In this work are obtained some criteria which guarantee the oscillatory behavior of the differential system of mixed type x′(t) = ∫⁰₋₁ d[ν(θ)]x(t−r(θ)) + ∫⁰₋₁d[η(θ)]x(t+τ(θ)), where x(t) ∈ Rⁿ, r(θ) and τ(θ) are real nonnegative continuous functions on [−1, 0], ν(θ) and η(θ) are real n-by-n matrix valued function of bounded variation on [−1, 0].

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This paper is concerned with the stationary problem for a bistable reaction-diffusion equation which possesses spatially inhomogeneous effects both in its diffusion and reaction terms. For this problem, we are interested in solutions with transition layers. Our main purpose is to study precise profiles of such solutions. In particular, we will determine the location of transition layers. Moreover, we will show some conditions under which one can observe multi-layers.

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We prove the global well-posedness for the Cauchy problem of the derivative nonlinear Schrödinger equation in H^s(T) for s > 1/2 with small data in L². We use the method of almost conserved energy or the I-method which was introduced by Colliander et al. and refine the bilinear estimate.

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We study the convergence or divergence of formal solutions of first order singular nonlinear ordinary differential equation (SE) f(x,u,u′) = 0, u(0)= 0, where the function f is holomorphic in a neighbourhood of the origin of C³. The equation (SE) is said to be singular if f(0,0,ξ) ≡ 0. M. Miyake and A. Shirai [2, 3] already studied this problem in case of partial differential equation and they gave a criterion of convergence or divergence of formal solutions by using the Taylor coefficients of order 1 of formal solution and equation. In this paper, we give a criterion by using the Taylor coefficients of higher order of formal solution and equation.

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Geometric study of a second-order Fuchsian differential equation u″−q(x)u = 0, where q is rational in x, has been made via the Schwarz map as well as via the hyperbolic and the derived Schwarz maps ([SYY]). When the equation admits an irregular singularity, such a study was first made in [SY] treating the confluent hypergeometric equation and the Airy equation. In this paper, we study the hyperbolic Schwarz map (note that this map governs the other Schwarz maps) of such an equation with any irregular singularity. We describe the asymptotic behavior of the map around the singular point: when the Poincaré rank is generic, it admits a uniform description; when the Poincaré rank is exceptional, a detailed study is made.

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Recently it was shown by Z. Guo and the author independently that the Cauchy problem for the nonperiodic Korteweg-de Vries equation is locally well-posed in the Sobolev space of the critical regularity. Their proofs were based on the iteration argument in the Besov endpoint of the Bourgain space with some additional modification in low frequency, and it was conjectured that the Bourgain space with the Besov-type modification only does not restore the bilinear estimate which is essential to the iteration. In the present article we give the answer to this conjecture by constructing an exact counterexample to the bilinear estimate.

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We study the Drinfeld-Sokolov hierarchies of type A_n⁽¹⁾ associated with the regular conjugacy classes of W(A_n). A class of fourth order Painlevé systems is derived from them by similarity reductions.

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