerical flux for the associated homogeneous system does numerical assessments involving dry areas and complex geometry are performed

numerical analysis of the rescaling method for parabolic problems with blow up in finite time abstract in this work we study the numerical solution for parabolic equations whose solutions have a common property of blowing up in finite time and the equations are invariant under the following scaling transformation u u x t 2 p 1 u x 2 t for that purpose we apply the rescaling method proposed by berger and kohn 1988 to such problems the convergence of the method is proved under some regularity assumption some numerical experiments are given to derive the blow up profile verifying henceforth the theoretical results

evolution of nonsoliton and quasi classical wavetrains in nonlinear schro dinger and korteweg de vries equations with dissipative perturbations abstract nonlinear schrodinger equations with attraction and with repulsion nse and nse and the korteweg de vries equation perturbed by small dissipative terms are considered near the linear instability threshold where nonlinear and spatially nonuniform dissipations dominate explicit solutions expressing the local amplitude and occupation numbers of a nonsoliton dispersive wavetrain in terms of arbitrary initial data are obtained at sufficiently large times in the logarithmic approximation for nse the particular case of the wavetrain near the soliton birth threshold is also considered the influence of the dissipation on the evolution of a quasi classical strongly nonlinear wavetrain is studied it is demonstrated that for nse the dissipation can be taken into account within the framework of the eikonal approximation while for the nse a perturbation theory based on the inverse scattering transform is relevant

exact solution of the riemann problem for the shallow water equations with discontinuous bottom geometry in this paper we present the exact solution of the riemann problem for the non linear shallow water equations with a step like bottom the solution has been obtained by solving an enlarged system that includes an additional equation for the bottom geometry and then using the principles of conservation of mass and momentum across the step the resulting solution is unique and satisfies the principle of dissipation of energy across the shock wave we provide examples of possible wave patterns numerical solution of a first order dissipative scheme as well as an implementation of our riemann solver in the second order upwind method are compared with the proposed exact riemann problem solution a practical implementation of the proposed exact riemann solver in the framework of a second order upwind tvd method is also illustrated

hole solutions in the 1d complex ginzburg landau equation abstract the cubic complex ginzburg landau equation cgle has a one parameter family of traveling localized source solutions these so called nozaki bekki holes are dynamically stable in some parameter range but always structurally unstable a perturbation of the equation in general leads to a positive or negative monotonic acceleration or an oscillation of the holes this confirms that the cubic cgle has an inner symmetry as a consequence small perturbations change some of the qualitative dynamics of the cubic cgle and enhance or suppress spatio temporal intermittency in some parameter range an analytic stability analysis of holes in the cubic cgle and a semianalytical treatment of the acceleration instability in the perturbed equation is performed by using matching and perturbation methods furthermore we treat the asymptotic hole shock interaction the results which can be obtained fully analytically in the nonlinear schro dinger limit are also used for the quantitative description of modulated solutions made up of periodic arrangements of traveling holes and shocks

finite volume schemes with equilibrium type discretization of source terms for scalar conservation laws we develop here a new class of finite volume schemes on unstructured meshes for scalar conservation laws with stiff source terms the schemes are of equilibrium type hence with uniform bounds on approximate solutions valid in cell entropy inequalities and exact for some equilibrium states convergence is investigated in the framework of kinetic schemes numerical tests show high computational efficiency and a significant advantage over standard cell centered discretization of source terms equilibrium type schemes produce accurate results even on test problems for which the standard approach fails for some numerical tests they exhibit exponential type convergence rate in two of our numerical tests an equilibrium type scheme with 441 nodes on a triangular mesh is more accurate than a standard scheme with 50002 grid points

index statistical properties of sparse random graphs using the replica method we develop an analytical approach to compute the characteristic function for the probability p n k that a large n n adjacency matrix of sparse random graphs has k eigenvalues below a threshold the method allows to determine in principle all moments of p n k from which the typical sample to sample fluctuations can be fully characterized for random graph models with localized eigenvectors we show that the index variance scales linearly with n 1 for 0 with a model dependent prefactor that can be exactly calculated explicit results are discussed for erdos renyi and regular random graphs both exhibiting a prefactor with a nonmonotonic behavior as a function of these results contrast with rotationally invariant random matrices where the index variance scales only as lnn with an universal prefactor that is independent of numerical diagonalization results confirm the exactness of our approach and in addition strongly support the gaussian nature of the index fluctuations

an exner based coupled model for two dimensional transient flow over erodible bed transient flow over erodible bed is solved in this work assuming that the dynamics of the bed load problem is described by two mathematical models the hydrodynamic model assumed to be well formulated by means of the depth averaged shallow water equations and the exner equation the exner equation is written assuming that bed load transport is governed by a power law of the flow velocity and by a flow sediment interaction parameter variable in time and space the complete system is formed by four coupled partial differential equations and a genuinely roe type first order scheme has been used to solve it on triangular unstructured meshes exact solutions have been derived for the particular case of initial value riemann problems with variable bed level and depending on particular forms of the solid discharge formula the model supplied with the corresponding solid transport formulae is tested by comparing with the exact solutions the model is validated against laboratory experimental data of different unsteady problems over erodible bed

asymptotic corrections to the wigner semicircular eigenvalue spectrum of a large real symmetric random matrix using the replica method the replica method has previously been used to calculate the semicircular averaged eigenvalue spectrum of the gaussian orthogonal ensemble of real symmetric n n random matrices in the limit where n to infinity the authors develop a perturbative scheme which within this same replica framework is used to calculate the corrections within this semicircular band of eigenvalues to order 1 n and 1 n2 a new and straightforward self consistency argument is presented and used to derive the shape of the averaged eigenvalue spectrum when n is large but finite and the scaling behaviour of this averaged eigenvalue spectrum near the band edges is demonstrated in a straightforward fashion some comments are made on the relation of the results to those of field theoretical calculations in zero dimensions

exact density of states for finite gaussian random matrix ensembles via supersymmetry we calculate the exact density of states dos for the three classical and two non classical random matrix ensembles for finite matrix size n using supersymmetric integrals the 1 n expansion yields already in lowest order good approximations to the exact result even for small values of n 5 we conjecture a connection between the n dependence of the oscillating part of the dos and the short distance behaviour of the two level correlation function

the local and global existence of solutions for a time fractional complex ginzburg landau equation abstract we study the following time fractional complex nonlinear ginzburg landau equation e i 0 c d t u u e i u p 1 u x r n t 0 u 0 x u 0 x x r n where 0 1 r 2 2 p 1 u 0 l q r n q q c n p 1 2 and q 1 is a complex valued function and d t 0 c u t 0 i t 1 u t x u 0 x where i t 1 0 denotes a left riemann liouville fractional integral of order 1 by defining two operators and establishing some estimates of them we prove the well posedness of the mild solution for this problem in c 0 t l q r n and l 2 r q n r q 0 t l r r n where r satisfies 1 q 1 r 2 n moreover we also obtain the existence of global solutions when u 0 l q c r n is sufficiently small

feedback control of subcritical oscillatory instabilities feedback control of a subcritical oscillatory instability is investigated in the framework of a globally controlled complex ginzburg landau equation that describes the nonlinear dynamics near the instability threshold the control is based on a feedback loop between the system linear growth rate and the maximum of the amplitude of the emerging pattern it is shown that such control can suppress the blow up and result in the formation of spatially localized pulses similar to oscillons in the one dimensional case depending on the values of the linear and nonlinear dispersion coefficients several types of the pulse dynamics are possible in which the computational domain contains i a single stationary pulse ii several coexisting stationary pulses iii competing pulses that appear one after another at random locations so that at each moment of time there is only one pulse in the domain iv spatiotemporally chaotic system of short pulses v spatially synchronized pulses similar dynamic behavior is found also in the two dimensional case the effect of the feedback delay is also studied it is shown that the increase of the delay leads to an oscillatory instability of the pulses and the formation of pulses with oscillating amplitude

the cubic complex ginzburg landau equation for a backward bifurcation abstract the one dimensional complex ginzburg landau equation cgle with a destabilizing cubic nonlinearity and no saturating higher order terms has stable bounded solutions we consider a simple pedagogical model exhibiting qualitatively the mechanism which may suppress the divergence of the solutions then we investigate the functional form of the blow up collapse solutions immediately before the divergence from this analysis we find analytic boundaries for the existence of collapse solutions in the parameter space of the cgle a comparison with numerical simulations demonstrates that for parameters without collapse solutions the solutions of the cgle remain bounded for all times finally we discuss the implications of our results for the solutions of the cgle when saturating higher order terms are included

computer simulation of wave collapses in the nonlinear schro dinger equation abstract the self tuning numerical scheme for simulation of collapse phenomena in the framework of the nonlinear schrodinger equation is developed the scheme allows to reach extremely high field amplitudes 2 0 2 10 15 without lack of validity numerical results are compared with analytical predictions

characteristic polynomials of random matrices number theorists have studied extensively the connections between the distribution of zeros of the riemann function and of some generalizations with the statistics of the eigenvalues of large random matrices it is interesting to compare the average moments of these functions in an interval to their counterpart in random matrices which are the expectation values of the characteristic polynomials of the matrix it turns out that these expectation values are quite interesting for instance the moments of order 2k scale for unitary invariant ensembles as the density of eigenvalues raised to the power k 2 the prefactor turns out to be a universal number i e it is independent of the specific probability distribution an equivalent behaviour and prefactor had been found as a conjecture within number theory the moments of the characteristic determinants of random matrices are computed here as limits at coinciding points of multi point correlators of determinants these correlators are in fact universal in dyson s scaling limit in which the difference between the points goes to zero the size of the matrix goes to infinity and their product remains finite

number statistics for β ensembles of random matrices applications to trapped fermions at zero temperature let mathcal p ensuremath beta v n mathcal i be the probability that a n ifmmode times else texttimes fi n phantom rule 4pt 0ex ensuremath beta ensemble of random matrices with confining potential v x has n mathcal i eigenvalues inside an interval mathcal i a b on the real line we introduce a general formalism based on the coulomb gas technique and the resolvent method to compute analytically mathcal p ensuremath beta v n mathcal i for large n we show that this probability scales for large n as mathcal p ensuremath beta v n mathcal i ensuremath approx exp left ensuremath ensuremath beta n 2 ensuremath psi v n mathcal i n right where ensuremath beta is the dyson index of the ensemble the rate function ensuremath psi v k mathcal i independent of ensuremath beta is computed in terms of single integrals that can be easily evaluated numerically the general formalism is then applied to the classical ensuremath beta gaussian mathcal i ensuremath l l ensuremath beta wishart mathcal i 1 l and ensuremath beta cauchy mathcal i ensuremath l l ensembles expanding the rate function around its minimum we find that generically the number variance var n mathcal i exhibits a nonmonotonic behavior as a function of the size of the interval with a maximum that can be precisely characterized these analytical results corroborated by numerical simulations provide the full counting statistics of many systems where random matrix models apply in particular we present results for the full counting statistics of zero temperature one dimensional spinless fermions in a harmonic trap

blow up profile for the complex ginzburg landau equation abstract we construct a solution to the complex ginzburg landau equation which blows up in finite time t only at one blow up point we also give a sharp description of its blow up profile the proof relies on the reduction of the problem to a finite dimensional one and the use of index theory to conclude two major difficulties arise in the proof the linearized operator around the profile is not self adjoint and it has a second neutral mode in the last section the interpretation of the parameters of the finite dimensional problem in terms of the blow up time and the blow up point gives the stability of the constructed solution with respect to perturbations in the initial data

multiwavelet based grid adaptation with discontinuous galerkin schemes for shallow water equations we provide an adaptive strategy for solving shallow water equations with dynamic grid adaptation including a sparse representation of the bottom topography a challenge in computing approximate solutions to the shallow water equations including wetting and drying is to achieve the positivity of the water height and the well balancing of the approximate solution a key property of our adaptive strategy is that it guarantees that these properties are preserved during the refinement and coarsening steps in the adaptation process the underlying idea of our adaptive strategy is to perform a multiresolution analysis using multiwavelets on a hierarchy of nested grids this provides difference information between successive refinement levels that may become negligibly small in regions where the solution is locally smooth applying hard thresholding the data are highly compressed and local grid adaptation is triggered by the remaining significant coefficients furthermore we use the multiresolution analysis of the underlying data as an additional indicator of whether the limiter has to be applied on a cell or not by this the number of cells where the limiter is applied is reduced without spoiling the accuracy of the solution by means of well known 1d and 2d benchmark problems we verify that multiwavelet based grid adaptation can significantly reduce the computational cost by sparsening the computational grids while retaining accuracy and keeping well balancing and positivity

stability and nonlinear dynamics of solitary waves generated by subcritical oscillatory instability under the action of feedback control we consider the influence of global feedback control which acts on an oscillatory system governed by a subcritical ginzburg landau equation exact solutions corresponding to solitary wave solutions are obtained a generalized variational approach is applied for the simplification of the whole problem and its reduction to a finite dimensional dynamical model the finite dimensional evolution model is used for studying the indirect interaction between solitary waves caused by the global control the stability analysis is held in the framework of the finite dimensional model the boundaries of monotonic and oscillatory instabilities are obtained the basic types of dynamics provided by the finite dimensional model are described and compared with the results of a direct numerical simulation of the original equation

enhancement of the labswe for shallow water flows in the lattice boltzmann method for the shallow water equations labswe the force term involves the first order derivative related to a bed slope which has to be determined using the centred scheme for an accurate solution however such calculation contradicts the spirit of only simple arithmetic calculations required in the lattice boltzmann method this paper shows how to remove this drawback from the labswe by incorporating the bed level into the lattice boltzmann equation in a consistent manner with the lattice boltzmann approach three numerical tests have been presented to verify the proposed method

dynamics of solitary waves generated by subcritical instability under the action of delayed feedback control abstract we consider the influence of a global delayed feedback control which acts on a system governed by a subcritical ginzburg landau equation the method based on a variational principle is applied for the derivation of a low dimensional evolution model in the framework of this model a one pulse solution is found and its linear and nonlinear stability analysis is carried out the existence region for a stable time periodic pulse solution is found between the boundaries in the parameter space corresponding to a hopf bifurcation and a saddle node bifurcation the obtained results are compared with the results of an analytical linear theory and direct numerical simulations of the original problem

replica method for wide correlators in gaussian orthogonal unitary and symplectic random matrix ensembles we calculate connected correlators in gaussian orthogonal unitary and symplectic random matrix ensembles by the replica method in the 1 n expansion we obtain averaged one point green s functions up to the next to leading order wide two level correlators up to the first non trivial order and wide three level correlators up to the first non trivial order by carefully treating fluctuations in saddle point evaluation

energy stable staggered schemes for the shallow water equations abstract we focus on the development and analysis of staggered schemes for the two dimensional non linear shallow water equations with varying bathymetry semi implicit and fully explicit time discretizations are proposed particular attention is paid on non linear stability results principally considered here through discrete energy dissipation arguments to address such an issue specific convective fluxes are employed implying diffusive terms relying on the pressure gradient in addition of providing an explicit control of the discrete energy budget the method is shown to preserve motionless steady states as well as the positivity of the water height these properties are still satisfied in a fully explicit context provided an appropriate discretization of the pressure gradient these stability results make the approach particularly robust and efficient for both coastal flows and low froude number regimes as a result in addition of a great ease of implementation the presented schemes meet the operational requirements attached to the simulation of large and small scale oceanic flows

how many eigenvalues of a gaussian random matrix are positive we study the probability distribution of the index n sub i e the number of positive eigenvalues of an nxn gaussian random matrix we show analytically that for large n and large n sub with the fraction 0 c n sub n 1 of positive eigenvalues fixed the index distribution p n sub cn n approx exp beta n sup 2 phi c where beta is the dyson index characterizing the gaussian ensemble the associated large deviation rate function phi c is computed explicitly for all 0 c 1 it is independent of beta and displays a quadratic form modulated by a logarithmic singularity around c 1 2 as a consequence the distribution of the index has a gaussian form near the peak but with a variance delta n of index fluctuations growing as delta n approx lnn beta pi sup 2 for large n for beta 2 this result is independently confirmed against an exact finite n formula yielding delta n lnn 2 pi sup 2 c o n sup 1 for large n where the constant c for even n has the nontrivial value c gamma 1 3ln2 2 pi sup 2 approx equal 0 185 248 and gamma 0 5772 is the euler constant we also determine for large n the probability that the interval zeta sub 1 zeta sub 2 is free of eigenvalues some of these results have been announced in a recent letter phys rev lett 103 220603 2009

weak solutions for partial differential equations with source terms application to the shallow water equations weak solutions of problems with m equations with source terms are proposed using an augmented riemann solver defined by m 1 states instead of increasing the number of involved equations these weak solutions use propagating jump discontinuities connecting the m 1 states to approximate the riemann solution the average of the propagated waves in the computational cell leads to a reinterpretation of the roe s approach and in the upwind treatment of the source term of vazquez cendon it is derived that the numerical scheme can not be formulated evaluating the physical flux function at the position of the initial discontinuities as usually done in the homogeneous case positivity requirements over the values of the intermediate states are the only way to control the global stability of the method also it is shown that the definition of well balanced equilibrium in trivial cases is not sufficient to provide correct results it is necessary to provide discrete evaluations of the source term that ensure energy dissipating solutions when demanded the one and two dimensional shallow water equations with source terms due to the bottom topography and friction are presented as case study the stability region is shown to differ from the one defined for the case without source terms and it can be derived that the appearance of negative values of the thickness of the water layer in the proximity of the wet dry front is a particular case of the wet wet fronts the consequence is a severe reduction in the magnitude of the allowable time step size if compared with the one obtained for the homogeneous case starting from this result 1d and 2d numerical schemes are developed for both quadrilateral and triangular grids enforcing conservation and positivity over the solution allowing computationally efficient simulations by means of a reconstruction technique for the inner states of the weak solution that allows a recovery of the time step size

construction of a blow up solution for the complex ginzburg landau equation in a critical case we construct a solution for the complex ginzburg landau equation in some critical case which blows up in finite time t only at one blow up point we also give a sharp description of its profile the proof relies on the reduction of the problem to a finite dimensional one and the use of index theory to conclude the interpretation of the parameters of the finite dimension problem in terms of the blow up point and time allows to prove the stability of the constructed solution

the surface gradient method for the treatment of source terms in the shallow water equations a novel scheme has been developed for data reconstruction within a godunov type method for solving the shallow water equations with source terms in contrast to conventional data reconstruction methods based on conservative variables the water surface level is chosen as the basis for data reconstruction this provides accurate values of the conservative variables at cell interfaces so that the fluxes can be accurately calculated with a riemann solver the main advantages are 1 a simple centered discretization is used for the source terms 2 the scheme is no more complicated than the conventional method for the homogeneous terms 3 small perturbations in the water surface elevation can be accurately predicted and 4 the method is generally suitable for both steady and unsteady shallow water problems the accuracy of the scheme has been verified by recourse to both steady and unsteady flow problems excellent agreement has been obtained between the numerical predictions and analytical solutions the results indicate that the new scheme is accurate simple efficient and robust

characterization of dispersive chaos and related states of binary fluid convection abstract we describe an extensive series of experiments on traveling wave convection in an ethanol water mixture with separation ratio 0 020 in a long narrow annular container as the stress parameter is quasi statically increased through the range 0 2 above onset this system evolves through a series of complex spatiotemporal states before settling into an extended pattern of steady spatially uniform convective rolls much of the dynamical behavior is influenced by strong nonlinear dispersion which causes the repeated and often irregular appearance and sudden collapse of spatially localized bursts of traveling waves small amplitude states seen very close to onset are analyzed quantitatively using a complex ginzburg landau equation complex dynamical states encountered further above onset are analyzed using a number of classical statistical techniques focusing mainly on correlation functions and fourier spectra in addition we use the technique of biorthogonal decomposition in an effort to extract the basic structures of the patterns and to characterize the complexty of their dynamics

wigner dyson statistics from the replica method we compute the correlation functions of the eigenvalues in the gaussian unitary ensemble using the fermionic replica method we show that non trivial saddle points which break replica symmetry must be included in the calculation in order to correctly reproduce the exact asymptotic behaviour of the correlation functions at a large distance

finite size corrections to the spectrum of regular random graphs an analytical solution we develop a thorough analytical study of the o 1 n correction to the spectrum of regular random graphs with n nodes the finite size fluctuations of the resolvent are given in terms of a weighted series over the contributions coming from loops of all possible lengths from which we obtain the isolated eigenvalue as well as an analytical expression for the o 1 n correction to the continuous part of the spectrum the comparison between this analytical formula and direct diagonalization results exhibits an excellent agreement confirming the correctness of our expression

fronts pulses sources and sinks in generalized complex ginzberg landau equations an important clement in the long time dynamics of pattern forming systems is a class of solutions we will call coherent structures these are states that are either themselves localized or that consist of domains of regular patterns connected by localized defects or interfaces this paper summarizes and extends recent work on such coherent structures in the one dimensional complex ginzburg landau equation and its generalizations for which rather complete information can be obtained on the existence and competition of fronts pulses sources and sinks for the special subclass of uniformly translating structures the solutions are derived from a set of ordinary differential equations that can be interpreted as a flow in a three dimensional phase space fixed points of the flow correspond to the two basic building blocks of coherent structures uniform amplitude states and evanescent waves whose amplitude decreases smoothly to zero a study of the stability of the fixed points under the flow leads to results on the existence and multiplicity of the different coherent structures the dynamical analysis of the original partial differential equation focusses on the competition between pulses and fronts and is expressed in terms of a set of conjectures for front propagation that generalize the marginal stability and pinch point approaches of earlier authors these rules together with an exact front solution whose dynamics plays an important role in the selection of patterns yield an analytic expression for the upper limit of the range of existence of pulse solutions as well as a determination of the regions of parameter space where uniformly translating fron t solutions can exist extensive numerical simulations show consistency with these rules and conjectures for the existence of fronts and pulses in the parameter ranges where no uniformly translating fronts can exist examples are shown of irregularly spreading fronts that generate strongly chaotic regions as well as nonuniformly translating fronts that lead to uniform amplitude states recent perturbative treatments based on expansions about the nonlinear schr6dinger equation are generalized to perturbations of the cubic quintic and derivative schr idinger equations for which both pulses and fronts exist in the unperturbed system comparison of the results with the exact solutions shows that the perturbation theory only yields a subset of the relevant solutions nevertheless those that are obtained are found to be consistent with the general conjectures and in particular they provide an analytic demonstration of front pulse competition while the discussion of the competition between fronts and pulses focusses on the complex ginzburg landau equation with quintic terms and a subcritical bifurcation a number of results are also presented for the cubic equation in particular the existence of a family of moving source solutions derived by bekki and nozaki for this equation contradicts the naive counting arguments we attribute this contradiction to a hidden symmetry of the solution but have not been able to show explicitly how this symmetry affects the phase space orbits

ader schemes for the shallow water equations in channel with irregular bottom elevation this paper deals with the construction of high order ader numerical schemes for solving the one dimensional shallow water equations with variable bed elevation the non linear version of the schemes is based on eno reconstructions the governing equations are expressed in terms of total water height instead of total water depth and discharge the eno polynomial interpolation procedure is also applied to represent the variable bottom elevation ader schemes of up to fifth order of accuracy in space and time for the advection and source terms are implemented and systematically assessed with particular attention to their convergence rates non oscillatory results are obtained for discontinuous solutions both for the steady and unsteady cases the resulting schemes can be applied to solve realistic problems characterized by non uniform bottom geometries

an approximate riemann solver for shallow water equations and heat advection in horizontal centrifugal casting an approximate riemann solver was developed for solving modified shallow water equations swe and energy transport describing the average flow dynamics of a single layer spreading inside a horizontally rotating cylinder the numerical model was particularly developed for simulating the horizontal centrifugal casting hsc of the outer shell of a work roll the swe were derived in the rotating frame of reference therefore fictitious forces the centrifugal force and the coriolis force were considered in addition other forces such as the bed shear force the force of gravity the wind shear force and forces arising from the variable liquid solid interface were taken into account the jacobian matrix of the nonlinear hyperbolic system of pdes was decomposed into a set of eigenvalues and corresponding eigenvectors using standard and corrected roe averages a harten hyman entropy fix was used to prevent expansion shocks entropy violating solutions typically occurring during transonic rarefactions source terms were applied as a stationary discontinuity and were physically bounded and well balanced for steady states producing non oscillatory solutions each wave was upwinded using the explicit godunov s method the high resolution corrections with flux limiters were used to achieve second order of accuracy and dispersion free solutions at discontinuities in addition to the riemann solver a central scheme fv model was used to solve the heat diffusion inside the cylinder mold and partially solidified liquid layer coupled with the solidification model several simulations were performed results were analyzed and discussed

a conservative high order discontinuous galerkin method for the shallow water equations with arbitrary topography a conservative high order godunov type scheme is presented for solving the balance laws of the 1d shallow water equations swe the scheme adopts a finite element runge kutta rk discontinuous galerkin dg framework based on an overall third order accurate formulation the model is referred to as rkdg3 treatment of topographic source term is built in the dg approximation simplified formulae for initializing bed data at a discrete level are derived by assuming a local linear bed function to ease practical flow simulations owing to the adverse effects caused by using an uncontrolled global limiting process in an rkdg3 scheme rkdg3 gl a new conservative rkdg3 scheme with user parameter free local limiting method rkdg3 ll is designed to gain better accuracy and conservativeness the advantages of the new rkdg3 ll model are demonstrated by applying to several steady and transient benchmark flow tests with irregular either differentiable or non differentiable topography copyright 2010 john wiley sons ltd

replica symmetric approach to the typical eigenvalue fluctuations of gaussian random matrices we discuss an approach to compute the first and second moments of the number of eigenvalues i n that lie in an arbitrary interval of the real line for gaussian random matrices the method combines the standard replica symmetric theory with a perturbative expansion of the saddle point action up to leading to the correct logarithmic scaling of the variance as well as to an analytical expression for the correction to the average standard results for the number variance at the local scaling regime are recovered in the limit of a vanishing interval the limitations of the replica symmetric method are unveiled by comparing our results with those derived through exact methods the present work represents an important step to study the fluctuations of i n in non invariant random matrix ensembles where the joint distribution of eigenvalues is not known