Funkcialaj Ekvacioj

Remarks on Nonlinear Smoothing under Randomization for the Periodic KdV and the Cubic Szegö Equation

Tadahiro Oh

2011 年 54 巻 3 号 p. 335-365
発行日: 2011年
公開日: 2011/11/30

DOIhttps://doi.org/10.1619/fesi.54.335

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We consider Cauchy problems of some dispersive PDEs with random initial data. In particular, we construct local-in-time solutions to the mean-zero periodic KdV almost surely for the initial data in the support of the mean-zero Gaussian measures on H^s(T), s > s₀ where s₀ = –11/6 + $\sqrt{61}$/6 ≈ –0.5316 < –1/2, by exhibiting nonlinear smoothing under randomization on the second iteration of the Duhamel formulation. We also show that there is no nonlinear smoothing for the dispersionless cubic Szegö equation under randomization of initial data.

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Non-Existence of Zero Resonances of Pauli Operators

Tsutomu Morita

2011 年 54 巻 3 号 p. 367-381
発行日: 2011年
公開日: 2011/11/30

DOIhttps://doi.org/10.1619/fesi.54.367

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We shall show that the Pauli operator P_A = (D–A)²I₂ –σ · B has no zero resonances if A(x) is smooth and |A_j(x)| + |∇A_j(x)| ≤ C<x>^–ρ with ρ ≥ 2.

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Resurgent Analysis of the Witten Laplacian in One Dimension

Alexander Getmanenko

2011 年 54 巻 3 号 p. 383-438
発行日: 2011年
公開日: 2011/11/30

DOIhttps://doi.org/10.1619/fesi.54.383

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The Witten Laplacian corresponding to a Morse function on the circle is studied using methods of complex WKB and resurgent analysis. It is shown that under certain assumptions the low-lying eigenvalues of the Witten Laplacian are resurgent.

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Radial and Nonradial Solutions for a Semilinear Elliptic System of Schrödinger Type

Dragoş-Pătru Covei

2011 年 54 巻 3 号 p. 439-449
発行日: 2011年
公開日: 2011/11/30

DOIhttps://doi.org/10.1619/fesi.54.439

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In this article we consider the system of equations Δu_i = p_i(x)f_i(u₁,...,u_d) for i = 1,...,d on R^N, N ≥ 3 and d ∈ {1,2,3,4,...}. We prove that the considered system has a bounded positive entire solution under some conditions on p_i and f_i. Also, we give a necessary condition as well as a sufficient condition for a positive radial solution to be large. The method of proving theorems is essentially based on a successive approximation. Furthermore, a non-radially symmetric solution is obtained by using a lower and upper solution method.

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Truncated Solutions of the Fifth Painlevé Equation

Shun Shimomura

2011 年 54 巻 3 号 p. 451-471
発行日: 2011年
公開日: 2011/11/30

DOIhttps://doi.org/10.1619/fesi.54.451

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The fifth Painlevé equation admits several families of solutions behaving exponentially in their proper sectors near infinity, which are called truncated solutions. For these truncated solutions, we discuss the frequency of a-points including poles outside the corresponding sectors. Except for some special cases, all the values are equally distributed, and for each a there exist infinitely many a-points.

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A Note on Zero Free Regions of the Stokes Multipliers for Second Order Ordinary Differential Equations with Cubic Polynomial Coefficients

Tatsuo Nishitani

2011 年 54 巻 3 号 p. 473-483
発行日: 2011年
公開日: 2011/11/30

DOIhttps://doi.org/10.1619/fesi.54.473

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In this note, by a completely elementary argument, we give a rough estimate about zero free regions of the Stokes multipliers for second order differential equations with cubic polynomial coefficients. We also prove that the Stokes multipliers indeed has zeros outside this zero free regions by resorting to some classical results about spectral problem for anharmonic oscillators.

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Classical Solutions to a Linear Schrödinger Evolution Equation Involving a Coulomb Potential with a Moving Center of Mass

Kentarou Yoshii

2011 年 54 巻 3 号 p. 485-493
発行日: 2011年
公開日: 2011/11/30

DOIhttps://doi.org/10.1619/fesi.54.485

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This paper is concerned with Cauchy problems for the linear Schrödinger evolution equation (i(∂/∂t) + Δ + |x – a(t)|^–1 + V₁(x,t))u(x,t) = f(x,t) in R^N × [0,T], subject to initial condition: u(·,0) ∈ H²(R^N) ∩ H₂(R^N), where i := $\sqrt{-1}$, N ≥ 3, T > 0 and a : [0,T] → R^N expresses the center of the Coulomb potential, V₁ and f are another real-valued potential and an inhomogeneous term, respectively, while H₂(R^N) := {v ∈ L²(R^N); |x|²v ∈ L²(R^N)}. We show that under some conditions on V₁ and f the equation has a classical solution u(·) ∈ C¹([0,T]; L²(R^N)) ∩ C([0,T]; H²(R^N) ∩ H₂(R^N)).

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A Connection Problem for Simpson's Even Family of Rank Four

Yoshishige Haraoka, Katsuhisa Mimachi

2011 年 54 巻 3 号 p. 495-515
発行日: 2011年
公開日: 2011/11/30

DOIhttps://doi.org/10.1619/fesi.54.495

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We solve a connection problem for the equation of rank four in Simpson's even family, which we call Even four. We use an integral representation of solutions, and consider the twisted homology group associated with it. The rank of the homology group is five. Its four dimensional subspace, consisting of regularizable cycles, corresponds to the solution space of Even four. The connection problem is reduced to solving the linear relations among twisted cycles. There are only few equations for which connection problems are solved in this way.

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