on the existence of periodic solutions to p laplacian rayleigh differential equation with a delay abstract in this paper the authors study the existence of periodic solutions to a p laplacian rayleigh differential equation with a delay as follows p y t f y t g y t t e t where p 1 is a constant p r r p u u p 2 u f g e c r r t t t with t 0 t 0 t and e t t e t t 0 is a constant by using mawhin s continuation theorem some new results are obtained

new singular solutions of the nonlinear schrodinger equation we present numerical simulations of a new type of singular solutions of the critical nonlinear schrodinger equation nls that collapse with a quasi self similar ring profile at a square root blowup rate we find and analyze the equation of the ring profile we observe that the self similar ring profile is an attractor for a large class of radially symmetric initial conditions but is unstable under symmetry breaking perturbations the equation for the ring profile admits also multi ring solutions that give rise to collapsing self similar multi ring solutions but these solutions are unstable even in the radially symmetric case and eventually collapse with a single ring profile collapsing ring solutions are also observed in the supercritical nls

existence and uniqueness of periodic solutions for rayleigh type p laplacian equation in this paper we use the coincidence degree theory to establish new results on the existence and uniqueness of t periodic solutions for the rayleigh type p laplacian equation of the form f p x t f t x t g t x t e t

existence and uniqueness of periodic solutions for a kind of lienard equation with two deviating arguments in this paper a kind of lienard equation with two deviating arguments of the formx t f x t x t g 1 t x email protected 1 t g 2 t x email protected 2 t p t is considered we derive some new sufficient conditions for checking uniqueness of t periodic solutions of this equation by using coincidence degree theory we also establish some new results on the existence and uniqueness of t periodic solutions for this equation which are new and complement previously known results

periodic solutions for p laplacian rayleigh equations abstract by employing mawhin s continuation theorem the existence of the p laplacian delay rayleigh equation p x t f x t g x t t e t under various assumptions is obtained

existence and uniqueness of periodic solutions for forced rayleigh type equations in this paper we use the coincidence degree theory to establish new results on the existence and uniqueness of t periodic solutions for a kind of forced rayleigh equation of the formx f x t g t x t e t

similarity structure of wave collapse abstract similarity transformations of the cubic schrodinger equation cse are investigated the transformations are used to remove the explicit time variation in the cse and reduce it to differential equations in the spatial variables only two different methods for similarity reduction are employed and the significance of similarity in the evolution of a collapsing wave packet is investigated numerical solutions in radial symmetry demonstrate that the similarity behaviour is local in space and time and that some similarity solutions must be classified as improper solutions the nature of the collapsing singularity is reexamined

periodic solutions for a kind of duffing type p laplacian equation abstract based on manasevich mawhin continuation theorem and some analysis skill some new sufficient conditions for the existence of periodic solutions for duffing type p laplacian differential equation are obtained an example is provided to demonstrate effectiveness of the proposed results

a model for nonlinear axisymmetric droplet vibrations abstract nonlinear inviscid droplet vibrations can be described by luke s variational principle the model equations for axisymmetric motion are derived from this principle with the rayleigh ritz method using four surface and four bulk modes the equations fulfill the energy mass and axial momentum conservation exactly poincare sections are shown for various energies several bifurcations are obtained a singularity of the type of a weak collapse which is associated to the splitting of the drop is also found for higher energies irregular motion associated to the collapse singularity occurs

periodic solutions for p laplacian neutral rayleigh equation with a deviating argument abstract by using topological degree theory and some analytical skill some criteria to guarantee the existence of periodic solutions are derived for p laplacian neutral rayleigh equation with a deviating argument of the following form p x t c x t f x t g x t t e t

numerical simulation of singular solutions to the two dimensional cubic schrodinger equation numerical simulations of the cubic nonlinear schrodinger equation are presented in dimension n 2 the emphasis is on a detailed mechanism of blow up the numerical results indicate that the blow up is not restricted to the case where the problem is posed in the entire r2 space but also occurs with periodic boundary conditions the structure of the solution as the singular time t is approached has been investigated using examples with and without radial symmetry and or periodicity in most cases we observe that the amplitude develops something like the peaked self similar profile predicted by zakharov and synakh 38 which has a t t 2 3 maximum amplitude and a t t 2 3 width

a variational method for extended nonlinear schrodinger systems abstract we develop a variational procedure for solving nonlinear schrodinger equations in the form i z u u q u 2 u f u 0 where f u is an arbitrary function of u being perturbative or not this method provides a general dynamical system describing the typical length scale of localized solutions u and it includes a relation for the power lost by these solutions in dissipative systems the complete set of dynamical equations is then applied to models describing the propagation of high power beams in gases which involve saturating nonlinearities multiphoton sources and nonlinear dissipation as well theoretical results are confronted with numerical simulations

some new results on the existence of periodic solutions to a kind of rayleigh equation with a deviating argument abstract by employing the continuation theorem of coincidence degree theory we study a kind of rayleigh equation with a deviating argument as follows x t f x t g x t t p t some new results on the existence of periodic solutions are obtained

periodic solutions for lienard type p laplacian equation with a deviating argument in this paper we use the coincidence degree theory to establish new results on the existence of t periodic solutions for the lienard type p laplacian equation with a deviating argument of the form f p x t f x t x t g t x t t t e t

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

existence and uniqueness of anti periodic solutions for a kind of rayleigh equation with two deviating arguments abstract in this paper we use the leray schauder degree theory to establish new results on the existence and uniqueness of anti periodic solutions for a kind of rayleigh equation with two deviating arguments of the form x f t x t g 1 t x t 1 t g 2 t x t 2 t e t

ergodic properties of invariant measures for piecewise monotonic transformations during the last decade a lot of research has been done on onedimensional dynamics different kinds of invariant measures for certain classes of piecewise monotonic transformations have been considered in most of these cases the perron frobenius operator plays an important role in this paper we try to unify these different examples which are discussed in detail below first we give a discription of the results proved in this paper

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

on existence of periodic solutions of rayleigh equation of retarded type existence of periodic solutions for a kind of non autonomous rayleigh equations of retarded typex t f t x t s g t x t t t p t is studied and some new results are obtained our work generalizes and improves the known results in the literature

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

piecewise invertible dynamical systems the aim of the paper is the investigation of piecewise monotonic maps t of an interval x the main tool is an isomorphism of x t with a topological markov chain with countable state space which is described by a 0 1 transition matrix m the behavior of the orbits of points in x under t is very similar to the behavior of the paths of the markov chain every irreducible submatrix of m gives rise to a t invariant subset l of x such that l is the set x of all limit points of the orbit of an x x the topological entropy of l is the logarithm of the spectral radius of the irreducible submatrix which is a l 1 operator besides these sets l there are two t invariant sets y and p such that for all x x the set x is either contained in one of the sets l or in y or in p the set p is a union of periodic orbits and y is contained in a finite union of sets y with y x and has topological entropy zero this isomorphism of x t with a topological markov chain is also an important tool for the investigation of t invariant measures on x results in this direction which are published elsewhere are described at the end of the paper furthermore a part of the proofs in the paper is purely topological without using the order relation of the interval x so that some results hold for more general dynamical systems x t

tracing complex singularities with spectral methods abstract a numerical method for investigating the possibility of blow up after a finite time is introduced for a large class of nonlinear evolution problems with initial data analytic in the space variable s the solutions have for any t 0 complex space singularities at the edge of an analyticity strip of width t loss of regularity corresponds to the vanishing of t numerical integration by high resolution spectral methods reveals the large wavenumber behavior of the fourier transform of the solutions from which t is readily obtained its time evolution can be traced down to about one mesh length by extrapolation of t such numerical experiments provide evidence suggesting finite time blow up or all time regularity the method is tested on the inviscid and viscous burgers equations and is applied to the one dimensional nonlinear schrodinger equation with quartic potential and to the two dimensional incompressible euler equation all with periodic boundary conditions in the latter case evidence is found suggesting that existing all time regularity results can be substantially sharpened

local structure of the self focusing singularity of the nonlinear schro dinger equation abstract for the cubic schrodinger equation in two dimensions we construct a family of singular solutions by perturbing slightly the dimension d 2 tod 2

periodic solutions for a kind of rayleigh equation with a deviating argument existence of periodic solutions for a kind of rayleigh equation with a deviating argument x t f x t g x email protected t p t is studied and some new results are obtained our work generalizes and improves the known results in the literature

decay of correlations for piecewise expanding maps this paper investigates the decay of correlations in a large class of non markov one dimensional expanding maps the method employed is a special version of a general approach recently proposed by the author explicit bounds on the rate of decay of correlations are obtained

periodic solutions for a kind of lienard equation with a deviating argument abstract by means of continuation theorem of coincidence degree theory we study a kind of lienard equation with a deviating argument as follows x t f x t x t g x t t x t e t some new results on the existence of periodic solutions are obtained

existence of asymptotically stable periodic solutions of a rayleigh type equation abstract in this work we investigate the following rayleigh type equation x t f x t g t x t e t some criteria for guaranteeing the existence of asymptotically stable in the sense of lyapunov periodic solutions of this equation is presented by using mawhin s continuation theorem floquet theory lyapunov stability theory and some analysis techniques moreover an example is provided to illustrate the validity of our results

a sharp condition for scattering of the radial 3d cubic nonlinear schrodinger equation we consider the problem of identifying sharp criteria under which radial h 1 finite energy solutions to the focusing 3d cubic nonlinear schrodinger equation nls i t u u u 2 u 0 scatter i e approach the solution to a linear schrodinger equation as t the criteria is expressed in terms of the scale invariant quantities u 0 l 2 nabla u 0 l 2 and m u e u where u 0 denotes the initial data and m u and e u denote the conserved in time mass and energy of the corresponding solution u t the focusing nls possesses a soliton solution e it q x where q is the ground state solution to a nonlinear elliptic equation and we prove that if m u e u q l 2 nabla q l 2 then the solution blows up in finite time the technique employed is parallel to that employed by kenig merle 17 in their study of the energy critical nls

integral boundary value problems for first order integro differential equations with impulsive integral conditions this paper is concerned with the integral boundary value problems for first order integro differential equations with impulsive integral conditions sufficient conditions are obtained for the existence of extremal solutions

sufficient conditions for the existence of periodic solutions to some second order differential equations with a deviating argument abstract by means of mawhin s continuation theorem we study some second order differential equations with a deviating argument x t f t x t x t t x t e t some new results on the existence of periodic solutions are obtained the interest is that we allow the degree with respect to the variables x 0 x 1 x 2 of f t x 0 x 1 x 2 to be greater than 1 and also the result theorem 3 2 is related to the deviating argument t meanwhile we give an example to demonstrate our result

focusing and multi focusing solutions of the nonlinear schro dinger equation abstract a method of dynamic rescaling of variables is used to investigate numerically the nature of the focusing singularities of the cubic and quintic schrodinger equations in two and three dimensions and describe their universal properties the same method is applied to simulate the multi focusing phenomena produced by simple models of saturating nonlinearities

periodic solutions for a kind of second order differential equation with multiple deviating arguments by means of continuation theorem of coincidence degree theory some new results on the existence and nonexistence of periodic solutions for a kind of second order functional differential equation with multiple deviating arguments are obtained

specification on the interval we study the consequences of discontinuities on the specification property for interval maps after giving a necessary and sufficient condition for a piecewise monotonic piecewise continuous map to have this property we show that for a large and natural class of families of such maps including the 3 transformations the set of parameters for which the specification property holds though dense has zero lebesgue measure thus regarding the specification property the general case is at the opposite of the continuous case solved by a m blokh russian math surveys 38 1983 133 134 for which we give a proof

singular ring solutions of critical and supercritical nonlinear schrodinger equations abstract we present new singular solutions of the nonlinear schrodinger equation nls i t t r r r d 1 r r 2 0 1 d 2 d 2 these solutions collapse with a quasi self similar ring profile q i e q where q 1 l 1 t q r r m t l exp i 0 t d s l 2 s i l t 4 l r 2 1 r r m t 2 l t is the ring width that vanishes at the singularity r m t r 0 l t is the ring radius and 2 d 1 the blowup rate of these solutions is 1 1 for 2 d 2 and 1 d 0 1 and a square root with a loglog correction the loglog law when 2 and 1 d 0 therefore the nls has solutions that collapse with any blowup rate p for 1 2 p 1 this study extends the results of g fibich n gavish x wang new singular solutions of the nonlinear schrodinger equation physica d 211 2005 193 220 for 1 and d 2 and of p raphael existence and stability of a solution blowing up on a sphere for a l 2 super critical non linear schrodinger equation duke math j 134 2 2006 199 258 for 2 and d 2 to all 2 d 2 and 1 d

eigenvalue cutoff in the cubic quintic nonlinear schrodinger equation using theoretical arguments we prove the numerically well known fact that the eigenvalues of all localized stationary solutions of the cubic quintic 2 1 dimensional nonlinear schr odinger equation exhibit an upper cutoff value the existence of the cutoff is inferred using gagliardo nirenberg and h older inequalities together with pohozaev identities we also show that in the limit of eigenvalues close to zero the eigenstates of the cubic quintic nonlinear schr odinger equation behave similarly to those of the cubic nonlinear schr odinger equation

chains of infinite order chains with memory of variable length and maps of the interval we show how to construct a topological markov map of the interval whose invariant probability measure is the stationary law of a given stochastic chain of infinite order in particular we characterize the maps corresponding to stochastic chains with memory of variable length the problem treated here is the converse of the classical construction of the gibbs formalism for markov expanding maps of the interval

existence and uniqueness of periodic solutions for a kind of lienard type p laplacian equation abstract by using topological degree theory and some analysis skills we obtain some sufficient conditions for the existence and uniqueness of periodic solutions for lienard type p laplacian differential equations

the effects of normal dispersion on collapse events abstract it is shown analytically that positive or normal dispersion arrests at least the initial stage of the critical self similar collapse events associated with the nonlinear schrodinger equation this is done by pertubing a wave packet that undergoes critical self similar collapse in d dimensions by adding weak positive dispersion in one additional direction singular perturbation analysis then yields a closed set of simple equations for the parameters of the collapsing mode from which it is clear that in the absence of normal dispersion the self similar collapse will continue while in its presence the collapse attractor disappears direct numerical integrations are in excellent agreement with the predictions of the simple reduced equations

problems of periodic solutions for a type of duffing equation with state dependent delay sufficient criteria are established for the existence of periodic solutions to a type of duffing equation with state dependent delay which improve and generalize some related results in the literature the approach is based on mawhin s continuation theorem the significance of the present paper is that our results are relevant to delay by lemma 2 1 which is different from the corresponding results of past work

new results of periodic solutions for forced rayleigh type equations in this paper we use the coincidence degree theory to establish new results on the existence and uniqueness of t periodic solutions for a kind of forced rayleigh equation of the form x f t x t g t x t e t

a grid redistribution method for singular problems many physical phenomena develop singular or nearly singular behavior in localized regions e g boundary layers or blowup solutions using uniform grids for such problems becomes computationally prohibitive as the solution approaches singularity ren and wang developed a semi static adaptive grid method w ren x p wang an iterative grid redistribution method for singular problems in multiple dimensions j comput phys 159 2000 246 273 for the solution of these problems known as the iterative grid redistribution igr method in this study we develop a theoretical basis for semi static adaptive grid method for singular problems based on this theory we obtain the key result of this study a methodology for designing robust weight functionals which ensures grid resolution in the singular region as well as control of the maximal grid spacing in the outer region using this methodology we introduce a semi static adaptive grid method which does not involve an iterative procedure for grid redistribution as in the igr method we demonstrate the efficacy of this method with numerical examples of solutions which localize by more than nine orders of magnitude

periodic solutions to a second order nonlinear neutral functional differential equation in the critical case abstract in this paper the authors study the existence of periodic solutions for a second order neutral functional differential equation x t cx t f x t x t g t x t t e t in the critical case c 1 by employing mawhin s continuation theorem and some analysis techniques sufficient conditions are given for the existence of periodic solutions

new results of periodic solutions for a kind of duffing type p laplacian equation by using mawhin manasevich continuation theorem some new sufficient conditions for the existence and uniqueness of periodic solutions of duffing type p laplacian differential equation are established which are complement of previously known results

blowup in the nonlinear schrodinger equation near critical dimension abstract this article contains an analysis of the cubic nonlinear schrodinger equation and solutions that become singular in finite time numerical simulations show that in three dimensions the blowup is self similar and symmetric in two dimensions the blowup still appears to be symmetric but is no longer self similar in the case that the dimension d is greater than and exponentially close to 2 in terms of a small parameter associated to the norm of the blow up solution a locally unique monotonically decreasing in modulus self similar solution that satisfies the boundary and global conditions associated with the blow up solution is constructed in kopell and landman 1995 siam j appl math 55 1297 1323 in this article it is shown that this locally unique solution also exists for d 2 and algebraically close to 2 in the same small parameter the central idea of the proof involves constructing a pair of manifolds of solutions to the nonautonomous ordinary differential equation satisfied by the self similar solutions that satisfy the conditions at r 0 and the asymptotic conditions respectively and then showing that these intersect transversally a key step involves tracking one of the manifolds over a midrange in which the ordinary differential equation has a turning point and hence obtaining good control over the solutions on the manifold

periodic solutions to neutral differential equation with deviating arguments by employing the continuation theorem of coincidence degree theory and some analysis techniques we study a kind of neutral differential equation with deviating arguments as follows x t cx t r f x t g x t t p t some new results on the existence of periodic solutions are obtained our result generalizes main work of wang app math lett 12 1999 41 44

anti periodic solutions for a class of nonlinear nth order differential equations with delays in this paper we use the leray schauder degree theory to establish new results on the existence and uniqueness of anti periodic solutions for a class of nonlinear nth order differential equations with delays of the form x n t f t x n 1 t g t x email protected t e t

a numerical scheme for the simulation of blow up in the nonlinear schrodinger equation we present a self adaptive multigrid version of a conservative finite difference scheme useful for the study of collapse processes in nonlinear schrodinger equations nlses as an example we study the character of the focusing singularity of the two dimensional critical nlse

on existence of periodic solutions of a kind of rayleigh equation with a deviating argument abstract existence of periodic solutions for a kind of rayleigh equation with a deviating argument x t f x t g t x t t p t is studied and some new results are obtained our work generalizes and improves the known results in the literature

a universal data compression system a universal data compression algorithm is described which is capable of compressing long strings generated by a finitely generated source with a near optimum per symbol length without prior knowledge of the source this class of sources may be viewed as a generalization of markov sources to random fields moreover the algorithm does not require a working storage much larger than that needed to describe the source generating parameters

existence and uniqueness of periodic solutions for a kind of first order neutral functional differential equations abstract in this paper we use the coincidence degree theory to establish new results on the existence and uniqueness of t periodic solutions for the first order neutral functional differential equation of the form x t b x t g 1 t x t g 2 t x t p t

existence and uniqueness of periodic solutions for a kind of rayleigh equation with finitely many deviating arguments abstract in this paper we consider a kind of rayleigh equation with finitely many deviating arguments of the form x f t x t k 1 n g k t x t k t p t by using the coincidence degree theory we establish the results on the existence and uniqueness of periodic solutions for the above equation the results we obtained include and improve the results existing in the literature

new results on the existence of periodic solutions to a p laplacian differential equation with a deviating argument by means of mawhin s continuation theorem a kind of p laplacian differential equation with a deviating argument as follows p x t f t x t x t t x t e t is studied some new results on the existence of periodic solutions are obtained the main results theorems 3 2 and 3 3 are all related to the deviating argument t meanwhile the degrees with respect to the variables x0 x1 of f t x0 x1 x2 are allowed to be grater than p 1 which is different from the corresponding conditions of known literature

on loglog blow up in higher order reaction diffusion and wave equations a formal geometric approach abstract in 1934 petrovskii s boundary regularity study of the heat equation gave the first appearance of the ln ln t t blow up factor in pde theory we discuss the origin of analogous non self similar blow up in higher order reaction diffusion parabolic or wave hyperbolic equations of the form u t u 2 u x x x x u or u t t u 2 u x x x x u in l l 0 t with zero dirichlet boundary conditions at x l where l l 0 2 we present formal arguments that the standard similarity blow up rate 1 t t acquires an extra universal ln ln t t factor the explanation is based on a geometric matching with the so called logarithmic travelling waves as group invariant solutions of the pdes we also discuss connections with log log blow up factors occurring in earlier studies of plasma physics parabolic equations and the nonlinear critical schrodinger equation

existence and uniqueness of periodic solutions for a kind of rayleigh equation with two deviating arguments in this paper we consider a kind of rayleigh equation with two deviating arguments of the form x f t x t g 1 t x email protected 1 t g 2 t x email protected 2 t p t by using the coincidence degree theory we establish new results on the existence and uniqueness of periodic solutions for the above equation

periodic solutions for a kind of lienard equation using inequality techniques and coincidence degree theory new results are provided concerning the existence and uniqueness of t periodic solutions for a lienard equations with delay an illustrative example is provided to demonstrate that the results in this paper hold under weaker conditions than existing results and are more effective

periodic solutions for a kind of rayleigh equation with a deviating argument abstract in this paper we use the coincidence degree theory to establish new results on the existence of t periodic solutions for the rayleigh equation with a deviating argument of the form x f x t g t x t t p t

numerical approximation of blow up of radially symmetric solutions of the nonlinear schrodinger equation we consider the initial value problem for the radially symmetric nonlinear schrodin ger equation with cubic nonlinearity nls in d 2 and 3 space dimensions to approximate smooth solutions of this problem we construct and analyze a numerical method based on a standard galerkin finite element spatial discretization with piecewise linear continuous functions and on an implicit crank nicolson type time stepping procedure we then equip this scheme with an adaptive spatial and temporal mesh refinement mechanism that enables the numerical technique to approximate well singular solutions of the nls equation that blow up at the origin as the temporal variable t tends from below to a finite value t star for the blow up of the amplitude of the solution we recover numerically the well known rate t star t 1 2 for d 3 for d 2 our numerical evidence supports the validity of the log log law ln ln frac 1 t star t t star t 1 2 for t extremely close to t star the scheme also approximates well the details of the blow up of the phase of the solution at the origin as t to t star

periodic solutions for p laplacian rayleigh equations with a deviating argument abstract by using the generalized borsuk theorem in coincidence degree theory some new criteria for guaranteeing the existence of periodic solutions for p laplacian rayleigh equations with a deviating argument are derived