classification of steady solutions of the full kinematic model abstract the geometric type of kinematic model governs any moving plane curve which propagates in the normal direction in the medium there have appeared some results in the literature via this model however these results are all based upon some simplification on the model or prior phenomenological assumptions on the solutions in this paper we use really full kinematic model to classify steady solutions i e curvature is independent of time with positive or negative curvature various type of solutions have been obtained such as rotating and translating waves especially for the rotating spiral waves spiral waves arise in many biological chemical and physiological systems the kinematical model can be used to describe the motion of the spiral arms approximated as curves in the plane in fact using our results theorem 3 8 we can answer the following questions is there any steadily rotating wave for a given weakly excitable medium if yes what kind of information we can know about these rotating waves e g the tip s curvature the tip s tangential velocity and the rotating frequency comparing our results with previous ones in the literature there are some differences between them the only evolving plane curves with positive or negative curvature must have monotonous curvatures via simplified model but full model admits solutions with any given oscillating number of the curvatures

spiral waves in reaction diffusion equations we consider the reaction diffusion system begin gathered r t nabla 2 r r left 1 r 2 vec nabla theta cdot vec nabla theta right hfill r theta t r nabla 2 theta 2 vec nabla r cdot vec nabla theta qr 3 hfill end gathered this system governs the solutions of reaction diffusion systems near a hopf bifurcation point in two spatial dimensions we use formal asymptotic expansions to contruct one armed and multi armed archimedean spiral waves for small values of the parameter q we then show that the one armed spiral waves are probably stable and the multi armed ones are unstable for q small next by numerical continuation methods we construct spiral waves for all q these calculations show that the one armed spiral waves are unstable when q 1 397 cdots we also find the explicit one dimensional analogues of spiral waves for all q and show that they are unstable for q small

the finite toda lattices connection is established between one dimensional toda lattices constructed on the basis of the systems of simple roots of classical and affine lie algebras and other integrable systems of interacting particles that connection allows us to find new lattices differing from the known ones by the interaction of particles near the ends some of the new lattices admit non abelian generalizations

hypermeander of spirals local bifurcations and statistical properties in both experimental studies and numerical simulations of waves in excitable media rigidly rotating spiral waves are observed to undergo transitions to complicated spatial dynamics with long term brownian like motion of the spiral tip this phenomenon is known as hypermeander in this paper we review a number of recent results on dynamics with noncompact group symmetries and make the case that hypermeander may occur at a codimension two bifurcation from a rigidly rotating spiral wave our predictions are based on center bundle reduction sandstede scheel and wulff and on central limit theorems and invariance principles for group extensions of hyperbolic dynamical systems these predictions are confirmed by numerical simulations of the center bundle equations

spiral selection as a free boundary problem abstract we present a new formulation of the spiral selection problem for the belousov zhabotinsky reaction in particular we focus on deriving an exact integro differential shape equation and discuss the possible behavior of solutions to this equation we also present some new results on the asymptotic far from core structure of spirals in the fife scaling regime

a geometrical theory for spiral waves in excitable media in this paper we develop a geometrical theory for waves in excitable reacting media using singular perturbation arguments and dispersion of traveling plane wave trains we derive an approximate theory of wave front propagation which has strong resemblance to the geometrical diffraction theory of high frequency waves in hyperbolic systems governed by the eikonal equation using this theory we study the effect of curvature on waves in excitable media specifically rotating spiral patterns in planar regions from this theory we are able to determine the frequency and wavelength for spiral patterns in excitable nonoscillatory media

the neatest many body problem amenable to exact treatments a goldfish various formulations findings and conjectures are reviewed which relate to a many body problem that is arguably the neatest nontrivial such model amenable to exact treatments

nonlinear optimization exclusion tests for finding all solutions of nonlinear equations abstract exclusion tests are a well known tool in the area of interval analysis for finding the zeros of a function over a compact domain recently k georg developed linear programming exclusion tests based on taylor expansions in this paper we modify his approach by choosing another objective function and using nonlinear constraints to make the new algorithm converges faster than the algorithm in k georg a new exclusion test j comput appl math 152 2003 147 160 in this way we reduce the number of subinterval in each level the computational complexity for the new tests are investigated also numerical results and comparisons will be presented

analysis of meandering in zykov kinetics abstract an analysis of computer results of zykov kinetics is given the meandering of the spiral tip is modeled by a composition of two circular motions one of them is the circle along which the spiral tip would move if the spirals were rotating rigidly the other one is the circle along which the spiral tip moves around a point of the previous circle this model suggests that the meandering of the spiral tip develops from the rigid rotation through a hopf bifurcation

integrable hydrodynamic chains a new approach for derivation of benney type moment chains and integrable hydrodynamic type systems is presented new integrable hydrodynamic chains are constructed all their hydrodynamical reductions are described and integrated new 2 1 integrable hydrodynamic type systems are found

on the stability of steady rotating waves in the free boundary formulation abstract we perform a linear stability analysis of the steadily rotating solutions found from the free boundary problem of wave propagation in single diffusive excitable media two kind of modes can be considered a tangential mode which moves the spiral tip tangentially to the circle limiting the unexcited region and a radial mode which moves the spiral tip outwards the same circle in the radial direction the study of the second kind of modes is complicated by a technical difficulty associated to the divergence of the refractory field perturbations at the spiral tip square root difficulty the tangential modes are found to be stable whatever the angular velocity of the steady solution

differential geometric approach to the integrability of hydrodynamic chains the haantjes tensor the integrability of an m component system of hydrodynamic type u t v u u x by the generalized hodograph method requires the diagonalizability of the m m matrix v u this condition is known to be equivalent to the vanishing of the corresponding haantjes tensor we generalize this approach to hydrodynamic chains infinite component systems of hydrodynamic type for which the matrix v u is sufficiently sparse for such systems the haantjes tensor is well defined and the calculation of its components involves finite summations only we illustrate our approach by classifying broad classes of conservative and hamiltonian hydrodynamic chains with the zero haantjes tensor we prove that the vanishing of the haantjes tensor is a necessary condition for a hydrodynamic chain to possess an infinity of semi hamiltonian hydrodynamic reductions thus providing an easy to verify necessary condition for the integrability

a class of integrable hamiltonian systems whose solutions are perhaps all completely periodic we show that the dynamical system characterized by the complex equations of motion qj i qj k 1 k jnqjqkf qj qk j 1 n with f x x x is hamiltonian and integrable and we conjecture that all its solutions qj t j 1 n are completely periodic with a period that is a finite integral multiple of t 2 here n is an arbitrary positive integer is an arbitrary nonvanishing real constant y y is the weierstrass function with arbitrary semiperiods and are two arbitrary constants special cases are f x 2 coth x 1 r2 sinh2 x f x 2 coth x f x 2 sinh x f x 2 x 1 2x2 and of course f x 2 x these findings as well as the conjecture which is shown to be true in some of these special cases are based on the possibility to recast these equations of motion in the modified lax form l i l l m with l and m appropriate n n matrix functions of the n dynamical variables qj and of their time derivatives qj

rigidity functional equations and the calogero moser model suppose we have a natural hamiltonian h of n particles on the line centre of mass momentum p and a further independent quantity q cubic in the momenta if these are each sn invariant and mutually poisson commute we have the calogero moser system with potential v 1 6 i j qi qj const

kinematical theory of spiral waves in excitable media comparison with numerical simulations abstract the kinematical theory of spiral waves is compared with the data of numerical simulations using complete partial differential equations of a reaction diffusion model we find good agreement for temporal periods of spiral waves which rotate around minimal holes and in the free medium no adjustable parameters were used in the calculations

discrete versions of some classical integrable systems and factorization of matrix polynomials discrete versions of several classical integrable systems are investigated such as a discrete analogue of the higher dimensional force free spinning top euler arnold equations the heisenberg chain with classical spins and a new discrete system on the stiefel manifold the integrability is shown with the help of a lax pair representation which is found via a factorization of certain matrix polynomials the complete description of the dynamics is given in terms of abelian functions the flow becomes linear on a prym variety corresponding to a spectral curve the approach is also applied to the billiard problem in the interior of ann dimensional ellipsoid

spiral waves in the belousov zhabotinskii reaction abstract the beautiful spiral waves of oxidation in the belousov zhabotinskii reaction are the source of many interesting and important questions about periods structures in excitable media it has long been known that these spirals are similar to involutes of circles at least some distance from the center but until now no way has been known to determine the correct wavelength and frequency in this paper we show that the parameters of a spiral wave can be viwed s eigenvalues of a problem with unique solution the critical ingredients of the theory are the effects of curvature on the propagation of wavefronts in two dimensional media and the dispersion of plane waves our analytical results are shown to be in good agreement with experimental data for the belousov zhabotinskii reagent

a technique to identify solvable dynamical systems and a solvable generalization of the goldfish many body problem a simple approach is discussed which associates to solvable matrix equations solvable dynamical systems generally interpretable as interesting many body problems possibly involving auxiliary dependent variables in addition to those identifying the positions of the moving particles we then focus on cases in which the auxiliary variables can be altogether eliminated reobtaining thereby via this unified approach well known solvable many body problems and moreover a solvable extension of the goldfish model

the lax pairs for elliptic cn and bcn ruijsenaars schneider models and their spectral curves we study the elliptic c n and bcn ruijsenaars schneider models which are elliptic generalization of systems given in previous paper by the present authors chen j math phys 41 8132 2000 the lax pairs for these models are constructed by hamiltonian reduction technology we show that the spectral curves can be parametrized by the involutive integrals of motion for these models taking nonrelativistic limit and scaling limit we verify that they lead to the systems corresponding to calogero moser and toda types c 2001 american institute of physics

finding all solution sets of piecewise linear interval equations using an integer programming solver abstract this paper presents an efficient and easily implementable method for finding all solution sets of piecewise linear interval equations using integer programming in this method the problem of finding all solution sets is formulated as a mixed integer programming problem and it is solved by a high performance integer programming solver such as cplex it is shown that all solution sets are obtained by solving a single mixed integer programming problem and related linear programming problems numerical examples are given to confirm the effectiveness of the proposed method

a new exclusion test exclusion tests are a well known tool for obtaining all solutions of a nonlinear system of equations see e g the work of krawczyk or moore recently yamamura and collaborators have developed linear programming exclusion tests that turned out to be highly successful for nonlinear problems in electrical networking the author has developed higher order exclusion tests based on taylor expansions in the present paper it is shown that the ideas behind both approaches can be combined we present a new class of higher order linear programming exclusion tests investigate their computational complexity and illustrate their performance on several examples

finding all solutions of nonlinear equations using the dual simplex method recently an efficient algorithm has been proposed for finding all solutions of systems of nonlinear equations using linear programming this algorithm is based on a simple test termed the lp test for nonexistence of a solution to a system of nonlinear equations using the dual simplex method in this letter an improved version of the lp test algorithm is proposed by numerical examples it is shown that the proposed algorithm could find all solutions of a system of 300 nonlinear equations in practical computation time

wave front interaction in steadily rotating spirals abstract we develop a self consistent model for excitable media the so called wave front interaction model wfm in which two waves the front and the back interact with a relaxational field c 2 we study numerically the solutions corresponding to spirals rotating steadily with an angular velocity with a tip rotating around an unexcited region of radius r 0 it is found that these two quantities are determined uniquely as a function of e the ratio between fast reaction time and refractory time and the excitability corresponding shapes for spirals are determined and in particular the shape and the size of the core a semi analytical solution for the spiral shape is found in the large r 0 limit

spirals in excitable media the free boundary limit with diffusion abstract we solve numerically for the steady state spiral in the thin interface limit including the effects of diffusion of the slow field the calculation is performed using a generalization of the hybrid scheme of keener in this method the diffusion equation is solved on a suitable mapped lattice while the eikonal equation relating the field on the interface to the interfacial velocity and curvature is solved independently we present results for the selected frequency and tip radius as a function of the various parameters we note that a stability analysis based on these results may be performed

lp narrowing a new strategy for finding all solutions of nonlinear equations an efficient algorithm is proposed for finding all solutions of systems of n nonlinear equations this algorithm is based on interval analysis and a new strategy called lp narrowing in the lp narrowing strategy boxes n dimensional rectangles in the solution domain containing no solution are excluded and boxes containing solutions are narrowed so that no solution is lost by using linear programming techniques since the lp narrowing is very powerful all solutions can be found very efficiently by numerical examples it is shown that the proposed algorithm could find all solutions of systems of 5000 50 000 nonlinear equations in practical computation time

a cell exclusion algorithm for determining all the solutions of a nonlinear system of equations abstract a new group of methods named cell exclusion algorithms ceas is developed for finding all the solutions of a nonlinear system of equations these types of algorithms different in principle from those of homotopy interval and cell mapping dynamical analysis approaches are based on cellular discretization and the use of a certain simple necessity test of the solutions the main advantages of the algorithms are their simplicity reliability and general applicability having all features of interval techniques but without using interval arithmetic and with complexity o log 1 e the algorithms improve significantly on both the interval algorithms and the cell mapping techniques theoretical analysis and numerical simulations both demonstrate that ceas are very efficient

dynamical r matrix for the elliptic ruijsenaars schneider system the classical r matrix structure for the generic elliptic ruijsenaars schneider model is presented it makes manifest the integrability of this model as well as of its discrete time version that was constructed in a recent paper

matrix factorizations and integrable systems on montre que les algorithmes qr lij et de cholesky pour calculer les valeurs propres de matrices reelles sont les evaluations a temps entier de flots hamiltoniens completement integrables

the lax representation for an integrable class of relativistic dynamical systems we exhibit the lax pair for the class of relativistic dynamical systems recently introduced by ruijsenaars and schneider whose equations of motion read whereb is an arbitrary constant and the weierstrass elliptic function

associated integrable systems new class of integrable hamiltonian systems is obtained as systems directly defined over jacobi varieties this approach enables one to establish a link between associated discrete and continuous systems in particular regular procedure is found for the discretization of the integrable continuous systems hierarchies of the continuous systems associated with toda lattices relativistic toda lattices and volterra lattices are found and studied

efficient and safe global constraints for handling numerical constraint systems numerical constraint systems are often handled by branch and prune algorithms that combine splitting techniques local consistencies and interval methods this paper first recalls the principles of tt quad a global constraint that works on a tight and safe linear relaxation of quadratic subsystems of constraints then it introduces a generalization of tt quad to polynomial constraint systems it also introduces a method to get safe linear relaxations and shows how to compute safe bounds of the variables of the linear constraint system different linearization techniques are investigated to limit the number of generated constraints tt quadsolver a new branch and prune algorithm that combines tt quad local consistencies and interval methods is introduced tt quadsolver has been evaluated on a variety of benchmarks from kinematics mechanics and robotics on these benchmarks it outperforms classical interval methods as well as constraint satisfaction problem solvers and it compares well with state of the art optimization solvers

dynamics of wave fronts in excitable media abstract the dynamics of wave fronts in two dimensional excitable media is analytically studied by means of the asymptotic perturbation method for wave fronts in reaction diffusion systems we derive a space derivative burgers equation which governs the shape of boundary layers and their propagation application of lie group analysis of differential equations yields group invariant solutions of this equation and their classification some classes of the group invariant solutions determine the remarkable patterns expanding circle waves rotating spiral waves initial value problems for the burgers equation are also studied and a spiral wave is shown to be asymptotically realized under some proper boundary conditions

the dn ruijsenaars schneider model the lax pair of the ruijsenaars schneider model with interaction potential of trigonometric type based on dn lie algebra is presented we give a general form for the lax pair and prove partial results for small n liouville integrability of the corresponding system follows a series of involutive hamiltonians generated by the characteristic polynomial of the lax matrix the rational case appears as a natural degeneration and the non relativistic limit exactly leads to the well known calogero moser system associated with dn lie algebra

a modified toda spectral problem and its hierarchy of bi hamiltonian lattice equations starting from a modified toda spectral problem a hierarchy of generalized toda lattice equations with two arbitrary constants is constructed through discrete zero curvature equations it is shown that the hierarchy possesses a bi hamiltonian structure and a hereditary recursion operator which implies that there exist infinitely many common commuting symmetries and infinitely many common commuting conserved functionals two cases of the involved constants present two specific integrable sub hierarchies one of which is exactly the toda lattice hierarchy

a closed form solution of the one dimensional born green yvon equation for a hard rod fluid using the method of the inverse scattering transform we can generate an infinite number of closed form solutions for the one dimensional nonlinear born green yvon differential difference equation that governs an approximate expression for the equilibrium pair correlation function of a homogeneous system of hard rods

theory of the spiral core in excitable media abstract a rigorous asymptotic spiral solution to an excitable reaction diffusion system is found by separating space into two scaling regions an outer region having the same spatial scaling as the overall spiral structure which exhibits a singularity at the spiral tip and a core region around the spiral tip where space is scaled so as to resolve the tip singularity the stability of the spiral structure is investigated for both the outer region which is found to possess no intrinsic instability and the core region which is found to be unstable both the zero and small diffusion systems are found to exhibit qualitatively similar behaviour the implication of these results for both experiment and simulation in particular within the context of the observed meandering instability are addressed

the core of the spiral the problem of finding the unique frequency and shape including the core of spirals in excitable media is studied a leading order approximation of the problem is formulated as a nonlinear eigenvalue problem involving only ordinary differential equations odes shooting and comparison arguments are used to show why solutions of the leading order approximate problem exist and some of their quantitative features are explored numerically because of inaccuracies of the results in certain parameter ranges an improved formulation of the problem is proposed when solved numerically as in the special case of symmetric spirals solutions of the improved formulation differ significantly from solutions of the leading order problem but agree closely with the numerical solution of the full partial differential equation pde system

integrable time discretisation of the ruijsenaars schneider model an exactly integrable symplectic correspondence is derived which in a continuum limit leads to the equations of motion of the relativistic generalization of the calogero moser system that was introduced for the first time by ruijsenaars and schneider for the discrete time model the equations of motion take the form of bethe ansatz equations for the inhomogeneous spin 1 2 xyz heisenberg magnet we present a lax pair the sympletic structure and prove the involutivity of the invariants exact solutions are investigated in the rational and hyperbolic trigonometric limits of the system that is given in terms of elliptic functions these solutions are connected with discrete soliton equations the results obtained allow us to consider the bethe ansatz equations as ones giving an integrable symplectic correspondence mixing the parameters of the quantum integrable system and the parameters of the corresponding bethe wavefunction

a technique to identify solvable dynamical systems and another solvable extension of the goldfish many body problem we take advantage of the simple approach recently discussed which associates to solvable matrix equations solvable dynamical systems interpretable as interesting many body problems possibly involving auxiliary dependent variables in addition to those identifying the positions of the moving particles starting from a solvable matrix evolution equation we obtain the corresponding many body model and note that in one case the auxiliary variables can be altogether eliminated obtaining thereby an also hamiltonian extension of the goldfish model the solvability of this novel model and of its isochronous variant is exhibited a related as well solvable model is also introduced as well as its isochronous variant finally the small oscillations of the isochronous models around their equilibrium configurations are investigated and from their isochronicity certain diophantine relations are evinced

finding all solutions of separable systems of piecewise linear equations using integer programming abstract finding all solutions of nonlinear or piecewise linear equations is an important problem which is widely encountered in science and engineering various algorithms have been proposed for this problem however the implementation of these algorithms are generally difficult for non experts or beginners in this paper an efficient method is proposed for finding all solutions of separable systems of piecewise linear equations using integer programming in this method we formulate the problem of finding all solutions by a mixed integer programming problem and solve it by a high performance integer programming software such as glpk scip or cplex it is shown that the proposed method can be easily implemented without making complicated programs it is also confirmed by numerical examples that the proposed method can find all solutions of medium scale systems of piecewise linear equations in practical computation time

singular perturbation theory of traveling waves in excitable media a review abstract waves of chemical or electrical activity traveling through space have been observed in several contexts chemical reaction mixtures in non convecting liquid phase cell suspensions nerve axons and neuromuscular tissues typically wave supporting preparations are excitable that is they respond and sensitivity to perturbations are rapidly damped out but suprathreshold disturbances trigger an abrupt and substantial response the abruptness of the response can be exploited by the methods of singular perturbation theory to obtain a mathematical description of wave propagation in spatially distributed excitable media singular perturbation analysis of propagating waves in one spatial dimension is straightforward and uncontentious but the analysis of propagating waves in one spatial dimension is straightforward and several fundamentally different ways we compare and contrast the approaches taken by greenberg zykov fife krinskii and his collaborators and ourselves with particular emphasis on the case of rotating spiral waves our intention is to bring some order to the important but difficult theory of propagating waves in two dimensional excitable media in conclusion we discuss briefly some possible extensions of the singular perturbation approach to propagating wave surfaces in three dimensional space

the non dynamical r matrix structure for the elliptic calogero moser model in this paper we construct a new lax operator for the elliptic calogero moser model with general from the classical dynamical twisting in which the corresponding r matrix is purely numerical non dynamic the non dynamical r matrix structure of this lax operator is obtained which is the elliptic symmetric r matrix

dynamics of spiral waves on unbounded domains using center manifold reductions abstract an equivariant center manifold reduction near relative equilibria ofg equivariant semiflows on banach spaces is presented in contrast to previous results the lie groupginduces a strongly continuous group action on the underlying function space in fact gmay act discontinuously the results are applied to bifurcations of stable patterns arising in reaction diffusion systems on the plane or in three space modeling chemical systems such as catalysis on platinum surfaces and belousov zhabotinsky reactions these systems are equivariant under the euclidean symmetry group hopf bifurcations from rigidly rotating spiral waves to meandering or drifting waves and from twisted scroll rings are investigated