a nonlinear elliptic equation arising from gauge field theory and cosmology we study radially symmetric solutions of a nonlinear elliptic partial differential equation in r 2 with critical sobolev growth i e the nonlinearity is of exponential type this problem arises from a wide variety of important areas in theoretical physics including superconductivity and cosmology our results lead to many interesting implications for the physical problems considered for example for the self dual chern simons theory we are able to conclude that the electric charge magnetic flux or energy of a non topological n vortex solution may assume any prescribed value above an explicit lower bound for the einstein matter gauge equations we find a necessary and sufficient condition for the existence of a self dual cosmic string solution such a condition imposes an obstruction for the winding number of a string in terms of the universal gravitational constant

on a liouville type equation with sign changing weight in this paper we study the existence nonexistence and multiplicity of non negative solutions for the family of problems u a x e f x u u h 0 where is a bounded domain in r2 and 0 is a parameter the coefficient a x is allowed to change sign the techniques used in the proofs are a combination of upper and lower solutions the trudingermoser inequality and variational methods note that when f x u 0 the equation is of liouville type

synchronization in lattices of coupled oscillators abstract we consider coupled nonlinear oscillators with external periodic forces and the dirichlet boundary conditions we prove that synchronization occurs provided that the coupling is dissipative and the coupling coefficients are sufficiently large the synchronization here is of an obvious type the size of an attractor is comparable to the difference of the subsystems

enriched finite elements for regions with multiple interacting singular fields enriched two dimensional finite elements are formulated for analysis of solids with interacting singular points these elements contain both the order and the angular variation of all singular stress fields emanating from each singular point so that the possible interaction of stress states present in such regions is represented the order and distribution of the singular fields are determined with a separate finite element eigenanalysis these field characterizations and the subsequent enrichment are carried out numerically rather than analytically for ease of extension of the technique to complex geometries for which analytical solutions would be difficult to obtain generalized stress intensity factors as well as the stresses and displacements result directly from models incorporating enriched elements enriched element models can be applied not only to crack problems but also to singular stress states not involving cracks and crack growth and are thus more versatile than j integral and virtual crack closure techniques used for calculating energy release rates in addition the proper angular distribution of the displacement fields leading to each singular stress state is used in formulating the element an advantage not shared by formulations such as the quarter point elements the advantages of using enriched elements containing information from interacting singular points are demonstrated with an example having two interacting singular points this tensile groove specimen is modeled using these new elements as well as enriched elements that do not take interaction into consideration attention is given to modeling issues when using interactive enriched elements

brief paper estimating the region of attraction for large scale systems with uncertainties in this paper a new approach is proposed for estimating regions of attraction for large scale dynamic systems in designing control laws for such systems it is essential to incorporate the underlying information structure constraints while keeping the number of optimization variables at a minimum the proposed method successfully accomplishes both of these objectives it is computationally efficient and can produce decentralized control laws without imposing structural constraints on the lyapunov function a feature that can considerably improve the quality of the estimate the design algorithm is based on linear matrix inequalities and can easily accommodate various types of uncertainties in the system model an example with 300 states is provided to demonstrate the suitability of this approach for large sparse systems

self dual maxwell chern simons theory on a cylinder in this paper we study the relativistic maxwell chern simons vortices on an asymptotically flat cylinder a topological multivortex solution is constructed by variational methods and the maxwell and the chern simons limits are verified

su 3 chern simons vortex theory and toda systems motivated by the study of the asymptotic properties of non topological condensates in the non abelian chern simons vortex theory see 26 we analyze the su 3 toda system p 1 2 zj j e 2i 1kijzi me i 12kijzi 1 on m mzj 0j 1 2 where m r2 z2 k kij 2 1 12 is the su 3 cartan matrix and j are positive parameters we study the variational problem associated with the system p 1 2 in a range of parameters where the trivial solution is a strict local minimum and the corresponding sobolev type inequality fails to apply in this situation a lack of compactness may occur due to concentration phenomena nonetheless we are able to establish the existence of a non trivial solution for p 1 2 which is not a minimizer

the relativistic non abelian chern simons equations we study ther xr system of nonlinear elliptic equations delta u a lambda sum limits b 1 r k ab e u b lambda sum limits b 1 r sum limits c 1 r e u b k ab e u c k bc 4 pi sum limits j 1 n a delta p aj a 1 2 r x r 2 where 0 is a constant parameter k kab is the cartan matrix of a semi simple lie algebra and p is the dirac measure concentrated atp r 2 this system of equations arises in the relativistic non abelian chern simons theory and may be viewed as a nonintegrable deformation of the integrable toda system we establish the existence of a class of solutions known as topological multivortices the crucial step in our method is the use of the decomposition theorem of cholesky for positive definite matrices so that a variational principle can be formulated

a system of elliptic equations arising in chern simons field theory we prove the existence of topological vortices in a relativistic self dual abelian chem simons theory with two higgs particles and two gauge fields through a study of a coupled system of two nonlinear elliptic equations over r 2 we present two approaches to prove existence of solutions on bounded domains via minimization of an indefinite functional and via a fixed point argument we then show that we may pass to the full r 2 limit from the bounded domain solutions to obtain a topological solution in r 2

development and applications of two singular points finite elements an element formulation is suggested to obtain variable order singularities simultaneously at the two neighbouring corner nodes of a 4 noded quadrilateral the element is useful for modelling a kinked crack with small kink length the accuracy is good and this is obtained at a substantially reduced computational cost case studies are presented to illustrate the usefulness and the accuracy obtainable

vortex condensations of a non relativistic maxwell chern simons theory abstract we study an elliptic system subject to periodic boundary conditions arising from the study of vortex condensates for a selfdual non relativistic model introduced by dunne et al phys rev d 43 1332 1345 1990 in the framework of chern simons gauge theory we prove the existence of a vortex condensate for all values of the relevant physical parameters involved in the model this completes a recent work of spruck and yang 16 who could deal only with a restricted set of those parameters uniqueness is also discussed

on the entire radial solutions of the chern simons su 3 system in this paper we study the entire radial solutions of the self dual equations arising from the relativistic su 3 chern simons model proposed by kao and lee phys rev d 50 6626 6632 1994 and dunne phys lett b 345 452 457 1995 nuclear phys b 433 333 348 1995 understanding the structure of entire radial solutions is one of the fundamental issues for the system of nonlinear equations in this paper we prove that any entire radial solutions must be one of topological non topological and mixed type solutions and completely classify the asymptotic behaviors at infinity of these solutions even for radial solutions this classification has remained an open problem for many years as an application of this classification we prove that the two components u and v have intersection at most finite times

existence of topological vortices in an abelian chern simons model in this paper we prove the existence of topological vortices by variational method applied on an abelian chern simons model with a generic renormalizable potential we also establish some properties of the solutions

new type of nontopological bubbling solutions in the su 3 chern simons model in r2 abstract we are concerned with su 3 chern simons model in r 2 nontopological solutions were shown to exist in 10 in a radially symmetric situation also in 10 there constructed a sequence of partially bubbling solutions of which one component only bubbles we here construct a sequence of radial nontopological bubbling solutions of another type explicitly one component of these bubbling solutions locally converges to a nice function but both components blow up partially in different regions at infinity at the same time this type of bubbling solutions not only is a new phenomenon to this type of systems but also shows a more complicated pattern of the nontopological solution structure than the abelian chern simons higgs equation proposed by 13 15

asymptotic behavior of condensate solutions in the chern simons higgs theory we study the asymptotic behavior of condensate solutions in the chern simons higgs model as the chern simons coupling parameter tends to zero using the variational method we prove that there exist condensate solutions which show concentration phenomena

on the chern simons higgs vortex equation without gauge fields abstract this paper is concerned with the nonself dual chern simons higgs model on r 2 with vanishing gauge fields we prove the existence of radial solutions with the topological boundary condition and the nonexistence of radial solutions with the nontopological boundary condition we also establish the asymptotic properties of solutions and derive the quantization of the potential energy

existence of the semilocal chern simons vortices we consider the bogomol nyi equations of the abelian chern simons higgs model with su n global u 1 local symmetry this is a generalization of the well known abelian chern simons higgs model with u 1 local symmetry we prove existence of both topological and nontopological multivortex solutions of the system on the plane

existence and uniqueness for mean field equations on multiply connected domains at the critical parameter we consider the mean field equation begin aligned left begin array l quad l vardelta u rho frac e u int varomega e u 0 hbox in varomega u 0 hbox on partial varomega end array right end aligned 1 where varomega subset mathbb r 2 is an open and bounded domain of class c 1 in his 1992 paper suzuki proved that if varomega is a simply connected domain then eq 1 admits a unique solution for rho in 0 8 pi this result for varomega a simply connected domain has been extended to the case rho 8 pi by chang chen and the second author however the uniqueness result for varomega a multiply connected domain has remained a long standing open problem which we solve positively here for rho in 0 8 pi to obtain this result we need a new version of the classical bol s inequality suitable to be applied on multiply connected domains our second main concern is the existence of solutions for 1 when rho 8 pi we obtain a necessary and sufficient condition for the solvability of the mean field equation at rho 8 pi which is expressed in terms of the robin s function gamma for varomega for example if eq 1 has no solution at rho 8 pi then gamma has a unique nondegenerate maximum point as a by product of our results we solve the long standing open problem of the equivalence of canonical and microcanonical ensembles in the onsager s statistical description of two dimensional turbulence on multiply connected domains

a uniform invariance principle for periodic systems with applications to synchronization abstract in this paper we propose an extension of the invariance principle which is uniform with respect to parameter uncertainties for the class of periodic ordinary differential equations this extension allows the derivative of the auxiliary function v commonly called a lyapunov function to be positive in some bounded sets this important feature has the potential to simplify the problem of exhibiting a function of lyapunov type and allows the application of the principle in systems that cannot be treated with the conventional principle either due to the nonexistence of a lyapunov type function or due to the difficulty in exhibiting it the extension of the invariance principle is useful to obtain estimates of attractors and regions of attraction that are uniform with respect to parameters the study of synchronization of periodic coupled systems illustrates an application of the principle

existence theorems for non abelian chern simons higgs vortices with flavor abstract in this paper we establish the existence of vortex solutions for a chern simons higgs model with gauge group su n u 1 and flavor su n these symmetries ensure the existence of genuine non abelian vortices through a color flavor locking under a suitable ansatz we reduce the problem to a 2 2 system of nonlinear elliptic equations with exponential terms we study this system over the full plane and over a doubly periodic domain respectively for the planar case we use a variational argument to establish the existence result and derive the decay estimates of the solutions over the doubly periodic domain we show that the system admits at least two gauge distinct solutions carrying the same physical energy by using a constrained minimization approach and the mountain pass theorem in both cases we get the quantized vortex magnetic fluxes and electric charges

the existence of chern simons vortices a new type of vortices called chern simons vortices is considered and the existence is established

the generalized invariance principle for dynamic equations on time scales the study of long term behavior of a solution generated by dynamics on time scales always gains a great amount of attentions both from engineers and from mathematicians in the paper a generalized invariance principle which not only includes the case described by the conventional invariance principle but also involves the case where the sign of the delta derivative of the lyapunov function along with the solution is not determined is established this consequently allows us to verify the stability and boundedness of the solution generated by a much wider class of dynamic equations on time scales by utilizing this generalized principle

the liouville equation with singular data a concentration compactness principle via a local representation formula abstract for a bounded domain r 2 we establish a concentration compactness result for the following class of singular liouville equations u e u 4 j 1 m j p j in where pj j 0 and pj denotes the dirac measure with pole at point pj j 1 m our result extends brezis merle s theorem comm partial differential equations16 1991 1223 1253 concerning solution sequences for the regular liouville equation where the dirac measures are replaced by lp data p 1 in some particular case we also derive a mass quantization principle in the same spirit of li shafrir indiana univ math j 43 1994 1255 1270 our analysis was motivated by the study of the bogomol nyi equations arising in several self dual gauge field theories of interest in theoretical physics such as the chern simons theory self dual chern simons theories lecture notes in physics vol 36 springer verlag berlin 1995 and the electroweak theory selected papers on gauge theory of weak and electromagnetic interactions world scientific singapore

upper semicontinuity of attractors and synchronization abstract in this paper we prove that diffusively coupled abstract semilinear parabolic systems synchronize we apply the abstract results obtained to a class of ordinary differential equations and to reaction diffusion problems the technique consists of proving that the attractors for the coupled differential equations are upper semicontinuous with respect to the attractor of a limiting problem explicitly exhibited in the diagonal

topological multivortex solutions of the self dual maxwell chern simons higgs system abstract we study the existence and various behaviors of topological multivortex solutions of the relativistic self dual maxwell chern simons higgs system we first prove the existence of general topological solutions by applying variational methods to the newly discovered minimizing functional then by an iteration method we prove the existence of topological solutions satisfying some extra conditions which we call admissible solutions we establish asymptotic exponential decay estimates for these topological solutions we also investigate the limiting behavior of the admissible solutions as parameters in our system go to some limits for the abelian higgs limit we obtain strong convergence result while for the chern simons limit we only obtained that our admissible solutions weakly approximate one of the chern simons solutions

an invariance principle for nonlinear discrete autonomous dynamical systems this note proposes an extension of lasalle s invariance principle for nonlinear discrete autonomous dynamical systems the invariance principle is extended to allow the first difference of the auxiliar scalar function usually a lyapunov function to be positive in some bounded regions moreover a uniform version is proposed to deal with nonlinear discrete dynamical systems that vary with parameters both extensions have the original invariance principle as a particular case as a consequence a larger class of systems can be treated with this new theory the extensions are very useful to obtain attractor estimates as well as their corresponding stability regions the uniform version in particular is useful to obtain estimates that are uniform regarding parameters

existence and uniqueness of topological multivortex solutions of the self dual chern simons cp 1 model abstract in this paper we study the existence and uniqueness of the topological chern simons vortices in the cp 1 model in r 2 after reducing the self dual equations to semilinear elliptic partial differential equations we show that a topological solution exists and it is unique up to a real smooth function if the chern simons coupling constant is sufficiently small

concentrating solutions for an exponential nonlinearity with robin boundary condition abstract we study the following robin boundary value problem 0 1 u 2 u p 1 e u p 0 u 0 in u n u 0 on where is a bounded domain in r 2 with smooth boundary as 0 and simultaneously 0 p 2 by using a lyapunov schmidt reduction procedure we extend for the whole range of exponents 0 p 2 the results of davila and topp 2012 who constructed bubbling solutions to problem 0 1 for p 1

on the use of one point and two points singularity elements in the analysis of kinked cracks a 3 noded triangular two singular points finite element is proposed to model two variable order singularities lying at close proximities as in the case of a kinked crack the element meets the rigid body mode and the interelement compatibility of the convergence criteria a degenerate form of the element is a one point variable order singularity element which meets all the convergence requirements a number of kinked crack problems have been analysed using the new element and its degenerate form the computed results are compared with analytical solutions the accuracy of the results is found to be good

existence theorems for a general 2 2 non abelian chern simons higgs system over a torus in this paper we study a general 2 2 non abelian chern simons higgs system of the form ui 1e2 j 12kjieuj j 12 k 12kkjkjieujeuk 4 j 1ni pij x i 1 2 over a flat 2 torus t2 where e 0 p is the dirac measure at p ni n i 1 2 k is a non degenerate 2 2 matrix of the form k 1 a a b1 b which may cover the physically interesting case when k is a cartan matrix of a rank 2 semisimple lie algebra concerning the existence results of this type system over t2 usually in the literature there is a requirement that a b 0 however it is an open problem so far for the solvability about such system with a b 0 such that this system admits a solution over the torus if 0 e e0 provided a b are suitably small furthermore if ab 0 in addition with suitable condition on a b n1 n2 this system admits a mountain pass solution our argument is based on a perturbation approach and the mountain pass lemma

nontopological n vortex condensates for the self dual chern simons theory we prove the existence of nontopological n vortex solutions for an arbitrary number n of vortex points for the self dual chern simons higgs theory with t hooft periodic boundary conditions we use a shadowing type lemma to glue together any number of single vortices obtained as a perturbation of a radially symmetric entire solution of the liouville equation 2003 wiley periodicals inc

existence of the self graviting chern simons vortices we prove existence of multivortex solutions of the self dual einstein chern simons higgs system proposed by clement phys rev d 54 1844 1847 1996 we consider both the topological and the nontopological boundary conditions for open conformally flat manifolds for nontopological boundary conditions we use perturbation argument from a solution of the liouville equation combined with the implicit function theorem using this argument we have existence for arbitrary positive number for the gravitational constant for topological boundary condition we construct solutions for small gravitational constant by using the super subsolution method for sufficiently large gravitational constant we have a nonexistence result for the radially symmetric topological solutions we also obtain the decay estimates near infinity for both of the topological and the nontopological solutions

uniform invariance principle and synchronization robustness with respect to parameter variation abstract the objective of this work is to obtain uniform estimates with respect to parameters of the attractor and of the basin of attraction of a dynamical system and to apply these results to analyze the roughness of the synchronization of two subsystems these estimates are obtained through a uniform version of the invariance principle of la salle which is stated and proved in this work

invariance principles allowing of non lyapunov functions for estimating attractor of discrete dynamical systems this technical note establishes several versions of invariance principles for describing the eventual dynamical behaviors of discrete dynamical systems instead of the requirement of the so called lyapunov functions in the classical lasalle invariance principle some more relaxed conditions are imported the established invariance principles thus can be applied to a more general class of discrete dynamical systems for classifying their orbits into two categories based on the eventual dynamical behaviors and the proposed classification scheme is suitable for theoretically and numerically estimating the local or global attractors produced by the discrete dynamical systems the practical usefulness of the analytical results is verified by systematically investigating several representative discrete systems

local exact boundary controllability of the boussinesq equation we study the local exact boundary controllability problem for the boussinesq equations that describe an incompressible fluid flow coupled to thermal dynamics the result that we get in this paper is as follows suppose that hat y t x is a given solution of the boussinesq equation where t in 0 t x in omega omega is a bounded domain with c infty boundary partial omega let y 0 x be a given initial condition and vert hat y 0 cdot y 0 vert epsilon where epsilon epsilon hat y is small enough then there exists boundary control u such that the solution y t x of the boussinesq equations satisfying y vert 0 t times partial omega u qquad y vert t 0 y 0 coincides with hat y t x at the instant t y t x equiv hat y t x

a multicorner variable order singularity triangle to model neighbouring singularities a six noded triangular finite element has been proposed to model variable order singularities in the first derivatives of field variables occurring at multiple locations up to three lying in close proximity the element satisfies the convergence requirements fully the element will be very useful in the analysis of neighbouring cracks zig zag cracks kinked cracks etc involving mechanical and or thermal loadings the first degenerate form which is a two point variable order singularity element can be used for example to cover the distance between two adjacent crack tips the second degenerate form which is a one point variable order singularity element can be used around a crack tip like any other crack tip elements six case studies have been reported three involving mechanical loadings and the other three thermal loadings these have helped to check the performance of the element and its two degenerate forms the accuracy of the results in all the examples is good

periodic maxwell chern simons vortices with concentrating property in order to study electrically and magnetically charged vortices in fractional quantum hall effect and anyonic superconductivity the maxwell chern simons mcs model was introduced by lee et al phys lett b 252 79 83 1990 as a unified system of the classical abelian higgs model ah and the chern simons cs model in this article the first goal is to obtain the uniform cs limit result of mcs model with respect to the chern simons parameter without any restriction on either a particular class of solutions or the number of vortex points as the chern simons mass scale tends to infinity the most important step for this purpose is to derive the relation between the higgs field and the neutral scalar field our cs limit result also provides the critical clue to answer the open problems raised by ricciardi and tarantello comm pure appl math 53 811 851 2000 and tarantello milan j math 72 29 80 2004 and we succeed to establish the existence of periodic maxwell chern simons vortices satisfying the concentrating property of the density of superconductive electron pairs furthermore we expect that the cs limit analysis in this paper would help to study the stability multiplicity and bubbling phenomena for solutions of the mcs model

non topological multivortex solutions to the self dual maxwell chern simons higgs systems in this paper we construct non topological multivortex solutions to the non relativistic self dual maxwell chern simons higgs system in r2 which make the energy functional finite moreover our proof of the existence of solutions reveals precise asymptotic behavior of solutions near spatial infinity using exactly the same method we also establish the existence of non topological multivortex solutions to the relativistic self dual maxwell chern simons higgs system

existence of mixed type solutions in the chern simons gauge theory of rank two in r2 abstract we consider the self dual equations arising from the chern simons gauge theory of rank 2 such as the s u 3 s o 5 and g 2 chern simons model in r 2 there are three possible types of solutions in these theories namely topological nontopological and mixed type solutions the existence of a mixed type solution for an arbitrary configuration of vortex points has been a long standing open problem we construct here a family of mixed type solutions u 1 e u 2 e e 0 for an arbitrary configuration of vortex points under a mild non degeneracy condition the main new idea of our construction of an approximate solution is to use different scalings for different components in the process of the finite dimensional reduction all estimates for this particular construction become rather delicate

uniqueness of the topological multivortex solution in the self dual chern simons theory we establish a uniqueness result for the topological multivortex solution to the self dual equations of the abelian relativistic self dual chern simons higgs model we prove that the topological multivortex solution is unique if the chern simons coupling parameter 0 is sufficiently small we also establish a uniqueness result for 0 sufficiently large

a finite element for variable order singularities based on the displacement formulation an element for analysis of variable order one point singularities in plane problems has been proposed the element meets all the convergence requirements viz the rigid body mode the constant strain condition and the interelement compatibility it can be easily incorporated in any package based on the displacement formulation four examples from fracture mechanics are presented to demonstrate its performance the examples involve mechanical or thermal loadings the accuracy of results in all the examples is very good

blow up at infinity in the su 3 chern simons model part i abstract we consider non topological solutions of a nonlinear elliptic system problem see 1 4 below derived from the s u 3 chern simons models in r 2 the existence of non topological solutions even for radial symmetric case has been a long standing open problem recently choe kim and lin in 7 8 showed the existence of radial symmetric non topological solution when the vortex points collapse however the arguments in 7 8 cannot work for an arbitrary configuration of vortex points in this paper we develop a new approach by using different scalings for different components of the system to construct a family of non topological solutions which blows up at infinity

existence and nonexistence in chern simons higgs theory with a constant electric charge density in this paper we are devoted to proving the existence and nonexistence of self dual equations arising in chern simons higgs theory with a constant electric charge density there are three kinds of boundary conditions that admit solitonic structures it is shown that there exist solutions in two cases of them in the other case we prove that there is a critical electric charge density with negative value such that above the value there exists a solution and below it we have no solution we also study asymptotic behaviors for solutions as the electric charge density goes to zero it is found that they converge to solutions of a topological chern simons system without constant electric charge density

the existence of non topological multivortex solutions in the relativistic self dual chern simons theory we construct a general type of multivortex solutions of the self duality equations the bogomol nyi equations of 2 1 dimensional relativistic chern simons model with the non topological boundary condition near infinity for such construction we use a perturbation argument around the explicit solutions of the liouville equation

bubbling solutions for a skew symmetric chern simons system in a torus abstract we establish the existence of bubbling solutions for the following skew symmetric chern simons system u 1 1 e 2 e u 2 1 e u 1 4 i 1 n 1 p i 1 u 2 1 e 2 e u 1 1 e u 2 4 i 1 n 2 p i 2 over a parallelogram with doubly periodic boundary condition where e 0 is a coupling parameter and p denotes the dirac measure concentrated at p we obtain that if n 1 1 n 2 1 1 there exists an e 0 0 such that for any e 0 e 0 the above system admits a solution u 1 e u 2 e satisfying u 1 e and u 2 e blow up simultaneously at the point p and 1 e 2 e u j e 1 e u i e 4 n i p 1 i j 2 i j as e 0 where the location of the point p defined by 1 12 satisfies the condition 1 13

concentrating solutions for a planar elliptic problem involving nonlinearities with large exponent abstract we consider the boundary value problem u u p 0 in a bounded smooth domain in r 2 with homogeneous dirichlet boundary condition and p a large exponent we find topological conditions on which ensure the existence of a positive solution u p concentrating at exactly m points as p in particular for a nonsimply connected domain such a solution exists for any given m 1

existence and asymptotics of topological solutions in the self dual maxwell chern simons o 3 sigma model we are concerned with the topological vortex equations arising in the self dual maxwell chern simons o 3 sigma model we prove the existence of solutions for any parameters and q by a variational method we also verify both the maxwell limit and the chern simons limit for the variational solutions

local and global solutions of the chern simons higgs system abstract we study low regularity solutions of the chern simons higgs equations the lorentz gauge condition makes them hyperbolic equations with the null form under the coulomb gauge condition they are formulated in the hyperbolic equation coupled with elliptic equation the div curl decomposition is used in the temporal gauge

the solution structure of the o 3 sigma model in a maxwell chern simons theory in this paper a system of semilinear elliptic equations arising from a relativistic self dual maxwell chern simons o 3 sigma model is considered we reveal the uniqueness aspect of the topological solutions for the model the uniqueness result is associated with a clear solution structure of the equations of the radially symmetric case we locate each solution set denoted by a planar diagram

nonself dual chern simons and maxwell chern simons vortices on bounded domains abstract we first show the existence of the multivortex solutions of maxwell chern simons higgs mcsh model on bounded domains and prove that the solutions of the euler lagrange equations of the mcsh energy functional converge to those of the abelian higgs ah model and the chern simons higgs csh model in suitable limits respectively we also show the existence of the multivortex solutions of the nonself dual csh model on bounded domains besides we study asymptotics for the minimizers of mcsh energy functional when the gauge field vanishes