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D, , pp. Chen, Generation, propagation, and annihilation of metastable patterns, J. April 23, 60 Master Review Vol. Pego Dai, Universal bounds on coarsening rates for some models of phase transitions, PhD thesis, University of Maryland, College Park, Dai and R. Davis, The new diamond age, Wired, Derrida, Coarsening phenomena in one dimension, in Complex systems and binary networks Guanajuato, , vol.

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Reports, 34 , pp. Kohn and F. Otto, Upper bounds on coarsening rates, Comm. Kohn and X. Yan, Upper bound on the coarsening rate for an epitaxial growth model, Comm. Math, 56 , pp. Yan, Coarsening rates for models of multicomponent phase separation, Interfaces Free Bound. Li and J. Liu, Epitaxial growth without slope selection: energetics, coarsening, and dynamic scaling, J. Lifshitz and V. Slyozov, The kinetics of precipitation from supersaturated solid solutions, J. Solids, 19 , pp. Mendoza, J. Alkemper, and P. Voorhees, The morphological evolution of dendritic microstructures during coarsening, Metall.

Menon and R. Moldovan and L. Golubovic, Interfacial coarsening dynamics in epitaxial growth with slope selection, Phys. E, 61 , pp. Nagai and K. Niethammer and F. Otto, Ostwald ripening: the screening length revisited, Calc. Niethammer and R. Novick-Cohen and R. Rump, and A. Otto, P. Penzler, A. Pego, Front migration in the nonlinear Cahn-Hilliard equation, Proc. Pego, Stabilization in a gradient system with a conservation law, Proc.

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Watson, Coarsening dynamics of growing facetted crystal surfaces: the annealing to growth transtion. Throughout the last few decades, both theoretical and computational studies have shed light on the characteristics of the quantized vortex nucleation and dynamics. In this short lecture notes, we intend to provide a concise description of the physical background, several relevant mathematical models, and the numerical methods developed for the study of the motion and interaction of quantized vortices in various contexts. Contents 1 Introduction 2 Superconductivity and mathematical models 2.

Introduction Quantized vortices have a long history that begins with the studies of liquid Helium and superconductors. In recent years, there have been many works on the mathematical analysis and numerical simulations of quantized vortices. It is truly remarkable that many of the phenomenological properties of quantized vortices have been well captured by relatively simple mathematical models, for example, the Ginzburg-Landau equations and the Gross-Pitaesvkii equations. The phenomenological model of Ginzburg and Landau, the center piece of Nobel physics prize winning work, has been widely used in the study of superconductivity, and it is also the focus of our study here.

The structures of quantized vortices have been studied through various approaches ranging from asymptotic analysis, numerical simulations and rigorous mathematical analysis. The vortex motion, unfortunately, induces electrical resistance and causes the loss of superconductivity. Despite the much progress made in the last decade, it should be pointed out that the rigorous mathematical study of a large part of the subject on vortex dynamics remains nearly non-existent.

Indeed, what become available in the literature are primarily studies of the various dynamical laws April 23, 66 Master Review Vol. On the other hand, numerical simulations have become useful tools that could help providing a more clear picture on the exotic vortex dynamics driven by various forces, even though there are also challenging computational issues to be tackled. In this chapter, we give some physical background to both problems in superconductivity and in Bose-Einstein condensation and the associated vortex phenomena.

We also present some related mathematical models and describe the mathematical and computational studies of these models. Some computational results given in the literature as well as open questions are also provided here. We refer to the list of the references given at the end of the chapter for more detailed and more complete studies on the subject. Superconductivity and mathematical models We begin with a brief account of the basic phenomena in superconductivity and an introduction to some basic terminologies.

What is superconductivity? The superconductivity of certain metals, such as mercury, lead and tin, at very low temperatures was discovered by H. Kamerlingh-Onnes in see [] for a historical account. Since then, superconductivity has become one of the most fascinating subject of modern science. Maple, a physicist at UCSB. The discovery of superconductivity was awarded the Nobel Prize in Meissner and R.

Furthermore, passage through the critical temperature is reversible which leads to the fact that April 23, Master Review Vol. The promise of exciting new applications of high-Tc superconductivity has naturally brought a resurgence in interest of superconductivity. It is now referring to an electronic state of matter characterized by zero resistance, perfect diamagnetism, and long-range quantum mechanical order. Their two dimensional planar cross-sections are referred as two dimensional vortices. One of the most challenging problems to mathematicians working on the superconductivity models is the understanding of vortex phenomena in type-II superconductors, which include the recently discovered hightemperature superconductors.

Quantized vortex structures have been studied extensively on the mezoscale using the well-known Ginzburg-Landau models of superconduc- April 23, Master Review Vol. The existence of vortex like solutions for the full nonlinear Ginzburg-Landau equations has been investigated by researchers using methods ranging from asymptotic analysis to numerical simulations.

Therefore, it is crucial to understand the dynamics of these vortex lines. Various such vortex pinning mechanisms have been advanced by physicists, engineers, and material scientists. For example, normal nonsuperconducting impurities in an otherwise superconducting sample are believed to provide sites at which vortices are pinned. Likewise, anisotropy and other material inhomogeneities such as the thickness variations in thin samples are also believed to provide pinning sites.

These mechanisms have been introduced into the general Ginzburg-Landau framework to derive various variants of the original Ginzburg-Landau models of superconductivity. As of today, applications of superconductivity that are currently being used include magnetic shielding devices, medical imaging systems, superconducting cyclotron, superconducting quantum interference devices SQUIDS , infrared sensors, analog signal processing devices, and microwave devices.

Applications that are being explored for the future include power transmission, superconducting magnets in generators, energy storage devices, particle accelerators, and magnetic levitated vehicle transportation. More recently, the concept of quantum computing has become important research directions in superconducting electronics [22].

The superconducting current controller Current Fault Limiter or CFL are being developed to control the reduction of fault currents surges in transmission lines. Superconducting magnets are already crucial components of several technologies. Magnetic resonance imaging MRI is playing an ever increasing role in diagnostic medicine. Superconductivity models and mathematical problems Along with the historical development of experimental discoveries in superconductivity, there have been numerous theoretical studies trying to decipher the mystery of this intriguing phenomena.

However, a good theoretical understanding of the low-temperature superconductivity was not arrived at until the s. Indeed, a completely acceptable microscopic theory did not exist until Bardeen-Cooper-Schriefer BCS published their landmark paper in This work was awarded the Nobel physics prize in Near Tc , two electrons not only experience a repulsive Coulomb force, but also an attractive force through electron-phonon interactions, thus forming the so called Cooper pairs.

Such electron pairs are coherent structures that can pass through the conductor in unison. According to the BCS theory, as a negatively charged electron passes by positively charged ions in the lattice of the superconductor, the lattice distorts. That is, the attraction between the negative electron and the positive ion causes a lattice vibration from ion to ion until the other electron of the pair absorbs the vibration. This causes phonons to be emitted, thus forming a trough of positive charges around the electron. Before the electron passes by and before the lattice springs back to its normal position, a second electron is drawn into the trough.

The exchange of phonon keeps the Cooper pairs together, though the pairs are constantly breaking and reforming. Such a scenario gives the basic mechanism of low Tc superconductivity. The BCS theory not only leads to fundamental understanding of the low Tc superconductivity, but also gives the so-called BCS gap equations that can be used to determine the critical transition temperature.

We refer to [75] for April 23, Master Review Vol. Historically, however, various mezoscopic theories had been proposed even earlier than the BCS theory, most notably the London theory in , the nonlocal theory of Pippard and the theory of Ginzburg and Landau in Despite of the recent progress on the mathematical analysis and numerical simulations of the various superconductivity models, there is still a diverse class of many interesting and challenging questions to be answered in the future.

The mathematical theory of Ginzburg-Landau models The phenomenological model of Ginzburg and Landau [65, ], and its various generalizations have been widely used in the mathematical and numerical studies of the vortex phenomena in superconductivity. April 23, 74 Master Review Vol. Since then, it has been formally derived from the microscopic BCS, generalized to many variations for high Tc superconductors.

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To this day, it remained as a standard initial approach to study problems in superconductivity. We refer to [65] for an earlier review on the basic mathematical theory. Numerical approximations of the Ginzburg-Landau have also been developed there systematicaly, see also [59] for a brief survey on the computational aspect. It was originally given to describe the equilibrium state and later generalized to the time-dependent cases.

A few comments about the free energy used in the Ginzburg-Landau models are in order. First, the choice of complex order parameter, though April 23, Master Review Vol. The Ginzburg-Landau model produced many interesting and valid results. One important consequence is its prediction of the existence of two characteristic lengths in a superconductor.

April 23, 76 Master Review Vol. To apply the Ginzburg-Landau theory in such situations, a coupled system of equations must be solved in both the sample and its exterior. In the literature, more general boundary conditions have also been studied. The energy functional and all the physically observable quantities are obviously invariant under the gauge transformation. However, nice remedy has been developed to avoid such pitfalls [65]. With the equivalent formulation, one can simply enforce the Coulomb gauge implicitly by solving for the variational problems with respect to F April 23, Master Review Vol.

In other words, the magnitude of the order parameter is bounded above by 1, the value of the superconducting state in our nondimensionalized setting. The TDGL equations 3. As the energy only gives the control on the curl A, a regularization was April 23, Master Review Vol. Hence, we get the existence and uniqueness of strong solutions of the original system without the regularization. Sharper results based on some better energy estimates and the long time solution behavior have later been studied, for instance, in [, ] and [85].

In particular, by utilizing the gauge invariance, it has been shown that Lemma 3. The above lemma, coupled with the theory of L. Naturally, the most interesting dynamics is when there is an applied current. More detailed discussions along this direction will be given later. Numerical algorithms for Ginzburg-Landau models The development of approximation methods of the Ginzburg-Landau model goes back to the s shortly after the inception of the model [89]. Particularly notable works include the seminal paper by Abrikosov [1] on the April 23, Master Review Vol.

The systematic studies of the G-L models from the numerical analysis point of view, to our knowledge, have not been seriously developed until the publication of [65]. The work in [65] was partly motivated by [2, 47] of casting the equilibrium models into a variational framework. Extensions to the dynamic models, i. By now, almost all aspects of modern numerical analysis have been utilized by people working on the numerical solutions of the G-L models, ranging from the design and applications of various discretization methods and fast algorithms, domain decomposition and parallelization techniques, and adaptive computation strategies.

Let us add that there are also mis-conceptions related to the computation of vortices based on the G-L models. For a recent review, we refer to [59]. As there have been a large amount of works on the numerical simulations of the G-L models in the last twenty years, we make no attempt to provide a comprehensive survey on all existing works on the subject due to limited space. In particular, our review of the works appeared in the vast physics literature is very much limited to those that have also received much attention in the numerical analysis community or have been examined more rigorously in the mathematics literature.

Then the nonlinear problems are shown to be the compact perturbations in the appropriate Sobolev spaces, so that on the nonsingular solution branches, the optimal order error estimates also hold. We note that the derivation of error estimates of the fully discrete scheme was, however, not rigorously provided there.

By using a mixed formulation, [32] has presented a more complete theory for the approximation of the TDGL along with optimal order error estimates in two space dimension. Later, [35] considered approximations to a related optimal control problem. Generalizations to the time-dependent Lawrence-Doniach model have been presented in [96]. The motivation has come from the fact that the underlying physical model enjoys the gauge invariance property. Many subsequent works have followed up on such an approach via an introduction of the so called link and bond variables, and various extensions have also been made [37, 38, 82, 87, 91, 93, , ].

For simplicity, let us consider the two dimensional setting with a uniform rectangular mesh of grid size h. See Figure4 for an illustration. The approximation preserves the variational structure at the discrete level: Theorem 4. It remains to see if a general higher order in time fully discrete gauge invariant scheme can be developed for the TDGL equations. We note here also that, in practical numerical simulations, explicit or semi-implicit in time difference schemes have been mostly employed.

In general, these schemes are only conditionally stable at best. The dual tessellation to a Voronoi tessellation consisting of triangles is referred to as a Delaunay triangulation. A staggered uniform grid and a Voronoi-Delaunay mesh. This can be seen by making the equivalence of the rectangular cells with the Voronoi cells and the equivalence of the dual cells with pairs of right Delaunay triangles sharing a common edge opposing the right angles. This, coupled with suitable energy estimates, leads to the convergence of the discrete approximations as the mesh size goes to zero, see [71] for details.

In addition to the basic convergence properties, a novel feature of the discussions in [68, 69] is the consideration of a special Voronoi-Delaunay pair, namely, the spherical centroidal Voronoi tessellations and the corresponding Delaunay triangulations [63]. For the standard CVTs in the Euclidean spaces, the generators of the Voronoi regions all coincide with the mass centers of the corresponding regions.

It has been shown that by using the SCVTs, the discrete approximations exhibit a higher order convergence comparing with the conventional Voronoi-Delaunay pair. Such approximations are particularly valid for high-kappa materials in such cases, some reduced G-L models have been proposed and used in numerical simulations of vortex lines [27, 41, 61]. Since J can be explicitly expressed with the help of Legendre functions, various orders of approximations can be constructed.

More on time-discretization For TDGL models, while most of the rigorous mathematical analysis have been focused on the fully implicit in time discretizations, explicit marching schemes and semi-implicit marching schemes [] have also been frequently used in numerical simulations due to their simplicity in implementation. For example, a linearized Crank-Nicolson scheme has been considered in [], similar to the semiimplicit approach. Analytical studies of an alternating marching scheme have also been made in [] where for the order parameter and magnetic April 23, Master Review Vol.

Multi-level, adaptive and parallel algorithms The numerical simulations of the vortex state in type-II superconductors based on the G-L models become computationally challenging when there is a need to resolve a large number of vortices. There have been a lot of interesting attempts made along this direction. Using a natural domain decomposition strategy, a number of parallel algorithms for the simulation of layered superconductors based on the Lawrence-Doniach model have been studied in [62].

Other methods For the time-discretization, there are also studies on various time-splitting methods. With the time-splitting scheme, it has become a popular approach to use spectral Galerkin or spectral collocation methods for the spatial discretization. One can employ FFT to perform the transformation between the real space and frequency space representations.

The detailed analysis and simulations are presented here. The phenomena of quantized vortices are well-known features of superconductivity. For type-II superconductors, the study of Abrikosov on the vortex lattices based on the G-L models has become a seminal work Nobel Physics Prize in that exhibits the great predictive power of the G-L theory. When H becomes larger than HC2 , the vortex structures lattices would generally be destroyed and the solution becomes one that corresponding to April 23, Master Review Vol.

For a two dimensional geometry, the bifurcation diagrams of the G-L models are quite complex depending on the parameters involved. What distinguishes them are features like the existence or the lack of existence of vortex solutions, the global and local stability of solutions, and the hysteresis phenomena. We refer to [3, ] for detailed calculations and some rigorous analysis. Vortex solutions In Fig. The integer n gives the topological degree of the vortex. The stability of such vortex solutions has been carried out both numerically and analytically in recent years [3, 65, ].

A phase diagram left with respect to the sample size horizontal axis and GL parameter vertical axis ; associated magnetization curves in four parameter regimes right. Given a bounded domain, it has been well understood both physically April 23, Master Review Vol. Then the global minimizer starts to nucleate vortices. Other works concerning the linearized problem include [, ]. This is what is called surface superconductivity.

A linearization of the GinzburgLandau equation has been done near the normal solution which is consistent with the work of Saint James and de Gennes []. In this limit, some rigorous mathematical results can be established. We outline one of the such result here. Quantum tunneling is a process arising from the wave nature of the electron.

This arrangement is called a Josephson junction. The key here is that the two superconductors act to preserve their long-range order across the insulating barrier. Rapid alternating currents occur within the insulator when a steady voltage is applied across the superconductors. Studies of SNS junctions are particularly useful in many applications including the design of microwave devices using high-Tc superconductors.

The Josephson junction can act as a super-fast switching devise that can perform switching functions such as switching voltages approximately ten times faster than ordinary semi-conducting circuits. The April 23, Master Review Vol. The nonlinear Josephson equations proposed by Josephson was partly motivated by the Ginzburg-Landau theory. For example, the G-L type models for SNS junctions was derived in [25] to account for both the superconducting layers and the normal layer.

A discussion of supercurrent across a one-dimensional junction is also presented there. Other related mathematical works include [, ]. In [43], the minimizers of the free energy functional 5. The more interesting issues are naturally related to the vortex solutions. A couple of sample results are provided below. An applied current exerts the Lorentz force on the vortices, and the motion of vortices unfortunately, induces electrical resistance and cause the loss of superconductivity.

Thus, understanding the dynamics of quantized vortices thus bears tremendous importance in the practical application of superconductivity. On the other hand, numerical simulations have become useful tools that could help providing a more clear picture on the exotic vortex dynamics driven by various forces.

We should note that other types of dynamic equations are also available, such as the wave dynamics which is believed to be valid near 0K , and the more general Glauber dynamics. Dynamics of vortex nucleation Typical snapshots of the solutions of TDGL for a two dimensional square domain are given in the Fig. Later, the front becomes unstable, and individual vortices start to nucleate near the edges, and move into the interior of sample. For more numerical simulations, see [41, 66, 67, 87, 91, ]. Neu, and later extended, and improved by many others April 23, Master Review Vol.

Such formal expansions have now been largely rigorously proved [, , ]. By examining the particular forms of the renormalized energy, it has been shown that isolated vortices of the same signs tend to repel while isolated vortices of opposite signs attract. The case of motion laws derived as the limit of the full time-dependent Ginzburg-Landau models has also been studied, for instance, in [30, 48]. In such case, a x is a conventional trapping potential while curl A0 represents the angular velocity of the rotating trap [4].

Dynamics driven by the applied current Since there is no loss in electrical energy when superconductors carry electrical current, relatively narrow wires made of superconducting materials can be used to carry huge currents. However, there is a certain maximum current the so called critical current that these materials can be made to carry, above which they stop being superconductors. In general, the value of critical current density is a function of temperature; i. Thus, in superconductivity, it is important to understand the interaction of the vortices with the applied current and study the critical values of the applied current which will dislodge the vortices from their equilibrium positions.

The TDGL equations may be used as a prototype model for the study of critical current. Beyond the basic well-posedness [61], there exists very little mathematical analysis on the HKHF equation. In fact, even many questions on its steady state solutions remain largely unanswered. In Fig. For detailed numerical simulations substantiating the discussion here, we refer to [61] and also more recent computations in [68, 69]. First of all, it is trivial to see that the solution beginning with the nor- April 23, Master Review Vol.

Left: An incomplete phase diagram left and an energy bifurcation diagram right for the HKHF model driven by current. These vortices will move through the sample, and the Lorentz force is again strong enough to push them over the barrier on the other side of the boundary, thus lead to possibly timeperiodic motion of single vortex or vortex arrays, see Fig. Such a current J is thought to be above the critical current. If J is exceedingly large, the time dependent solution would eventually collapse to the normal state regardless of its initial state.

Motion of vortex array in the presence of an applied current. We note that similar discussions for the full TDGL also remain to be carried out. Numerical simulations can again be very helpful. Vortex state in a thin superconducting spherical shell The geometry of spherical shell is not only used in superconductivity applications, but also provides an ideal setting for one to examine the vortex state. Plots of energy in time: evidence of time-periodic solution. SCVT meshes top and a computed vortex lattice.

We refer to [68] for more detailed descriptions of the corresponding parameter values. The nucleation and the splitting of vortex pairs near equator. Other simulations on vortex annihilation can be found in [61] in the planar domain, aided by an applied current. In the study of stochastic vortex dynamics given in [95], the average position of vortices is determined by averaging the equations that determine the movement of vortices, i. To model such phenomena, we may use stochastic GL models following some Langevin dynamics [86, 48, 49].

The above equations belong to the class of stochastic G-L models with an additive noise. Snapshots of the vortex tubes in a cubic sample with increasing additive noises. Theorem 6. We also observe that while the additive noise tends to make the positions of vortices vibrate, multiplicative noises tends to alter the core sizes of the vortices which is even more apparent when discrete lattice random functions are used.

Variants of G-L models: Lawrence-Doniach and d-wave models The great success of the Ginzburg-Landau models for low Tc superconductivity generated tremendous interests in extending them to other settings including layered materials and high Tc superconductors. The Lawrence-Doniach LD model is a derivative of the basic G-L model for a layered superconductor with G-L energy given in individual layers and the Josephson-like coupling between the layers []. Such pancake vortices may interact both magnetically and through Josephson coupling. They may align under certain conditions into elastic vortex lines similar to those in three dimensional bulk isotropic superconductors.

Vortex torsion due to the pinning of the normal inclusions in a three dimensional superconductor sample. The numerical algorithms provided in [62] for the LD model were based on codes developed for the two-dimensional Ginzburg-Landau equations. This interdependence of the variables between layers may be solved for iteratively.

The results of [62] show that with a judiciously-chosen iterative scheme, the coupling of the variables between the layers may be broken which allows for a straight-forward parallelization of the solution. The convergence of those iterative schemes can be studied rigorously. Three dimensional vortex tubes in a three dimensional layered sample pinned due to spatially distributed inhomogeneities were computed via the Lawrence-Doniach [62].

Other numerical studies can be found, for example, in []. The basic feature of these models is the use of multi-component order parameters. The free energy densities are ex- April 23, Master Review Vol. A rigorous mathematical framework was established and comparison with the conventional Ginzburg-Landau model for the low-Tc superconductors were made. The numerical results provided various new and exotic structures in the vortex solutions of the d wave Ginzburg-Landau model. The simulations in [55] illustrated that the vortex solutions there typically display a four fold symmetry.

In a limiting regime, the d-wave model may be viewed as a perturbation April 23, Master Review Vol. More analytical and numerical studies on the d-wave G-L models can be found in [, , ]. Dynamics of the merger of two d wave vortices. Numerical studies of other extensions of the G-L models have also been performed, for instance, see [7] for simulations based on the SO 5 model. Vortex density models The number of vortices present in a superconducting sample of dimension, say, a millimeter, will contain a huge number vortices as the separation of individual vortices occurs on a typical length scale of or so angstrom.

An approximate solution sequence to the above system 6. The approximate solution sequence to 6. In [, ], the critical current was cast as a constraint in a variational inequality formulation. Macroscopic models like the critical state models are particularly relevant to the device design using superconductors.

There have been more studies made both from analytical and computational aspects on macroscopic models, see [34, 56, 58, 76, 79, 80, ] and the references listed there. In recent BEC experiments, vortices have been nucleated with the help of laser stirring and rotating magnetic traps. Remarkably, many of the phenomenological properties of quantized vortices have been well captured by mathematical models such as the Gross-Pitaevskii G-P equations.

The breakthrough development in BEC has attracted the attention of many mathematicians and computational scientists, in particular, a large community of researchers who have worked on NLS and related problems. In recent years, the studies of quantized vortices in BECs have also becoming increasingly important. Mathematical and computational studies on the vortex state in such experiments have been carried out based on the Gross-Pitaevskii theory.

For the case of rotating magnetic traps, the mathematical form of the GP equations have close resemblance with the high-kappa Ginzburg-Landau G-L model we have studied in the context of superconductivity modeling. Utilizing the mathematical theory and the numerical codes developed for the G-L, we were able to build up a similar mathematical framework for rigorously characterizing the critical velocities for the vortex nucleation in a BEC cloud which is subject to a rotating magnetic trap. This also allowed us to make qualitative studies of the quantized vortex state [4].

The main ingredient of the analysis lies in the decoupling of the energy into three sources: a part coming from the state without vortices, another part from contribution of individual vortices and an additional part produced due to the rotation. Based on the energy expansion, we get estimates on the critical angular velocity for the nucleation of n-vortices:! In addition, it was shown that near the critical angular April 23, Master Review Vol.

Vortex tubes in a BEC with a rotating magnetic trap. We refer to [4] for more detailed analysis and additional references. In [12], such methodology has been developed to solve the Gross-Pitaevskii equation with a rotation term, the same idea can also be applied to solve the time-dependent Ginzburg-Landau equations. To preserve the long-time stability, we have also discussed symplectic and multisymplectic schemes in [57]. Pomeau, goes beyond the rotating magnetic trap. With a blue tune laser beam stirring the BEC cloud, energy dissipation has also been observed in recent experiments [].

The above equations provided good models for the study of the onset of dissipation, and the vortex-sound interaction. A typical two-dimensional simulation is shown in Fig. In three dimensional BECs, due to the spatial inhomogeneities of the vortex density distribution, the onset of vortex shedding becomes much more complex. In [5], we found that there is always a drag around the laser beam for whatever values of the velocity of the stirrer and we analyzed the mechanism of vortex nucleation. The critical velocity computed through our 3D simulations of the NLS dynamics is lower than the critical velocity obtained for the corresponding 2D problem at the center of the cloud and agrees well with experimental results.

This is often regarded as one of the most fundamental problems in modern physics, and progress along this direction certainly will generate new issues to be studied. In addition, it also helps to study the interplay between superconductivity and ferromagnetism and antiferro-magnetism. Simulations of G-L models have been performed recently on structures like bucky balls []. Such studies are also related to the understanding of the vortex state in junction arrays and networks.

In addition, the interactions of vortices with other matters, such as BEC vortices in optical traps, and those interacting with lasers, are also very interesting subjects to study. In , a team of Japanese physicists discovered superconductivity in an abundant Magnesium-Boron compound M gB2 , with a transition temperature about 40K, which is believed to be the limit of the conventional BCS superconductors. A transition temperature that high can be achieved by technologies that cost much less than those needed to bring about superconductivity in April 23, Master Review Vol.

Due to the multi-band feature, this naturally brings out new analytical and numerical issues to be examined. A buckyball and the atomic structure of MgB2. Though numerical simulations have been conducted, a rigorous mathematical theory is still limited. Preliminary studies on the vortex states in the presence of an applied current has been made, numerical simulations of periodic vortex motion have also been carried out.

A complete mathematical characterization is still lacking. Vortices in 3D : the motion, tangling, orientation, pinning, etc. As for problems in the technological applications of superconductivity, one may consider the optimal design of composite materials. On the computational side, there are still number of questions related April 23, Master Review Vol. For the vortex density models, most of the numerical analysis have been limited to the two dimensional cases.

Generalizations to the three dimensional models have not been explored so far. For large n and even moderate values of m, this can become very demanding computationally even for two dimensional problems, not to mention the more challenging three dimensional cases. It is easy to see that the variation in the phase variable starts to become increasingly dramatic when getting closer to the boundary equator.

Moreover, we observe an interesting phemenonon in terms of the dimension mismatch, namely, for a lattice like distribution of the vortices, as the diameter of the domain doubles, the number of vortices quadraples.

## Bao Weizhu

Hence, even though the boundary size gets only doubled, the resolution needs to be improved four-folds. On the other hand, for high-temperature superconductors, codes for mezoscale G-L models cannot hope to be of direct use in the design of devices due to the presence of large number of vortices.

The number of vortices present in a superconducting sample of dimension, say, a millimeter, April 23, Master Review Vol. In this paper, various methods for the numerical approximations of the Ginzburg-Landau models of superconductivity are discussed, with an emphasis on the application to the study of vortex dynamics. From a practical point of view, large-scale numerical simulations of the magnetic vortices complement physical experiments due to the complex three-dimensional, time-dependent, stochastic and multi-scale nature of the phenomena.

For the last 50 years, April 23, Master Review Vol. Acknowledgment The author wishes to thank all of his collaborators for joint works in the area covered by this lecture notes over the years, most of their names are provided in the references. Shi and Y. Optical Soc. General neck condition for the limit shape of budding vesicles , with P. Tu, Physical Review E , 95, , A conservative nonlocal convection-diffusion model and asymptotically compatible finite difference discretization , with L. Ju and H. Tian, Comp. Nonlocal diffusion and peridynamic models with Neumann type constraints and their numerical approximations , with Y.

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