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.     

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.     

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.     

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.     

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.     

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.     

Solving the non-isospectral Ablowitz–Ladik hierarchy via the inverse scattering transform and reductions

The non-isospectral Ablowitz–Ladik hierarchy is integrated by the inverse scattering transform. In contrast with the isospectral Ablowitz–Ladik hierarchy, the eigenvalues of the non-isospectral Ablowitz–Ladik equations in the scattering data are time-dependent. The multi-soliton solution for the hierarchy is presented. The reductions to the non-isospectral discrete NLS hierarchy and the non-isospectral discrete mKdV hierarchy and their solutions are considered.     

On consistency of stochastic dominance and mean–semideviation models

We analyze relations between two methods frequently used for modeling the choice among uncertain outcomes: stochastic dominance and mean–risk approaches. New necessary conditions for stochastic dominance are developed. These conditions compare values of a certain functional, which contains two components: the expected value of a random outcome and a risk term represented by the central semideviation of the corresponding degree. If the weight of the semideviation in the composite objective does not exceed the weight of the expected value, maximization of such a functional yields solutions which are efficient in terms of stochastic dominance. The results are illustrated graphically. Received: September 15, 1998 / Accepted: October 1, 2000?Published online December 15, 2000     

Comparing predator–prey models with hidden and explicit resources

In this paper two mathematical models are proposed and analyzed to elucidate the influence on a generalist predator of its hidden and explicit resources. Boundedness of the system’s trajectories, feasibility, local and global stability of the equilibria for both models are established, as well as possible local bifurcations. The findings indicate that the relevant behaviour of the system, including switching of stability, extinction and persistence of the involved populations, depends mainly on the reproduction rate of the favorite prey. To achieve full ecosystem survival some balance between the respective grazing pressures exerted by the predator on the prey populations needs to be maintained, while higher grazing pressure just on one species always leads to its extinction.     

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

Received July 16, 1998; accepted August 7, 1999