Self-similar solutions of the one-dimensional Landau-Lifshitz-Gilbert equation

We consider the one-dimensional Landau-Lifshitz-Gilbert (LLG) equation, a model describing the dynamics for the spin in ferromagnetic materials. Our main aim is the analytical study of the bi-parametric family of self-similar solutions of this model. In the presence of damping, our construction provides a family of global solutions of the LLG equation which are associated to a discontinuous initial data of infinite (total) energy, and which are smooth and have finite energy for all positive times. Special emphasis will be given to the behaviour of this family of solutions with respect to the Gilbert damping parameter. We would like to emphasize that our analysis also includes the study of self-similar solutions of the Schr\"odinger map and the heat flow for harmonic maps into the 2-sphere as special cases. In particular, the results presented here recover some of the previously known results in the setting of the 1d-Schr\"odinger map equation

On the numerical evaluation of algebro-geometric solutions to integrable equations

Physically meaningful periodic solutions to certain integrable partial differential equations are given in terms of multi-dimensional theta functions associated to real Riemann surfaces. Typical analytical problems in the numerical evaluation of these solutions are studied. In the case of hyperelliptic surfaces efficient algorithms exist even for almost degenerate surfaces. This allows the numerical study of solitonic limits. For general real Riemann surfaces, the choice of a homology basis adapted to the anti-holomorphic involution is important for a convenient formulation of the solutions and smoothness conditions. Since existing algorithms for algebraic curves produce a homology basis not related to automorphisms of the curve, we study symplectic transformations to an adapted basis and give explicit formulae for M-curves. As examples we discuss solutions of the Davey-Stewartson and the multi-component nonlinear Schr\"odinger equations.Comment: 29 pages, 20 figure

On linear instability of solitary waves for the nonlinear Dirac equation

We consider the nonlinear Dirac equation, also known as the Soler model: i\p\sb t\psi=-i\alpha \cdot \nabla \psi+m \beta \psi-f(\psi\sp\ast \beta \psi) \beta \psi, $\psi(x,t)\in\mathbb{C}^{N}$, $x\in\mathbb{R}^n$, $n\le 3$, f\in C\sp 2(\R), where $\alpha_j$, $j = 1,...,n$, and $\beta$ are $N \times N$ Hermitian matrices which satisfy $\alpha_j^2=\beta^2=I_N$, $\alpha_j \beta+\beta \alpha_j=0$, $\alpha_j \alpha_k + \alpha_k \alpha_j =2 \delta_{jk} I_N$. We study the spectral stability of solitary wave solutions $\phi(x)e^{-i\omega t}$. We study the point spectrum of linearizations at solitary waves that bifurcate from NLS solitary waves in the limit $\omega\to m$, proving that if $k>2/n$, then one positive and one negative eigenvalue are present in the spectrum of the linearizations at these solitary waves with $\omega$ sufficiently close to $m$, so that these solitary waves are linearly unstable. The approach is based on applying the Rayleigh--Schroedinger perturbation theory to the nonrelativistic limit of the equation. The results are in formal agreement with the Vakhitov--Kolokolov stability criterion.Comment: 17 pages. arXiv admin note: substantial text overlap with arXiv:1203.3859 (an earlier 1D version

Quantum Bound States in Yang-Mills-Higgs Theory

We give rigorous proofs for the existence of infinitely many (non-BPS) bound states for two linear operators associated with the Yang-Mills-Higgs equations at vanishing Higgs self-coupling and for gauge group SU(2): the operator obtained by linearising the Yang-Mills-Higgs equations around a charge one monopole and the Laplace operator on the Atiyah-Hitchin moduli space of centred charge two monopoles. For the linearised system we use the Riesz-Galerkin approximation to compute upper bounds on the lowest 20 eigenvalues. We discuss the similarities in the spectrum of the linearised system and the Laplace operator, and interpret them in the light of electric-magnetic duality conjectures.Comment: minor corrections implemented; to appear in Communications in Mathematical Physic

Frozen Gaussian approximation with surface hopping for mixed quantum-classical dynamics: A mathematical justification of fewest switches surface hopping algorithms

We develop a surface hopping algorithm based on frozen Gaussian approximation for semiclassical matrix Schr\"odinger equations, in the spirit of Tully's fewest switches surface hopping method. The algorithm is asymptotically derived from the Schr\"odinger equation with rigorous approximation error analysis. The resulting algorithm can be viewed as a path integral stochastic representation of the semiclassical matrix Schr\"odinger equations. Our results provide mathematical understanding to and shed new light on the important class of surface hopping methods in theoretical and computational chemistry.Comment: 35 page