Darboux theory of integrability for a class of nonautonomous vector fields

The goal of this paper is to extend the classical Darboux theory of integrability from autonomous polynomial vector fields to a class of nonautonomous vector fields. We also provide sufficient conditions for applying this theory of integrability and we illustrate this theory in several examples.Postprint (published version

Darboux parameter for empty FRW quantum universes and quantum cosmological singularities

I present the factorization(s) of the Wheeler-DeWitt equation for vacuum FRW minisuperspace universes of arbitrary Hartle-Hawking factor ordering, including the so-called strictly isospectral supersymmetric method. By the latter means, one can introduce an infinite class of singular FRW minisuperspace wavefunctions characterized by a Darboux parameter that mathematically speaking is a Riccati integration constant, while physically determines the position of these strictly isospectral singularities on the Misner time axisComment: 3 pages, LaTe

On the algebraic invariant curves of plane polynomial differential systems

We consider a plane polynomial vector field $P(x,y)dx+Q(x,y)dy$ of degree $m>1$. To each algebraic invariant curve of such a field we associate a compact Riemann surface with the meromorphic differential $\omega=dx/P=dy/Q$. The asymptotic estimate of the degree of an arbitrary algebraic invariant curve is found. In the smooth case this estimate was already found by D. Cerveau and A. Lins Neto [Ann. Inst. Fourier Grenoble 41, 883-903] in a different way.Comment: 10 pages, Latex, to appear in J.Phys.A:Math.Ge

Darboux transformation and multi-soliton solutions of Two-Boson hierarchy

We study Darboux transformations for the two boson (TB) hierarchy both in the scalar as well as in the matrix descriptions of the linear equation. While Darboux transformations have been extensively studied for integrable models based on $SL(2,R)$ within the AKNS framework, this model is based on $SL(2,R)\otimes U(1)$. The connection between the scalar and the matrix descriptions in this case implies that the generic Darboux matrix for the TB hierarchy has a different structure from that in the models based on $SL(2,R)$ studied thus far. The conventional Darboux transformation is shown to be quite restricted in this model. We construct a modified Darboux transformation which has a much richer structure and which also allows for multi-soliton solutions to be written in terms of Wronskians. Using the modified Darboux transformations, we explicitly construct one soliton/kink solutions for the model.Comment:

Higher-order Abel equations: Lagrangian formalism, first integrals and Darboux polynomials

A geometric approach is used to study a family of higher-order nonlinear Abel equations. The inverse problem of the Lagrangian dynamics is studied in the particular case of the second-order Abel equation and the existence of two alternative Lagrangian formulations is proved, both Lagrangians being of a non-natural class (neither potential nor kinetic term). These higher-order Abel equations are studied by means of their Darboux polynomials and Jacobi multipliers. In all the cases a family of constants of the motion is explicitly obtained. The general n-dimensional case is also studied

Integrability and explicit solutions in some Bianchi cosmological dynamical systems

The Einstein field equations for several cosmological models reduce to polynomial systems of ordinary differential equations. In this paper we shall concentrate our attention to the spatially homogeneous diagonal G_2 cosmologies. By using Darboux's theory in order to study ordinary differential equations in the complex projective plane CP^2 we solve the Bianchi V models totally. Moreover, we carry out a study of Bianchi VI models and first integrals are given in particular cases

An extended scaling analysis of the S=1/2 Ising ferromagnet on the simple cubic lattice

It is often assumed that for treating numerical (or experimental) data on continuous transitions the formal analysis derived from the Renormalization Group Theory can only be applied over a narrow temperature range, the "critical region"; outside this region correction terms proliferate rendering attempts to apply the formalism hopeless. This pessimistic conclusion follows largely from a choice of scaling variables and scaling expressions which is traditional but which is very inefficient for data covering wide temperature ranges. An alternative "extended caling" approach can be made where the choice of scaling variables and scaling expressions is rationalized in the light of well established high temperature series expansion developments. We present the extended scaling approach in detail, and outline the numerical technique used to study the 3d Ising model. After a discussion of the exact expressions for the historic 1d Ising spin chain model as an illustration, an exhaustive analysis of high quality numerical data on the canonical simple cubic lattice 3d Ising model is given. It is shown that in both models, with appropriate scaling variables and scaling expressions (in which leading correction terms are taken into account where necessary), critical behavior extends from Tc up to infinite temperature.Comment: 16 pages, 17 figure

Explicit solution of the linearized Einstein equations in TT gauge for all multipoles

We write out the explicit form of the metric for a linearized gravitational wave in the transverse-traceless gauge for any multipole, thus generalizing the well-known quadrupole solution of Teukolsky. The solution is derived using the generalized Regge-Wheeler-Zerilli formalism developed by Sarbach and Tiglio.Comment: 9 pages. Minor corrections, updated references. Final version to appear in Class. Quantum Gra

Generating Complex Potentials with Real Eigenvalues in Supersymmetric Quantum Mechanics

In the framework of SUSYQM extended to deal with non-Hermitian Hamiltonians, we analyze three sets of complex potentials with real spectra, recently derived by a potential algebraic approach based upon the complex Lie algebra sl(2, C). This extends to the complex domain the well-known relationship between SUSYQM and potential algebras for Hermitian Hamiltonians, resulting from their common link with the factorization method and Darboux transformations. In the same framework, we also generate for the first time a pair of elliptic partner potentials of Weierstrass $\wp$ type, one of them being real and the other imaginary and PT symmetric. The latter turns out to be quasiexactly solvable with one known eigenvalue corresponding to a bound state. When the Weierstrass function degenerates to a hyperbolic one, the imaginary potential becomes PT non-symmetric and its known eigenvalue corresponds to an unbound state.Comment: 20 pages, Latex 2e + amssym + graphics, 2 figures, accepted in Int. J. Mod. Phys.

Searching for degeneracies of real Hamiltonians using homotopy classification of loops in SO(nn)

Topological tests to detect degeneracies of Hamiltonians have been put forward in the past. Here, we address the applicability of a recently proposed test [Phys. Rev. Lett. {\bf 92}, 060406 (2004)] for degeneracies of real Hamiltonian matrices. This test relies on the existence of nontrivial loops in the space of eigenbases SO$(n)$. We develop necessary means to determine the homotopy class of a given loop in this space. Furthermore, in cases where the dimension of the relevant Hilbert space is large the application of the original test may not be immediate. To remedy this deficiency, we put forward a condition for when the test is applicable to a subspace of Hilbert space. Finally, we demonstrate that applying the methodology of [Phys. Rev. Lett. {\bf 92}, 060406 (2004)] to the complex Hamiltonian case does not provide any new information.Comment: Minor changes, journal reference adde