discrete solitons in nonlinear schrodinger lattices with a power law nonlinearity abstract we study the discrete nonlinear schrodinger lattice model with the onsite nonlinearity of the general form u 2 u we systematically verify the conditions for the existence and stability of discrete solitons in the one dimensional version of the model predicted by means of the variational approximation va and demonstrate the following monostability of fundamental solitons fss in the case of the weak nonlinearity 2 1 3 68 bistability in a finite range of values of the soliton s power for 3 68 2 1 5 and the presence of a threshold minimum norm of the fs for 2 1 5 we also perform systematic numerical simulations to study higher order solitons in the same general model i e bound states of the fss while all in phase bound states are unstable stability regions are identified for antisymmetric double solitons and their triple counterparts these numerical findings are supplemented by an analytical treatment of the stability problem which allows quantitively accurate predictions for the stability features of such multipulses when these waveforms are found to be unstable we show by means of direct simulations that they self trap into a persistent lattice breather or relax into a stable fs or sometimes decay completely

pattern analysis of a modified leslie gower predator prey model with crowley martin functional response and diffusion abstract in this paper we study a modified leslie gower predator prey model with crowley martin functional response and spatial diffusion under homogeneous neumann boundary condition and obtain some important qualitative properties including the existence of the global positive solution the dissipation and persistence of the two species the local and global asymptotic stability of the constant equilibria and hopf bifurcation around the interior constant equilibrium in addition we establish the existence and nonexistence of nonconstant positive steady states

global solutions to a class of multi species reaction diffusion systems with cross diffusions arising in population dynamics in this paper an n species strongly coupled cooperating diffusive system is considered in a bounded smooth domain subject to homogeneous neumann boundary conditions employing the method of energy estimates we obtain some conditions on the diffusion matrix and inter specific cooperatives to ensure the global existence and uniform boundedness of a nonnegative solution the globally asymptotical stability of the constant positive steady state is also discussed as a consequence all the results hold true for multi species lotka volterra type competition model and prey predator model

dynamics in a diffusive predator prey system with ratio dependent predator influence abstract a diffusive predator prey system with predator functional response subject to neumann boundary conditions is considered existence nonexistence and boundedness of positive steady state solutions are shown to identify the ranges of parameters of spatial pattern formation hopf and steady state bifurcation analyses are carried out in detail these results provide theoretical evidences to the complex spatio temporal dynamics found by numerical simulation

analysis of a parabolic cross diffusion population model without self diffusion abstract the global existence of non negative weak solutions to a strongly coupled parabolic system arising in population dynamics is shown the cross diffusion terms are allowed to be arbitrarily large whereas the self diffusion terms are assumed to disappear the last assumption complicates the analysis since these terms usually provide h 1 estimates of the solutions the existence proof is based on a positivity preserving backward euler galerkin approximation discrete entropy estimates and l 1 weak compactness arguments furthermore employing the entropy entropy production method we show for special stationary solutions that the transient solution converges exponentially fast to its steady state as a by product we prove that only constant steady states exist if the inter specific competition parameters disappear no matter how strong the cross diffusion constants are

breather solutions of the discrete nonlinear schrodinger equations with unbounded potentials in this paper i investigate the existence of nontrivial breather solutions of the discrete nonlinear schrodinger equation with the unbounded potential at infinity first i derive a discrete version of compact embedding theorem then combining the nehari manifold approach and the compact embedding theorem i show the existence of breather solutions without palais smale condition the results on the exponential decay of breather solutions are also provided in this paper

smooth solutions to a quasi linear system of diffusion equations for a certain population model abstract the system of diffusion equations proposed by kawasaki shigesada and teramoto describes a population model of two competing species with self and cross population pressures the densities of the two species are denoted by u and v in this paper the author studies the initial boundary value problem associated with the neumann boundary condition corresponds to the case where the flux is zero at the boundary many investigators have considered nonlinear diffusion systems arising from various physical and biological problems these equations however have a special structure the highest order derivatives are not coupled or at least the coefficient matrix for the highest order derivatives is positive definite this is not the case for the system and hence some of the techniques which are effective for the system are no longer applicable to our case nevertheless we can still use sobolevski s method see reference 2 to establish the local existence of solutions under the special assumption c1 c2 in we can also prove the global existence of solutions by energy estimates the unusual structure of seems to make it difficult to settle the question of asymptotic stability of solutions

on the existence of homoclinic solutions of a class of discrete nonlinear periodic systems abstract in this paper we obtain a new sufficient condition on the existence of homoclinic solutions of a class of discrete nonlinear periodic systems by using critical point theory in combination with periodic approximations we prove that it is also necessary in some special cases

multiple solutions for discrete periodic nonlinear schrodinger equations in this paper we study the existence of multibump solutions for discrete nonlinear schrodinger equations with periodic potentials we first reduce the existence of multibump homoclinic solutions to the existence of an isolated homoclinic solution with a nontrivial critical group then we study the existence of homoclinics with nontrivial critical groups for both superlinear and asymptotically linear discrete periodic nonlinear schrodinger equations and we provide simple sufficient conditions for the existence of homoclinics with nontrivial critical groups in the positive definite case as an application we get without any symmetry assumptions infinitely many geometrically distinct homoclinic solutions with exponential decay at infinity

uniform boundedness and convergence of solutions to cross diffusion systems abstract uniform boundedness and convergence of global solutions are proved for cross diffusion systems in population dynamics gagliardo nirenberg type inequalities are used in the estimates of solutions in order to establish w 2 1 bounds uniform in time in each step of estimates the contribution of the diffusion coefficients are emphasized and it is concluded that the uniform bound remains independent of the growth of the diffusion coefficient in the system hence convergence of solutions are established for systems with large diffusion coefficients

positive steady states for a nonlinear diffusion beddington deangelis model the main purpose of this work is to investigate the effects of nonlinear diffusion on positive steady states in a beddington deangelis model by the energy method and leray schauder degree theory we consider the non existence and existence of concerning non constant positive steady states of the model we demonstrate that nonlinear diffusion can create non constant positive steady state solutions even when the random diffusion fails to do so

global attractor for a chemotaxis model with prevention of overcrowding abstract chemotaxis model with volume filling introduced by painter and hillen is studied under no flux or dirichlet boundary conditions existence of global in time solution to a full model is proved for some cases existence of a global attractor in the space w 1 p r 2 p n r n is shown

traveling waves for a class of cross diffusion systems with small parameters abstract this paper is concerned with the existence of the travelling wave solutions for some cross diffusion systems with small parameters by using singular perturbation method we prove the existence of the travelling wave solutions with a speed c which extend the results of 22 and 23

an elliptic cross diffusion system describing two species models on a bounded domain with different natural conditions this paper is concerned with an elliptic cross diffusion system describing two species models on a bounded domain where consists of a finite number of subdomains i i 1 m separated by interfaces j j 1 m 1 and the natural conditions of the subdomains i are different this system is strongly coupled and the coefficients of the equations are allowed to be discontinuous on interfaces j the main goal is to show the existence of nonnegative solutions for the system by schauder s fixed point theorem furthermore as applications the existence of positive solutions for some lotka volterra models with cross diffusion self diffusion and discontinuous coefficients are also investigated

on a time dependent bio reactor model with chemotaxis long time dynamics of solutions to a strongly coupled system of parabolic equations modeling the competition in bio reactors with chemotaxis will be studied if the parameters of the system are periodic sufficient conditions for positive periodic solutions will be derived in terms of parabolic eigenvalue problems

dynamics in a parabolic elliptic two species population competition model with cross diffusion for one species abstract this work deals with a simplified parabolic elliptic population competition model with cross diffusion for one species we first claim that this model has at least a non trivial global smooth solution and then we detect explicit parameter ranges under which the coexistence steady state and the two semi trivial steady states are globally and asymptotically stable

qualitative analysis for a diffusive predator prey model with a transmissible disease in the prey population in this paper we study a diffusive predator prey model with a transmissible disease in the prey population we offer a complete discussion of the dynamical properties under the homogeneous neumann boundary condition we analyze local and global stabilities of nonnegative constant steady states and long time behaviors of the positive solutions moreover we study the existence nonexistence and bifurcation of nonconstant positive stationary solutions finally some numerical simulations are presented to support and strengthen our analytical analysis

dynamics of a bio reactor model with chemotaxis long time dynamics of solutions to a strongly coupled system of parabolic equations modeling the competition in bio reactors with chemotaxis will be studied in particular we show that the dynamical system possesses a global attractor and that it is strongly uniformly persistent if the trivial steady state is unstable using a result of smith and waltman on perturbation of global attractors we also show that the positive steady state is unique and globally attracting

global existence analysis of cross diffusion population systems for multiple species the existence of global in time weak solutions to reaction cross diffusion systems for an arbitrary number of competing population species is proved the equations can be derived from an on lattice random walk model with general transition rates in the case of linear transition rates it extends the two species population model of shigesada kawasaki and teramoto the equations are considered in a bounded domain with homogeneous neumann boundary conditions the existence proof is based on a refined entropy method and a new approximation scheme global existence follows under a detailed balance or weak cross diffusion condition the detailed balance condition is related to the symmetry of the mobility matrix which mirrors onsager s principle in thermodynamics under detailed balance and without reaction the entropy is nonincreasing in time but counter examples show that the entropy may increase initially if detailed balance does not hold

on stability of two degenerate reaction diffusion systems abstract in this paper we construct a partially degenerate reaction diffusion equation subject to the neumann boundary condition and show that the zero solution is asymptotically stable but not exponentially asymptotically stable in this way we solve an open problem proposed by casten and holland 1977 4 moreover we give the exponential asymptotic stability of the zero solution to a totally degenerate system with cross diffusion effects which cannot be determined by a simple spectral analysis based on the well developed semigroup theory

breathers in nonlinear lattices existence linear stability and quantization abstract it has been proved by the principle of anti integrable limit renamed here anti continuous limit and the implicit function theorem that under rather weak hypotheses there exist spatially localized and time periodic solutions breathers in arrays of nonlinear coupled classical oscillators provided the coupling is not too strong the models can be translationally invariant or not and be of any dimension there are also time periodic solutions corresponding to arbitrary distributions on the lattice of such solutions multibreathers the whole set of these solutions can be labeled by a discrete coding sequence the condition that the frequency of the breather or multibreather solution has no harmonics in the phonon spectrum can be discarded for a large subset of spatially extended time periodic solutions called phonobreather which correspond to special codes which can be also spatially chaotic and where no oscillator is at rest these results extend almost without change to models with coupled rotors the existence of rotobreathers which are exact solutions consisting of a single rotor rotating while the other rotors oscillate is proved there are also multirotobreather solutions and phono rotobreathers etc it is also shown that the phase of the single breathers constituting a multibreather or multirotobreather solution can be submitted to a phase torsion these new dynamical structures carry a nonzero energy flow in models with two dimensions and more the phase torsion of multibreather solutions can generate new dynamical solutions with vortices in the energy flow the dynamics of a quantum electron coupled to a classical lattice is also studied within the same approach from the anti continuous limit the existence of two kinds of polarobreathers which are spatially exponentially localized dynamical solutions is proved for a small enough electronic transfer integral the polarobreathers of the first kind include the standard static polaron the lattice configuration is static time independent when the electron is in an excited state i e the electronic oscillator oscillates periodically they can be spatially chaotic a polarobreather of the second kind corresponds to a localized periodic lattice oscillation associated with a localized quasiperiodic electronic oscillation the most simple of these solutions are those for which both the lattice oscillation and the electron wave function are mostly concentrated close to the same single site the standard floquet analysis of the linear stability of the breather and multibreather solutions is shown to be related with the band spectrum property of the newton operator involved in the implicit function theorem the krein theory of bifurcations is reinterpreted from this point of view the linear stability of the single breather and rotobreather solutions is then proved at least at small enough coupling finally it is shown that in the quantum version of translationally invariant classical models where there cannot exist in principle any strictly localized excitation there are narrow bands of excitations consisting of bound states of many bosons which are the quantum counterpart of the classical breathers the bandwidth of these excitations goes exponentially to zero with the number of bound phonons this theory provides simultaneously existence theorems and a new powerful and accurate numerical method for calculating practically any of the solutions which has been formally predicted to exist these numerical applications are currently written elsewhere

spatio temporal complexity of a delayed diffusive model for plant invasion abstract in this paper we develop a diffusive plant invasion model with delay under the homogeneous neumann boundary condition first the existence and uniqueness of a non negative solution persistence property and local asymptotic stability of the constant steady states are established then by analyzing the associated characteristic equation the stability of steady states and the existence of hopf bifurcation are demonstrated under special circumstance the discontinuous hopf bifurcation is also investigated furthermore the existence and non existence of nonconstant positive steady states of this model are studied through considering the effect of large diffusivity finally in order to verify our theoretical results some numerical simulations are also included it shows that the numerically observed behaviors are in good agreement with the theoretically proposed results on the basis of numerical simulations we provide some useful comparisons for the readers between our results and the existing ones and discuss the effects of the delay and the diffusion term on dynamic behaviors numerical results indicate that diffusion can make the system unstable and increasing delay may cause the plant to go extinct

discrete breathers localization and transfer of energy in discrete hamiltonian nonlinear systems we present a partial review of results concerning the theory of discrete breathers in nonlinear hamiltonian and discrete systems these special time periodic solutions gained much interest during the last decade because they may be involved in numerous and various phenomena in physics and biophysics where they could produce nontrivial effects of energy focusing and transfer we first review the principles which govern their existence and which are used in the existence proofs available up to now next we discuss their linear stability and the interaction of discrete breathers with small amplitude waves showing also how they could grow or decay we also briefly discuss the existence of intraband dbs in systems with a linear discrete spectrum which are not spatially periodic random or otherwise and where linear waves cannot propagate we also show that nonlinearity may restore the existence of solutions that propagate energy we review some results concerning energy transportation by dbs and especially the main features associated with their mobility we briefly discuss new perspectives opened up by the theory for applications that in particular look especially interesting for biophysics

a strongly coupled predator prey system with non monotonic functional response abstract the predator prey system with non monotonic functional response is an interesting field of theoretical study in this paper we consider a strongly coupled partial differential equation model with a non monotonic functional response a holling type iv function in a bounded domain with no flux boundary condition we prove a number of existence and non existence results concerning non constant steady states patterns of the underlying system in particular we demonstrate that cross diffusion can create patterns when the corresponding model without cross diffusion fails

homoclinic solutions of discrete nonlinear schrodinger equations with asymptotically or super linear terms abstract in this paper we study the following periodic discrete nonlinear schrodinger equation u n n u n n g n u n n z where the nonlinearity g n is asymptotically or super linear the sufficient conditions on the existence and on the nonexistence of nontrivial solitons was established by using critical point theory

existence of gap solitons in periodic discrete nonlinear schrodinger equations in this paper we discuss how to use the critical point theory to study the existence of gap solitons for periodic discrete nonlinear schrodinger equations an open problem proposed by professor alexander pankov is solved

discrete solitons for periodic discrete nonlinear schrodinger equations in this paper we study a class of periodic discrete nonlinear schrodinger equations with superlinear nonlinearities and periodic potential by using the critical point theory we establish results on the existence of discrete solitons our results improve some recent ones

holder regularity for certain strongly coupled parabolic systems abstract we present a shorter proof to show holder continuity of bounded solutions to a general class of quasilinear parabolic equations the proof will be extended to obtain regularity results for bounded solutions to certain strongly coupled or cross diffusion quasilinear parabolic systems

positive steady state solutions of a competing reaction diffusion system with large cross diffusion coefficients abstract this paper deals with a coupled system of reaction diffusion equations modeling the competition of two species with cross diffusion we discuss the existence of positive steady state solutions in relation to the intrinsic birth rates a 1 a 2 of species and the cross diffusion coefficients 12 21 our results show that positive steady state solutions exist if a 1 and a 2 lie in certain range or if 12 and 21 are sufficiently large

discrete nonlinear schrodinger equations with superlinear nonlinearities abstract in this paper we study the discrete nonlinear equation lu n u n n g n u n n z where belongs to a finite spectral gap of the operator l or it is a lower edge of a finite spectral gap by using a variant generalized weak linking theorem for strongly indefinite problem developed by schechter and zou we will study the existence of nontrivial solitons for this equation with gn is super linear

diffusion self diffusion and cross diffusion u1 x 0 u 1 0 x u2 x 0 u 2 0 x i n 0 where 2 n 1 2 x 2 is the laplace operator 0 is a bounded smooth domain of r n with n1 0 and 0 are the boundary and the closure of 0 respectively 0t 0 0 t and 0t 0 0 t for some t 0 is the outward unit normal vector on 0 di ai bi ci i 1 2 are all positive constants while ij i j 1 2 denote non negative constants the initial values u1 0 and u2 0 are non negative smooth func tions which are not identically zero the mathematical model 1 1 was proposed by shigesada et al 27 in their study of spatial segregation of interacting species where u1 and u2 represent the densities of two competing species d1 and d2 are their diffu sion rates a1 and a2 denote the intrinsic growth rates b1 and c2 account for intra specific competitions b2 and c1 are the coefficients of inter specific competitions 11 and 22 are usually referred as self diffusion pressures and 12 and 21 are cross diffusion pressures an interesting feature of 1 1 is that the movements of these two competing species are affected by the population pressures due to the mutual interference between the individuals and it was hoped that the nonlinear dispersive force due to the

optimality for a diffusive predator prey system in a spatially heterogeneous environment incorporating a prey refuge abstract in this paper we consider the optimal harvesting controls for a diffusive predator prey system which is formulated by prey refuge and modified leslie gower functional response the locally and globally asymptotic stability of unique positive constant steady state is discussed under homogeneous neumann boundary condition the nonexistence and existence of nonconstant steady states are obtained by leray schauder degree theory also the optimal harvesting strategies to get maximum sustainable yield msy and maximum economic yield mey are correspondingly given

global existence of solutions for a strongly coupled quasilinear parabolic system we are interested in existence uniqueness and a priori estimates of non negative solutions to the following problem begin gathered u t delta left left 1 alpha v right u right uf u v hfill v t delta v vg u v quad in 0 infty times omega hfill u v 0 quad in 0 infty times partial omega hfill u 0 u 0 v 0 v 0 quad in omega hfill end gathered 1 0

strongly coupled elliptic systems and applications to lotka volterra models with cross diffusion abstract the aim of this paper is to investigate the existence and method of construction of solutions for a general class of strongly coupled elliptic systems by the method of upper and lower solutions and its associated monotone iterations the existence problem is for nonquasimonotone functions arising in the system while the monotone iterations require some mixed monotone property of these functions applications are given to three lotka volterra model problems with cross diffusion and self diffusion which are some extensions of the classical competition prey predator and cooperating ecological systems the monotone iterative schemes lead to some true positive solutions of the competition system and to quasisolutions of the prey predator and cooperating systems also given are some sufficient conditions for the existence of a unique positive solution to each of the three model problems

periodic solutions for hamiltonian systems without ambrosetti rabinowitz condition and spectrum 0 abstract in this paper we consider the superquadratic second order hamiltonian system u t a t u t h t u t 0 t r our main results here allow the classical ambrosetti rabinowitz superlinear condition to be replaced by a general superquadratic condition and 0 lies in a gap of b where b d 2 d t 2 a t we will study the ground state periodic solutions for this problem the main idea here lies in an application of a variant generalized weak linking theorem for strongly indefinite problem developed by schechter and zou

periodic and decaying solutions in discrete nonlinear schrodinger with saturable nonlinearity we demonstrate the existence of solutions in the discrete nonlinear schrodinger equation dnls with saturable nonlinearity we consider two types of solutions to dnls periodic and vanishing at infinity calculus of variations and the nehari manifolds are employed to establish the existence of these solutions we present some extensions of our results combining the nehari manifold approach and the mountain pass argument

the structure of hyperfinite borel equivalence relations we study the structure of the equivalence relations induced by the orbits of a single borel automorphism on a standard borel space we show that any two such equivalence relations which are not smooth i e do not admit borel selectors are borel embeddable into each other this utilizes among other things work of effros and weiss using this and also results of dye varadarajan and recent work of nadkarni we show that the cardinality of the set of ergodic invariant measures is a complete invariant for borel isomorphism of aperiodic nonsmooth such equivalence relations in particular since the only possible such cardinalities are the finite ones countable infinity and the cardinality of the continuum there are exactly countably infinitely many isomorphism types canonical examples of each type are also discussed

standing waves for discrete schrodinger equations in infinite lattices with saturable nonlinearities abstract in this paper we consider the periodic discrete nonlinear equation l u n u n g n u n n z lim n u n 0 where l is a jacobi operator and the nonlinearities g n s are asymptotically linear as s in the two different cases is a spectral endpoint of l or it belongs to a finite spectral gap of l we obtain the existence of nontrivial solitons of this equation by using variational methods in particular a necessary and sufficient condition is obtained for the existence of gap solitons of the nonlinear equation here solitons appear when we look for standing waves of some discrete nonlinear schrodinger equations

generic dynamics and monotone complete c algebras soit r une relation d equivalence generique denombrable ergodique sur un espace parfait polonais x modulo un sous ensemble maigre de x r peut s identifier a la relation d equivalence d orbite suivant une action canonique de z

large diffusivity and asymptotic behavior in parabolic systems abstract for systems of reaction diffusion equations with neumann boundary conditions it is shown that the solutions are asymptotic to the solutions of an ordinary differential equation if the diffusivity is large the methods apply also to reaction diffusion systems with time delays

analysis of a population model with strong cross diffusion in unbounded domains we study a parabolic population model in the full space and prove the global in time existence of a weak solution this model consists of two strongly coupled diffusion equations describing the population densities of two competing species the system features intrinsic growth inter and intra specific competition of the species as well as diffusion cross diffusion and self diffusion and drift terms related to varying environment quality the cross diffusion terms can be large making the system non parabolic for large initial data for the first time solutions in unbounded domains are studied the method of our proof is a combination of a time semi discretization a transformation of the system guaranteeing the positivity of the solution a special entropy symmetrizing the system and compactness arguments 2000 mathematics subject classification 35k55 primary 92d25 35d05 secondary

stationary patterns for a prey predator model with prey dependent and ratio dependent functional responses and diffusion abstract we study a reaction diffusion system arising from a three species prey predator model with prey dependent and ratio dependent functional responses the main aim of this paper is to study the global existence and bifurcation of non constant positive steady states in particular we will show that even though the unique positive constant steady state is stable for the ode dynamics non constant positive steady states exist for the reaction diffusion system this demonstrates that stationary patterns arise as a result of diffusion

qualitative analysis of a prey predator model with stage structure for the predator in this paper we propose a diffusive prey predator model with stage structure for the predator we first analyze the stability of the nonnegative steady states for the reduced ode system and then study the same question for the corresponding reaction diffusion system with homogeneous neumann boundary conditions we find that a hopf bifurcation occurs in the ode system but no turing pattern happens in the reaction diffusion system however when a natural cross diffusion term is included in the model we can prove the emergence of stationary patterns i e nonconstant positive stationary solutions for this system moreover these stationary patterns do not exist in the considered parameter regime when there is no cross diffusion

semi discretization in time and numerical convergence of solutions of a nonlinear cross diffusion population model a positivity preserving numerical scheme for a strongly coupled cross diffusion model for two competing species is presented based on a semi discretization in time the variables are the population densities of the species existence of strictly positive weak solutions to the semidiscrete problem is proved moreover it is shown that the semidiscrete solutions converge to a non negative solution of the continuous system in one space dimension the proofs are based on a symmetrization of the problem via an exponential transformation of variables and the use of an entropy functional

qualitative analysis of steady states to the sel kov model in this work we are concerned with a reaction diffusion system well known as the sel kov model which has been used for the study of morphogenesis population dynamics and autocatalytic oxidation reactions we derive some further analytic results for the steady states to this model in particular we show that no nonconstant positive steady state exists if 0 1 and large where nonconstant positive steady states can occur thus these conclusions indicate that the parameter p plays a crucial role in leading to spatially nonhomogeneous distribution of the two reactants the a priori estimates are fundamental to our mathematical approaches

discrete schrodinger equations with sign changing nonlinearities infinitely many homoclinic solutions abstract we obtain infinitely many homoclinic solutions for a class of discrete nonlinear schrodinger equations where nonlinearities are superlinear at infinity and primitive functions of nonlinearities are allowed to be sign changing by using some weaker conditions our results extend and improve some existed results in the literature besides some examples are given to illuminate our results

an initial boundary value problem for a certain density dependent diffusion system on considere le systeme parabolique quasilineaire u t x c 1 d 1 v v e 1 a 1 v b 1 u u v t x c 2 d 2 u v e 2 a 2 u b 2 o v dans o t u x s 0 v x s sur o t u x o w 1 x v x o w 2 x sur ou r n est un domaine borne a frontiere lisse