Finite-difference effects in quantum field theory and quantization of classical solutions

Scientific-Research Institute of Nuclear Physics at the Moscow State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 82, No. 1, pp. 55–65, January, 1990.     

非守恒相场模型解的blow-up

讨论了一类非守恒相场模型解的性态,证明当ａ２ｐ － １ ＜ ０ 及初值充分大时解在有限时刻 ｂｌｏｗ ｕｐ．     

Using singular classical solutions for calculating multiparticle cross sections in field theory

A method for reducing the problem of calculating multiparticle tree cross sections in ϕ⁴ theory to the solution of a singular classical Euclidean boundary-value problem is introduced. The solutions are obtained numerically in terms of the spherical harmonic decomposition, and the corresponding estimates of the tree cross sections are found at arbitrary energies. Numerical analysis agrees with analytic results obtained earlier in the limiting cases of high and low energies. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 115, No. 3, pp. 358–372, June, 1998     

Blow-up of solutions for a time-space fractional evolution system

In this paper, firstly, we study the local existence and uniqueness of mild solutions for fractional evolution systems with nonlocal in time nonlinearity. Then, we claim that such a mild solution is weak solution of this system. Finally, we prove a blow-up result under some conditions.     

Blow-up solutions to a system of nonlinear Volterra equations

A system of nonlinear Volterra integral equations with convolution kernels is considered. Estimates are given for the blow-up time when conditions are such that the solution is known to become unbounded in finite time. For two examples that arise in combustion problems, numerical estimates of blow-up time are presented. Additionally, the asymptotic behavior of the blow-up solution in the key limit is established for the power-law and exponential nonlinearity cases.     

Blow-up properties of solutions for a multi-coupled parabolic system

This paper deals with the blow-up behavior of radial solutions to a parabolic system multi-coupled via inner sources and boundary flux. We first obtain a necessary and sufficient condition for the existence of non-simultaneous blow-up, and then find five regions of exponent parameters where both non-simultaneous and simultaneous blow-up may happen. In particular, nine simultaneous blow-up rates are established for different regions of parameters. It is interesting to observe that different initial data may lead to different simultaneous blow-up rates even with the same exponent parameters.     

Blow-up of solutions to a parabolic system with nonlocal source

This paper deals with a parabolic system with different diffusion coefficients and coupled nonlocal sources, subject to homogeneous Dirichlet boundary conditions. The conditions on global existence, simultaneous or non-simultaneous blow-up, blow-up set, uniform blow-up profiles and boundary layer are got using comparison principle and asymptotic analysis methods.     

Blow-up criteria for the three dimensional nonlinear dissipative system modeling electro-hydrodynamics

In this paper, we investigate some sufficient conditions for the breakdown of local smooth solutions to the three dimensional nonlinear nonlocal dissipative system modeling electro-hydrodynamics. This model is a strongly coupled system by the well-known incompressible Navier–Stokes equations and the classical Poisson–Nernst–Planck equations. We show that the maximum of the vorticity field alone controls the breakdown of smooth solutions, which reveals that the velocity field plays a more dominant role than the density functions of charged particles in the blow-up theory of the system. Moreover, some Prodi–Serrin type blow-up criteria are also established.     

Blow-up of solutions for a system of nonlocal singular viscoelastic equations

The main propose of this paper is to study the blow-up of solutions of an initial boundary value problem with a nonlocal boundary condition for a system of nonlinear singular viscoelastic equations. where the blow-up of solutions in finite time with nonpositive initial energy combined with a positive initial energy are shown.     

On the connection between discontinuous step-like and smooth kink-type classical solutions in quantum field theory

A nonperturbative procedure for subtracting singularities caused by finite discontinuities of the field configurations is suggested. This procedure is applied to the Lorentz-covariant quantization of(1+1)-dimensional topological kinks. In the course of this procedure, quantum copies of a classical kink naturally appear. These copies have the same topological charge but have negligibly small sizes and masses because the subtraction procedure eliminates divergences caused by the field differentiations at the discontinuity points. These effects are investigated in detail for those(1+1)-dimensional scalar field models where classical kinks exist.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 108, No. 2, pp. 212–248, August, 1996.     

Discontinuous solutions in the classical theory of bending of plates

We present necessary and sufficient conditions for existence of a class of analytic solutions of the equations of classical theory of bending of plates in a domain formed by removing from the infinite plane a set of points belonging to an arbitrary piecewise-smooth contour. Here we assume that all the components of a solution of a given class possess the property of continuous extendability onto almost all points of the boundary of the domain and that the boundary values are locally summable functions on the boundary.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 12, pp. 1598–1605, December, 1990.     

Blow-up Dynamics of L~2 Solutions for the Davey–Stewartson System

We study the blow-up solutions for the Davey–Stewartson system(D–S system, for short)in L2x(R2). First, we give the nonlinear profile decomposition of solutions for the D–S system. Then, we prove the existence of minimal mass blow-up solutions. Finally, by using the characteristic of minimal mass blow-up solutions, we obtain the limiting profile and a precisely mass concentration of L2 blow-up solutions for the D–S system.     

Blow-up of solutions of a system of equations with double nonlinearities and nonlocal sources

We consider a nonlocal system of equations of parabolic type with double nonlinearities. A modified Levine method is used to prove the blow-up of solutions.     

Optimal system and approximate solutions of nonlinear dissipative media

In this paper, the problem of approximate symmetries of a class of nonlinear wave equations with a small nonlinear dissipation has been investigated. In order to compute the first-order approximate symmetry, we have applied the method that was proposed by Valenti basically based on the expansion of the dependent variables in perturbation series but removing the drawback of the impossibility to work in hierarchy in calculating symmetries. The algebraic structure of the approximate symmetries is discussed, an optimal system of one-dimensional subalgebras is defined and constructed, and, finally, some invariant solutions corresponding to the resulted symmetries are obtained.     

Peculiarities of relativistic particle evolution in classical scalar field theory

The interaction of particles in the model of a scalar field linearly coupled with the source is demonstrated to have the following peculiarities. First, the particles are attracted to one another at large interparticle distances and are repelled at small. Second, in the process of evolution of the system of particles, its components gain the maximum possible velocity, the velocity of light. Third, after the head-on collision of two identical particles, the pumping effect may probably arise. This effect consists of the fact that after experiencing a few oscillations between the walls of a squeezing potential well, the particles fly out of the finite motion region to the region of infinite motion. Fourth, on the trajectory of the interacting particles, a plateau region of motion with a high and relatively constant velocity appears.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 109, No. 3, pp. 464–473, December, 1996.     

Statistical field theory of a nonadditive system

Based on quantum field methods, we develop a statistical theory of complex systems with nonadditive potentials. Using the Martin-Siggia-Rose method, we find the effective system Lagrangian, from which we obtain evolution equations for the most probable values of the order parameter and its fluctuation amplitudes. We show that these equations are unchanged under deformations of the statistical distribution while the probabilities of realizing different phase trajectories depend essentially on the nonadditivity parameter. We find the generating functional of a nonadditive system and establish its relation to correlation functions; we introduce a pair of additive generating functionals whose expansion terms determine the set of multipoint Green’s functions and their self-energy parts. We find equations for the generating functional of a system having an internal symmetry and constraints. In the harmonic approximation framework, we determine the partition function and moments of the order parameter depending on the nonadditivity parameter. We develop a perturbation theory that allows calculating corrections of an arbitrary order to the indicated quantities.