Integrable nonlinear Klein-Gordon equations and Toda lattices

We present a class of nonlinear Klein-Gordon systems which are soluble by means of a scattering transform. More specifically, for eachN2 we present a system of (N–1) nonlinear Klein-Gordon equations, together with the correspondingN ×N matrix scattering problem which can be used to solve it. We illustrate these with some special examples. The general system is shown to be closely related to the equations of the periodic Toda lattice. We present a Bäcklund transformation and superposition formula for the general system.     

Stable standing waves of nonlinear Klein-Gordon equations

In this paper we give sufficient conditions for the stability of the standing waves of least energy for nonlinear Klein-Gordon equations.     

非线性Klein-Gordon方程新的精确解

将行波变换替换为更一般的函数变换,推广了修正的Jacobi椭圆函数展开方法.给出了非线性 Klein-Gordon方程新的周期解.当模m→1或m→0时,这些解退化成相应的孤立波解、三 角函数解和奇异的行波解.对于某些非线性方程,在一定条件下一般变换退化为行波约化. 关键词： Jacobi椭圆函数 非线性发展方程 精确解     

The cauchy problem for non-linear Klein-Gordon equations

We consider in ⁿ⁺¹,n2, the non-linear Klein-Gordon equation. We prove for such an equation that there is a neighbourhood of zero in a Hilbert space of initial conditions for which the Cauchy problem has global solutions and on which there is asymptotic completeness. The inverse of the wave operator linearizes the non-linear equation. If, moreover, the equation is manifestly Poincaré covariant then the non-linear representation of the Poincaré Lie algebra, associated with the non-linear Klein-Gordon equation is integrated to a non-linear representation of the Poincaré group on an invariant neighbourhood of zero in the Hilbert space. This representation is linearized by the inverse of the wave operator. The Hilbert space is, in both cases, the closure of the space of the differentiable vectors for the linear representation of the Poincaré group, associated with the Klein-Gordon equation, with respect to a norm defined by the representation of the enveloping algebra.     

Clifford algebra structure and Einstein's equations

We point out that a Clifford algebra representation for the Riemannian curvature leads to an equation for gravity similar to the Yang-Mills equation, in a gauge model for gravity with the Lorentz group. Einstein's equation of general relativity emerges as a natural solution in this approach.     

Solutions of Klein-Gordon and Dirac equations on quantum Minkowski spaces

Covariant differential calculi and exterior algebras on quantum homogeneous spaces endowed with the action of inhomogeneous quantum groups are classified. In the case of quantum Minkowski spaces they have the same dimensions as in the classical case. Formal solutions of the corresponding Klein-Gordon and Dirac equations are found. The Fock space construction is sketched.This research was supported in part by NSF grant DMS-9508597 and in part by Polish KBN grant No. 2 P301 02007.     

Separation of variables in the Klein-Gordon equations. II

Two kinds of external nonstationary electromagnetic fields are found containing arbitrary functions which admit of total separation of variables in the Klein-Gordon equations by using two differential symmetry operators and one second order operator. Curvilinear coordinates are presented in which the variables are divided, and equations are written down in the separated variables.Translated from Izvestiya VUZ, Fizika, No. 12, pp. 45–52, December, 1973.     

Quantum Einstein’s equations and constraints algebra

In this paper we shall address this problem: Is quantum gravity constraints algebra closed and what are the quantum Einstein’s equations. We shall investigate this problem in the de-Broglie-Bohm quantum theory framework. It is shown that the constraint algebra is weakly closed and the quantum Einstein’s equations are derived.     

Quadratic algebras and noncommutative integration of Klein-Gordon equations in non-steckel Riemann spaces

The method of noncommutative integration of linear partial differential equations is used to solve the Klein-Gordon equations in Riemann space, in the case when the set of noncommutating symmetry operators of this equation for a quadratic algebra consists of one second-order operator and several first-order operators. Solutions that do not permit variable separation are presented.Tomsk State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 83–87, May, 1995.     

Dirac and Klein-Gordon equations: Convergence of solutions in the nonrelativistic limit

The convergence of solutions of the Dirac and Klein-Gordon equations to solutions of the Pauli and Schrödinger equations in the non-relativistic limit is discussed. An abstract theory of these equations is developed which is general enough to allow physical space to be an arbitrary complete Riemannian manifold.Research partially supported by National Science Foundation grant MCS-77-13070     

Relations between quantum defects and phase shifts for the Klein-Gordon and Dirac equations

Seaton's relation, which forms the basis of the quantum defect method, is generalized to the relativistic case. A connection is established between the quantum defects of discrete levels and the phase shift of the continuum wave function for the Klein-Gordon and Dirac equations. The possibility of applying these relations, in particular to the study of mesoatoms, is discussed.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 59–62, August, 1977.     

Discrete and continuous Bogomolny equations through the deformed algebra

We show that there exists one-parameter algebra connecting the discrete and continuous Bogomolny equations. The algebra is the deformation of the extended conformal algebra. This shows that the deformed algebra plays a role of the link between the matrix valued model and the model with one more space dimension higher.     

Additional symmetries for integrable equations and conformal algebra representation

We present a regular procedure for constructing an infinite set of additional (spacetime variables explicitly dependent) symmetries of integrable nonlinear evolution equations (INEEs). In our method, additional symmetry equations arise together with their L-A pairs, so that they are integrable themselves. This procedure is based on a modified dressing method. For INEEs in 1+1 dimensions, some appropriate symmetry equations are shown to form the vector fields on a circle S ¹ algebra representation. In contrast to the so-called isospectral deformations, these symmetries result from conformal transformations of the associated linear problem spectrum. For INEEs in 2+1 dimensions, the commutation relations for symmetry equations are shown to coincide with operators , with integer m, p. Some additional results about Kac-Moody algebra applications are presented.     

Klein-Gordon equations for energy-momentum of the relativistic particle in rapidity space

The notion of four-rapidity is defined as a four-vector with one time-like and three space-like coordinates. It is proved, the energy and momentum defined in the space of four-rapidity obey Klein-Gordon equations constrained by the classical trajectory of a relativistic particle. It is shown, for small values of a proper mass influence of the constraint is weakened and the classical motion gains features of a wave motion.     

Internal degrees of freedom, long-range interactions and nonlocal effects in perturbed Klein-Gordon equations

We investigate different mechanisms for the excitation of soliton internal degrees of freedom and for the existence of long-range interactions between solitons. We will study a nonlocal Klein-Gordon equation that is used as a model for Josephson junctions in thin films. We will show the connections between nonlocality, nonlinearity, internal degrees of freedom, long-range interactions and power-law behaviors.     

Applications of dynamical algebra of invariance of integrodifferential equations

General results on the algebraic properties of integrodifferential equations are used to obtain coherent and squeezed states and Green functions for the matrixdifferential models of condensed matter theory.     

Space-time algebra for the generalization of gravitational field equations

The Maxwell–Proca-like field equations of gravitolectromagnetism are formulated using space-time algebra (STA). The gravitational wave equation with massive gravitons and gravitomagnetic monopoles has been derived in terms of this algebra. Using space-time algebra, the most generalized form of gravitoelectromagnetic Klein–Gordon equation has been obtained. Finally, the analogy in formulation between massive gravitational theory and electromagnetism has been discussed.     

A generalized variational algebra and conserved densities for linear evolution equations

The symbolic algebra of Gel'fand and Dikii is generalized to the case of n variables. Using this algebraic approach we give a rigorous characterization of the polynomial kernel of the variational derivative. This is applied to classify all the conservation laws for linear polynomial evolution equations of arbitrary order.Partially supported by the Junta de Energía Nuclear, Madrid.