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We showcase the combination of experimental neutron scattering data and molecular dynamics (MD) simulations for exemplary phospholipid membrane systems. Neutron and X-ray reflectometry and small-angle scattering measurements are determined by the scattering length density profile in real space, but it is not usually possible to retrieve this profile unambiguously from the data alone. MD simulations predict these density profiles, but they require experimental control. Both issues can be addressed simultaneously by cross-validating scattering data and MD results. The strengths and weaknesses of each technique are discussed in detail with the aim of optimizing the opportunities provided by this combination.

We use the group-based discrete moving frame method to study invariant evolutions in a $$n$$ -dimensional centro-affine space. We derive the induced integrable equations for invariants, which can be transformed to local and nonlocal multi-component Toda lattices under a Miura transformation, and thus establish their geometric realizations in centro-affine space.

Ginzburg-Landau vortices in superconductors attract or repel depending on whether the value of the coupling constant is less than 1 or larger than 1. At critical coupling it was previously observed that a strongly localised magnetic impurity behaves very similarly to a vortex. This remains true for axially symmetric configurations away from critical coupling. In particular, a delta function impurity of a suitable strength is related to a vortex configuration without impurity by singular gauge transformation. However, the interaction of vortices and impurities is more subtle and depends not only on the coupling constant and the impurity strength, but also on how broad the impurity is. Furthermore, the interaction typically depends on the distance and may be attractive at short distances and repulsive at long distances. Numerical simulations confirm moduli space approximation results for the scattering of one and two vortices with an impurity. However, a double vortex will split up when scattering with an impurity, and the direction of the split depends on the sign of the impurity. Head-on collisions of a single vortex with different impurities away from critical coupling is also briefly discussed., 39 pages, 18 figures, this version is accepted for publication in PRD

Two recently found coupled BPS submodels of the Skyrme model are further analyzed. Firstly, we provide a geometrical formulation of the submodels in terms of the eigenvalues of the strain tensor. Secondly, we study their thermodynamical properties and show that the mean-field equations of state coincide at high pressure and read $p=\bar{\rho}/3$. We also provide evidence that matter described by the first BPS submodel has some similarity with a Bose-Einstein condensate. Moreover, we show that extending the second submodel to a non-BPS model by including certain additional terms of the full Skyrme model does not spoil the respective ansatz, leading to an ordinary differential equation for the profile of the Skymion, for any value of the topological charge. This allows for an almost analytical description of the properties of Skyrmions in this model. In particular, we analytically study the breaking and restoration of the BPS property. Finally, we provide an explanation of the success of the rational map ansatz., Comment: LaTex file, 27 pages, 5 figures. v2: An error in Section V (old version) corrected, paper slightly reorganised, main conclusions unchanged. v3: version accepted for publication in PRD, minor changes

It is well known that the low-energy classical dynamics of solitons of Bogomol'nyi type is well approximated by geodesic motion in M_n, the moduli space of static n-solitons. There is an obvious quantization of this dynamics wherein the wavefunction evolves according to the Hamiltonian H_0 equal to (half) the Laplacian on M_n. Born-Oppenheimer reduction of analogous mechanical systems suggests, however, that this simple Hamiltonian should receive corrections including k, the scalar curvature of M_n, and C, the n-soliton Casimir energy, which are usually difficult to compute, and whose effect on the energy spectrum is unknown. This paper analyzes the spectra of H_0 and two corrections to it suggested by work of Moss and Shiiki, namely H_1=H_0+k/4 and H_2=H_1+C, in the simple but nontrivial case of a single CP^1 lump moving on the two-sphere. Here M_1=TSO(3), a noncompact kaehler 6-manifold invariant under an SO(3)xSO(3) action, whose geometry is well understood. The symmetry gives rise to two conserved angular momenta, spin and isospin. A hidden isometry of M_1 is found which implies that all three energy spectra are symmetric under spin-isospin interchange. The Casimir energy is found exactly on the zero section of TSO(3), and approximated numerically on the rest of M_1. The lowest 19 eigenvalues of H_i are found for i=0,1,2, and their spin-isospin and parity compared. The curvature corrections in H_1 lead to a qualitatively unchanged low-level spectrum while the Casimir energy in H_2 leads to significant changes. The scaling behaviour of the spectra under changes in the radii of the domain and target spheres is analyzed, and it is found that the disparity between the spectra of H_1 and H_2 is reduced when the target sphere is made smaller., Comment: 35 pages, 3 figures

The Dirac equation in a rotating frame of reference is derived from first principles within a linear approximation. This equation is employed to exhibit an equivalence between a particle in a Dirac oscillator potential and a free particle in a rotating frame of reference. A zero-point contribution to the energy of the particle, resulting from its spin, is also noted.

We investigate numerically kink collisions in a $1+1$ dimensional scalar field theory with multiple vacua. The domain wall model we are interested in involves two scalar fields and a potential term built from an asymmetric double well and (double) sine-Gordon potential together with an interaction term. Depending on the initial kink setup and impact velocities, the model allows for a wide range of scattering behaviours. Kinks can repel each other, annihilate, form true or false domain walls and reflect off each other., Matches version published in J. Phys. A. 32 pages, 20 figures and 9 kink scattering videos included as ancillary files

Lipkin’s zilches are a set of little-known conserved quantities in classical\ud electromagnetic theory. Here we report a systematic calculation of the\ud zilches for topologically non-trivial vacuum electromagnetic fields and their\ud interpretation in terms of both the physical and mathematical properties of\ud the fields. Several families of electromagnetic fields have been explored and\ud examined computationally. In these cases it is found that the zilches can be\ud written in terms of more familiar conserved quantities: energy, momentum\ud and angular momentum. Furthermore we demonstrate that the zilches also\ud contain information about the topology of the field lines for the fields we\ud have examined, thus providing a previously unsuspected aspect to their\ud interpretation. We conjecture that these properties generalise to all integrable\ud fields.

We investigate the role of pressure in a class of generalised Skyrme models. We introduce pressure as the trace of the spatial part of the energy-momentum tensor and show that it obeys the usual thermodynamical relation. Then, we compute analytically the mean-field equation of state in the high and medium pressure regimes by applying topological bounds on compact domains. The equation of state is further investigated numerically for the charge one skyrmions. We identify which term in a generalised Skyrme model is responsible for which part in the equation of state. Further, we compare our findings with the corresponding results in the Walecka model., 29 pages, 8 figures; updated to match published version in Phys. Rev. C

In this paper, we present a detailed study of Skyrmion-Skyrmion scattering for two $B=1$ Skyrmions in the attractive channel where we observe two different scattering regimes. For large separation, the scattering can be approximated as interacting dipoles. We give a qualitative estimate when this approximation breaks down. For small separations we observe an additional short-range repulsion which is qualitatively similar to monopole scattering. We also observe the interesting effect of "rotation without rotating" whereby two Skyrmions, whose orientations remain constant while well-separated, change their orientation after scattering. We can explain this effect by following preimages through the scattering process, thereby measuring which part of an in-coming Skyrmion forms part of an out-going Skyrmion. This leads to a new way of visualising Skyrmions. Furthermore, we consider spinning Skyrmions and find interesting trajectories., 25 pages, 16 figures, version accepted for publication

Euler's equations for a two-dimensional system can be written in Hamiltonian form, where the Poisson bracket is the Lie-Poisson bracket associated to the Lie algebra of divergence free vector fields. We show how to derive the Poisson brackets of 2d hydrodynamics of ideal fluids as a reduction from the one associated to the full algebra of vector fields. Motivated by some recent results about the deformations of Lie-Poisson brackets of vector fields, we study the dispersive deformations of the Poisson brackets of Euler's equation and show that, up to the second order, they are trivial., Comment: 11 pages

We calculate reduced $B(E2)$ electromagnetic transition strengths for light nuclei of mass numbers $B=8,12,16,20,24$ and $32$ within the Skyrme model. We find that the predicted transition strengths are of the correct order of magnitude and the computed intrinsic quadrupole moments match the experimentally observed effective nuclear shapes. For the Hoyle state we predict a large $B(E2)\!\uparrow$ value of $0.0521\, \rm{e}^2\rm{b}^2$. For Oxygen-16, we can obtain a quantitative understanding of the ground state rotational band and the rotational excitations of the second spin-0 state, $0_2^+$., Comment: 23 pages, 2 figures; updated to match published version in Phys. Rev. C

We study the decay of "false" domain walls, which are metastable states of the quantum theory where the true vacuum is trapped inside the wall, with the false vacuum outside. We consider a theory with two scalar fields, a shepherd field and a field of sheep. The shepherd field serves to herd the solitons of the sheep field so that they are nicely bunched together. However, quantum tunnelling of the shepherd field releases the sheep to spread out uncontrollably. We show how to calculate the tunnelling amplitude for such a disintegration., Comment: 20 pages, 9 figures

We develop a one-parameter family of static baby Skyrme models that do not require a potential term to admit topological solitons. This is a novel property as the standard baby Skyrme model must contain a potential term in order to have stable soliton solutions, though the Skyrme model does not require this. Our new models satisfy an energy bound that is linear in terms of the topological charge and can be saturated in an extreme limit. They also satisfy a virial theorem that is shared by the Skyrme model. We calculate the solitons of our new models numerically and observe that their form depends significantly on the choice of parameter. In one extreme, we find compactons while at the other there is a scale invariant model in which solitons can be obtained exactly as solutions to a Bogomolny equation. We provide an initial investigation into these solitons and compare them with the baby Skyrmions of other models., Comment: 28 pages, 8 figures; updated to match published version in Phys. Rev. D

We investigate the dynamics of BPS vortices in the presence of magnetic impurities taking the form of axially-symmetric localised lumps and delta-functions. We present numerical results for vortices on flat space, as well as exact results for vortices on hyperbolic space in the presence of delta-function impurities. In fact, delta-function impurities of appropriate strength can be captured within the moduli space approximation by keeping one or more of the vortices fixed. We also show that previous work on vortices on the 2-sphere extends naturally to the inclusion of delta-function impurities., 17 pages, 8 figures

The theory of Poisson Vertex Algebras (PVAs) is a good framework to treat Hamiltonian partial differential equations. A PVA consists of a pair $(\mathcal{A},\{\cdot_\lambda\cdot\})$ of a differential algebra $\mathcal{A}$ and a bilinear operation called the $\lambda$-bracket. We extend the definition to the class of algebras $\mathcal{A}$ endowed with $d\geq1$ commuting derivations. We call this structure a \emph{multidimensional PVA}: it is a suitable setting to study Hamiltonian PDEs with $d$ spatial dimensions. We apply this theory to the study of deformations of the Poisson brackets of hydrodynamic type for $d=2$., Comment: Revision with shorter exposition of the content of Sec 2 and new results about first cohomology groups. 50 pages. Reference and equation numbers fixed with respect to version 3

In this Brief Report we discuss a solution of the free-particle Schr\"odinger equation in which the time and space dependence are not separable. The wave function is written as a product of exponential terms, Hermite polynomials, and a phase. The peaks in the wave function decelerate and then accelerate around $t=0$. We analyze this behavior within both a quantum and a semiclassical regime. We show that the acceleration does not represent true acceleration of the particle but can be related to the envelope function of the allowed classical paths. Comparison with other ``accelerating'' wave functions is also made. The analysis provides considerable insight into the meaning of the quantum wave function.

We perform full two-dimensional (2D) numerical relaxations of isospinning soliton solutions in the baby Skyrme model in which the global $O(3)$ symmetry is broken by the 2D analogue of the pion mass term in the Skyrme model. In our calculations we explicitely allow the isospinning solitons to deform and to break the symmetries of the static configurations. We find that stable isospinning baby Skyrme solutions can be constructed numerically for all angular frequencies $\omega\le \text{min}(\mu,1)$, where $\mu$ is the mass parameter of the model. Stable, rotationally-symmetric baby Skyrmion solutions for higher angular velocities are simply an artefact of the hedgehog approximation. Isospinning multisoliton solutions of topological charge $B$ turn out to be unstable to break up into their $B$ charge-1 constituents at some critical breakup frequency value. Furthermore, we find that for $\mu$ sufficiently large the rotational symmetry of charge-2 baby Skyrmions becomes broken at a critical angular frequency $\omega$., Comment: 24 pages, 15 figures, Published version

The Skyrme model is a non-linear field theory whose solitonic solutions, once quantised, describe atomic nuclei. The classical static soliton solutions, so-called Skyrmions, have interesting discrete symmetries and can only be calculated numerically. Mathematically, these Skyrmions can be viewed as maps between to two three-manifolds and, as such, their stable singularities can only be folds, cusps and swallowtails. Physically, the occurrence of singularities is related to negative baryon density. In this paper, we calculate the charge three Skyrmion to a high resolution in order to examine its singularity structure in detail. Thereby, we explore regions of negative baryon density. We also discuss how the negative baryon density depends on the pion mass., Comment: 21 pages, 8 figures

We make explicit in terms of categories a number of statements from the theory of partial inner product spaces (PIP spaces) and operators on them. In particular, we construct sheaves and cosheaves of operators on certain PIP spaces of practical interest., Comment: 21 pages

Quantum revivals are now a well-known phenomena within nonrelativistic quantum theory. In this\ud Letter we display the effects of relativity on revivals and quantum carpets. It is generally believed that\ud revivals do not occur within a relativistic regime. Here we show that while this is generally true, it is\ud possible, in principle, to set up wave packets with specific mathematical properties that do exhibit exact\ud revivals within a fully relativistic theory.

This paper investigates a background charge one Skyrme field chirally coupled to light fermions on the 3-sphere. The Dirac equation for the system commutes with a generalised angular momentum or grand spin. It can be solved explicitly for a Skyrme configuration given by the hedgehog form. The energy spectrum and degeneracies are derived for all values of the grand spin. Solutions for non-zero grand spin are each characterised by a set of four polynomials. The paper also discusses the energy of the Dirac sea using zeta function regularization., 19 pages, 2 figures

The link between relativistic Hermite polynomials and Lorentz beams is shown. That suggests introducing new optical fields. The paraxial propagation properties of such fields are studied in detail. They are finally put in relation to the so-called Weber-Hermite beams, which emerged within a certain class of general solutions of the 1D paraxial wave equation in Cartesian coordinates as a result of a recent re-analysis of such an equation.

The question of the dimensional reduction of two-dimensional (2d) quantum models on a sphere to one-dimensional (1d) models on a circle is adressed. A possible application is to look at a relation between the 2d anyon model and the 1d Calogero-Sutherland model, which would allow for a better understanding of the connection between 2d anyon exchange statistics and Haldane exclusion statistics. The latter is realized microscopically in the 2d LLL anyon model and in the 1d Calogero model. In a harmonic well of strength \omega or on a circle of radius R - both parameters \omega and R have to be viewed as long distance regulators - the Calogero spectrum is discrete. It is well known that by confining the anyon model in a 2d harmonic well and projecting it on a particular basis of the harmonic well eigenstates, one obtains the Calogero-Moser model. It is then natural to consider the anyon model on a sphere of radius R and look for a possible dimensional reduction to the Calogero-Sutherland model on a circle of the same radius. First, the free one-body case is considered, where a mapping from the 2d sphere to the 1d chiral circle is established by projection on a special class of spherical harmonics. Second, the N-body interacting anyon model is considered : it happens that the standard anyon model on the sphere is not adequate for dimensional reduction. One is thus lead to define a new spherical anyon-like model deduced from the Aharonov-Bohm problem on the sphere where each flux line pierces the sphere at one point and exits it at its antipode., Comment: 10 pages, 1 figure

Exact metrics on some totally geodesic submanifolds of the moduli space of static hyperbolic N-vortices are derived. These submanifolds, denoted as Sigma_{n,m}, are spaces of C_n-invariant vortex configurations with n single vortices at the vertices of a regular polygon and m=N?n coincident vortices at the polygon's center. The geometric properties of Sigma_{n,m} are investigated, and it is found that Sigma_{n,n?1} is isometric to the hyperbolic plane of curvature ?(3 pi n)^{?1}. The geodesic flow on Sigma_{n,m} and a geometrically natural variant of geodesic flow recently proposed by Collie and Tong [“The dynamics of Chern-Simons vortices,” Phys. Rev. D Part. Fields Gravit. Cosmol. 78, 065013 (2008);e-print arXiv:hep-th/0805.0602] are analyzed in detail.

There exists, in general, no unique definition of the size (volume, area, etc., depending on dimension) of a soliton. Here we demonstrate that the geometric volume (area etc.) of a soliton is singled out in the sense that it exactly coincides with the thermodynamical or continuum-mechanical volume. In addition, this volume may be defined uniquely for rather arbitrary solitons in arbitrary dimensions., 16 pages, LaTex, no figures; published version; a discussion section and some references added