New non-linear equations and modular form expansion for double-elliptic Seiberg-Witten prepotential

Integrable N-particle systems have an important property that the associated Seiberg-Witten prepotentials satisfy the WDVV equations. However, this does not apply to the most interesting class of elliptic and double-elliptic systems. Studying the commutativity conjecture for theta-functions on the families of associated spectral curves, we derive some other non-linear equations for the perturbative Seiberg-Witten prepotential, which turn out to have exactly the double-elliptic system as their generic solution. In contrast with the WDVV equations, the new equations acquire non-perturbative corrections which are straightforwardly deducible from the commutativity conditions. We obtain such corrections in the first non-trivial case of N=3 and describe the structure of non-perturbative solutions as expansions in powers of the flat moduli with coefficients that are (quasi)modular forms of the elliptic parameter.Comment: 25 page

Unitary matrix integrals in the framework of Generalized Kontsevich Model. I. Brezin-Gross-Witten Model

We advocate a new approach to the study of unitary matrix models in external fields which emphasizes their relationship to Generalized Kontsevich Models (GKM) with non-polynomial potentials. For example, we show that the partition function of the Brezin-Gross-Witten Model (BGWM), which is defined as an integral over unitary $N\times N$ matrices, $\int [dU] e^{\rm{Tr}(J^\dagger U + JU^\dagger)}$, can also be considered as a GKM with potential ${\cal V}(X) = \frac{1}{X}$. Moreover, it can be interpreted as the generating functional for correlators in the Penner model. The strong and weak coupling phases of the BGWM are identified with the "character" (weak coupling) and "Kontsevich" (strong coupling) phases of the GKM, respectively. This sort of GKM deserves classification as $p=-2$ one (i.e. $c=28$ or $c=-2$) when in the Kontsevich phase. This approach allows us to further identify the Harish-Chandra-Itzykson-Zuber (IZ) integral with a peculiar GKM, which arises in the study of $c=1$ theory and, further, with a conventional 2-matrix model which is rewritten in Miwa coordinates. Inspired by the considered unitary matrix models, some further extensions of the GKM treatment which are inspired by the unitary matrix models which we have considered are also developed. In particular, as a by-product, a new simple method of fixing the Ward identities for matrix models in an external field is presented.Comment: FIAN/TD-16/93, ITEP-M6/93, UBC/S-93/93 (39 pages

Phase behaviour of block copolymer melts with arbitrary architecture

The Leibler theory [L. Leibler, Macromolecules, v.13, 1602 (1980)] for microphase separation in AB block copolymer melts is generalized for systems with arbitrary topology of molecules. A diagrammatic technique for calculation of the monomeric correlation functions is developed. The free energies of various mesophases are calculated within the second-harmonic approximation. Model highly-branched tree-like structures are considered as an example and their phase diagrams are obtained. The topology of molecules is found to influence the spinodal temperature and asymmetry of the phase diagrams, but not the types of phases and their order. We suggest that all model AB block-copolymer systems will exhibit the typical phase behaviour.Comment: Submitted to J. Chem. Phys., see also http://rugmd4.chem.rug.nl/~morozov/research.htm

Continuum Limits of ``Induced QCD": Lessons of the Gaussian Model at d=1 and Beyond

We analyze the scalar field sector of the Kazakov--Migdal model of induced QCD. We present a detailed description of the simplest one dimensional {($d$$=$$1$)} model which supports the hypothesis of wide applicability of the mean--field approximation for the scalar fields and the existence of critical behaviour in the model when the scalar action is Gaussian. Despite the ocurrence of various non--trivial types of critical behaviour in the $d=1$ model as $N\rightarrow\infty$, only the conventional large-$N$ limit is relevant for its {\it continuum} limit. We also give a mean--field analysis of the $N=2$ model in {\it any} $d$ and show that a saddle point always exists in the region $m^2>m_{\rm crit}^2(=d)$. In $d=1$ it exhibits critical behaviour as $m^2\rightarrow m_{\rm crit}^2$. However when $d$$>$$1$ there is no critical behaviour unless non--Gaussian terms are added to the scalar field action. We argue that similar behaviour should occur for any finite $N$ thus providing a simple explanation of a recent result of D. Gross. We show that critical behaviour at $d$$>$$1$ and $m^2>m^2_{\rm crit}$ can be obtained by adding a $logarithmic$ term to the scalar potential. This is equivalent to a local modification of the integration measure in the original Kazakov--Migdal model. Experience from previous studies of the Generalized Kontsevich Model implies that, unlike the inclusion of higher powers in the potential, this minor modification should not substantially alter the behaviour of the Gaussian model.Comment: 31 page

On non existence of tokamak equilibria with purely poloidal flow

It is proved that irrespective of compressibility tokamak steady states with purely poloidal mass flow can not exist in the framework of either magnetohydrodynamics (MHD) or Hall MHD models. Non-existence persists within single fluid plasma models with pressure anisotropy and incompressible flows.Comment: The conclusion reported in the last sentence of the first paragraph of Sec. V in the version of the paper published in Physics of Plasmas is incorrect. The correct conclusion is given here (15 pages

Revisiting the stability of spatially heterogeneous predator-prey systems under eutrophication

We employ partial integro-differential equations to model trophic interaction in a spatially extended heterogeneous environment. Compared to classical reaction-diffusion models, this framework allows us to more realistically describe the situation where movement of individuals occurs on a faster time scale than the demographic (population) time scale, and we cannot determine population growth based on local density. However, most of the results reported so far for such systems have only been verified numerically and for a particular choice of model functions, which obviously casts doubts about these findings. In this paper, we analyse a class of integro-differential predator-prey models with a highly mobile predator in a heterogeneous environment, and we reveal the main factors stabilizing such systems. In particular, we explore an ecologically relevant case of interactions in a highly eutrophic environment, where the prey carrying capacity can be formally set to 'infinity'. We investigate two main scenarios: (i) the spatial gradient of the growth rate is due to abiotic factors only, and (ii) the local growth rate depends on the global density distribution across the environment (e.g. due to non-local self-shading). For an arbitrary spatial gradient of the prey growth rate, we analytically investigate the possibility of the predator-prey equilibrium in such systems and we explore the conditions of stability of this equilibrium. In particular, we demonstrate that for a Holling type I (linear) functional response, the predator can stabilize the system at low prey density even for an 'unlimited' carrying capacity. We conclude that the interplay between spatial heterogeneity in the prey growth and fast displacement of the predator across the habitat works as an efficient stabilizing mechanism.Comment: 2 figures; appendices available on request. To appear in the Bulletin of Mathematical Biolog