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.     

Extracting the Maxwell charge from the Wheeler–DeWitt equation

We consider the Wheeler–DeWitt equation as a device for finding eigenvalues of a Sturm–Liouville problem. In particular, we will focus our attention on the electric (magnetic) Maxwell charge. In this context, we interpret the Maxwell charge as an eigenvalue of the Wheeler–De Witt equation generated by the gravitational field fluctuations. A variational approach with Gaussian trial wave functionals is used as a method to study the existence of such an eigenvalue. We restrict the analysis to the graviton sector of the perturbation. We approximate the equation to one loop in a Schwarzschild background and a zeta function regularization is involved to handle with divergences. The regularization is closely related to the subtraction procedure appearing in the computation of Casimir energy in a curved background. A renormalization procedure is introduced to remove the infinities together with a renormalization group equation.     

Semiclassical approximation of the Wheeler–DeWitt equation: arbitrary orders and the question of unitarity

We extend the Born–Oppenheimer type of approximation scheme for the Wheeler–DeWitt equation of canonical quantum gravity to arbitrary orders in the inverse Planck mass squared. We discuss in detail the origin of unitarity violation in this scheme and show that unitarity can be restored by an appropriate modification which requires back reaction from matter onto the gravitational sector. In our analysis, we heavily rely on the gauge aspects of the standard Born–Oppenheimer scheme in molecular physics.     

On the possibility of dark energy from corrections to the Wheeler–DeWitt equation

We present a method for approximating the effective consequence of generic quantum gravity corrections to the Wheeler–DeWitt equation. We show that in many cases these corrections can produce departures from classical physics at large scales and that this behaviour can be interpreted as additional matter components. This opens up the possibility that dark energy (and possible dark matter) could be large scale manifestations of quantum gravity corrections to classical general relativity. As a specific example we examine the first order corrections to the Wheeler–DeWitt equation arising from loop quantum cosmology in the absence of lattice refinement and show how the ultimate breakdown in large scale physics occurs.     

Wheeler–DeWitt Equation with a Screened-Coulomb Dilation Potential

Wheeler–DeWitt equation for anisotropically expanding homogeneous high-dimension spaces is approximately solved under a screened-coulomb dilation potential via an appropriate approximation. The wave function is reported in terms of the Jacobi polynomials and eigenvalues and eigenfunctions are reported via the Nikiforov–Uvarov technique.     

Bianchi type I cosmology in generalized Saez–Ballester theory via Noether gauge symmetry

In this paper, we investigate the generalized Saez–Ballester scalar–tensor theory of gravity via Noether gauge symmetry (NGS) in the background of Bianchi type I cosmological spacetime. We start with the Lagrangian of our model and calculate its gauge symmetries and corresponding invariant quantities. We obtain the potential function for the scalar field in the exponential form. For all the symmetries obtained, we determine the gauge functions corresponding to each gauge symmetry which include constant and dynamic gauge. We discuss cosmological implications of our model and show that it is compatible with the observational data.     

Lie symmetries, perturbation to symmetries and adiabatic invariants of Poincaré equations

Based on the invariance of differential equations under infinitesimal transformations, Lie symmetry, laws of conservations, perturbation to the symmetries and adiabatic invariants of Poincaré equations are presented.The concepts of Lie symmetry and higher order adiabatic invariants of Poincar\'{e} equations are proposed. The conditions for existence of the exact invariants and adiabatic invariants are proved, and their forms are also given. In addition, an example is presented to illustrate these results.     

Covariance in the Batalin–Vilkovisky formalism and the Maurer–Cartan equation for curved Lie algebras

We express covariance of the Batalin–Vilkovisky formalism in classical mechanics by means of the Maurer–Cartan equation in a curved Lie superalgebra, defined using the formal variational calculus and Sullivan’s Thom–Whitney construction. We use this framework to construct a Batalin–Vilkovisky canonical transformation identifying the Batalin–Vilkovisky formulation of the spinning particle with an AKSZ field theory.

     

The Gibbons–Tsarev equation: symmetries,invariant solutions,and applications

In this paper we present the full classification of symmetry-invariant solutions for the Gibbons–Tsarev equation. Then we use these solutions to construct explicit expressions for two-component reductions of Benney’s moments equations, to get solutions of Pavlov’s equation, and to find integrable reductions of the Ferapontov–Huard–Zhang system, which describes implicit two-phase solutions of the dKP equation.     

On nonlocal symmetries for the Krichever–Novikov equation

We construct new infinite hierarchies of nonlocal symmetries and cosymmetries for the Krichever–Novikov equation using the inverse of the fourth-order recursion operator of the latter.     

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.     

Lie point symmetries and ODEs passing the Painlevé test

The Lie point symmetries of ordinary differential equations (ODEs) that are candidates for having the Painlevé property are explored for ODEs of order n = 2,?. . .?, 5. Among the 6 ODEs identifying the Painlevé transcendents only P_III, P_V and P_VI have nontrivial symmetry algebras and that only for very special values of the parameters. In those cases the transcendents can be expressed in terms of simpler functions, i.e. elementary functions, solutions of linear equations, elliptic functions or Painlevé transcendents occurring at lower order. For higher order or higher degree ODEs that pass the Painlevé test only very partial classifications have been published. We consider many examples that exist in the literature and show how their symmetry groups help to identify those that may define genuinely new transcendents.     

Lie symmetries and conserved quantities of controllable nonholonomic dynamical systems

This paper concentrates on studying the Lie symmetries and conserved quantities of controllable nonholonomic dynamical systems. Based on the infinitesimal transformation, we establish the Lie symmetric determining equations and restrictive equations and give three definitions of Lie symmetries before the structure equations and conserved quantities of the Lie symmetries are obtained. Then we make a study of the inverse problems. Finally, an example is presented for illustrating the results.     

Soliton solutions and symmetries of the 2+1 dimensional Kaup–Kupershmidt equation

The 2+1 dimensional Kaup–Kupershmidt (KK) equation is considered. A bilinear form for the equation is found and then 3-soliton solutions are obtained with the assistance of Mathematica. Six symmetries of the bilinear 2+1 dimensional KK equation are given and their symmetry algebra is identified.     

The four-dimensional Martínez Alonso–Shabat equation: Reductions and nonlocal symmetries

We consider the four-dimensional integrable Martínez Alonso–Shabat equation and list three integrable three-dimensional reductions thereof. We also present a four-dimensional integrable modified Martínez Alonso–Shabat equation together with its Lax pair.We further construct an infinite hierarchy of commuting nonlocal symmetries (and not just the shadows, as it is usually the case in the literature) for the Martínez Alonso–Shabat equation.     

A New Way of Solving the Wheeler–DeWitt Equation as Applied to Point Sources

A new way of solving the Wheeler–DeWitt equation is proposed which is based on quantization over free parameters of metrics satisfying the Einstein equations. This technique is applied to two point sources described in the classical case by the Tangherlini metric (in an n-dimensional space) and the Reissner–Nordström metric (in the case of the presence of a charge). The results obtained clarify the sense of the Wheeler hypothesis about statistical weights of small dimensionalities and make possible a new approach to the problem of variation of fundamental constants.     

Lie point symmetries,conservation laws and exact solutions of ($$1+ n$$)-dimensional modified Zakharov–Kuznetsov equation describing the waves in plasma physics

An approximate solution for the nuclear Hulthén plus atomic Hulthén potentials is constructed by solving the associated Volterra integral equation by series substitution method. Within the framework of supersymmetry-inspired factorisation method, this solution is exploited to construct higher partial wave interactions. The merit of our approach is examined by computing elastic scattering phases of the \(\alpha {-}\alpha \) system by the judicious use of phase function method. Reasonable agreements in phase shifts are obtained with standard data.