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Hideshi Yamane

Publications and source records attributed to Hideshi Yamane.

14 recordsLinked to original sources
arXiv2024

Asymptotic Expansions of Finite Hankel Transforms and the Surjectivity of Convolution Operators

A compactly supported distribution is called invertible in the sense of Ehrenpreis-Hörmander if the convolution with it induces a surjection from $\mathcal{C}^{\infty}(\mathbb{R}^{n})$ to itself. We give sufficient conditions for radial functions to be invertible. Our analysis is based on the asymptotic expansions of finite Hankel transforms. The dominant term may be the contribution from the origin or from the boundary of the support of the function. For the proof, we propose a new method to calculate the asymptotic expansions of finite Hankel transforms of functions with singularities at a point other than the origin.

math.FA↗
arXiv2020

Local and global analyticity for a generalized Camassa-Holm system

We solve the analytic Cauchy problem for the generalized two-component Camassa-Holm system introduced by R. M. Chen and Y. Liu. We show the existence of a unique local/global-in-time analytic solution under certain conditions. This is the first result about global analyticity for a Camassa-Holm-like system. The method of proof is basically that developed by Barostichi, Himonas and Petronilho. The main differences of their proof and ours are twofold: (i) the system of Chen and Liu is not symmetic in the two unknowns and our estimates are not trivial generalization of those in their articles, (ii) we have simplified their argument by using fewer function spaces and the main result is stated in a simple and natural way.

math.AP↗
arXiv2020

Generalized spherical mean value operators on Euclidean space

We consider the Neumann version of the spherical mean value operator and its variants in the space of smooth functions, distributions and compactly supported ones. Surjectivity and range characterization issues are addressed from the viewpoint of convolution equations.

math.FA↗
arXiv2019

Local and global analyticity for $μ$-Camassa-Holm equations

We solve Cauchy problems for some $μ$-Camassa-Holm integro-partial differential equations in the analytic category. The equations to be considered are $μ$CH of Khesin-Lenells-Misiołek, $μ$DP of Lenells-Misiołek-Tiğlay, the higher-order $μ$CH of Wang-Li-Qiao and the non-quasilinear version of Qu-Fu-Liu. We prove the unique local solvability of the Cauchy problems and provide an estimate of the lifespan of the solutions. Moreover, we show the existence of a unique global-in-time analytic solution for $μ$CH, $μ$DP and the higher-order $μ$CH. The present work is the first result of such a global nature for these equations. AMS subject classification: 35R09, 35A01, 35A10, 35G25

math.AP↗
arXiv2019

Analytic Cauchy problem for the $μ$-Camassa-Holm equation and its non-quasilinear version

We solve the Cauchy problems for the $μ$-Camassa-Holm integro-partial differential equation of Khesin-Lenells-Misiołek and its non-quasilinear version introduced by Qu-Fu-Liu in the complex-analytic framework. These equations have nonlocal nature at two levels: they involve a pseudo-differential operator of negative order and this operator is defined in terms of an integral of the unknown function. We prove unique solvability of the Cauchy problems and provide an estimate of the lifespan of the solutions. Our method is the Ovsyannikov type argument by Batrostichi-Himonas-Petronilho about a scale of Banach spaces of analytic functions, but the non-quasilinearity is dealt with in a different way from theirs. Indeed, we use a simple reduction which is just a nonlocal version of a classical trick.

math-ph↗
arXiv2018

Some trigonometric integrals and the Fourier transform of a spherically symmetric exponential function

We calculate the Fourier transform of a spherically symmetric exponential function. Our evaluation is much simpler than the known one. We use the polar coordinates and reduce the Fourier transform to the integral of a rational function of trigonometric functions. Its evaluation turns out to be much easier than expected because of homogeneity and a hidden symmetry. Relationship with a Fourier integral representation formula for harmonic functions is explained.

math.CA↗
arXiv2018

Riemann-Hilbert factorization of matrices invariant under inversion in a circle

We consider matrix functions with certain invariance under inversion in the unit circle. If such a function satisfies a positivity assumption on the unit circle, then only zero partial indices appear in its Riemann-Hilbert (Wiener-Hopf) factorization. It implies the unique solvability of a certain class of Riemann-Hilbert boundary value problems. It includes the ones associated with the inverse scattering transform of the focusing/defocusing integrable discrete nonlinear Schrödinger equations.

math-ph↗
arXiv2016

Long-Time Asymptotics for the integrable discrete nonlinear Schrödinger equation: The Focusing Case

We investigate the long-time asymptotics for the focusing integrable discrete nonlinear Schrödinger equation. Under generic assumptions on the initial value, the solution is asymptotically a sum of 1-solitons. We find different phase shift formulas in different regions. Along rays away from solitons, the behavior of the solution is decaying oscillation. This is one way of stating the soliton resolution conjecture. The proof is based on the nonlinear steepest descent method.

math-ph↗
arXiv2015

Long-Time Asymptotics for the Defocusing Integrable Discrete Nonlinear Schrödinger Equation II

We investigate the long-time asymptotics for the defocusing integrable discrete nonlinear Schrödinger equation. If $|n|<2t$, we have decaying oscillation of order $O(t^{-1/2})$ as was proved in our previous paper. Near $|n|=2t$, the behavior is decaying oscillation of order $O(t^{-1/3})$ and the coefficient of the leading term is expressed by the Painlevé II function. In $|n|>2t$, the solution decays more rapidly than any negative power of $n$.

math-ph↗
arXiv2007

Logarithmic singularities of solutions to nonlinear partial differential equations

We construct a family of singular solutions to some nonlinear partial differential equations which have resonances in the sense of a paper due to T. Kobayashi. The leading term of a solution in our family contains a logarithm, possibly multiplied by a monomial. As an application, we study nonlinear wave equations with quadratic nonlinearities. The proof is by the reduction to a Fuchsian equation with singular coefficients.

math.AP↗
arXiv2006

Nonlinear wave equations and singular solutions

We construct solutions to nonlinear wave equations that are singular along a prescribed noncharacteristic hypersurface which is the graph of a function satisfying not the Eikonal but another partial differential equation of the first order. The method of Fuchsian reduction is employed.

math.AP↗