The second Painlevé equation with a large parameter (P_II) is analyzed from the viewpoint of the exact WKB analysis. The purpose of this study is to investigate a new kind of Stokes phenomena which we call "parametric Stokes phenomenon", occurring to transseries solutions of (P_II). To analyze the parametric Stokes phenomenon, we introduce the "Voros coefficient" for (P_II) and derive the connection formula for transseries solutions of (P_II) through the analysis of the Voros coefficient. We also show that the connection formula can be derived by completely different method: the isomonodromic deformation.

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The aim of this paper is to give an extension of the improved Sobolev embedding theorem for single-valued functions to the case of vector-valued functions which is involved with the three-dimensional massless Dirac operator together with the three- or two-dimensional Weyl-Dirac (or Pauli) operator, the Cauchy-Riemann operator and also the four-dimensional Euclidian Dirac operator.

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We are concerned with the asymptotic behavior of the ground state of quasilinear Schrödinger equations in R². We show the asymptotic non-degeneracy and uniqueness of the ground state for a wide class of nonlinearities.

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Consider the initial-boundary value problem for coupled degenerate dissipative hyperbolic systems of Kirchhoff type: ρu_tt − (||∇u(t)||² + ||∇v(t)||²)^γΔu + u_t = 0, ρv_tt − (||∇u(t)||² + ||∇v(t)||²)^γΔv + v_t = 0, with homogeneous Dirichlet boundary condition and ρ > 0 and γ > 0. When either the coefficient ρ or the initial data are appropriately small, we prove the global existence theorem by using several identities and the energy decay. Moreover, under the same assumption for ρ and the initial data, we derive the decay estimates of the solutions and their second order derivatives.

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In this paper, we study the stability of a coexistence equilibrium in a prey-predator system with dormancy of predators. The results suggest that dormancy of predators can stabilize the population dynamics of prey-predator systems if the switch to predators entering dormancy is very sharp or if both the hatching and mortality rates of dormant predators are sufficiently low.

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