卷 190, 编号 1 (2017)

We classify all four-dimensional real Lie bialgebras of symplectic type and obtain the classical r-matrices for these Lie bialgebras and Poisson structures on all the associated four-dimensional Poisson–Lie groups. We obtain some new integrable models where a Poisson–Lie group plays the role of the phase space and its dual Lie group plays the role of the symmetry group of the system.

We propose a linearizable version of a multidimensional system of n-wave-type nonlinear partial differential equations (PDEs). We derive this system using the spectral representation of its solution via a procedure similar to the dressing method for nonlinear PDEs integrable by the inverse scattering transform method. We show that the proposed system is completely integrable and construct a particular solution.

We consider an effective potential model consisting of an electromagnetic part plus a nuclear part as the ground state interaction for an α–p system. The next few higher partial wave interactions are generated using the formalism of supersymmetric quantum mechanics. We adapt the phase function method to compute α–p elastic scattering phases up to 12 MeV, including the effect of the electromagnetic interaction quite rigorously in our phase shift calculation. With the further incorporation of some energy-dependent correction factors to our interactions, we obtain a good agreement with the experimental data.

Using a three- and four-dimensional Pauli–Villars regularization scheme, we investigate quark–antiquark and diquark condensation in the framework of the Nambu–Jona-Lasinio model. Using the particle Fermi momentum as a cutoff parameter, we study the energy gap width and coherence length for the meson condensate 〈\(q\bar q\) 〉. We also study the energy gap width and critical coherence length (the distance over which there would be no diquark condensation) for the diquark 〈qq〉 and the dependence on the Fermi momentum. We obtain an estimate of the Fermi momentum value for meson and diquark condensates with an energy gap width of the order of 100 MeV.

We use a non-gauge-invariant modification of the exact Hamiltonian to obtain a new Hamiltonian-like operator for a simple exactly solvable boson model. The eigenvalues of the new operator are close to those of the original Hamiltonian. We make a one-body approximation of the new two-body operator in the spirit of the Bogoliubov approximation. Because only the number operator appears, the c-number approximation is not required individually for the creation or annihilation operators in the ground state. For the simple model, the results using the new approximation are closer to the exact results than the usual Bogoliubov results over a wide range of parameters. The improvement increases dramatically as the model interaction strength increases.

We investigate the stability of the Einstein static universe under linear scalar, vector, and tensor perturbations in the context of a deformed Hoˇrava-Lifshitz (HL) cosmology related to entropic forces. We obtain a general stability condition under linear scalar perturbations. Using this general condition, we show that there is no stable Einstein static universe in the case of a flat universe (k = 0). In the special case of large values of the parameter ω of HL gravity in a positively curved universe (k > 0), the domination of the quintessence and phantom matter fields with a barotropic equation of state parameter β < −1/3 is necessary, while for a negatively curved universe (k < 0), matter fields with β > −1/3 must be the dominant fields of the universe. We also demonstrate a neutral stability under vector perturbations. We obtain an inequality including the cosmological parameters of the Einstein static universe for stability under tensor perturbations. It turns out that for large values of ω, there is stability under tensor perturbations.

In a very general setting, we discuss possibilities of applying p-adics to geophysics using a p-adic diffusion representation of the master equations for the dynamics of a fluid in capillaries in porous media and formulate several mathematical problems motivated by such applications. We stress that p-adic wavelets are a powerful tool for obtaining analytic solutions of diffusion equations. Because p-adic diffusion is a special case of fractional diffusion, which is closely related to the fractal structure of the configuration space, p-adic geophysics can be regarded as a new approach to fractal modeling of geophysical processes.