Coverings and Integrability of the Gauss–Mainardi–Codazzi Equations

Using the covering theory approach (zero-curvature representations with the gauge group SL), we insert the spectral parameter into the Gauss–Mainardi–Codazzi equations in Chebyshev and geodesic coordinates. For each choice, four integrable systems are obtained.     

Integrability and continuity of solutions to Fokker–Planck–Kolmogorov equations

New results concerning the local integrability of any order and continuity of solution densities of Fokker–Planck–Kolmogorov equations with nondifferentiable coefficients are obtained.     

Integrability and exact solutions of the nonautonomous mixed mKdV–sinh–Gordon equation

In this paper, a nonautonomous mixed mKdV–sinh–Gordon equation with one arbitrary time-dependent variable coefficient is discussed in detail. It is proved that the equation passes the Painlevé test in the case of positive and negative resonances, respectively. Furthermore, a dependent variable transformation is introduced to get its bilinear form. Then, soliton, negaton, positon and interaction solutions are introduced by means of the Wronskian representation. Velocities are found to depend on the time-dependent variable coefficient appearing in the equation and this leads to a wide range of interesting behaviours. The singularities and asymptotic estimate of these solutions are discussed. At last, the superposition formulae for these solutions are also constructed.     

Reflection positivity and Levin–Wen models

We give a transparent algebraic formulation of our pictorial approach to the reflection positivity (RP), that we introduced in a previous paper. We apply this quantization to the Levin–Wen model to obtain anyonic/quantum spin chain theory on the boundary, possibly entangled in the bulk. The reflection positivity property has played a central role in both mathematics and physics, as well as providing a crucial link between the two subjects. In a previous paper we gave a new geometric approach to understanding reflection positivity in terms of pictures. Here we give a transparent algebraic formulation of our pictorial approach. We use insights from this translation to establish the reflection positivity property for the fashionable Levin–Wen models with respect both to vacuum and to bulk excitations. We believe these methods will be useful for understanding a variety of other problems.     

Integrability of the Tolman–Oppenheimer–Volkoff equation

In this work we study the global analytic integrability, Louville integrability and Puiseux integrability of the Tolman–Oppenheimer–Volkoff equation.     

Integrability aspects of NLS–MB system with variable dispersion and nonlinear effects

In this paper, we analyze the integrability aspects of the NLS–MB system with variable dispersion and nonlinear effects. We obtain the constraints for which the above system becomes integrable by using the Painlevé singularity analysis. Obtained results are in agreement with the known results.     

On Integrability of the Camassa–Holm Equation and Its Invariants

Using geometrical approach exposed in (Kersten et al. in J. Geom. Phys. 50:273–302, [2004] and Acta Appl. Math. 90:143–178, [2005]), we explore the Camassa–Holm equation (both in its initial scalar form, and in the form of 2×2-system). We describe Hamiltonian and symplectic structures, recursion operators and infinite series of symmetries and conservation laws (local and nonlocal). This work was supported in part by the NWO–RFBR grant 047.017.015 and RFBR–Consortium E.I.N.S.T.E.I.N. grant 06-01-92060.     

Hybrid competitive Lotka–Volterra ecosystems: countable switching states and two-time-scale models

This work is concerned with competitive Lotka–Volterra model with Markov switching. A novelty of the contribution is that the Markov chain has a countable state space. Our main objective of the paper is to reduce the computational complexity by using the two-time-scale systems. Because existence and uniqueness as well as continuity of solutions for Lotka–Volterra ecosystems with Markovian switching in which the switching takes place in a countable set are not available, such properties are studied first. The two-time scale feature is highlighted by introducing a small parameter into the generator of the Markov chain. When the small parameter goes to 0, there is a limit system or reduced system. It is established in this paper that if the reduced system possesses certain properties such as permanence and extinction, etc., then the complex system also has the same properties when the parameter is sufficiently small. These results are obtained by using the perturbed Lyapunov function methods.     

Sigma models as Gross–Neveu models

Theoretical and Mathematical Physics - We review the correspondence between integrable sigma models with complex homogeneous target spaces and the chiral bosonic (and possibly mixed...     

Maurer–Cartan moduli and models for function spaces

We set up a formalism of Maurer–Cartan moduli sets for _L∞$L_{\infty }$ algebras and associated twistings based on the closed model category structure on formal differential graded algebras (a.k.a. differential graded coalgebras). Among other things this formalism allows us to give a compact and manifestly homotopy invariant treatment of Chevalley–Eilenberg and Harrison cohomology. We apply the developed technology to construct rational homotopy models for function spaces.     

Hamiltonian Oscillators in 1—1—1 Resonance: Normalization and Integrability

Received July 16, 1998; accepted August 7, 1999     

Stability and bifurcation in two species predator–prey models

Changes in the number and stability of equilibrium points in the Lotka–Volterra model as well as some of its generalizations that lead to qualitative changes in the behavior of the system as a function of some of its parameters are studied by bifurcation analysis. A generalization involving a cubic interaction as proposed by Nutku is shown to change the stability properties in a simple way and in particular cases introduce additional stability. Numerical methods and the approach provided by approximate techniques near equilibrium points are used in the analysis.     

Generalized models and local invariants of Kohn–Nirenberg domains

The main obstruction for constructing holomorphic reproducing kernels of Cauchy type on weakly pseudoconvex domains is the Kohn–Nirenberg phenomenon, i.e. nonexistence of supporting functions and local nonconvexifiability. In this paper we give an explicit verifiable characterization of weakly pseudoconvex but locally nonconvexifiable hypersurfaces of finite type in dimension two. It is expressed in terms of a generalized model, which captures local geometry of the hypersurface both in the complex tangential and nontangential directions. As an application we obtain a new class of nonconvexifiable pseudoconvex hypersurfaces with convex models. The author was supported by a grant of the GA ČR no. 201/05/2117.