gamma convergence of gradient flows with applications to ginzburg landau we present a method to prove convergence of gradient flows of families of energies that converge to a limiting energy it provides lower bound criteria to obtain the convergence that correspond to a sort of c1 order convergence of functionals we then apply this method to establish the limiting dynamical law of a finite number of vortices for the heat flow of the ginzburg landau energy in dimension 2 retrieving in a different way the existing results for the case without magnetic field and obtaining new results for the case with magnetic field 2004 wiley periodicals inc

invariance of white noise for kdv on the line we consider the korteweg de vries equation with white noise initial data posed on the whole real line and prove the almost sure existence of solutions moreover we show that the solutions obey the group property and follow a white noise law at all times past or future as an offshoot of our methods we also obtain a new proof of the existence of solutions and the invariance of white noise measure in the torus setting

the γ limit of the two dimensional ohta kawasaki energy droplet arrangement via the renormalized energy this is the second in a series of papers in which we derive a expansion for the two dimensional non local ginzburg landau energy with coulomb repulsion known as the ohta kawasaki model in connection with diblock copolymer systems in this model two phases appear which interact via a nonlocal coulomb type energy here we focus on the sharp interface version of this energy in the regime where one of the phases has very small volume fraction thus creating small droplets of the minority phase in a sea of the majority phase in our previous paper we computed the limit of the leading order energy which yields the averaged behavior for almost minimizers namely that the density of droplets should be uniform here we go to the next order and derive a next order limit energy which is exactly the coulombian renormalized energy obtained by sandier and serfaty as a limiting interaction energy for vortices in the magnetic ginzburg landau model the derivation is based on the abstract scheme of sandier serfaty that serves to obtain lower bounds for 2 scale energies and express them through some probabilities on patterns via the multiparameter ergodic theorem thus without appealing to the euler lagrange equation we establish for all configurations which have almost minimal energy the asymptotic roundness and radius of the droplets and the fact that they asymptotically shrink to points whose arrangement minimizes the renormalized energy in some averaged sense via a kind of equivalence the obtained results also yield an expansion of the minimal energy and a characterization of the zero super level sets of the minimizers for the original ohta kawasaki energy this leads to the expectation of seeing triangular lattices of droplets as energy minimizers

higher dimensional coulomb gases and renormalized energy functionals we consider a classical system of n charged particles in an external confining potential in any dimension d larger than 2 the particles interact via pairwise repulsive coulomb forces and the coupling parameter scales like the inverse of n mean field scaling by a suitable splitting of the hamiltonian we extract the next to leading order term in the ground state energy beyond the mean field limit we show that this next order term which characterizes the fluctuations of the system is governed by a new renormalized energy functional providing a way to compute the total coulomb energy of a jellium i e an infinite set of point charges screened by a uniform neutralizing background in any dimension the renormalization that cuts out the infinite part of the energy is achieved by smearing out the point charges at a small scale as in onsager s lemma we obtain consequences for the statistical mechanics of the coulomb gas next to leading order asymptotic expansion of the free energy or partition function characterizations of the gibbs measures estimates on the local charge fluctuations and factorization estimates for reduced densities this extends results of sandier and serfaty to dimension higher than two by an alternative approach

periodic nonlinear schrodinger equation and invariant measures in this paper we continue some investigations on the periodic nlseiu u iu xx u u p 2 p 6 started in lrs we prove that the equation is globally wellposed for a set of data of full normalized gibbs measrue e beta h phi hd phi x h phi tfrac 1 2 int left phi right 2 tfrac 1 p int left phi right p after suitablel 2 truncation the set and the measure are invariant under the flow the proof of a similar result for the kdv and modified kdv equations is outlined the main ingredients used are some estimates from b1 on periodic nls and kdv type equations

limiting behavior of the ginzburg landau functional we continue our study of the functional ee u u12 u 2 14e2 1 u 2 2dx for u h1 u r2 where u is a bounded open subset of r2 compactness results for the scaled jacobian of ue are proved under the assumption that ee ue is bounded uniformly by a function of e in addition the gamma limit of ee ue ln e 2 is shown to be e v 12 v 22 v m where v is the limit of j ue ln e j ue ue due and m is the total variation of a radon measure these results are applied to the ginzburg landau functionalfe u a hext u12 au 2 14e2 1 u 2 2 12 a hext dx with external magnetic field hext h ln e the gamma limit of fe ln e 2 is calculated to bef v a h 12 v a 22 v m a h 22 where v is as before and a is the limit of ae ln e

construction of classical superintegrable systems with higher order integrals of motion from ladder operators we construct integrals of motion for multidimensional classical systems from ladder operators of one dimensional systems this method can be used to obtain new systems with higher order integrals we show how these integrals generate a polynomial poisson algebra we consider a one dimensional system with third order ladder operators and found a family of superintegrable systems with higher order integrals of motion we obtain also the polynomial algebra generated by these integrals we calculate numerically the trajectories and show that all bounded trajectories are closed

a quantization property for static ginzburg landau vortices for any critical point of the complex ginzburg landau functional in dimension 3 we prove that for large coupling constants if the energy of this critical point on a ball of a given radius r is relatively small compared to r log then the ball of half radius contains no vortex the modulus of the solution is larger than we then show how this property can be applied to describe limiting vortices as e 0 2001 john wiley sons inc

statistical mechanics of nonlinear wave equations 4 cubic schrodinger the cubic schrodinger equation is considered on the circle both in the de focussing and the focussing case the existence of the flow is proved together with the invariance of the appropriate gibbsian measure namely the petit canonical measure in the defocussing case and the micro canonical measure in the focussing case

exact solvability of superintegrable systems it is shown that all four superintegrable quantum systems on the euclidean plane possess the same underlying hidden algebra sl 3 the gauge rotated hamiltonians as well as their integrals of motion once rewritten in appropriate coordinates preserve a flag of polynomials this flag corresponds to highest weight finite dimensional representations of the sl 3 algebra realized by first order differential operators

concentration of the invariant measures for the periodic zakharov kdv nls and gross piatevskii equations in 1d and 2d abstract this paper concerns gibbs measures for some nonlinear pde over the d torus t d the hamiltonian h t d u 2 t d u p has canonical equations with solutions in n u l 2 t d u 2 n this n is a parameter in quantum field theory analogous to the number of particles in a classical system for d 1 and 2 p 6 n supports the gibbs measure d u z 1 e h u x t d u x which is normalized and formally invariant under the flow generated by the pde the paper proves that n l 2 is a metric probability space of finite diameter that satisfies the logarithmic sobolev inequalities for the periodic kdv the focussing cubic nonlinear schrodinger equation and the periodic zakharov system for suitable subset of n a logarithmic sobolev inequality also holds in the critical case p 6 for d 2 the gross piatevskii equation has h t 2 u 2 t 2 v u 2 u 2 for a suitable bounded interaction potential v and the gibbs measure lies on a metric probability space h s which satisfies lsi in the above cases d is the limit in l 2 transportation distance of finite dimensional n n given by fourier sums

classical and quantum superintegrability with applications a superintegrable system is roughly speaking a system that allows more integrals of motion than degrees of freedom this review is devoted to finite dimensional classical and quantum superintegrable systems with scalar potentials and integrals of motion that are polynomials in the momenta we present a classification of second order superintegrable systems in two dimensional riemannian and pseudo riemannian spaces it is based on the study of the quadratic algebras of the integrals of motion and on the equivalence of different systems under coupling constant metamorphosis the determining equations for the existence of integrals of motion of arbitrary order in real euclidean space e2 are presented and partially solved for the case of third order integrals a systematic exposition is given of systems in two and higher dimensional space that allow integrals of arbitrary order the algebras of integrals of motions are not necessarily quadratic but close polynomially or rationally the relation between superintegrability and the classification of orthogonal polynomials is analyzed

invariant measures for some partial differential equations we construct invariant measures for hamiltonian systems such as the nonlinear schrrdinger equation or the wave equation in order to prove poisson s recurrence the particular case of schemes finite dimensional spaces is also treated in order to explain the recurrence phenomenon which is observed during numerical simulations

a product estimate for ginzburg landau and corollaries abstract we prove a new inequality for the jacobian or vorticity associated to the ginzburg landau energy in any dimension it allows to retrieve existing lower bounds on the energy to extend them to the case of unbounded vorticity and to get a few other corollaries it also provides a new estimate on the time variation for time dependent families which has applications for the study of ginzburg landau dynamics

superintegrability in three dimensional euclidean space potentials for which the corresponding schrodinger equation is maximally superintegrable in three dimensional euclidean space are studied the quadratic algebra which is associated with each of these potentials is constructed and the bound state wave functions are computed in the separable coordinates

the diffusive phase of a model of self interacting walks we consider simple random walk onzd perturbed by a factor exp t pjt wheret is the length of the walk and j t sum nolimits 0 leqslant i 0 ford 2 the diffusion constant is equal to 1 but ford 2 it is renormalized ford 1 andp 3 2 we prove diffusion for all real positive or negative ford 2 the scaling limit is brownian motion but ford 2 it is the edwards model with the wrong sign of the coupling when 0 which governs the limiting behaviour the latter arises since for p frac 4 d 2 t p j t is the discrete self intersection local time this establishes existence of a diffusive phase for this model existence of a collapsed phase for a very closely related model has been proven in work of bolthausen and schmock

group theory of the smorodinsky winternitz system the three degrees of freedom smorodinsky winternitz system is a degenerate or super integrable hamiltonian that possesses five functionally independent globally defined and single valued integrals of the motion in both classical and quantum mechanics this is explained in terms of a forced degeneracy imposed as a consequence of the invariance of the hamiltonian under a group of symmetry transformations isomorphic to the three dimensional unitary unimodular group su 3 in turn this degeneracy group is embedded in a larger group of transformations that maps all the bound energy levels among each other the so called dynamical group all the bound state eigenfunctions act as basis functions for a single irreducible representation of the dynamical group so in common with the hydrogen atom and the harmonic oscillator the quantum mechanics of the smorodinsky winternitz system may be completely solved within the framework of group theory alone

on the shape of interlayer vortices in the lawrence doniach model we consider the lawrence doniach model for layered superconductors in which stacks of parallel superconducting planes are coupled via the josephson effect to model experiments in which the superconductor is placed in an external magnetic field oriented parallel to the superconducting planes we study the structure of isolated vortices for a doubly periodic problem we consider a singular limit which simulates certain experimental regimes in which isolated vortices have been observed corresponding to letting the interlayer spacing of the superconducting planes tend to zero and the ginzburg landau parameter simultaneously but at a fixed relative rate

integrable and superintegrable systems with spin a system of two particles with spin s 0 and s 12 respectively moving in a plane is considered it is shown that such a system with a nontrivial spin orbit interaction can allow an eight dimensional lie algebra of first order integrals of motion the pauli equation is solved in this superintegrable case and reduced to a system of ordinary differential equations when only one first order integral exists

separation of variables and subgroup bases on n dimensional hyperboloids a graphical formalism is introduced for describing subgroup type coordinates on n dimensional lorentzian hyperboloids imbedded into n 1 dimensional minkowski spaces the o n 1 group element is parametrized according to different subgroup chains involving lorentz rotation and euclidean subgroups the coordinates are then induced by the corresponding group action eigenfunctions of the laplace beltrami operator are obtained as products of jacobi functions associated legendre functions and modified bessel functions

a boundary value problem related to the ginzburg landau model we analyze the ginzburg landau equation for a superconductor in the case of a 2 dimensional model a cylindrical conductor with a magnetic field parallel to the axis this amounts to find the extrema of the free energy mathcal a kappa 1 2 int limits omega nabla ia phi 2 b a 2 kappa 4 phi 2 1 2 dx where is a bounded domain with smooth boundary in 2 a a1 a2 the vector potential ba 1a2 2a1 the magnetic field a complex field we describe the connected components of the maximal configuration space i e of the set of all a with components in the sobolev spaceh1 and such that 1 on the boundary modulo the action of the gauge group in the critical case 1 we give a complete description of the minimal configurations in each component

defects in nematic shells a γ convergence discrete to continuum approach in this paper we rigorously investigate the emergence of defects on nematic shells with a genus different from one this phenomenon is related to a non trivial interplay between the topology of the shell and the alignment of the director field to this end we consider a discrete xy system on the shell m described by a tangent vector field with unit norm sitting at the vertices of a triangulation of the shell defects emerge when we let the mesh size of the triangulation go to zero namely in the discrete to continuum limit in this paper we investigate the discrete to continuum limit in terms of convergence in two different asymptotic regimes the first scaling promotes the appearance of a finite number of defects whose charges are in accordance with the topology of shell m via the poincare hopf theorem the second scaling produces the so called renormalized energy that governs the equilibrium of the configurations with defects

completeness of superintegrability in two dimensional constant curvature spaces we classify the hamiltonians h px2 py2 v x y of all classical superintegrable systems in two dimensional complex euclidean space with two additional second order constants of the motion we similarly classify the superintegrable hamiltonians h j12 j22 j32 v x y z on the complex two sphere where x2 y2 z2 1 this is achieved in all generality using properties of the complex euclidean group and the complex orthogonal group

nonlinear schrodinger equations and sharp interpolation estimates a sharp sufficient condition for global existence is obtained for the nonlinear schrodinger equation begin array 20 c nls 2i phi t delta phi left phi right 2 sigma phi 0 x in mathbb r n t in mathbb r end array in the case 2 n this condition is in terms of an exact stationary solution nonlinear ground state of nls it is derived by solving a variational problem to obtain the best constant for classical interpolation estimates of nirenberg and gagliardo

lorentz space estimates for the ginzburg landau energy abstract in this paper we prove novel lower bounds for the ginzburg landau energy with or without magnetic field these bounds rely on an improvement of the vortex balls construction estimates by extracting a new positive term in the energy lower bounds this extra term can be conveniently estimated through a lorentz space norm on which it thus provides an upper bound the lorentz space l 2 we use is critical with respect to the expected vortex profiles and can serve to estimate the total number of vortices and get improved convergence results

supersymmetry as a method of obtaining new superintegrable systems with higher order integrals of motion the main result of this article is that we show that from supersymmetry we can generate new superintegrable hamiltonians we consider a particular case with a third order integral and apply mielnik s construction in supersymmetric quantum mechanics we obtain a new superintegrable potential separable in cartesian coordinates with a quadratic and quintic integrals and also one with a quadratic integral and an integral of order of 7 we also construct a superintegrable system written in terms of the fourth painleve transcendent with a quadratic integral and an integral of order of 7

weak anchoring for a two dimensional liquid crystal abstract we study the weak anchoring condition for nematic liquid crystals in the context of the landau de gennes model we restrict our attention to two dimensional samples and to nematic director fields lying in the plane for which the landau de gennes energy reduces to the ginzburg landau functional and the weak anchoring condition is realized via a penalized boundary term in the energy we study the singular limit as the length scale parameter e 0 assuming the weak anchoring parameter e at a prescribed rate we also consider a specific example of a bulk nematic liquid crystal with an included oil droplet and derive a precise description of the defect locations for this situation for e k e with 0 1 we show that defects lie on the weak anchoring boundary for 0 1 2 or for 1 2 and k small but they occur inside the bulk domain for 1 2 or 1 2 with k large

statistical mechanics of the nonlinear schrodinger equation we investigate the statistical mechanics of a complex fieldo whose dynamics is governed by the nonlinear schrodinger equation such fields describe in suitable idealizations langmuir waves in a plasma a propagating laser field in a nonlinear medium and other phenomena their hamiltonian h phi int omega frac 1 2 nabla phi 2 1 p phi p dx is unbounded below and the system will under certain conditions develop self focusing singularities in a finite time we show that when is the circle and thel 2 norm of the field which is conserved by the dynamics is bounded byn the gibbs measure obtained is absolutely continuous with respect to wiener measure and normalizable if and only ifp andn are such that classical solutions exist for all time no collapse of the solitons this measure is essentially the same as that of a one dimensional version of the more realisitc zakharov model of coupled langmuir and ion acoustic waves in a plasma we also obtain some properties of the gibbs state by both analytic and numerical methods asn and the temperature are varied

quasiseparation of variables in the schrodinger equation with a magnetic field we consider a two dimensional integrable hamiltonian system with a vector and scalar potential in quantum mechanics contrary to the case of a pure scalar potential the existence of a second order integral of motion does not guarantee the separation of variables in the schrodinger equation we introduce the concept of quasiseparation of variables and show that in many cases it allows us to reduce the calculation of the energy spectrum and wave functions to linear algebra

lotka volterra systems integrable in quadratures hamiltonian n dimensional lotka volterra systems are introduced that have n 1 conserved quantities the explicit integrability in quadratures is demonstrated

superintegrable 3d systems in a magnetic field corresponding to cartesian separation of variables we consider three dimensional superintegrable systems in a magnetic field we study the class of such systems which separate in cartesian coordinates in the limit when the magnetic field vanishes i e possess two second order integrals of motion of the cartesian type for such systems we look for additional integrals up to second order in momenta which make these systems minimally or maximally superintegrable and construct their polynomial algebras of integrals and their trajectories we observe that the structure of the leading order terms of the cartesian type integrals should be considered in a more general form than for the case without magnetic field

on minimizers of a landau de gennes energy functional on planar domains we study tensor valued minimizers of the landau de gennes energy functional on a simply connected planar domain with non contractible boundary data here the tensorial field represents the second moment of a local orientational distribution of rod like molecules of a nematic liquid crystal under the assumption that the energy depends on a single parameter a dimensionless elastic constant varepsilon 0 we establish that as varepsilon to 0 the minimizers converge to a projection valued map that minimizes the dirichlet integral away from a single point in we also provide a description of the limiting map

non existence of vortices in the small density region of a condensate abstract in this paper we answer a question raised by lev pitaevskii and prove that the ground state of the gross pitaevskii energy describing a bose einstein condensate in a rotationally symmetric trap at low rotation does not have vortices in the low density region therefore the first ground state with vortices has its vortices in the bulk in fact we prove something stronger which is that the ground state for the model at low and moderate rotations is equal to the ground state in a condensate with no rotation this is obtained by proving that for small rotational velocities the ground state is multiple of the ground state with zero rotation we rely on sharp bounds of the decay of the wave function combined with weighted jacobian estimates

statistical mechanics of the 2 dimensional focusing nonlinear schrodinger equation we study a natural construction of an invariant measure for the 2 dimensional periodic focusing nonlinear schrodinger equation with the critical cubic nonlinearity we find that a phase transition occurs as the coupling constant defining the strength of the nonlinearity is increased but that the natural construction successful for the 1 dimensional case and for the 2 dimensional defocusing case cannot produce an invariant measure our methods rely on an analysis of a statistical mechanical model closely related to the spherical model of berlin and kac

probabilistic methods for discrete nonlinear schrodinger equations we show that the thermodynamics of the focusing cubic discrete nonlinear schrodinger equation are exactly solvable in dimension 3 and higher a number of explicit formulas are derived

higher order superintegrability of separable potentials with a new approach to the post winternitz system the higher order superintegrability of separable potentials is studied it is proved that these potentials possess in addition to the two quadratic integrals a third integral of higher order in the momenta that can be obtained as the product of powers of two particular rather simple complex functions some systems related to the harmonic oscillator such as the generalized sw system and the ttw system were studied in previous papers now a similar analysis is presented for superintegrable systems related to the kepler problem in this way a new proof of the superintegrability of the post winternitz system is presented and the explicit expression for the integral is obtained finally the relations between the superintegrable systems with quadratic constants of motion separable in several different coordinate systems and the superintegrable systems with higher order constants of motion are analyzed

metamorphoses sudden jumps in basin boundaries in some invertible maps of the plane that depend on a parameter boundaries of basins of attraction are extremely sensitive to small changes in the parameter a basin boundary can jump suddenly and as it does change from being smooth to fractal such changes are calledbasin boundary metamorphoses we prove under certain non degeneracy assumptions that a metamorphosis occurs when the stable and unstable manifolds of a periodic saddle on the boundary undergo a homoclinic tangency

analysis of nematic liquid crystals with disclination lines we investigate the structure of nematic liquid crystal thin films described by the landau de gennes tensor valued order parameter model with dirichlet boundary conditions on the sides of nonzero degree we prove that as the elasticity constant goes to zero in the energy a limiting uniaxial nematic texture forms with a finite number of defects all of degree frac 1 2 or all of degree frac 1 2 corresponding to vertical disclination lines at those locations we also state a result on the limiting behavior of minimizers of the chern simons higgs model without magnetic field that follows from a similar proof

vortices and pinning effects for the ginzburg landau model in multiply connected domains we consider the two dimensional ginzburg landau model with magnetic fi eld for a superconductor with a multiply connected cross section we study energy minimizers in the london limit as the ginzburg landau parameter 1 to determine the number and asymptotic location of vortices we show that the holes act as pinning sites acquiring nonzero winding for bounded fields and attracting all vortices away from the interior for fields up to a critica l value hex o ln at the critical level the pinning effect breaks down and vortices appear in the interior of the superconductor at locations that we id entify explicitly in terms of the solutions of an elliptic boundary value problem the method involves sharp upper and lower energy estimates and a careful analysis of the limiting problem that captures the interaction between the vortices and the holes c 2005 wiley periodicals inc

the line tension approximation as the dilute limit of linear elastic dislocations we prove that the classical line tension approximation for dislocations in crystals that is the approximation that neglects interactions at a distance between dislocation segments and accords dislocations energy in proportion to their length follows as the limit of regularized linear elasticity as the lattice parameter becomes increasingly small or equivalently as the dislocation measure becomes increasingly dilute we consider two regularizations of the theory of linear elastic dislocations a core cutoff and a mollification of the dislocation measure we show that both regularizations give the same energy in the limit namely an energy defined on matrix valued divergence free measures concentrated on lines the corresponding self energy per unit length b t which depends on the local burgers vector and orientation of the dislocation does not however necessarily coincide with the self energy per unit length 0 b t obtained from the classical theory of the prelogarithmic factor of linear elastic straight dislocations indeed microstructure can occur at small scales resulting in a further relaxation of the classical energy down to its h1 elliptic envelope

invariant measures and the soliton resolution conjecture the soliton resolution conjecture for the focusing nonlinear schrodinger equation nls is the vaguely worded claim that a global solution of the nls for generic initial data will eventually resolve into a radiation component that disperses like a linear solution plus a localized component that behaves like a soliton or multisoliton solution considered to be one of the fundamental open problems in the area of nonlinear dispersive equations this conjecture has eluded a proof or even a precise formulation to date this paper proves a statistical version of this conjecture at mass subcritical nonlinearity in the following sense the uniform probability distribution on the set of all functions with a given mass and energy if such a thing existed would be a natural invariant measure for the nls flow and would reflect the long term behavior for generic initial data with that mass and energy unfortunately such a probability measure does not exist we circumvent this problem by constructing a sequence of discrete measures that in principle approximate this fictitious probability distribution as the grid size goes to 0 we then show that a continuum limit of this sequence of probability measures does exist in a certain sense and in agreement with the soliton resolution conjecture the limit measure concentrates on the unique ground state soliton combining this with results from ergodic theory we present a tentative formulation and proof of the soliton resolution conjecture in the discrete setting the above results following in the footsteps of a program of studying the long term behavior of nonlinear dispersive equations through their natural invariant measures initiated by lebowitz rose and speer and carried forward by bourgain mckean tzvetkov oh and others are proved using a combination of techniques from large deviations pdes harmonic analysis and bare hands probability theory it is valid in any dimension 2014 wiley periodicals inc

integrable hamiltonian systems with velocity dependent potentials the integrability of a two dimensional hamiltonian in which the potential depends explicitly on the momenta is investigated hamiltonians of this kind are encountered in the description of the motion of a particle in a magnetic field two integrable classes of potentials are identified and the second integral of motion is constructed for each of them the singularity analysis of the equations of motion is also performed confirming once more the relation between the weak painleve property and integrability

gamma convergence of maxwell chern simons higgs energy in this article we derive the limit of the maxwell chern simons higgs energy with log 2 energy bound our interests focus on the case with external electro magnetic potentials similarly to the abelian ginzburg landau model the corresponding obstacle problems are derived

on approximation of ginzburg landau minimizers by s1 valued maps in domains with vanishingly small holes abstract we consider a two dimensional ginzburg landau problem on an arbitrary domain with a finite number of vanishingly small circular holes a special choice of scaling relation between the material and geometric parameters ginzburg landau parameter vs hole radius is motivated by a recently discovered phenomenon of vortex phase separation in superconducting composites we show that for each hole the degrees of minimizers of the ginzburg landau problems in the classes of s 1 valued and c valued maps respectively are the same the presence of two parameters that are widely separated on a logarithmic scale constitutes the principal difficulty of the analysis that is based on energy decomposition techniques

superintegrable systems in darboux spaces almost all research on superintegrable potentials concerns spaces of constant curvature in this paper we find by exhaustive calculation all superintegrable potentials in the four darboux spaces of revolution that have at least two integrals of motion quadratic in the momenta in addition to the hamiltonian these are two dimensional spaces of nonconstant curvature it turns out that all of these potentials are equivalent to superintegrable potentials in complex euclidean 2 space or on the complex 2 sphere via coupling constant metamorphosis or equivalently via stackel multiplier transformations we present a table of the results

classical versus quantum integrability classical integrability and quantum integrability are compared for two degrees of freedom hamiltonian systems we use c number representatives for quantum operators and the moyal bracket for the commutator three different cases are found i the c number representative of the quantum mechanical second invariant is identical to the classical second invariant ii o 2 corrections are needed in the classical second invariant to obtain the quantum invariant and iii also the potential must be deformed by an o 2 term several examples from the henon heiles and holt families of integrable potentials are included

hamiltonians separable in cartesian coordinates and third order integrals of motion we present in this article all hamiltonian systems in e 2 that are separable in cartesian coordinates and that admit a third order integral both in quantum and in classical mechanics many of these superintegrable systems are new and it is seen that there exists a relation between quantum superintegrable potentials invariant solutions of the korteweg de vries equation and the painleve transcendents

statistical mechanics of the nonlinear schrodinger equation ii mean field approximation we investigate a mean field approximation to the statistical mechanics of complex fields with dynamics governed by the nonlinear schrodinger equation such fields whose hamiltonian is unbounded below may model plasmas lasers and other physical systems restricting ourselves to one dimensional systems with periodic boundary conditions we find in the mean field approximation a phase transition from a uniform regime to a regime in which the system is dominated by solitons we compute explicitly as a function of temperature and density l2 norm the transition point at which the uniform configuration becomes unstable to local perturbations static and dynamic mean field approximations yield the same result

on self attracting d dimensional random walks let x t t geq 0 be a symmetric nearest neighbor random walk on mathbb z d with exponential holding times of expectation 1 d starting at the origin for a potential v mathbb z d to 0 infty with finite and nonempty support define transformed path measures by d hat mathbb p t equiv exp t 1 int 0 t int 0 t v x s x t ds dt d mathbb p z t for t 0 where z t is the normalizing constant if d 1 or if the self attraction is sufficiently strong then x t infty has an exponential moment under mathbb p t which is uniformly bounded for t 0 and t in 0 t we also prove that x t t geq 0 under suitable subsequences of hat mathbb p t t 0 behaves for large t asymptotically like a mixture of space inhomogeneous ergodic random walks for special cases like a sufficiently strong dirac type interaction we even prove convergence of the transformed path measures and the law of x t as well as of the law of the empirical measure l t under hat mathbb p t t 0

pinning effects and their breakdown for a ginzburg landau model with normal inclusions we study a ginzburg landau model for an inhomogeneous superconductor in the singular limit as the ginzburg landau parameter 1 the inhomogeneity is represented by a potential term v 14 a x 2 2 with a given smooth function a x which is assumed to become negative in finitely many smooth subdomains the normally included regions for bounded applied fields independent of the ginzburg landau parameter 1 we show that the normal regions act as giant vortices acquiring large vorticity for large fixed applied field hex for hex o ln we show that this pinning effect eventually breaks down and free vortices begin to appear in the superconducting region where a x 0 at a point set which is determined by solving an elliptic boundary value problem the associated operators are strictly but not uniformly elliptic leading to some regularity questions to be resolved near the boundaries of the normal regions

energy estimates and cavity interaction for a critical exponent cavitation model we consider the minimization of in a perforated domain of among maps that are incompressible det and invertible and satisfy a dirichlet boundary condition u g on if the volume enclosed by g is greater than any such deformation u is forced to map the small holes be ai onto macroscopically visible cavities which do not disappear as e 0 we restrict our attention to the critical exponent p n where the energy required for cavitation is of the order of and the model is suited therefore for an asymptotic analysis v1 vm denote the volumes of the cavities in the spirit of the analysis of vortices in ginzburg landau theory we obtain estimates for the renormalized energy showing its dependence on the size and the shape of the cavities on the initial distance between the cavitation points a1 am and on the distance from these points to the outer boundary based on those estimates we conclude for the case of two cavities that either the cavities prefer to be spherical in shape and well separated or to be very close to each other and appear as a single equivalent round cavity this is in agreement with existing numerical simulations and is reminiscent of the interaction between cavities in the mechanism of ductile fracture by void growth and coalescence 2012 wiley periodicals inc

integrable and superintegrable quantum systems in a magnetic field integrable quantum mechanical systems with magnetic fields are constructed in two dimensional euclidean space the integral of motion is assumed to be a first or second order hermitian operator contrary to the case of purely scalar potentials quadratic integrability does not imply the separation of variables in the schrodinger equation moreover quantum and classical integrable systems do not necessarily coincide the hamiltonian can depend on the planck constant in a nontrivial manner

exponential relaxation to equilibrium for a one dimensional focusing non linear schrodinger equation with noise we construct generalized grand canonical and canonical gibbs measures for a hamiltonian system described in terms of a complex scalar field that is defined on a circle and satisfies a nonlinear schrodinger equation with a focusing nonlinearity of order p 6 key properties of these gibbs measures in particular absence of phase transitions and regularity properties of field samples are established we then study a time evolution of this system given by the hamiltonian evolution perturbed by a stochastic noise term that mimics effects of coupling the system to a heat bath at some fixed temperature the noise is of ornstein uhlenbeck type for the fourier modes of the field with the strength of the noise decaying to zero as the frequency of the mode tends to we prove exponential approach of the state of the system to a grand canonical gibbs measure at a temperature and chemical potential determined by the stochastic noise term

lower bounds for the energy of unit vector fields and applications we prove lower bounds for the dirichlet energy of a unit vector field defined in a perforated domain of r2with nonzero degree on the outer boundary in terms of the total diameter of the holes we use this to derive lower bounds and then compactness results for sequences ue of minimizers or almost minimizers of ginzburg landau functionals with coupling constant 1 e2tending to

on the variations of the vortex number in a periodic ginzburg landau model in this paper we study the variations of the number of vortices contained in the minimizer of a two dimensional ginzburg landau functional describing a type ii superconductor in the london limit with periodic conditions on the boundary of the sample we prove that under the assumption that the sample is rectangular with height small enough the number of vortices contained in the minimizer of the periodic ginzburg landau functional jumps by unit step as the applied magnetic field increases we convert the problem into the study of corresponding renormalized energy by using the convergence we reduce the two dimensional renormalized energy to one dimensional one then the result is obtained by analyzing the one dimensional model

quantum superintegrable systems with quadratic integrals on a two dimensional manifold there are two classes of quantum integrable systems on a manifold with quadratic integrals the liouville and the lie integrable systems as it happens in the classical case the quantum liouville quadratic integrable systems are defined on a liouville manifold and the schrodinger equation can be solved by separation of variables in one coordinate system the lie integrable systems are defined on a lie manifold and are not generally separable ones but can be solved therefore there are superintegrable systems with two quadratic integrals of motion not necessarily separable in two coordinate systems the quantum analogs of the two dimensional superintegrable systems with quadratic integrals of motion on a manifold are classified by using the quadratic associative algebra of the integrals of motion there are six general fundamental classes of quantum superintegrable systems corresponding to the classical ones analytic formulas for the involved integrals are calculated in all the cases all the known quan

generalized micz kepler system duality polynomial and deformed oscillator algebras we present the quadratic algebra of the generalized micz kepler system in three dimensional euclidean space e3 and its dual the four dimensional singular oscillator in four dimensional euclidean space e4 we present their realization in terms of a deformed oscillator algebra using the daskaloyannis construction the structure constants are in these cases functions not only of the hamiltonian but also of other integrals commuting with all generators of the quadratic algebra we also present a new algebraic derivation of the energy spectrum of the micz kepler system on the three sphere s3 using a quadratic algebra these results point out also that results and explicit formula for structure functions obtained for quadratic cubic and higher order polynomial algebras in the context of two dimensional superintegrable systems may be applied to superintegrable systems in higher dimensions with and without monopoles

γ convergence analysis of systems of edge dislocations the self energy regime this paper deals with the elastic energy induced by systems of straight edge dislocations in the framework of linearized plane elasticity the dislocations are introduced as point topological defects of the displacement gradient fields following the core radius approach we introduce a parameter varepsilon 0 representing the lattice spacing of the crystal we remove a disc of radius varepsilon around each dislocation and compute the elastic energy stored outside the union of such discs namely outside the core region then we analyze the asymptotic behaviour of the elastic energy as varepsilon rightarrow 0 in terms of convergence we focus on the self energy regime of order log frac 1 varepsilon we show that configurations with logarithmic diverging energy converge up to a subsequence to a finite number of multiple dislocations and we compute the corresponding limit

from the ginzburg landau model to vortex lattice problems we introduce a coulombian renormalized energy w which is a logarithmic type of interaction between points in the plane computed by a renormalization we prove various of its properties such as the existence of minimizers and show in particular using results from number theory that among lattice configurations the triangular lattice is the unique minimizer its minimization in general remains open

quantum superintegrability and exact solvability in n dimensions a family of maximally superintegrable systems containing the coulomb atom as a special case is constructed in n dimensional euclidean space two different sets of n commuting second order operators are found overlapping in the hamiltonian alone the system is separable in several coordinate systems and is shown to be exactly solvable it is solved in terms of classical orthogonal polynomials the hamiltonian and n further operators are shown to lie in the enveloping algebra of a hidden affine lie algebra

invariant measures for the 2d defocusing nonlinear schrodinger equation consider the2d defocusing cubic nlsiu t u u u 2 0 with hamiltonian smallint left left nabla phi right 2 tfrac 1 2 left phi right 4 right it is shown that the gibbs measure constructed from the wick ordered hamiltonian i e replacing 4 by 4 is an invariant measure for the appropriately modified equationiu t u u u 2 2 u 2 dx u 0 there is a well defined flow on thesupport of the measure in fact it is shown that for almost all data the solutionu u 0 satisfiesu t e it c hs for somes 0 first a result local in time is established and next measure invariance considerations are used to extend the local result to a global one cf b2

gamma limit of the nonself dual chern simons higgs energy abstract we consider the gamma limit of the abelian chern simons higgs energy g csh 1 2 u a e u e 2 2 4 curl a e h ex 2 u e 2 1 e 2 u e 2 1 u e 2 2 d x on a bounded simply connected two dimensional domain under the e 0 limit as a first step we study the gamma limit of e csh 1 2 u u e 2 1 e 2 u e 2 1 u e 2 2 d x under two different scalings e csh log e and e csh log e 2 we apply the log e 2 scaling result to the full chern simons higgs energy g csh and as a consequence we are able to compute the first critical field h 1 h 1 u for the nucleation of a vortex the method entails estimating in certain weak topologies the jacobian j u e det u e in terms of the chern simons higgs energy e csh

the critical velocity for vortex existence in a two dimensional rotating bose einstein condensate we investigate a model corresponding to the experiments for a two dimensional rotating bose einstein condensate it consists in minimizing a gross pitaevskii functional defined in r 2 under the unit mass constraint we estimate the critical rotational speed 1 for vortex existence in the bulk of the condensate and we give some fundamental energy estimates for velocities close to 1

boundary regularity and the dirichlet problem for harmonic maps in a previous paper 10 we developed an interior regularity theory for energy minimizing harmonic maps into riemannian manifolds in the first two sections of this paper we prove boundary regularity for energy minimizing maps with prescribed dirichlet boundary condition we show that such maps are regular in a full neighborhood of the boundary assuming appropriate regularity on the manifolds the boundary and the data the reader may refer to theorem 2 7 for a statement of the precise result it is not surprising that the boundary regularity is actually stronger than the partial regularity we obtained for the interior this is due to the fact that there are no nontrivial smooth harmonic maps from hemispheres s which map the boundary s 9s to a point i j n 2 and is analogous to the fact that in certain cases we were able to obtain complete regularity in the interior many authors have worked on boundary regularity for this general type of problem we mention hildebrandt and widman 5 and hamilton 4 as having obtained important results specifically for harmonic maps morrey had obtained the boundary regularity for domain dimension n 2 in conjunction with his investigation of the plateau problem in riemannian manifolds 8 in 3 of this paper we observe that the direct method gives solvability of the dirichlet problem under reasonable hypotheses on the manifolds we give as an application an amusing proof of a theorem of sacks and uhlenbeck 9 on the existence of minimal 2 spheres representing the second homotopy group of a manifold the same method gives smooth harmonic representations for k n for a certain class of manifolds n these are characterized by the nonexistence of lower dimensional harmonic spheres whose homogeneous extensions are minimal see proposition 3 4 in the last section of the paper we discuss approximation of l maps by smooth maps we give a simple example of an l map from the three dimensional ball to the two sphere which is not an l limit of continuous maps we