应用多项式完全判别系统方法求解时空分数阶复Ginzburg-Landau方程

研究了时空分数阶复Ginzburg-Landau方程.首先通过分数阶复变换将时空分数阶复Ginzburg-Landau方程转化为一个常微分方程.然后将常微分方程化为初等积分形式.最后用多项式完全判别系统法求得一系列精确解,其中包含有孤立波解、有理函数解、三角函数周期解、Jacobi椭圆函数双周期解.     

Equivalent and Nonequivalent Barotropic Modes for Rotating Stratified Flows

All the possible equivalent barotropic (EB) laminar solutions are firstly explored,and all the possible non-EB elliptic circulations and hyperbolic laminar modes of rotating stratified fluids are disco...     

A novel hydrodynamic approach to superconductivity. Appearance of the Meissner effect

From the point of view of spin interactions, considering the electron a charged quantised vortex-type object (QVTO) with vortex strength Γ=±h/2m, we study a two-dimensional system of electrons with antiferromagnetic arrangement of spins. In the conditions of an applied magnetic field some of the electrons will flip the spin and the equivalent QVTO system will start to move due to corroborated action of the vortex population. The developed currents will create a magnetic field opposed to the applied magnetic field, leading to the appearance of Meissner effect. As a function of the intrinsic pinning, the velocity field yields two behaviours, identified with Type I and Type II superconductors. The critical values of the magnetic field arise naturally from the balance between the Lorentz and Coulombian forces acting upon a moving QVTO. A temperature dependence of the distance between the QVTO and critical field is derived.     

Catastrophe of coronal magnetic flux ropes in fully open magnetic field

The catastrophe of coronal magnetic flux ropes is closely related to solar explosive phenomena, such as prominence eruptions, coronal mass ejections, and two-ribbon solar flares. Using a 2-dimensional, 3-component ideal MHD model in Cartesian coordinates, numerical simulations are carried out to investigate the equilibrium property of a coronal magnetic flux rope which is embedded in a fully open background magnetic field. The flux rope emerges from the photosphere and enters the corona with its axial and annular magnetic fluxes controlled by a single "emergence parameter". For a flux rope that has entered the corona, we may change its axial and annular fluxes artificially and let the whole system reach a new equilibrium through numerical simulations. The results obtained show that when the emergence parameter, the axial flux, or the annular flux is smaller than a certain critical value, the flux rope is in equilibrium and adheres to the photosphere. On the other hand, if the critical value is exceeded, the flux rope loses equilibrium and erupts freely upward, namely, a catastrophe takes place. In contrast with the partly-opened background field, the catastrophic amplitude is infinite for the case of fully-opened background field.     

Catastrophe of coronal magnetic flux ropes in fully open magnetic field

The catastrophe of coronal magnetic flux ropes is closely related to solar explosive phenomena, such as prominence eruptions, coronal mass ejections, and two-ribbon solar flares. Using a 2-dimensional, 3-component ideal MHD model in Cartesian coordinates, numerical simulations are carried out to investigate the equilibrium property of a coronal magnetic flux rope which is embedded in a fully open background magnetic field. The flux rope emerges from the photosphere and enters the corona with its axial and annular magnetic fluxes controlled by a single “emergence parameter”. For a flux rope that has entered the corona, we may change its axial and annular fluxes artificially and let the whole system reach a new equilibrium through numerical simulations. The results obtained show that when the emergence parameter, the axial flux, or the annular flux is smaller than a certain critical value, the flux rope is in equilibrium and adheres to the photosphere. On the other hand, if the critical value is exceeded, the flux rope loses equilibrium and erupts freely upward, namely, a catastrophe takes place. In contrast with the partly-opened background field, the catastrophic amplitude is infinite for the case of fully-opened background field     

Theoretical analysis of the velocity field, stress field and vortex sheet of generalized second order fluid with fractional anomalous diffusion

The velocity field of generalized second order fluid with fractional anomalous diiusion caused by a plate moving impulsively in its own plane is investigated and the anomalous diffusion problems of the stress field and vortex sheet caused by this process are studied. Many previous and classical results can be considered as particular cases of this paper, such as the solutions of the fractional diffusion equations obtained by Wyss; the classical Rayleigh’s time-space similarity solution; the relationship between stress field and velocity field obtained by Bagley and co-worker and Podlubny’s results on the fractional motion equation of a plate. In addition, a lot of significant results also are obtained. For example, the necessary condition for causing the vortex sheet is that the time fractional diffusion index β must be greater than that of generalized second order fluid α; the establiihment of the vorticity distribution function depends on the time history of the velocity profile at a given point, and the time history can be described by the fractional calculus.     

Lp-solvability of a full superconductive model

In this article the mean-field vortex model arising from the II-type superconductivity is investigated. The vortex model is reduced to a nonlinear hyperbolic–elliptic system of PDEs in a bounded domain. Motivated by experiments, we consider physical boundary conditions, which describe a flux of superconducting vortices through the boundary of the domain. We prove the global solvability for the system. To show the solvability result we take a vanishing “viscosity” limit in an approximated parabolic–elliptic system. Since the approximated solutions do not have a compactness property, we justify this limit transition, using a kinetic formulation of our problem. The main trick is that instead of the nonlinear system, we have to investigate a linear transport equation.     

Hamiltonian reduced fluid model for plasmas with temperature and heat flux anisotropies

For an arbitrary number of species, we derive a Hamiltonian fluid model for strongly magnetized plasmas describing the evolution of the density, velocity, and electromagnetic fluctuations and also of the temperature and heat flux fluctuations associated with motions parallel and perpendicular to the direction of a background magnetic field. We derive the model as a reduction of the infinite hierarchy of equations obtained by taking moments of a Hamiltonian drift-kinetic system with respect to Hermite–Laguerre polynomials in velocity–magnetic-moment coordinates. We show that a closure relation directly coupling the heat flux fluctuations in the directions parallel and perpendicular to the background magnetic field provides a fluid reduction that preserves the Hamiltonian character of the parent drift-kinetic model. We find an alternative set of dynamical variables in terms of which the Poisson bracket of the fluid model takes a structure of a simple direct sum and permits an easy identification of the Casimir invariants. Such invariants in the limit of translational symmetry with respect to the direction of the background magnetic field turn out to be associated with Lagrangian invariants of the fluid model. We show that the coupling between the parallel and perpendicular heat flux evolutions introduced by the closure is necessary for ensuring the existence of a Hamiltonian structure with a Poisson bracket obtained as an extension of a Lie–Poisson bracket.     

Mathematical model of the Galathea-belt toroidal magnetic trap

We consider a mathematical model of equilibrium configurations of plasma, magnetic field, and electric field in a toroidal trap with two ring conductors with current loaded into plasma. We present the mathematical apparatus of the model based on the numerical solution of boundary value problems for the Grad–Shafranov equation (a differential equation of elliptic type for the magnetic flux function), solution methods for these problems, and numerically obtained properties of equilibrium configurations. We indicate the differences in configurations in the toroidal trap and in its analog straightened into a cylinder.     

Zero level of a purely magnetic two-dimensional nonrelativistic Pauli operator for SPIN-1/2 particles

We study the manifold of complex Bloch-Floquet eigenfunctions for the zero level of a two-dimensional nonrelativistic Pauli operator describing the propagation of a charged particle in a periodic magnetic field with zero flux through the elementary cell and a zero electric field. We study this manifold in full detail for a wide class of algebraic-geometric operators. In the nonzero flux case, the Pauli operator ground state was found by Aharonov and Casher for fields rapidly decreasing at infinity and by Dubrovin and Novikov for periodic fields. Algebraic-geometric operators were not previously known for fields with nonzero flux because the complex continuation of “magnetic” Bloch-Floquet eigenfunctions behaves wildly at infinity. We construct several nonsingular algebraic-geometric periodic fields (with zero flux through the elementary cell) corresponding to complex Riemann surfaces of genus zero. For higher genera, we construct periodic operators with interesting magnetic fields and with the Aharonov-Bohm phenomenon. Algebraic-geometric solutions of genus zero also generate soliton-like nonsingular magnetic fields whose flux through a disc of radius R is proportional to R (and diverges slowly as R → ∞). In this case, we find the most interesting ground states in the Hilbert space L ₂ (ℝ ² ).     

Multiplicity results of nonlinear fractional magnetic Schrödinger equation with steep potential

In this article, we prove the existence and multiplicity of non-trivial solutions for an indefinite fractional elliptic equation with magnetic field and concave–convex nonlinearities. Our multiplicity results are based on studying the decomposition of the Nehari manifold.     

Fractional (space–time) diffusion equation on comb-like model

In this work, a directed connection between the fractal structure and the fractional calculus has been achieved. The fractional space–time diffusion equation is derived using the comb-like structure as a background model. The solution of the obtained equation will be established for three different interesting cases.     

Bose–Einstein Condensates with Non-classical Vortex

Coupled systems of nonlinear Schrödinger equations have been used extensively to describe Bose–Einstein condensates. In this paper, we study a two-component Bose–Einstein condensate (BEC) with an external driving field in a three-dimensional space. This model gives rise to a new kind of vortex–filaments, with fractional degree and nontrivial core structure. We show that vortex–filaments is 1-rectifiable set, and calculate its mean curvature in the strong coupling (Thomas–Fermi) limit. In particular, we show that large strength of the external driving field causes vortex–filaments for a two-component BEC.     

Tanh-travelling wave solutions, truncated Painlevé expansion and reduction of Bullough–Dodd equation to a quadrature in magnetohydrodynamic equilibrium

The equations of magnetohydrodynamic (MHD) equilibria for a plasma in gravitational field are investigated. For equilibria with one ignorable spatial coordinate, the MHD equations are reduced to a single nonlinear elliptic equation for the magnetic potential , known as the Grad–Shafranov equation. Specifying the arbitrary functions in this equation, the Bullough–Dodd equation can be obtained. The truncated Painlevé expansion and reduction of the partial differential equation to a quadrature problem (RQ method) are described and applied to obtain the travelling wave solutions of the Bullough–Dodd equation for the case of isothermal magnetostatic atmosphere, in which the current density J is proportional to the exponentially of the magnetic flux and moreover falls off exponentially with distance vertical to the base, with an “e-folding” distance equal to the gravitational scale height.     

Normal and tangential continuous elements for least‐squares mixed finite element methods

We consider a finite element discretization of the primal first‐order least‐squares mixed formulation of the second‐order elliptic problem. The unknown variables are displacement and flux, which are approximated by equal‐order elements of the usual continuous element and the normal continuous element, respectively. We show that the error bounds for all variables are optimal. In addition, a field‐based least‐squares finite element method is proposed for the 3D‐magnetostatic problem, where both magnetic field and magnetic flux are taken as two independent variables which are approximated by the tangential continuous and the normal continuous elements, respectively. Coerciveness and optimal error bounds are obtained. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004.     

The commutator of the Kato square root for second order elliptic operators on ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

Let L=-div(A▽) be a second order divergence form elliptic operator, and A be an accretive, n×n matrix with bounded measurable complex coefficients in Rⁿ. We obtain the L^p bounds for the commutator generated by the Kato square root √L and a Lipschitz function, which recovers a previous result of Calderón, by a different method. In this work, we develop a new theory for the commutators associated to elliptic operators with Lipschitz function. The theory of the commutator with Lipschitz function is distinguished from the analogous elliptic operator theory.     

Coupled vortex equations and moduli: deformation theoretic approach and Kähler geometry

We investigate differential geometric aspects of moduli spaces parametrizing solutions of coupled vortex equations over a compact Kähler manifold X. These solutions are known to be related to polystable triples via a Kobayashi–Hitchin type correspondence. Using a characterization of infinitesimal deformations in terms of the cohomology of a certain elliptic double complex, we construct a Hermitian structure on these moduli spaces. This Hermitian structure is proved to be Kähler. The proof involves establishing a fiber integral formula for the Hermitian form. We compute the curvature tensor of this Kähler form. When X is a Riemann surface, the holomorphic bisectional curvature turns out to be semi-positive. It is shown that in the case where X is a smooth complex projective variety, the Kähler form is the Chern form of a Quillen metric on a certain determinant line bundle.     

Quantum flux-based resonances in solitonic vortices

Working with the Nielsen–Olesen Lagrangian in static cylindric coordinates, we derive the system of coupled field equations and perform a first-order perturbative approach, pointing out an interesting contribution connected to the London–Heitler current. For an r,θ-depending scalar boson, evolving in a constant or zero magnetic field we get, besides the flux quantization and the Landau energy levels, a less expected structure of the scalar modes whose radial and azimuthal parts are decoupled by the presence of the quantized magnetic flux.     

Multiple vortices for a self-dual CP(1) Maxwell-Chern-Simons model

We prove the existence of at least two doubly periodic vortex solutions for a self-dual CP(1) Maxwell-Chern-Simons model. To this end we analyze a system of two elliptic equations with exponential nonlinearities. Such a system is shown to be equivalent to a fourth-order elliptic equation admitting a variational structure. Tonia Ricciardi: Partially supported by the MIUR National Project Variational Methods and Nonlinear Differential Equations