Soliton-like excitations in the strong spin-phonon interaction

Soliton-like solutions are shown to exist in a one-dimensional chain of spins $\text{1}\text{2}$ strongly coupled to the lattice phonons. The features of these solutions are investigated and discussed.     

Non-string two-magnon configurations in the isotropic Heisenberg magnet

By means of the $\text{1}\text{N}$ expansion we study the Bethe-ansatz equations for two-magnon states in the one-dimensional isotropic Heisenberg spin chain of N spins $\text{1}\text{2}$. A qualitative picture of complex solutions for N → ∞ is obtained which substantially disagrees with the string hypothesis. For example, the solutions $\text{λ}{}_{\mathrm{<mtext>1,2</mtext>}}\text{= x ±}\text{i}\text{y, x ～ N,y ～}\text{N}$ are found, whereas according to the string hypothesis $\text{y →}\text{1}\text{2}$ if N → ∞.     

Orbital classical solutions,non-perturbative phenomena and singularity at the zero coupling constant point

We try to extend previous arguments on orbital classical solutions in non-relativistic quantum mechanics to the $\text{1}\text{4}\text{λ|φ|}{}^4$ complex relativistic field theory. The single valuedness of the Green function in the semiclassical $\text{(}\text{h}\text{?}\text{→ 0)}$ limit leads to a Bohr-Sommerfeld quantization. A path integral formalism for the Green functions analogous to that in non-relativistic quantum mechanics is employed and a semiclassical approach which uses our classical solutions indicates non-perturbative effects. They reflect and $\text{e}{}^{\mathrm{<mtext>1</mtext><mtext>\lambda </mtext>}}$ singularity at the zero coupling constant point.     

Exact solutions in λϕn theory

Families of exact solutions of the λ?ⁿ theory in d + 1-dimensional space—time are obtained. Certain finite-energy static solutions are found in the case when $\text{n =}\text{2d}\text{(d ? 2)}$.     

Spontaneous symmetry breaking and fermion chirality in higher-dimensional gauge theory

The number of chiral fermions may change in the course of spontaneous symmetry breaking. We discuss solutions of a six-dimensional Einstein-Yang-Mills theory based on SO(12). In the resulting effective four-dimensional theory they can be interpreted as spontaneous breaking of a gauge group SO(10) to $\text{H}\text{=}\text{SU}\text{(3)}{}_{\mathrm{<mtext>C</mtext>}}\text{×}\text{SU}\text{(2)}{}_{\mathrm{<mtext>L</mtext>}}\text{×}\text{U}\text{(1)}{}_{\mathrm{<mtext>R</mtext>}}\text{×}\text{U}\text{(10)}{}_{\mathrm{B?L}}$. For all solutions, the fermions which are chiral with respect to $\text{H}$ form standard generations. However, the number of generations for the solutions with broken SO(10) may be different compared to the symmetric solutions. All solutions considered here exhibit a local generation group SU(2)_G × U(1)_G. For the solutions with broken SO(10) symmetry, the leptons and quarks within one generation transform differently with respect to SU(2)_G × U(1)_G. Spontaneous symmetry breaking also modifies the SO(10) relations among Yukawa couplings. All this has important consequences for possible fermion mass relations obtained from higher-dimensional theories.     

A classification of compactifying solutions for d =11 supergravity

Supergravity in eleven dimensions is known to have classical solutions of the type (anti-de Sitter space-time) × (7-dimensional Einstein space). We give a list of all homogeneous 7-manifolds which admit an Einstein metric. Known solutions are reviewed, with some emphasis on the SU(3) × SU(2) × U(1) compactifications. Their topology is discussed in detail.The list includes three new solutions, with symmetry groups SU(3) × SU(2), SO(5) and SO(5) × U(1). The first solution has no supersymmetry, while the second and third yield respectively N = 1 and N = 2 supersymmetry in four dimensions. The last two solutions may be extended to solutions with nonzero internal photon curl, breaking all supersymmetry.The existence of a spin structure on homogeneous manifolds $\text{G}\text{H}$ is discussed and related to topological properties of $\text{G}\text{H}$. As an illustration, we treat the coset spaces SU(m + 1) × SU(n + 1)/SU(m) × SU(n) × U(1), which include the spaces with SU(3) × SU(2) × U(1) symmetry.     

Petrov classification of supergravity

N = 1 supergravity is analyzed in light of the Petrov classification scheme. It is found that the ($\text{2,}\text{3}\text{2}$) chiral multiplet obtained must be recognized as the Weyl submultiplet, W_abα, where only $\text{(}\text{3}\text{2}\text{, 0) + (0,}\text{3}\text{2}\text{)}$ components contribute. Due to the SUSY transformation laws the classes of solutions allowed for various Petrov types are shown to be restrictive — indeed, in certain instances there can be no consistent solution. The freedom left in the solutions and their possible interpretation are discussed. One-loop invariants are also considered by resorting to non-covariant forms.     

Multiple meron solutions of the classical Yang-Mills equation

We give a construction for multiple meron-antimeron solutions to the classical SU(2) Yang-Mills field equations. The corresponding charge density is $\text{Q(x) =}\text{1}\text{2}\text{∑ ± γ(x?x}{}_{\mathrm{i}}\text{)}$, with the x_i distinct points on a line. The problem is reduced to the solution of a scalar φ⁴ equation. We obtain a model with merons connected in pairs.     

On the Schrödinger equation for the x2 + λx2(1 + gx2) interaction

We present an infinite set of exact solutions and eigenvalues for the one-dimensional Schrödinger equation involving the potential $\text{x}{}^2\text{+}\text{λx}{}^2\text{(1 + gx}{}^2\text{)}$. Comparison with numerical methods is made.     

On possible semiconducting and superconducting phases in hexaboride alloys

Based on the fact that hexaborides exist within a certain range of valence-electron concentration possible solid solutions are proposed that allow a wide variation of physical parameters. Phases of the type Ln_{$\text{2}\text{3}$}³⁺B₆ and Na_{$\text{1}\text{2}$}Ln_{$\text{1}\text{2}$}³⁺B₆ should be magnetic semiconductors.     

Power-like decreasing solutions of the linearized Boltzmann equation and conservation of mass or energy

We study the power-like solutions of the spatially homogeneous linearized Boltzmann equation for a class of binary cross-sections proportional to $\text{|g|}{}^{\mathrm{<mtext>1?</mtext><mtext>4</mtext><mtext>s</mtext>}}$, s=4 or s < - 1, g being the relative speed. We show that these solutions violate the physical requirement of conservation of energy. A similar study for the associate thermalization problem leads to a violation of the conservation law of mass. We study the asymptotic behaviours of the eigenfunctions associated to non-discrete eigenvalues and corresponding to the regular spectrum. The main point, which was already present in our previous study of the hard sphere case is the link between a critical power like decreasing behaviour and conservation of energy. We proved that there exists a solution ($\text{R≈v}{}^{\mathrm{<mtext>-(6?</mtext><mtext>4</mtext><mtext>s</mtext><mtext>)</mtext>}}$) associated to this behaviour (as conjecture by Ernst, Hellesoe, Hauge) and it is the only one living outside the standard Hilbert space. A very interesting tool is provided by asymptotic kernels which carry the dominant part of the asymptotic behaviour of the solutions.     

Self pulsing inm dispersive optical bistability

We analyze the stability of the stationary solutions of dispersive optical bistability in a ring cavity for a homogeneously broadened atomic system. The analysis is performed in the limit αL ? 1, with $\text{C =}\text{αL}\text{2}\text{T}$ fixed. It is shown that in the dispersive case one can yield instability also when bistability is absent. This feature suggests that the dispersive case is more suitable than the purely absorptive one with respect to the observation of the self-pulsing behavior.     

Cosmological solutions of a quadratic theory of gravity with torsion

Following the general approach of Hehl, and Hayashi and Shirafuji, we give the gravity equations for the lagrangian $\text{L}\text{=(}\text{e}\text{/2L}{}^2\text{)(F+}\text{1}\text{2}\text{×F}{}^2\text{) +}\text{L}{}_{\mathrm{M}}$. We have found the explicit Einstein-de Sitter solutions for a spinless dust. We have discussed in this case the singularity problem for the metric and for the torsion.     

The Cauchy problem for the O(N), CP(N − 1), and GC(N,p) models

We study the Cauchy problem for the O(N), $\text{C}$P(N ? 1) and G$\text{C}$(N, p) models in n + 1 dimensional space-time. We prove the existence and uniqueness of solutions for small time intervals and for any n. In space-time dimension two, the previous solutions can be extended to all times by the method of a priori estimates. In space-time dimensions three and four, our estimates yield only partial results on the global existence problem. In all cases the solutions are required only to belong to local spaces, which means that they satisfy local regularity conditions but have no restrictions on their behaviour at infinity in space.     

A new method for determining K-matrix elements

Formal solutions to the integral equation for the K operator are difficult to obtain because the principal value Green function does not have a Lippmann-Schwinger iteration. We introduce a new Hermitian operator $\text{K = V+V[}\text{P}\text{(E?H)}\text{]V}$ in this paper and show that its on-shell and half-off-shell partial wave matrix elements are equal to $\text{cos}{}^2\text{δ}{}_{\mathrm{l}}$ times those of the usual K operator. From these the fully off-shell values of the K can be determined.     

SU (4) × SU (4) symmetry breaking and its implications for SU (3) × SU (3) breaking

We investigate models where the SU (4) × SU (4) symmetry breaking Hamiltonian, $\text{H}$_SB, belongs to the (15, 15) and $\text{(10,}\text{10}\text{) + (}\text{10}\text{, 10)}$ representations, and show how they are equivalent to models of SU (3) × SU (3) breaking where $\text{H}$_SB belongs to a mixture of different representations. The results for the ππ scattering lengths in the (15, 15) model are outside the experimental limits, but the $\text{(10,}\text{10}\text{) + (}\text{10}\text{, 10)}$ model yields solutions with a wide range of values for the scattering lengths within the experimental bounds.     

A slice theorem for the space of solutions of Einstein's equations

A slice for the action of a group G on a manifold X at a point x ? X is, roughly speaking, a submanifold S_x which is transverse to the orbits of G near x. Ebin and Palais proved the existence of a slice for the diffeomorphism group of a compact manifold acting on the space of all Riemannian metrics. We prove a slice theorem for the group $\text{D}$ of diffeomorphisms of spacetime acting on the space $\text{E}$ of spatially compact, globally hyperbolic solutions of Einstein's equations. New difficulties beyond those encountered by Ebin and Palais arise because of the Lorentz signature of the spacetime metrics in $\text{E}$ and because $\text{E}$ is not a smooth manifold- it is known to have conical singularities at each spacetime metric with symmetries. These difficulties are overcome through the use of the dynamic formulation of general relativity as an infinite dimensional Hamiltonian system (ADM formalism) and through the use of constant mean curvature foliations of the spacetimes in $\text{E}$. (We devote considerable space to a review and extension of some special properties of constant mean curvature surfaces and foliations that we need.) The conical singularity structure of $\text{E}$, the sympletic aspects of the ADM formalism, and the uniqueness of constant mean curvature foliations play key roles in the proof of the slice theorem for the action of $\text{D}$ on $\text{E}$. As a consequence of this slice theorem, we find that the space $\text{D}$ = $\text{E}$/$\text{D}$ of gravitational degrees of freedom is a stratified manifold with each stratum being a sympletic manifold. The spaces for homogeneous cosmologies of particular Bianchi types give rise to special finite dimensional symplectic strata in this space $\text{G}$. Our results should extend to such coupled field theories as the Einstein-Yang-Mills equations, since the Yang-Mills system in a given background spacetime admits a slice theorem for the action of the gauge transformation group on the space of Yang-Mills solutions, since there is a satisfactory Hamiltonian treatment of the Einstein-Yang-Mills system, and since the singularity structure of the solution set is known.     

The response of one-degree-of-freedom systems with cubic and quartic non-linearities to a harmonic excitation

The method of multiple scales is used to analyze the response of single-degree-of freedom systems with cubic and quartic non-linearities to a harmonic excitation. Two first-order ordinary differential equations describing the evolution of the amplitude and the phase are derived for superharmonic resonances of order two and four, subharmonic resonances of order one-half and one-fourth, and the supersubharmonic resonances of order $\text{3}\text{2}$ and $\text{2}\text{3}$. In all cases, the steady state solutions and their stability are determined and representative numerical results are included.