A non-homogeneous boundary-value problem for the Korteweg-de Vries equation in a quarter plane

The Korteweg-de Vries equation was first derived by Boussinesq and Korteweg and de Vries as a model for long-crested small-amplitude long waves propagating on the surface of water. The same partial differential equation has since arisen as a model for unidirectional propagation of waves in a variety of physical systems. In mathematical studies, consideration has been given principally to pure initial-value problems where the wave profile is imagined to be determined everywhere at a given instant of time and the corresponding solution models the further wave motion. The practical, quantitative use of the Korteweg-de Vries equation and its relatives does not always involve the pure initial-value problem. Instead, initial-boundary-value problems often come to the fore. A natural example arises when modeling the effect in a channel of a wave maker mounted at one end, or in modeling near-shore zone motions generated by waves propagating from deep water. Indeed, the initial-boundary-value problem

studied here arises naturally as a model whenever waves determined at an entry point propagate into a patch of a medium for which disturbances are governed approximately by the Korteweg-de Vries equation. The present essay improves upon earlier work on (0.1) by making use of modern methods for the study of nonlinear dispersive wave equations. Speaking technically, local well-posedness is obtained for initial data in the class for \frac34$"> and boundary data in , whereas global well-posedness is shown to hold for when , and for when . In addition, it is shown that the correspondence that associates to initial data and boundary data the unique solution of (0.1) is analytic. This implies, for example, that solutions may be approximated arbitrarily well by solving a finite number of linear problems.

     

On Solutions of the q-Hypergeometric Equation with q N = 1

We consider the q-hypergeometric equation with q ^N = 1 and , , . We solve this equation on the space of functions given by a power series multiplied by a power of the logarithmic function. We prove that the subspace of solutions is two-dimensional over the field of quasi-constants. We get a basis for this space explicitly. In terms of this basis, we represent the q-hypergeometric function of the Barnes type constructed by Nishizawa and Ueno. Then we see that this function has logarithmic singularity at the origin. This is a difference between the q-hypergeometric functions with 0 < |q| < 1 and at |q| = 1.     

Note: An Unusual Route to General Relativity Equation in Presence of Dust in Irrotational Motion

We show that the general relativity theory equation, in presence of pressureless matter (dust) in irrotational motion, can be recovered from a scalar-tensor like variational approach. In this approach, the kinetic energy, , of a dynamical scalar field , couples directly to gravity. The lagrangian, exempt of explicit matter term, is varied in the framework of the first order formalism, and a conformal transformation, restoring riemannian geometry, is made. In this approach, it turns out that a non-empty spacetime is necessarily four-dimensional.     

Letter: Bounds on the Deficit Solid Angle Parameter from Solar System Tests

We analyze the Mercury perihelion precession, the bending of starlight, and the radar echo delay in the Schwarzschild field modified by the presence of a solid angle deficit. By using the experimental data we obtain that the parameter characterizing the solid angle deficit is less than 10^–9, 10^–8 and 10^–9, respectively, on the length scales associated with such phenomena. In particular, if the solid angle deficit is generated by a global monopole, it is shown that within the length scales associated with such phenomena, the observational data available constrain the monopole energy scale to lower bounds which are in considerable agreement with the limits derived from grand unified theory and cosmology.     

Pseudoinstantons in Metric-Affine Field Theory

In abstract Yang–Mills theory the standard instanton construction relies on the Hodge star having real eigenvalues which makes it inapplicable in the Lorentzian case. We show that for the affine connection an instanton-type construction can be carried out in the Lorentzian setting. The Lorentzian analogue of an instanton is a spacetime whose connection is metric compatible and Riemann curvature irreducible (pseudoinstanton). We suggest a metric-affine action which is a natural generalization of the Yang–Mills action and for which pseudoinstantons are stationary points. We show that a spacetime with a Ricci flat Levi-Civita connection is a pseudoinstanton, so the vacuum Einstein equation is a special case of our theory. We also find another pseudoinstanton which is a wave of torsion in Minkowski space. Analysis of the latter solution indicates the possibility of using it as a model for the neutrino.     

Quantum Cosmological Perfect Fluid Models

Perfect fluid Friedmann-Robertson-Walker quantum cosmological models for an arbitrary barotropic equation of state p = are constructed using Schutz's variational formalism. In this approach the notion of time can be recovered. By superposition of stationary states, finite-norm wave-packet solutions to the Wheeler-DeWitt equation are found. The behaviour of the scale factor is studied by applying the many-worlds and the ontological interpretations of quantum mechanics. Singularity-free models are obtained for < 1. Accelerated expansion at present requires –1/3 > > – 1.     

New Quantization Method for Evaluation of Eigenenergies

A quantum dynamical equation is constructed as the limit of a sequence of functions (called Semiquantum momentum functions or SQMF). The quantum action variable J is defined as the limit of the sequence of contour integrals of SQMFs such that the quantization condition is J = n, where n is a nonnegative integer for eigenvalues and a noninteger for off eigenvalues. This quantization condition is exact and J is an analytic function of energy. Based on new definitions, an accurate numerical method is developed for obtaining eigenenergies. The method can be applied to both real and PT symmetric complex potentials. The validity and the accuracy of this new method is demonstrated with three illustrations.     

The Homogeneous Hamilton–Jacobi and Bernoulli Equations Revisited, II

It is shown that the admissible solutions of the continuity and Bernoulli or Burgers' equations of a perfect one-dimensional liquid are conditioned by a relation established in 1949–1950 by Pauli, Morette, and Van Hove, apparently, overlooked so far, which, in our case, stipulates that the mass density is proportional to the second derivative of the velocity potential. Positivity of the density implies convexity of the potential, i.e., smooth solutions, no shock. Non-elementary and symmetric solutions of the above equations are given in analytical and numerical form. Analytically, these solutions are derived from the original Ansatz proposed in Ref. 1 and from the ensuing operations which show that they represent a particular case of the general implicit solutions of Burgers' equation. Numerically and with the help of an ad hoc computer program, these solutions are simulated for a variety of initial conditions called compatible if they satisfy the Morette–Van Hove formula and anti-compatible if the sign of the initial velocity field is reversed. In the latter case, singular behaviour is observed. Part of the theoretical development presented here is rephrased in the context of the Hopf–Lax formula whose domain of applicability for the solution of the Cauchy problem of the homogeneous Hamilton–Jacobi equation has recently been enlarged.     

Majorana Transformation for Differential Equations

We present a method for reducing the order of ordinary differential equations satisfying a given scaling relation (Majorana scale-invariant equations). We also develop a variant of this method, aimed to reduce the degree of nonlinearity of the lower order equation. Some applications of these methods are carried out and, in particular, we show that second-order Emden–Fowler equations can be transformed into first-order Abel equations. The work presented here is a generalization of a method used by Majorana in order to solve the Thomas–Fermi equation.     

On Effective Conductivity on ℤ d Lattice

We study the effective conductivity _e for a random wire problem on the d-dimensional cubic lattice ^d , d2 in the case when random conductivities on bonds are independent identically distributed random variables. We give exact expressions for the expansion of the effective conductivity in terms of the moments of the disorder parameter up to the 5th order. In the 2D case using the duality symmetry we also derive the 6th order expansion. We compare our results with the Bruggeman approximation and show that in the 2D case it coincides with the exact solution up to the terms of 4th order but deviates from it for the higher order terms.