On a reduction of the generalized Darboux–Halphen system

The equations for the general Darboux–Halphen system obtained as a reduction of the self-dual Yang–Mills can be transformed to a third-order system which resembles the classical Darboux–Halphen system with a common additive terms. It is shown that the transformed system can be further reduced to a constrained non-autonomous, non-homogeneous dynamical system. This dynamical system becomes homogeneous for the classical Darboux–Halphen case, and was studied in the context of self-dual Einstein's equations for Bianchi IX metrics. A Lax pair and Hamiltonian for this reduced system is derived and the solutions for the system are prescribed in terms of hypergeometric functions.     

On the Darboux integrability of a three–dimensional forced–damped differential system

In 2011 Pehlivan proposed a three–dimensional forced–damped autonomous differential system which can display simultaneously unbounded and chaotic solutions. This untypical phenomenon has been studied recently by several authors. In this paper we study the opposite to its chaotic motion, i.e. its integrability, mainly the existence of polynomial, rational and Darboux first integrals through the analysis of its invariant algebraic surfaces and its exponential factors.     

On colour non-singlet representations of the quark–gluon system at finite temperature

We use a group theoretical technique to project out the partition function for a system of quarks, antiquarks and gluons onto a particular representation of the internal symmetry group SU(3): the colour singlet, colour octet and colour 27-plet, at finite temperature. We do this to calculate the thermodynamic quantities for those representations. We also calculate the change in free energy of the plasma droplet formed from the hot hadronic gas. We find that the size of the droplet in the colour-octet representation is smaller than that in the colour-singlet representations at different temperatures in the vicinity of the critical temperatures of the phase transitions. Received: 1 February 1999 / Revised version: 24 February 2000 / Published online: 18 May 2000     

Exact solutions and residual symmetries of the Ablowitz–Kaup–Newell–Segur system

The residual symmetries of the Ablowitz–Kaup–Newell–Segur(AKNS)equations are obtained by the truncated Painleve′analysis.The residual symmetries for the AKNS equations are proved to be nonlocal and the nonlocal residual symmetries are extended to the local Lie point symmetries of a prolonged AKNS system.The local Lie point symmetries of the prolonged AKNS equations are composed of the residual symmetries and the standard Lie point symmetries,which suggests that the residual symmetry method is a useful complement to the classical Lie group theory.The calculation on the symmetries shows that the enlarged equations are invariant under the scaling transformations,the space–time translations,and the shift translations.Three types of similarity solutions and the reduction equations are demonstrated.Furthermore,several types of exact solutions for the AKNS equations are obtained with the help of the symmetry method and the Bcklund transformations between the AKNS equations and the Schwarzian AKNS equation.     

Darboux transformation of the Drinfeld–Sokolov–Satsuma–Hirota system and exact solutions

A Darboux transformation for the Drinfeld–Sokolov–Satsuma–Hirota system of coupled equations is constructed with the aid of gauge transformations between the Lax pairs. As an application, several types of solutions of the Drinfeld–Sokolov–Satsuma–Hirota system are obtained, including soliton solutions, periodic solutions, rational solutions and others.     

On the computation of ground state and dynamics of Schrödinger–Poisson–Slater system

In this paper, we deal with the computation of ground state and dynamics of the Schrödinger–Poisson–Slater (SPS) system. To this end, backward Euler and time-splitting pseudospectral methods are proposed for the nonlinear Schrödinger equation with the nonlocal Hartree potential approximated by solving a Poisson equation. The approximation approaches for the Hartree potential include fast convolution algorithms, which are accelerated by using FFT in 1D and fast multipole method (FMM) in 2D and 3D, and sine/Fourier pseudospectral methods. The inconsistency in 0-mode in Fourier pseudospectral approach is pointed out, which results in a significant loss of high-order of accuracy as expected for spectral methods. Numerical comparisons show that in 1D the fast convolution and sine pseudospectral approaches are compatible. While, in 3D the fast convolution approach based on FMM is second-order accurate and the Fourier pseudospectral approach is better than it from both efficiency and accuracy point of view. Among all these approaches, the sine pseudospectral one is the best candidate in the numerics of the SPS system. Finally, we apply the backward Euler and time-splitting sine pseudospectral methods to study the ground state and dynamics of 3D SPS system in different setups.     

On symmetries,conservation laws and exact solutions of the nonlinear Schrödinger–Hirota equation

In this paper, conservation laws and exact solution are found for nonlinear Schrödinger–Hirota equation. Conservation theorem is used for finding conservation laws. We get modified conservation laws for given equation. Modified simple equation method is used to obtain the exact solutions of the nonlinear Schrödinger–Hirota equation. It is shown that the suggested method provides a powerful mathematical instrument for solving nonlinear equations in mathematical physics and engineering.     

The Maxwell–Einstein system,Ward identities and the Vilkovisky construction

The gauge fixing dependence of the one-loop effective action of quantum gravity in the proper-time representation is investigated for a space of arbitrary curvature, and the investigation is extended to Maxwell–Einstein theory. The construction of Vilkovisky and DeWitt for removal of this dependence is then considered in general gauges, and it is shown that nontrivial criteria arising from a Ward identity of the theory must be obeyed by the regularization scheme, if the construction is to remove the gauge dependence of quadratic and quartic divergences. The results apply also to non-Abelian gauge theories; they are used to address the question of gauge dependence of asymptotic freedom arising through internal graviton lines at one-loop order as suggested by Robinson and Wilczek.     

On the spectra of Schrödinger operators

We give two formulas for the lowest point in the spectrum of the Schrödinger operatorL=–(d/dt)p(d/dt)+q, where the coefficientsp andq are real-valued, bounded, uniformly continuous functions on the real line. We determine whether or not is an eigenvalue forL in terms of a set of probability measures on the maximal ideal space of theC ^*-algebra generated by the translations ofp andq.Research supported in part by the National Science Foundation under Grant DMS-910496     

On eigenvalue estimates for the Dirac–Witten-type operators

We give optimal lower bounds for the eigenvalues of the Dirac–Witten-type operators associated with the _e0$e_0$-Killing connection and imaginary Killing connection, in terms of the mean curvature and the scalar curvature. The limiting cases are then studied and lead to interesting geometric situations.     

Localization of nonlocal symmetries and interaction solutions of the Sawada–Kotera equation

The nonlocal symmetry of the Sawada–Kotera(SK) equation is constructed with the known Lax pair. By introducing suitable and simple auxiliary variables, the nonlocal symmetry is localized and the finite transformation and some new solutions are obtained further. On the other hand, the group invariant solutions of the SK equation are constructed with the classic Lie group method.In particular, by a Galileo transformation some analytical soliton-cnoidal interaction solutions of a asymptotically integrable equation are discussed in graphical ways.     

Exact solutions of the Wheeler–DeWitt equation and the Yamabe construction

Exact solutions of the Wheeler–DeWitt equation of the full theory of four dimensional gravity of Lorentzian signature are obtained. They are characterized by Schrödinger wavefunctionals having support on 3-metrics of constant spatial scalar curvature, and thus contain two full physical field degrees of freedom in accordance with the Yamabe construction. These solutions are moreover Gaussians of minimum uncertainty and they are naturally associated with a rigged Hilbert space. In addition, in the limit the regulator is removed, exact 3-dimensional diffeomorphism and local gauge invariance of the solutions are recovered.     

On the phase diagram of a two-dimensional electron–hole system

The phase diagram for a system of spatially separated electrons and holes in coupled quantum wells or graphene double layers is studied in the framework of a BCS-like mean-field approach and a Landau expansion in terms of the pairing order parameter. We find a second order transition between an electron–hole plasma and a BCS phase, as well as a first-order transition between the BCS phase and a bosonic Mott phase of tightly bound electron–hole pairs without phase coherence. The electron–hole plasma exists at low and at high densities for weak interaction, the BCS phase at moderate density and the Mott phase at high density and strong interaction.     

On hidden symmetries of extremal Kerr–NUT–AdS–dS black holes

It is well known that the Kerr–NUT–AdS–dS black hole admits two linearly independent Killing vectors and possesses a hidden symmetry generated by a rank-2 Killing tensor. The near-horizon geometry of an extremal Kerr–NUT–AdS–dS black hole admits four linearly independent Killing vectors, and we show how the hidden symmetry of the black hole itself is carried over by means of a modified Killing–Yano potential which is given explicitly. We demonstrate that the corresponding Killing tensor of the near-horizon geometry is reducible as it can be expressed in terms of the Casimir operators formed by the four Killing vectors.     

The four-dimensional Martínez Alonso–Shabat equation: Differential coverings and recursion operators

We apply Cartan’s method of equivalence to find a contact integrable extension for the structure equations of the symmetry pseudo-group of the four-dimensional Martínez Alonso–Shabat equation. From this extension we derive two differential coverings including coverings with one and two non-removable parameters. Then we apply the same approach to the tangent covering and construct a recursion operator for symmetries of the equation under study.