Solutions of the Second-order Nonlinear Parabolic System Modeling the Diffusion Wave Motion

The paper continues a long series of our research and considers a secondorder nonlinear evolutionary parabolic system. The system can be a model of various convective and diffusion processes in continuum mechanics, including mass transfer in a binary mixture. In hydrology, ecology, and mathematical biology, it describes the propagation of pollutants in water and air, as well as population dynamics, including the interaction of two different biological species. We construct solutions that have the type of diffusion (heat) wave propagating over a zero background with a finite velocity. Note that the system degenerates on the line where the perturbed and zero (unperturbed) solutions are continuously joined. A new existence and uniqueness theorem is proved in the class of analytical functions. In this case, the solution has the desired type and is constructed in the form of characteristic series, the convergence of which is proved by the majorant method. We also present two new classes of exact solutions, the construction of which, due to ansatzes of a specific form, reduces to integrating systems of ordinary differential equations that inherit a singularity from the original formulation. The obtained results are expected to be helpful in modeling the evolution of the Baikal biota and the propagation of pollutants in the water of Lake Baikal near settlement

Summing up the perturbation series in the Schwinger Model

Perturbation series for the electron propagator in the Schwinger Model is summed up in a direct way by adding contributions coming from individual Feynman diagrams. The calculation shows the complete agreement between nonperturbative and perturbative approaches.Comment: 10 pages (in REVTEX

The Harris-Luck criterion for random lattices

The Harris-Luck criterion judges the relevance of (potentially) spatially correlated, quenched disorder induced by, e.g., random bonds, randomly diluted sites or a quasi-periodicity of the lattice, for altering the critical behavior of a coupled matter system. We investigate the applicability of this type of criterion to the case of spin variables coupled to random lattices. Their aptitude to alter critical behavior depends on the degree of spatial correlations present, which is quantified by a wandering exponent. We consider the cases of Poissonian random graphs resulting from the Voronoi-Delaunay construction and of planar, ``fat'' $\phi^3$ Feynman diagrams and precisely determine their wandering exponents. The resulting predictions are compared to various exact and numerical results for the Potts model coupled to these quenched ensembles of random graphs.Comment: 13 pages, 9 figures, 2 tables, REVTeX 4. Version as published, one figure added for clarification, minor re-wordings and typo cleanu

Extension to order β23\beta^{23} of the high-temperature expansions for the spin-1/2 Ising model on the simple-cubic and the body-centered-cubic lattices

Using a renormalized linked-cluster-expansion method, we have extended to order $\beta^{23}$ the high-temperature series for the susceptibility $\chi$ and the second-moment correlation length $\xi$ of the spin-1/2 Ising models on the sc and the bcc lattices. A study of these expansions yields updated direct estimates of universal parameters, such as exponents and amplitude ratios, which characterize the critical behavior of $\chi$ and $\xi$. Our best estimates for the inverse critical temperatures are $\beta^{sc}_c=0.221654(1)$ and $\beta^{bcc}_c=0.1573725(6)$. For the susceptibility exponent we get $\gamma=1.2375(6)$ and for the correlation length exponent we get $\nu=0.6302(4)$. The ratio of the critical amplitudes of $\chi$ above and below the critical temperature is estimated to be $C_+/C_-=4.762(8)$. The analogous ratio for $\xi$ is estimated to be $f_+/f_-=1.963(8)$. For the correction-to-scaling amplitude ratio we obtain $a^+_{\xi}/a^+_{\chi}=0.87(6)$.Comment: Misprints corrected, 8 pages, latex, no figure

Theta-point behavior of diluted polymer solutions: Can one observe the universal logarithmic corrections predicted by field theory?

In recent large scale Monte-Carlo simulations of various models of Theta-point polymers in three dimensions Grassberger and Hegger found logarithmic corrections to mean field theory with amplitudes much larger than the universal amplitudes of the leading logarithmic corrections calculated by Duplantier in the framework of tricritical O(n) field theory. To resolve this issue we calculate the universal subleading correction of field theory, which turns out to be of the same order of magnitude as the leading correction for all chain lengths available in present days simulations. Borel resummation of the renormalization group flow equations also shows the presence of such large corrections. This suggests that the published simulations did not reach the asymptotic regime. To further support this view, we present results of Monte-Carlo simulations on a Domb-Joyce like model of weakly interacting random walks. Again the results cannot be explained by keeping only the leading corrections, but are in fair accord with our full theoretical result. The corrections found for the Domb-Joyce model are much smaller than those for other models, which clearly shows that the effective corrections are not yet in the asymptotic regime. All together our findings show that the existing simulations of Theta-polymers are compatible with tricritical field theory since the crossover to the asymptotic regime is very slow. Similar results were found earlier for self avoiding walks at their upper critical dimension d=4.Comment: 15 pages,6 figure

On Analytic Solutions of the Problem of Heat Wave Front Movement for the Nonlinear Heat Equation with Source

We continue our investigation of the special boundary-value problems for the nonlinear parabolic heat equation (the porous medium equation) in the article. In the case of a power-law dependence of the heat conductivity coefficient on temperature the equation is used for describing high-temperature processes, filtration of gas through porous media, migration of biological populations, etc. Moreover, the equation has specific nonlinear properties, which may be interesting from the point of view of physics as well as mathematics. For example, it is well known, that the described disturbances may have a finite velocity of propagation. The heat waves (waves of filtration) compose an important class of solutions to the equation under consideration. Geometrically, these solutions are constructed from two integral surfaces, which are continuously connected on a curve - heat wave front. We consider a boundary-value problem, which has such solutions. The research is carried out in the class of analytic functions by the characteristic series method. This method was suggested by R. Courant and then it was adapted for nonlinear parabolic equations in A.F. Sidorov’s scientific school. We have already researched similar problems in case of closed front without source. For each problem we constructed the solution in form of characteristic series and proved the exist theorem, which guaranteed the convergence. The paper deals with the flat-symmetrical problem with given front and source. The theorem of existence of the analytic solution(heat wave’s nonnegative part) was proved and the solution in form of the power series was constructed. Also we considered an interesting case in which the source is a power function (such cases are common in applications). It was shown that the original problem may be reduced to the Cauchy problem for nonlinear ordinary differential equation of the second order