nonlinear scattering theory at low energy abstract we study the scattering theory of a conservative nonlinear one parameter group of operators on a hilbert space x relative to a group of linear unitary operators under certain hypotheses the scattering operator carries a neighborhood of 0 in x into x the theory is designed to apply to the semilinear schrodinger and klein gordon equations

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

a fitting algorithm for markov modulated poisson processes having two arrival rates abstract point processes with fluctuating arrival rates arise in many current applications of queueing theory notably in communications modeling markov modulated poisson processes mmpp have been used to describe such processes because they incorporate a mechanism to account for the temporal inhomogeneity of the arrival rates yet yield analytically tractable queueing results an iterative statistical procedure is developed for fitting mmpp having two arrival rates to observational data the procedure is largely numerical experimental and is motivated by maximum likelihood estimation

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

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

stability and asymptotic stability in the energy space of the sum of n solitons for subcritical gkdv equations we prove in this paper the stability and asymptotic stability in h1 of a decoupled sum of n solitons for the subcritical generalized kdv equations the proof of the stability result is based on energy arguments and monotonicity of the local l2 norm note that the result is new even for p 2 the kdv equation the asymptotic stability result then follows directly from a rigidity theorem in 16

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

interaction equations for short and long dispersive waves abstract we show the time local well posedness for a system of nonlinear dispersive equations for the water wave interaction formula it is shown that for any initial data u0 v0 hs r hs 1 2 r s 0 the solution for the above equation uniquely exists in a subset ofc t t hs c t t hs 1 2 and depends continuously on the data by virtue of a special structure of the nonlinear coupling the solution is stable under a singular limiting process

dilation method and smoothing effects of solutions to the benjamin ono equation in this paper we study smoothing effects of solutions to the benjamin ono equation where h is the hilbcrt transform defined by we prove that if h 2 and x x 4 then the solution u of bo belongs to where

a strong type 2 2 estimate for a maximal operator associated to the schrodinger equation soit t f x sup t 0 t t f x ou t t f 1 exp it 2 f 1 4 on montre que etant donne un intervalle fini i i t f 2 dx c i r f x 2 dx

well posedness and existence of bound states for a coupled schrodinger gkdv system abstract we derive some new results concerning the cauchy problem and the existence of bound states for a class of coupled nonlinear schrodinger gkdv systems in particular we obtain the existence of strong global solutions for initial data in the energy space h 1 r h 1 r generalizing previous results obtained in tsutsumi 1993 11 corcho and linares 2007 13 and dias et al submitted for publication 14 for the nonlinear schrodinger kdv system

local smoothing properties of dispersive equations is it possible for time evolution partial differential equations which are reversible and conservative to smooth locally the initial data for the linear wave equation for instance the answer is no however in 10 t kato found a local smoothing property of the korteweg de vries equation the solution of the initial value problem is locally one derivative smoother than the initial datum kato s proof uses in a curcial way the algebraic properties of the symbol for the korteweg de vries equation and the fact that the underlying spatial dimension is one actually judging from the way several integrations by parts and cancellations conspire to reveal a smoothing effect one would be inclined to believe this was a special property of the k dv equation this is not however the case in this paper we attempt to describe a general local smoothing effect for dispersive equations and systems the smoothing effect is due to the dispersive nature of the linear part of the equation all the physically significant dispersive equations and systems known to us have linear parts displaying this local smoothing property to mention only a few the k dv benjamin ono intermediate long wave various boussinesq and schrodinger equations are included we study thus equations and systems of the form

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

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

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

discrete time expectation maximization algorithms for markov modulated poisson processes in this paper we consider parameter estimation markov modulated poisson processes via robust filtering and smoothing techniques using the expectation maximization algorithm framework our filters and smoothers can be applied to estimate the parameters of our model in either an online configuration or an offline configuration further our estimator dynamics do not involve stochastic integrals and our new formulas in terms of time integrals are easily discretized and are written in numerically stable forms in w p malcolm r j elliott and j van der hoek ldquoon the numerical stability of time discretized state estimation via clark transformations rdquo presented at the ieee conf decision control mauii hi dec 2003

universal classification for hidden markov models binary hypotheses testing using empirically observed statistics is studied in the neyman pearson formulation for the hidden markov model hmm an asymptotically optimal decision rule is proposed and compared to the generalized likelihood ratio test glrt which has been shown in earlier studies to be asymptotically optimal for simpler parametric families the proof of the main theorem is provided the result can be applied to several types of hmms commonly used in speech recognition and communication applications several applications are demonstrated

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

why saturated probability spaces are necessary abstract an atomless probability space a p is said to have the saturation property for a probability measure on a product of polish spaces x y if for every random element f of x whose law is marg x there is a random element g of y such that the law of f g is a p is said to be saturated if it has the saturation property for every such we show each of a number of desirable properties holds for every saturated probability space and fails for every non saturated probability space these include distributional properties of correspondences such as convexity closedness compactness and preservation of upper semi continuity and the existence of pure strategy equilibria in games with many players we also show that any probability space which has the saturation property for just one good enough measure or which satisfies just one good enough instance of the desirable properties must already be saturated our underlying themes are 1 there are many desirable properties that hold for all saturated probability spaces but fail everywhere else 2 any probability space that out performs the lebesgue unit interval in almost any way at all is already saturated

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

smoothing estimates for non dispersive equations this paper describes an approach to global smoothing problems for non dispersive equations based on ideas of comparison principle and canonical transformation established in authors previous paper ruzhansky and sugimoto proc lond math soc 105 393 423 2012 where dispersive equations were treated for operators a d x of order m satisfying the dispersiveness condition nabla a xi ne 0 for xi not 0 the global smoothing estimate begin aligned left left langle x right rangle s d x m 1 2 e ita d x varphi x right l 2 left mathbb r t times mathbb r n x right le c left varphi right l 2 left mathbb r n x right quad s 1 2 end aligned is well known while it is also known to fail for non dispersive operators for the case when the dispersiveness breaks we suggest the estimate in the form begin aligned left left langle x right rangle s nabla a d x 1 2 e it a d x varphi x right l 2 left mathbb r t times mathbb r n x right le c left varphi right l 2 left mathbb r n x right quad s 1 2 end aligned which is equivalent to the usual estimate in the dispersive case and is also invariant under canonical transformations for the operator a d x we show that this estimate and its variants do continue to hold for a variety of non dispersive operators a d x where nabla a xi may become zero on some set moreover other types of such estimates and the case of time dependent equations are also discussed

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

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

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

recursive estimation in mixture models with markov regime a recursive algorithm is proposed for estimation of parameters in mixture models where the observations are governed by a hidden markov chain the often badly conditioned information matrix is estimated and its inverse is incorporated into the algorithm the performance of the algorithm is studied by simulations of a symmetric normal mixture the algorithm seems to be stable and produce approximately normally distributed estimates provided the adaptive matrix is kept well conditioned some numerical examples are included

uniqueness properties of solutions to the benjamin ono equation and related models abstract we prove that if u 1 u 2 are real solutions of the benjamin ono equation defined in x t r 0 t which agree in an open set r 0 t then u 1 u 2 we extend this uniqueness result to a general class of equations of benjamin ono type in both the initial value problem and the initial periodic boundary value problem this class of 1 dimensional non local models includes the intermediate long wave equation we relate our uniqueness results with those for a water wave problem finally we present a slightly stronger version of our uniqueness results for the benjamin ono equation

description of the inelastic collision of two solitary waves for the bbm equation we prove that the collision of two solitary waves of the bbm equation is inelastic but almost elastic in the case where one solitary wave is small in the energy space we show precise estimates of the nonzero residue due to the collision moreover we give a precise description of the collision phenomenon change of size of the solitary waves and shifts in their trajectories to prove these results we extend the method introduced in martel and merle description of two soliton collision for the quartic gkdv equation submitted preprint http arxiv org abs 0709 2672 commun math phys 286 39 79 2009 for the generalized kdv equation in particular in the quartic case the main argument is the construction of an explicit approximate solution in the collision region

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

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

smoothing and dispersive estimates for 1d schrodinger equations with bv coefficients and applications we prove smoothing estimates for schrodinger equations i t x a x x 0 with a x bv real and bounded from below we then bootstrap these estimates to obtain optimal strichartz and maximal function estimates all of which turn out to be identical to the constant coefficient case we also provide counterexamples showing a bv to be in a sense a minimal requirement finally we provide an application to sharp well posedness for a generalized benjamin ono equation

hidden markov processes an overview of statistical and information theoretic aspects of hidden markov processes hmps is presented an hmp is a discrete time finite state homogeneous markov chain observed through a discrete time memoryless invariant channel in recent years the work of baum and petrie 1966 on finite state finite alphabet hmps was expanded to hmps with finite as well as continuous state spaces and a general alphabet in particular statistical properties and ergodic theorems for relative entropy densities of hmps were developed consistency and asymptotic normality of the maximum likelihood ml parameter estimator were proved under some mild conditions similar results were established for switching autoregressive processes these processes generalize hmps new algorithms were developed for estimating the state parameter and order of an hmp for universal coding and classification of hmps and for universal decoding of hidden markov channels these and other related topics are reviewed

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

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

purely nonlinear instability of standing waves with minimal energy we consider hamiltonian systems with u 1 symmetry we prove that in the generic situation the standing wave that has the minimal energy among all other standing waves is unstable in spite of the absence of linear instability essentially the instability is caused by higher algebraic degeneracy of the zero eigenvalue in the spectrum of the linearized system we apply our theory to the non

dispersion of small amplitude solutions of the generalized korteweg de vries equation abstract we study the long time behavior of small solutions of the initial value problem for the generalized korteweg de vries equation t u x 3 u x f u 0 gkdv u x 0 g x for the case where f w w s with s 1 4 23 57 3 8625 our results imply that if g l11 g l22 is sufficiently small then sup r 1 t 1 3 u t l in particular the solution tends to zero in the supremum norm the proofs make use of duhamel s formula and dispersion estimates for the linear propagator as well as chain and leibniz rules for fractional derivatives of compositions d f u lp and products d fg lp 0

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

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

the cauchy problem and stability of solitary wave solutions for rlw kp type equations abstract the kadomtsev petviashvilli kp equation ut ux uux uxxx x euyy 0 arises in various contexts where nonlinear dispersive waves propagate principally along the x axis but with weak dispersive effects being felt in the direction parallel to the y axis perpendicular to the main direction of propagation we propose and analyze here a class of evolution equations of the form ut ux upux lut x euyy 0 which provides an alternative to eq in the same way the regularized long wave equation is related to the classical korteweg de vries kdv equation the operator l is a pseudo differential operator in the x variable p 1 is an integer and e 1 after discussing the underlying motivation for the class a local well posedness theory for the initial value problem is developed with assumptions on l and p that include conditions appertaining to models of interesting physical phenomenon the solutions defined locally in time t are shown to be smoothly extendable to the entire time axis in the particularly interesting case where l x2 and e 1 possesses travelling wave solutions u x y t c x ct y provided c 1 and 0 0 4 3 and c 1 and for 4 3 if c 4p 4 p the paper concludes with commentary on extensions of the present theory to more than two space dimensions

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

on the generalized benjamin ono equation we study well posedness of the initial value problem for the generalized benjamin ono equation tu ukdxu 0 k e in sobolev spaces hs r for small data and higher nonlinearities k 2 new local and global including scattering results are established our method of proof is quite general it combines several estimates concerning the associated linear problem with the contraction principle hence it applies to other dispersive models in particular it allows us to extend the results for the generalized benjamin ono to nonlinear schrodinger equations or systems of the form atu i2u p u oxu ui axi 0

smoothing properties and existence of solutions for the generalized benjamin ono equation abstract we study the cauchy problem for the generalized benjamin ono equation tu d d 2 u dv u where d d dx u 0 and v e 1 r r with v 0 v 0 0 using commutator identities and estimates we prove that the equation exhibits smoothing properties similar to those of the generalized korteweg de vries equation gkdv which is the special case 1 of if v satisfies suitable estimates at zero and at infinity we prove that the cauchy problem for the equation with initial data u 0 u0 has a solution u l r l2 l2loc r h loc if u0 l2 and has a solution u l r h l2loc r h2 loc if u0 h finite energy solution in addition we prove that the usual benjamin ono equation 1 2 v u u 2 has a solution u l r h1 l2loc r h 3 2 loc for u0 h1 those results are easily extended to related equations such as the intermediate long wave equation or the smith equation

smoothing properties of evolution equations via canonical transforms and comparison principle this paper describes a new approach to global smoothing problems for dispersive and non dispersive evolution equations based on the global canonical transforms and the underlying global microlocal analysis for this purpose the egorov type theorem is established with canonical transformations in the form of a class of fourier integral operators and their weighted l 2 boundedness properties are derived this allows us to globally reduce general dispersive equations to normal forms in one or two dimensions then a new comparison principle for evolution equations is introduced in particular it allows us to relate different smoothing estimates by comparing certain expressions involving their symbols as a result it is shown that the majority of smoothing estimates for different equations are equivalent to each other moreover new estimates as well as several refinements of known results are obtained the proofs are considerably simplified a comprehensive analysis is presented for smoothing estimates for dispersive equations applications are given to the detailed description of smoothing properties of the schrodinger relativistic schrodinger wave klein gordon and other equations

well posedness in energy space for the periodic modified benjamin ono equation abstract we prove that the periodic modified benjamin ono equation is locally well posed in the energy space h 1 2 this ensures the global well posedness in the defocusing case the proof is based on an x s b analysis of the system after a gauge transform

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

a convexity result for the range of vector measures with applications to large economies on a boolean algebra we consider the topology u induced by a finitely additive measure mu with values in a locally convex space and formulate a condition on u that is sufficient to guarantee the convexity and weak compactness of the range of mu this result a la lyapunov extends those obtained in khan sagara 2013 to the finitely additive setting through a more direct and less involved proof we will then give an economical interpretation of the topology u in the framework of coalitional large economies to tackle the problem of measuring the bargaining power of coalitions when the commodity space is infinite dimensional and locally convex we will show that our condition on u plays the role of the many more agents than commodities condition introduced by rustichini and yannelis in 1991 as a consequence of the convexity theorem we will obtain two straight generalizations of schmeidler s and vind s theorems on the veto power of coalitions of arbitrary economic weight

large time asymptotics of solutions to the generalized benjamin ono equation we study the asymptotic behavior for large time of solutions to the cauchy problem for the generalized benjamin ono bo equation ut u 1u x huxx 0 where h is the hilbert transform x t r when the initial data are small enough if the power of the nonlinearity is greater than 3 then the solution of the cauchy problem has a quasilinear asymptotic behavior for large time in the case 3 critical for the asymptotic behavior i e for the modified benjamin ono equation we prove that the solutions have the same l time decay as in the corresponding linear bo equation also we find the asymptotics for large time of the solutions of the cauchy problem for the bo equation in the critical and noncritical cases for the critical case we prove the existence of modified scattering states if the initial function is sufficiently small in certain weighted sobolev spaces these modified scattering states differ from the free scattering states by a rapidly oscillating factor

a comment about the paper on the instability of elliptic traveling wave solutions of the modified camassa holm equation abstract we discuss the recent paper by a daros and l k arruda 2019 6 our intention is to correct some imperfections left by the authors and present the orbital stability of periodic snoidal waves in the energy space

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

on the propagation of regularity for solutions of the fractional korteweg de vries equation abstract we consider the initial value problem ivp for the fractional korteweg de vries equation fkdv 0 1 t u d x x u u x u 0 x t r 0 1 u x 0 u 0 x it has been shown that the solutions to certain dispersive equations satisfy the propagation of regularity phenomena more precisely it deals in determine whether regularity of the initial data on the right hand side of the real line is propagated to the left hand side by the flow solution this property was found originally in solutions of korteweg de vries kdv equation and it has been also verified in other dispersive equations as the benjamin ono bo equation recently it has been shown that the solutions of the dispersive generalized benjamin ono dgbo equation this is 2 3 in 0 1 also satisfy the propagation of regularity phenomena this is achieved by introducing a commutator decomposition to handle the dispersive part in the equation following the approach used in the dgbo we prove that the solutions of the fkdv also satisfies the propagation of regularity phenomena consequently this type of regularity travels with infinite speed to its left as time evolves

recursive estimation of parameters in markov modulated poisson processes a hidden markov regime is a markov process that governs the time or space dependent distributions of an observed stochastic process recursive algorithms can be used to estimate parameters in mixed distributions governed by a markov regime the authors derive a recursive algorithm for estimation of parameters in a markov modulated poisson process also called a cox point process by this the authors mean a doubly stochastic poisson process with a time dependent intensity that can take on a finite number of different values the intensity switches randomly between the possible values according to a markov process the authors consider two different ways to observe the markov modulated poisson process in the first model the observations consist of the observed time intervals between events and in the second model they use the total number of events in successive intervals of fixed length they derive an algorithm for recursive estimation of the poisson intensities and the switch intensities between the two states and illustrate the algorithm in a simulation study the estimates of the switch intensities are based on the observed conditional switch probabilities

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

on the propagation of regularities in solutions of the benjamin ono equation abstract we shall deduce some special regularity properties of solutions to the ivp associated to the benjamin ono equation mainly for datum u 0 h 3 2 r whose restriction belongs to h m b for some m z m 2 and b r we shall prove that the restriction of the corresponding solution u t belongs to h m for any r and any t 0 therefore this type of regularity of the datum travels with infinite speed to its left as time evolves

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

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

some new function spaces and their applications to harmonic analysis abstract in this paper a family of spaces is introduced which seems well adapted for the study of a variety of questions related to harmonic analysis and its applications these spaces are the tent spaces they provide the natural setting for the study of such things as maximal functions the relevant space here is t p and also square functions where the space t2p is relevant as such these spaces lead to unifications and simplifications of some basic techniques in harmonic analysis thus they are closely related to lp and hardy spaces important parts of whose theory become corollaries of the description of tent spaces also as proc conf harmonic analysis cortona lect notes in math vol 992 springer verlag berlin new york 1983 already indicated where these spaces first appeared explicitly the tent spaces can be used to simplify some of the results related to the cauchy integral on lipschitz curves and multilinear analysis in retrospect one can recognize that various ideas important for tent spaces had been used if only implicitly for quite some time here one should mention carleson s inequality its simplifications and extensions the theory of hardy spaces and atomic decompositions

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

well posedness of the initial value problem for the korteweg de vries equation 1 1 itu axu u1xu o x t e r u x 0 uo x the kdv equation which was first derived as a model for unidirectional propagation of nonlinear dispersive long waves 21 has been considered in different contexts namely in its relation with the inverse scattering method in plasma physics and in algebraic geometry see 24 and references therein our purpose is to study local and global well posedness of the ivp 1 1 in classical sobolev spaces hs r we shall say that the ivp 1 1 is locally resp globally well posed in the function space x if it induces a dynamical system on x by generating a continuous local resp global flow it was established in the works of bona and smith 3 bona and scott 2 saut and temam 30 and kato 1 5 that the ivp 1 1 is locally resp globally well posed in hs with s 3 2 resp s 2 roughly speaking global well posedness in hs depends on the available local theory and on the conservation laws satisfied by solutions of 1 1 namely

construction of a minimal mass blow up solution of the modified benjamin ono equation we construct a minimal mass blow up solution of the modified benjamin ono equation mbo begin aligned u t u 3 d 1 u x 0 end aligned mbo which is a standard mass critical dispersive model let q in h frac 1 2 q 0 be the unique ground state solution of d 1 q q q 3 constructed using variational arguments by weinstein commun part differ equations 12 1133 1173 1987a j differ equations 69 192 203 1987b and albert et al proc r soc lond a 453 1233 1260 1997 and whose uniqueness was recently proved by frank and lenzmann acta math 210 261 318 2013 we show the existence of a solution s of mbo satisfying vert s vert l 2 vert q vert l 2 and begin aligned s t frac 1 lambda frac 1 2 t q left frac cdot x t lambda t right rightarrow 0 quad text in h frac 1 2 mathbb r text as t downarrow 0 end aligned where begin aligned lambda t sim t quad x t sim ln t quad hbox and quad vert s t vert dot h frac 1 2 sim t frac 1 2 vert q vert dot h frac 1 2 quad hbox as t downarrow 0 end aligned this existence result is analogous to the one obtained by martel et al j eur math soc 17 1855 1925 2015 for the mass critical generalized korteweg de vries equation however in contrast with the gkdv equation for which the blow up problem is now well understood in a neighborhood of the ground state s is the first example of blow up solution for mbo the proof involves the construction of a blow up profile energy estimates as well as refined localization arguments developed in the context of benjamin ono type equations by kenig et al ann inst h poincare anal non lin 28 853 887 2011 due to the lack of information on the mbo flow around the ground state the energy estimates have to be considerably sharpened in the present paper

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

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

burgers poisson a nonlinear dispersive model equation a dispersive model equation is considered which has been proposed by whitham linear and nonlinear waves john wiley sons new york 1974 as a shallow water model and which can also be seen as a caricature of two species euler poisson problems a number of formal properties as well as similarities to other dispersive equations are derived a travelling wave analysis and some numerical tests are carried out the equation features wave breaking in finite time a local existence result for smooth solutions and a global existence result for weak entropy solutions are proved finally a small dispersion limit is carried out for situations where the solution of the limiting equation is smooth

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