Multiplication for solutions of the equation

Linear first-order systems of partial differential equations (PDEs) of the form f=Mg, where M is a constant matrix, are studied on vector spaces over the fields of real and complex numbers. The Cauchy–Riemann equations belong to this class. We introduce on the solution space a bilinear *-multiplication, playing the role of a nonlinear superposition principle, that allows for algebraic construction of new solutions from known solutions. The gradient equation f=Mg is a simple special case of a large class of systems of PDEs, admitting a *-multiplication of solutions. We prove that any gradient equation has the exceptional property that the general analytic solution can be expressed as *-power series of certain simple solutions.     

Inertial reference frames in Einstein's theory of gravitation

A physical definition of the inertial reference frame (IRF) is given, and the properties of solutions of the Einstein equation (with cosmological constant), which admit an IRF (IRF solutions) are investigated. Their Petrov type is uniquely determined by the viscous stress tensor. Only the typesI, D or 0 are possible. The unique vacuum IRF solution is the Minkowski space-time. The unique IRF solution belonging to a perfect fluid is the Einstein universe. is of special importance. For=0, the only physically admissible IRF solution is the Minkowski space-time. For0, only interior solutions with strong restrictions for density and pressure are possible.     

Harmonics on hyperspheres,separation of variables and the bethe ansatz

A class of integrable evolution systems in the spacetimeR ²ⁿ⁺¹ (n 2) based on the generalized self-dual Yang-Mills equations are constructed. It is proved that the Darboux matrix method is applicable to these systems and a lot of explicit solutions are obtained. Starting with the trivial solutions, single soliton solutions and multi-soliton solutions are constructed. They are almost localized and the interaction between solitons is almost elastic.Supported by Chinese national basic research project Nonlinear Science and Japanese Inoue Scientific Foundation.     

Non-linear multi-plane wave solutions of self-dual Yang-Mills theory

New solutions of self-dual Yang-Mills (SDYM) equations are constructed in Minkowski space-time for the gauge groupSL(2, ). After proposing a Lorentz covariant formulation of Yang's equations, a set of Ansätze for exact non-linear multiplane wave solutions are proposed. The gauge fields are rational functions ofe ^x·ki (K _i ² =0, 1iN) for these Ansätze. At least, three families of multisoliton type solutions are derived explicitly. Their asymptotic behaviour shows that non-linear waves scatter non-trivially in Minkowski SDYM.On leave from LPTHE Université Paris VI, 4, Place Jussieu, Tour 16, ler étage, F-75230 Paris Cedex 05, France     

Periodic nonlinear waves on a half-line

Nontrivial solutions of the equationu _tt=u _xx–g(u) which are 2-periodic int and which decay asx are shown to exist ifg(a)=0 andg(0)>1. Breather-like solutions, which also decay asx –, can be interpreted as homoclinic solutions in thex-dynamics; their existence is still in question for generalg.     

Bianchi V imperfect fluid cosmology

Bianchi V, spatially homogeneous imperfect fluid cosmological models which contain both viscosity and heat flow are investigated. The Einstein field equations are established in the case that the equations of state are given byp-(-1),=o ^m, and=o ⁿ (where, o, o,m andn are constants). The physical constraints on the solutions of the Einstein field equations, and, in particular, the thermodynamical laws and energy conditions that govern such solutions, are discussed in some detail. Simple power law solutions and solutions in which there is no heat conduction are studied first. Exact solutions are then investigated in more generality, and it is shown that there exist two first integrals of the field equations for certain values of the physical parameters, m andn. Finally, it is shown that in a special case of interest (in whichm =n = 1/2) the imperfect fluid Bianchi V field equations can be written as a plane-autonomous system, thus facilitating the qualitative analysis of these cosmological models.     

Self-consistent solutions of the semiclassical Einstein equations with cosmological constant

The general solution to the semiclassical backreaction equation is found for conformally invariant free quantum fields in spatially flat homogeneous and isotropic spacetime with Cosmological constant and with no classical source when the ratio of the renormalisation parameters/=9/4. It contains a two-parameter family of bouncing solutions that avoid the singularity. There are several one-parameter families which do not have particle horizons. The stability of these solutions is investigated and it is found that they are stable when and have different signs. However, when both parameters have the same sign the set of stable solutions is restricted by the condition 0 < < 1/9. In both cases these solutions have a final de Sitter stage.     

Evolution of plane-symmetric self-similar space-times containing perfect fluid

Space-times with plane symmetry are considered which also admit a conformal group such that the metric depends on the variablez=x ¹/x ⁰ (homothetic or self-similar solutions). Different regions depending on the space-time character ofz can be discerned in each solution: thet region whenz is timelike and thes-region whenz is spacelike. Exact solutions of the Einstein equations are given for an energy-momentum tensor corresponding to an ideal fluid with equation of statep=(-1)e. Different types of solutions occur depending upon the value of in the equation of state, and these are studied using phase-plane methods. The solutions of considered type in the Minkowski space-time are also given.     

Exact (1+1)-dimensional solutions of discrete planar velocity Boltzmann models

For discrete velocity Boltzmann models we have found (1+1)-dimensional shock waves and periodic solutions that are rational solutions with two exponential variables exp(_ix + _it) (spacex, timet). These exact solutions are sums of two rational solutions, each with one exponential variable (similarity solutions). We study the planar velocity models and explicitly write the results for the square 4-velocity and the hexagonal 6-velocity models introduced by Gatignol.     

An investigation of axially symmetric electrovac solutions

For the axially symmetric, electrostatic vacuum problem, there are three unknowns:g ₀₀,g ₁₁, and , the electrostatic field. Herlt, using his generation methods, has presented several new solutions by explicitly givingg ₀₀ and only. Theg ₁₁ is now determined in two cases and a detailed discussion of these solutions is given. The solutions do not, in general, possess the overlapping equipotential structure [g ₀₀=g ₀₀()] which characterizes the Weyl class. One of the classes contains metrics which can be physically interpreted as representing exterior space-times of non-Weyl two-body-type charged point sources. With the specialization of the arbitrary parameters, the above solutions do reduce to known Weyl solutions. The Bonnor solution is a member of one of the above classes and consequently a possible physical reinterpretation of this solution is given. Kinnersley transformations applied to the above classes yield stationary space-times with line singularities and NUT-like asymptotic behavior.     

Yang-Baxter equation and representation theory: I

The problem of constructing the GL(N,) solutions to the Yang-Baxter equation (factorizedS-matrices) is considered. In caseN=2 all the solutions for arbitrarily finite-dimensional irreducible representations of GL(2,) are obtained and their eigenvalues are calculated. Some results for the caseN>2 are also presented.     

A one-parameter family of cylindrically symmetric perfect fluid cosmologies

Non-stationary cylindrically symmetric one-parameter solutions to Einstein's equations are given for a perfect fluid. There is a time singularity (t=0) at which the pressurep and density are equal to + throughout the radial coordinate range 0 r < , but the solutions are well behaved fort > 0,p and decreasing steadily to zero asr increases through the range 0r<, or as t increases through the range 0<t<. The motion is irrotational with shear, expansion and acceleration. The family of solutions, of Petrov type I, are generally spatially inhomogeneous, of class B(ii), having two spacelike Killing vectors which are mutually orthogonal and hypersurface orthogonal, associated with an orthogonally transitive groupG ₂. The particular members for which there are equations of statep=/3 andp= are specially considered.     

Cosmological models in a Kaluza-Klein theory with variable rest mass

A general class of solutions is obtained for a homogeneous, spatially isotropic five-dimensional (5D) Kaluza-Klein theory with variable rest mass. These solutions generalize in the algebraic and physical sense the previously found solutions in the literature. The 4D spacetime sections of the solutions reduce to the Minkowski metric, K=0 Robertson-Walker metric with the equation of statep=np (p=pressure,n=constant sound speed,=energy density), and to the Robertson-Walker spacetime with steady-state metric. Some of the solutions, in different limits, show compactification of the fifth dimension. Some extensions of the model are discussed.     

Exact solutions for the discrete Boltzmann models with specular reflection

We construct exact (1+1)-dimensional solutions (space x, time t), in the presence of a purely reflecting well, for both the four velocity discrete Boltzmann model and the Broadwell model. These exact solutions, sums of two similarity shock waves, are positive for x0, t0.     

Type-D vacuum solutions with cosmological constant

All type-D vacuum (nonnull orbit and null orbit) solutions with are exhibited in canonical coordinates. The nonnull orbit metrics with contain four families of solutions: the static Levi-Cività metrics, the nondivergingD's, the divergingD's, and the diverging and twisting solutions. The null orbit metrics subdivide into two subclasses of solutions: the divergenceless null orbitD's, and the diverging and twisting null orbit solution.     

Self-Similar Solutions of the Boltzmann Equation and Their Applications

We consider a class of solutions of the Boltzmann equation with infinite energy. Using the Fourier-transformed Boltzmann equation, we prove the existence of a wide class of solutions of this kind. They fall into subclasses, labelled by a parameter a, and are shown to be asymptotic (in a very precise sense) to the self-similar one with the same value of a (and the same mass). Specializing to the case of a Maxwell-isotropic cross section, we give evidence to the effect that the only self-similar closed form solutions are the BKW mode and the two solutions recently found by the authors. All the self-similar solutions discussed in this paper are eternal, i.e., they exist for –<t<, which shows that a recent conjecture cannot be extended to solutions with infinite energy. Eternal solutions with finite moments of all orders, and different from a Maxwellian, are also studied. It is shown that these solutions cannot be positive. Moreover all such solutions (partly negative) must be asymptotically (for large negative times) close to the exact eternal solution of BKW type.     

Periodic solutions of massive φ 4 and Yang-Mills theories

The well-known connection between the SU(2) gauge theory and the massless ⁴ theory is extended to theories with nonzero mass. Elliptic solutions of these massive theories are given. These are of the plane-wave variety, with independent variable u=p·x where p is a constant four-vector. They depend on a free parameter k. Two of the solutions are generalized plane-waves while the third describes fluctuations about the vacuum solution.     

Multiple-mode diffusion waves for viscous nonstrictly hyperbolic conservation laws

We study the large-time behaviors of solutions of viscous conservation laws whose inviscid part is a nonstrictly hyperbolic system. The initial data considered here is a perturbation of a constant state. It is shown that the solutions converge to single-mode diffusion waves in directions of strictly hyperbolic fields, and to multiple-mode diffusion waves in directions of nonstrictly hyperbolic fields. The multiple-mode diffusion waves, which are the new elements here, are the self-similar solutions of the viscous conservation laws projected to the nonstrictly hyperbolic fields, with the nonlinear fluxes replaced by their quadratic parts. The convergence rate to these diffusion waves isO(t ^–3/4+1/2p+) inL ^p , 1p, with >0 being arbitrarily small.This work was supported by the Applied Mathematical Sciences subprogram of the Office of Energy Research, U.S. Department of Energy, under Contract W-31-109-Eng-38     

Local theory of solutions for the 0(2k+1) σ-model

We develop a theory of solutionsn for the Euclidean nonlinear 0(2k+1)-model for arbitraryk and for a regionG². We consider a subclass of solutions characterized by a condition which is fulfilled, forG=², by thosen that live on the Riemann sphere S²². We are able to characterize this class completely in terms ofk meromorphic functions, and we give an explicit inductive procedure which allows us to calculate all 0(2k+1) solutions from the trivial 0(1) solutions.     

Shell sources of stationary axisymmetric gravitational fields

A number of solutions for material shell sources of stationary axisymmetric gravitational fields are presented. Explicit solutions are found for shells lying on equipotential hypersurfaces (g _tt = const) and generating static monopole fields in prolate and oblate spheroidal coordinates (Zipoy-Voorhees fields). Numerical solutions are found for shells lying on hypersurfaces of constantg /g and generating Kerr- and Tomimatsu-Sato ( = 2) fields. The shells have minimum areas allowed by the energy conditions of Hawking and Ellis.