on the asymptotic behavior for a certain nonlinear evolution equation let x be a real banach space we shall say that an operator a with domain d a and range r a in x is accretive if lx x2 11 0 and yi e axi i 1 2 and that a satisfies the range condition if r z i 1 d a the closure of d a for all a 0 where 1 denotes the identity let g 0 co 0 co be a nonincreasing continuous function such that lim r13 g t 0 let x e x and consider the following initial value problem

unbounded behavior and convergence of solutions of nonlinear volterra equations in banach spaces soit x un espace de banach reel de norme et a un operateur m accretif dans x on etudie le comportement asymptotique des solutions de l equation de volterra non lineaire g t u t b au t t r 0 ou b r r g r x sont donnees et est la convolution

darboux transformation with dihedral reduction group we construct the darboux transformation with dihedral reduction group for the 2 dimensional generalisation of the periodic volterra lattice the resulting backlund transformation can be viewed as a nonevolutionary integrable differential difference equation we also find its generalised symmetry and the lax representation for this symmetry using formal diagonalisation of the darboux matrix we obtain local conservation laws of the system

cyclical interactions with alliance specific heterogeneous invasion rates we study a six species lotka volterra type system on different two dimensional lattices when each species has two superior and two inferior partners the invasion rates from predator sites to a randomly chosen neighboring prey site depend on the predator prey pair whereby cyclic symmetries within the two three species defensive alliances are conserved monte carlo simulations reveal an unexpected nonmonotonous dependence of alliance survival on the difference of alliance specific invasion rates this behavior is qualitatively reproduced by a four point mean field approximation the study addresses fundamental problems of stability for the competition of two defensive alliances and thus has important implications in natural and social sciences

asymptotic behavior of solutions to nonlinear volterra integral equations in section 2 we establish the existence of solutions to 1 1 and to related equations theorems 2 1 and 2 3 in contrast with previous papers e g 7 we do not assume that a is m accretive instead we require a to satisfy a weaker cone condition certain other facts are also included in this section throughout this paper we also assume that b is completely positive 6 in the third section we present two mean ergodic theorems for the solutions of 1 1 in hilbert space the first theorem theorem 3 1 deals with weak almost convergence and the second theorem 3 4 with strong almost convergence these results improve upon those of 1 and include the known results for nonlinear semigroups 191 it is important to establish almost convergence because it then follows that the solution is summed by every strongly regular kernel corollaries 3 3 and 3 5 a weak convergence criterion corollary 3 2 is also obtained if the banach space x is reflexive and strictly convex and 0 e r a then

asymptotic behavior of solutions of nonlinear volterra equations and mean points abstract in this paper we study the asymptotic behavior of solutions to a nonlinear volterra equation in banach spaces first we deal with the mean point concerning an invariant mean on 0 for the unbounded behavior of solutions in a reflexive banach space using the mean point we obtain the weak and strong convergences for the unbounded behavior of solutions in a reflexive and strictly convex banach space and in the dual space which has a frechet differentiable norm respectively

lenard scheme for two dimensional periodic volterra chain we prove that for compatible weakly nonlocal hamiltonian and symplectic operators hierarchies of infinitely many commuting local symmetries and conservation laws can be generated under some easily verified conditions no matter whether the generating nijenhuis operators are weakly nonlocal or not we construct a recursion operator of the two dimensional periodic volterra chain from its lax representation and prove that it is a nijenhuis operator furthermore we show that this system is a generalized bi hamiltonian system rather surprisingly the product of its weakly nonlocal hamiltonian and symplectic operators gives rise to the square of the recursion operator

automorphic lie algebras with dihedral symmetry the concept of automorphic lie algebras arises in the context of reduction groups introduced in the early 1980s in the field of integrable systems automorphic lie algebras are obtained by imposing a discrete group symmetry on a current algebra of krichever novikov type past work shows remarkable uniformity between algebras associated to different reduction groups for example if the base lie algebra is sl2 c and the poles of the automorphic lie algebra are restricted to an exceptional orbit of the symmetry group changing the reduction group does not affect the lie algebra structure in this research we fix the reduction group to be the dihedral group and vary the orbit of poles as well as the group action on the base lie algebra we find a uniform description of automorphic lie algebras with dihedral symmetry valid for poles at exceptional and generic orbits

on the asymptotic behavior of nonlinear semigroups and the range of accretive operators ii abstract it is known that certain problems in partial differential equations may be interpreted as initial value problems for ordinary differential equations in banach spaces when such an evolution equation is governed by an accretive operator then its solutions give rise to a nonlinear contraction semigroup in this paper we study certain aspects of the asymptotic behavior of nonlinear semigroups and or resolvents of accretive operators we also derive new results on their behavior at the origin it turns out that the behavior of a nonlinear semigroup resembles that of the resolvent of its generator both at infinity and at the origin author

on the asymptotic behavior of solutions of nonlinear volterra equations where a is an m accretive operator in x b is a given real kernel and g is a given x valued function in what follows we make the same assumptions on b and g as in 2 5 71 hl beac r r b o o b ebv r r h2 ge w r y g el r x g o ed a h3 b is completely positive on r and b m 0 we may assume without loss of generality that b 0 1 sine b t and a may be replaced by b t b o and b o a respectively note that if h 1 holds the first statement in h3 is equivalent to the fact that there exists a k e l r r nonnegative and nonincreasing satisfying b t k b t 1 for t e r moreover in this case lim o b t b cc exists and b m 0 if and only if kel r r see 2 5 71 now define g u for u e c r x or c 0 t x t 0 as follows

interplay between partnership formation and competition in generalized may leonard games in order to better understand the interplay of partnership and competition in population dynamics we study a family of generalized may leonard models with n species these models have a very rich structure characterized by different types of space time patterns interesting partnership formations emerge following the maxim that the enemy of my enemy is my friend in specific cases cyclic dominance within coarsening clusters yields a peculiar coarsening behavior with intriguing pattern formation we classify the different types of dynamics through the analysis of the square of the adjacency matrix the dependence of the population densities on emerging pattern and propagating wave fronts is elucidated through a fourier analysis finally after having identified collaborating teams we study interface fluctuations where we initially populate different parts of the system with different teams

darboux polynomial matrices the classical massive thirring model as a study case one way of constructing explicit expressions of solutions of integrable systems of partial differential equations goes via the darboux method this requires the construction of darboux matrices here we introduce a novel algorithm to obtain such matrices in polynomial form our method is illustrated by applying it to the classical massive thirring model and by combining it with the dihedral group of symmetries possessed by this model

reductions of n wave interactions related to low rank simple lie algebras i z2 reductions the analysis and the classification of all reductions for the nonlinear evolution equations solvable by the inverse scattering method is an interesting and still open problem we show how the second order reductions of the n wave interactions related to low rank simple lie algebras g can be embedded also in the weyl group of g this allows us to display along with the well known ones a number of new types of integrable n wave systems some of the reduced systems find applications to nonlinear optics

twisted hierarchies associated with the generalized sine gordon equation twisted u and twisted u k hierarchies are soliton hierarchies introduced by terng to find higher flows of the generalized sine gordon equation twisted o j j o j o j hierarchies are among the most important classes of twisted hierarchies in this paper we derive explicit interesting first and higher flows of twisted o j j o j o j hierarchies justify that the one dimensional systems of twisted o j j o j o j hierarchies for j iq n q 1 q n 1 called the generalized sinh gordon equations are the gauss codazzi equations for n dimensional timelike submanifolds with constant sectional curvature 1 and index q in pseudo euclidean 2n 1 dimensional space r2q 12n 1 with index 2q 1 furthermore a unified treatment of the inverse scattering theory for twisted o j j o j o j hierarchies is provided

coevolutionary dynamics with clustering behaviors on cyclic competition abstract we propose a dynamic model for describing clustering behaviors on a cyclic game in which the same species form a cluster to compete the rates of consuming the prey depend not only on the individual competing ability v but also on the two interacting cluster s sizes the fragmentation and coagulation rates of the clusters are related to the cohesive strength among the individuals a new parameter u is introduced to indicate the uniting degree we find that the probability distribution of the clustering sizes is almost a power law in a large regime specified by the two parameters which reflects the scale free behavior in complex systems in addition the exponential magnitudes are mostly in the range of real social systems our simulation shows that clustering promotes biodiversity at steady state the amounts about the three species evolve tempestuously with asymmetric period the aggregations about big size s clusters to compete are obvious and on off intermittence

on the classification of automorphic lie algebras the problem of reduction of integrable equations can be formulated in a uniform way using the theory of invariants this provides a powerful tool of analysis and it opens the road to new applications of automorphic lie algebras beyond the con text of integrable systems in this paper it is shown that sl2 c based automorphic lie algebras associated to the icosahedral group i the octahedral group o the tetrahedral group t and the dihedral group dn are isomorphic the proof is based on techniques from classical invariant theory and makes use of clebsch gordan decomposition and transvectants molienfunctionsandthetrace form thisresultprovidesacompleteclas sification of sl2 c based automorphic lie algebras associated to finite groups when the group representations are chosen to be the same and it is a crucial step towards the complete classification of automorphic lie algebras

junctions and spiral patterns in generalized rock paper scissors models we investigate the population dynamics in generalized rock paper scissors models with an arbitrary number of species n we show that spiral patterns with n arms may develop both for odd and even n in particular in models where a bidirectional predation interaction of equal strength between all species is modified to include one n cyclic predator prey rule while the former case gives rise to an interface network with y type junctions obeying the scaling law l ensuremath propto t 1 2 where l is the characteristic length of the network and t is the time the latter can lead to a population network with n armed spiral patterns having a roughly constant characteristic length scale we explicitly demonstrate the connection between interface junctions and spiral patterns in these models and compute the corresponding scaling laws this work significantly extends the results of previous studies of population dynamics and could have profound implications for the understanding of biological complexity in systems with a large number of species

reductions of integrable equations on a iii type symmetric spaces we study a class of integrable nonlinear differential equations related to the a iii type symmetric spaces these spaces are realized as factor groups of the form su n s u n k u k we use the cartan involution corresponding to this symmetric space as an element of the reduction group and restrict generic lax operators to this symmetric space the symmetries of the lax operator are inherited by the fundamental analytic solutions and give a characterization of the corresponding riemann hilbert data

fixation in a cyclic lotka volterra model we study a cyclic lotka volterra model of n interacting species populating a d dimensional lattice in the realm of a kirkwood approximation a critical number of species nc d above which the system fixates is determined analytically we find nc d 5 14 23 in dimensionsdd 1 2 3 in remarkably good agreement with simulation results in two dimensions a cyclic variant of the lotka volterra model of interacting populations originally introduced by vito volterra for the description of the struggle for existence among species 1 2 has then appeared in a number of apparently unrelated fields ranging from plasma physics 3 to integrable systems 4 5 recently the cyclic lotka volterra model also known as the cyclic voter model has attracted considerable interest as it was realized that introduction of the spatial structure drastically enriches the dynamics 6 11 namely if species live on a one dimensional 1d lattice a homogeneous initial state evolves into a coarsening mosaic of interacting species this heterogeneous spatial structure spontaneously develops when the number of species is sufficiently small n nc fixation occurs i e the system approaches a frozen state little is known in higher dimensions not even the existence of nc has yet been established theoretically or numerically in simulations on two dimensional 2d lattices with n 6 10 species no sign of fixation has been found and instead a reactive steady state has been observed 6 11 in this letter we investigate the cyclic lotka volterra model in the framework of a kirkwood like approximation this approach predicts a finite nc in all spatial dimensions in the following we shall use the language of the voter model 12 consider the cyclic voter model with n possible opinions each site of a d dimensional cubic lattice is occupied by a voter which has an opinion labelled by with d 1 n voters can change their opinions in a cyclic manner 1 modulo n according to the opinions of their neighbourhood specifically the following sequential dynamics is implemented i we randomly choose a site of opinion say and one of its 2d nearest neighbours of opinion ii if d 1 then the chosen site changes its opinion from to d 1 and iii otherwise opinion does not change we set the timescale so that in unit time each site of the lattice is chosen once on average when nd 2 the cyclic voter model is

geometrical properties of the potts model during the coarsening regime we study the dynamic evolution of geometric structures in a polydegenerate system represented by a q state potts model with nonconserved order parameter that is quenched from its disordered into its ordered phase the numerical results obtained with monte carlo simulations show a strong relation between the statistical properties of hull perimeters in the initial state and during coarsening the statistics and morphology of the structures that are larger than the averaged ones are those of the initial state while the ones of small structures are determined by the curvature driven dynamic process we link the hull properties to the ones of the areas they enclose we analyze the linear von neumann mullins law both for individual domains and on the average concluding that its validity for the later case is limited to domains with number of sides around 6 while presenting stronger violations in the former case

a twisted integrable hierarchy with d2 symmetry a loop algebra approach to the gerdjikov mikhailov valchev gmv equation is provided to exploit the associated twisted integrable structure and a new twisted integrable hierarchy is discovered using the twisted loop algebra structure we obtain a transparent treatment of the associated scattering and inverse scattering theory and solve the initial value problem for the gmv equation

wave fronts and cascades of soliton interactions in the periodic two dimensional volterra system in the paper we develop the dressing method for the solution of the two dimensional periodic volterra system with a period n we derive soliton solutions of arbitrary rank k and give a full classification of rank 1 solutions we have found a new class of exact solutions corresponding to wave fronts which represent smooth interfaces between two nonlinear periodic waves or a periodic wave and a trivial zero solution the wave fronts are non stationary and they propagate with a constant average velocity the system also has soliton solutions similar to breathers which resembles soliton webs in the kp theory we associate the classification of soliton solutions with the schubert decomposition of the grassmannians grr k n and grc k n

darboux transformations with tetrahedral reduction group and related integrable systems in this paper we derive new two component integrable differential difference and partial difference systems by applying a lax darboux scheme to an operator formed from an x1d530 x1d529 3 based automorphic lie algebra the integrability of the found systems is demonstrated via lax pairs and generalised symmetries

reduction groups and automorphic lie algebras we study a new class of infinite dimensional lie algebras which has important applications to the theory of integrable equations the construction of these algebras is very similar to the one for automorphic functions and this motivates the name automorphic lie algebras for automorphic lie algebras we present bases in which they are quasigraded and all structure constants can be written out explicitly these algebras have useful factorisations on two subalgebras similar to the factorisation of the current algebra on the positive and negative parts

on the recursion operators for the gerdjikov mikhailov and valchev system we consider the recursion operator approach to the soliton equations related to an auxiliary linear system introduced recently by gerdjikov mikhailov and valchev gmv system we discuss the recursion operators obtained for the gmv system and show that they are particular cases of the generating operators related to a zakharov shabat type linear system on the algebra sl 3 c in pole gauge

curvature driven coarsening in the two dimensional potts model we study the geometric properties of polymixtures after a sudden quench in temperature we mimic these systems with the q states potts model on a square lattice with and without weak quenched disorder and their evolution with monte carlo simulations with nonconserved order parameter we analyze the distribution of hull enclosed areas for different initial conditions and compare our results with recent exact and numerical findings for q 2ising case our results demonstrate the memory of the presence or absence of long range correlations in the initial state during the coarsening regime and exhibit superuniversality properties

the surface statistics of a granular aggregate the problem of the surface fluctuations in a settled granular material is posed a simple model is given which describes the process by which a particle settles and comes to rest on the existing surface of the packing and from this a set of langevin equations for the fourier modes of the surface are derived these equations imply that the fourier amplitudes behave like the velocities of a set of independent brownian particles we show that this results in logarithmically divergent surface fluctuations if the flux of particles onto the surface is random the divergence being removed by a more accurate description of the settling material for example by having the granules fall through a sieve

reductions of integrable equations dihedral group we discuss the algebraic and analytic structure of rational lax operators with algebraic reductions of lax equations we associate a reduction group a group of automorphisms of the corresponding infinite dimensional lie algebra we present a complete study of dihedral reductions for sl 2 c lax operators with simple poles and corresponding integrable equations in the last section we give three examples of dihedral reductions for sl n c lax operators

on the asymptotic behavior of solutions of evolution equations associated with nonlinear volterra equations soit x un espace de banach reel on considere le probleme aux valeurs initiales f t du t dt au t g u t r 0 u 0 u 0 ou a est un operateur m accretif dans x f l loc 1 r x u 0 d a et g une application donnee