David M. A. Stuart

Dynamics of Abelian Higgs vortices in the near Bogomolny regime

The aim of this paper is to give an analytical discussion of the dynamics of the Abelian Higgs multi-vortices whose existence was proved by Taubes ([JT82]). For a particular value of a parameter of the theory, λ, called the Higgs self-coupling constant, there is no force between two vortices and there exist static configurations corresponding to vortices centred at any set of points in the plane. This is known as the Bogomolny regime. We will develop some formal asymptotic expansions to describe the dynamics of these multi-vortices for λ close, but not equal to, this critical value. We shall then prove the validity of these asymptotic expansions. These expansions allow us to give a finite dimensional Hamiltonian system which describes the vortex dynamics. The configuration space of this system is the “moduli space”—the space of solutions of the static equations modulo gauge equivalence. The kinetic energy term in the Hamiltonian is obtained from the natural metric on the moduli space given by theL2 inner product of the tangent vectors. The potential energy gives the intervortex potential which is non-zero when λ is not given by its critical value. Thus the reduced equations for the evolution of the vortex parameters take the form of geodesics, with force terms to express the departure from the Bogomolny regime. The geodesics are geodesics on the moduli space with respect to the metric defined by theL2 inner product of the tangent vectors, in accordance with Mantons suggestion ([Man82]). This allows an understanding of the two main phenomenological issues—first of all there is the right angle scattering phenomenon, according to which two vortices passing through one another scatter through ninety degrees. Secondly there is the conjecture from numerical calculations that vortices repel for λ greater than the critical value, and attract for λ less than this value. The results of this paper allow a rigorous understanding of the right angle scattering phenomenon ([Sam92, Hit88]) and reduce the question of attraction or repulsion in the near Bogomolny regime to an understanding of the potential energy term in the Hamiltonian ([JR79]).

The geodesic approximation for the Yang-Mills-Higgs equations

In this paper we consider the dynamics of the monopole solutions of Yang-Mills-Higgs theory on Minkowski space. The monopoles are solutions of the Yang-Mills-Higgs equations on three dimensional Euclidean space. It is of interest to understand how they evolve in time when considered as solutions of the Yang-Mills-Higgs equations on Minkowski space-i.e. the time dependent equations. It was suggested by Manton that in certain situations the monopole dynamics could be understood in terms of geodesics with respect to a certain, metric on the space of guage equivalence classes of monopoles-the moduli space. The metric is defined by taking theL2 inner product of tangent vectors to this space. In this paper we will prove that Mantons approximation is indeed valid in the right circumstances, which correspond to the slow motion of monopoles. The metric on the moduli space of monopoles was analysed in a book by Atiyah and Hitchin, so together with the results of this paper a detailed and rigorous understanding of the low energy dynamics of monopoles in Yang-Mills-Higgs theory is obtained. The strategy of the proof is to develop asymptotic expansions using appropriate gauge conditions, and then to use energy estimates to prove their validity. For the case of monopoles to be considered here there is a technical obstacle to be overcome-when the equations are linearised about the monopole the continuous spectrum extends all the way to the origin. This is overcome by using a norm introduced by Taubes in a discussion of index, theory for the Yang-Mills-Higgs functional.

Weak–Strong Uniqueness of Dissipative Measure-Valued Solutions for Polyconvex Elastodynamics

For the equations of elastodynamics with polyconvex stored energy, and some related simpler systems, we define a notion of a dissipative measure-valued solution and show that such a solution agrees with a classical solution with the same initial data, when such a classical solution exists. As an application of the method we give a short proof of strong convergence in the continuum limit of a lattice approximation of one dimensional elastodynamics in the presence of a classical solution. Also, for a system of conservation laws endowed with a positive and convex entropy, we show that dissipative measure-valued solutions attain their initial data in a strong sense after time averaging.

On Asymptotic Stability of Solitary Waves in Schrödinger Equation Coupled to Nonlinear Oscillator

The long-time asymptotics is analyzed for finite energy solutions of the 1D Schrödinger equation coupled to a nonlinear oscillator. The coupled system is invariant with respect to the phase rotation group U(1). For initial states close to a solitary wave, the solution converges to a sum of another solitary wave and dispersive wave which is a solution to the free Schrödinger equation. The proofs use the strategy of Buslaev and Perelman (1993): the linerization of the dynamics on the solitary manifold, the symplectic orthogonal projection, method of majorants, etc.

MODULATIONAL APPROACH TO STABILITY OF NON-TOPOLOGICAL SOLITONS IN SEMILINEAR WAVE EQUATIONS

Abstract Stability properties of a class of solitary wave solutions of the equation □φ+m2φ=β(|φ|)φ, where φ : R 1+n → C , are studied. The solitary waves, of the form eiωtf(x), are called non-topological solitons. A modulational approach to stability is developed along the lines of that for the non-linear Schrodinger equation; the novel features of the present analysis arise from the ω-dependence of the stability condition. This is explained at the linear level as a condition for positivity of the Hessian on the subspace symplectically orthogonal to the tangent space. In the case β(|φ|)=|φ|p−1 the stability interval for ω can be determined precisely from the Gagliardo–Nirenberg inequality. The main theorem provides a strenthening, for this equation, of existing very general stability results for solitary waves in Hamiltonian systems proved by Grillakis, Shatah and Strauss.

Existence and Newtonian limit of nonlinear bound states in the Einstein-Dirac system

An analysis is given of particlelike nonlinear bound states in the Newtonian limit of the coupled Einstein–Dirac system introduced by Finster et al. [“Particle-like solutions of the Einstein-Dirac-Maxwell equations,” Phys. Lett. A 259, 431 (1999)]. A proof is given of the existence of these bound states in the almost Newtonian regime, and it is proven that they may be approximated by the energy minimizing solution of the Newton–Schrodinger system obtained by Lieb.

Construction of entropy solutions for one dimensional elastodynamics via time discretisation

Abstract It is shown that the variational approximation scheme for one-dimensional elastodynamics given by time discretisation converges, subsequentially, weakly and a.e. to a weak solution which satisfies the entropy inequalities. We also prove convergence under the restriction of positive spatial derivative (for longitudinal motions).

EFFECTIVE DYNAMICS FOR SOLITONS IN THE NONLINEAR KLEIN–GORDON–MAXWELL SYSTEM AND THE LORENTZ FORCE LAW

We consider the nonlinear Klein–Gordon–Maxwell system derived from the Lagrangian on four-dimensional Minkowski space-time, where ϕ is a complex scalar field and Fμν = ∂μ𝔸ν - ∂ν𝔸μ is the electromagnetic field. For appropriate nonlinear potentials , the system admits soliton solutions which are gauge invariant generalizations of the non-topological solitons introduced and studied by Lee and collaborators for pure complex scalar fields. In this article, we develop a rigorous dynamical perturbation theory for these solitons in the small e limit, where e is the electromagnetic coupling constant. The main theorems assert the long time stability of the solitons with respect to perturbation by an external electromagnetic field produced by the background current 𝕁B, and compute their effective dynamics to O(e). The effective dynamical equation is the equation of motion for a relativistic particle acted on by the Lorentz force law familiar from classical electrodynamics. The theorems are valid in a scaling regime in which the external electromagnetic fields are O(1), but vary slowly over space-time scales of , and δ = e1 - k for as e → 0. We work entirely in the energy norm, and the approximation is controlled in this norm for times of .

Adiabatic Limit and the Slow Motion of Vortices in a Chern-Simons-Schrödinger System

We study a nonlinear system of partial differential equations in which a complex field (the Higgs field) evolves according to a nonlinear Schrödinger equation, coupled to an electromagnetic field whose time evolution is determined by a Chern-Simons term in the action. In two space dimensions, the Chern-Simons dynamics is a Galileo invariant evolution for A, which is an interesting alternative to the Lorentz invariant Maxwell evolution, and is finding increasing numbers of applications in two dimensional condensed matter field theory. The system we study, introduced by Manton, is a special case (for constant external magnetic field, and a point interaction) of the effective field theory of Zhang, Hansson and Kivelson arising in studies of the fractional quantum Hall effect. From the mathematical perspective the system is a natural gauge invariant generalization of the nonlinear Schrödinger equation, which is also Galileo invariant and admits a self-dual structure with a resulting large space of topological solitons (the moduli space of self-dual Ginzburg-Landau vortices). We prove a theorem describing the adiabatic approximation of this system by a Hamiltonian system on the moduli space. The approximation holds for values of the Higgs self-coupling constant λ close to the self-dual (Bogomolny) value of 1. The viability of the approximation scheme depends upon the fact that self-dual vortices form a symplectic submanifold of the phase space (modulo gauge invariance). The theorem provides a rigorous description of slow vortex dynamics in the near self-dual limit.

Perturbation Theory for Kinks

In this paper we prove the validity of formal asymptotic results on perturbation theory for kind solutions of the sine-Gordon equation, originally obtained by McLaughlin and Scott. We prove that for appropriate perturbations, of size ε in an appropriate norm, slowly varying in time in the rest frame of the kink, the shape of the kink is unaltered in theL∞ norm toO(ε) for a time ofO(1/ε). The kink parameters, which represent its velocity and centre, evolve slowly in time in the way predicted by the asymptotics. The method of proof uses an orthogonal decomposition of the solution into an oscillatory part and a one-dimensional “zero-mode” term. The slow evolution of the kink parameters is chosen so as to suppress secular evolution of the zero-mode.