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Hideo Takaoka

Publications and source records attributed to Hideo Takaoka.

14 recordsLinked to original sources
arXiv2026

On Inverse scattering for the nonlocal nonlinear Schrodinger equation with slowly decaying data

This paper discusses about an inverse scattering problem to the nonlocal nonlinear Schrodinger equation. We extend the previous result treated by Y. Zhao and E. Fan of inverse scattering result at the point with slowly decaying data. More precisely, we show the global existence of solution in $H^{1,s}$ for $s>1/2$ by assuming small data in $L^1$. The proof relies on the inverse scattering method based on the ZS-AKNS (Zakharov-Shabat, Ablowitz-Kaup-Newell-Segur) eigenvalue problem on the spectral theory of ordinary differential equations upon the framework of slowly decaying data.

math.AP↗
arXiv2025

Large-time existence results for the nonlocal NLS around ground state solutions

This paper discusses about solutions of the nonlocal nonlinear Schrodinger equation. We prove that the solution remains close to the orbit of the soliton for a large-time, if the initial data is close to the ground state solitons. The proof uses the hyperbolic dynamics near ground state, which exhibits properties of local structural stability of solutions with respect to the flows of the nonlocal nonlinear Schrodinger equation.

math.AP↗
arXiv2023

Bilinear Strichartz estimates for the KdV equation on the torus

In this paper, we consider the bilinear Strichartz estimates for the periodic KdV equation. We give a concrete counterexample to the false $L^p$ Strichartz estimates for $p=8$, at least for a subset of the range of Lebesgue exponents $p$.Moreover, we prove the bilinear Strichartz estimate in different frequency regions, which represents a kind of a smoothing effects. When the period tends to infinity, the frequency localized version of Strichartz estimate on the torus matches to the one in the real line setting. In addition, the quartic generalized KdV equation is considered as an application.

math.AP↗
arXiv2023

On the growth of Sobolev norm for the cubic NLS on two dimensional product space

We obtain polynomial bounds on the growth in time of Sobolev norm of solutions to the cubic defocusing nonlinear Schrodinger equation on two dimensional product space. We also give the angular improved bilinear Strichartz estimates for frequency localized functions, which estimates are used for enhancement of a smoothing estimates. Such upper bounds for the growth of Sobolev norms measure the transfer of energy from low to high modes as time grows on.

math.AP↗
arXiv2019

Energy transfer model and large periodic boundary value problem for the quintic nonlinear Schrodinger equations

We study a dynamics and energy exchanges between a linear oscillator and a nonlinear interaction state for the one dimensional, quintic nonlinear Schrodinger equation. Grebert and Thomann proved that there exist solutions with initial data built on four Fourier modes, that confirms the conservative exchange of wave energy. Captured multi resonance in multiple Fourier modes, we simulate a similar energy exchange in long-period waves.

math.AP↗
arXiv2016

Energy transfer model for the derivative nonlinear Schrodinger equations on the torus

We consider the nonlinear derivative Schrodinger equation with a quintic nonlinearity, on the one dimensional torus. We exhibit that the nonlinear dynamic properties consist of four frequency modes initially excited, whose frequencies include the resonant clusters and phase matched resonant interactions of nonlinearities. This phenomena arrests energy transfers between low and high modes, which are quantified by a growth in the Sobolev norm.

math.AP↗
arXiv2015

A priori estimates and weak solutions for the derivative nonlinear Schrödinger equation on torus below $H^{1/2}$

We propose a priori estimates for a weak solution to the derivative nonlinear Schrödinger equation (DNLS) on torus with small $L^2$-norm datum in low regularity Sobolev spaces. These estimates allow us to show the existence of solutions in $H^s(\mathbb{T})$ with some $s<1/2$ in a relatively weak sense. Furthermore we make some remarks on the error estimates arising from the finite dimensional approximation solutions of DNLS using the Fourier-Lesbesgue type as auxiliary spaces, despite the fact that Nahmod, Oh, Rey-Bullet and Staffilani \cite{nors} have already seen them.

math.AP↗
arXiv2006

Multilinear estimates for periodic KdV equations and applications

We prove an endpoint multilinear estimate for the $X^{s,b}$ spaces associated to the periodic Airy equation. As a consequence we obtain sharp local well-posedness results for periodic generalized KdV equations, as well as some global well-posedness results below the energy norm. In particular we prove a multilinear estimate which completes the proof of global well-posedness for periodic KdV in a preceding paper (math.AP/0110045) down to the optimal regularity H^{-1/2}.

math.AP↗
arXiv2006

Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in R^3

We obtain global well-posedness, scattering, and global $L^{10}_{t,x}$ spacetime bounds for energy-class solutions to the quintic defocusing Schrödinger equation in $\R^{1+3}$, which is energy-critical. In particular, this establishes global existence of classical solutions. Our work extends the results of Bourgain and Grillakis, which handled the radial case. The method is similar in spirit to the induction-on-energy strategy of Bourgain, but we perform the induction analysis in both frequency space and physical space simultaneously, and replace the Morawetz inequality by an interaction variant. The principal advantage of the interaction Morawetz estimate is that it is not localized to the spatial origin and so is better able to handle nonradial solutions. In particular, this interaction estimate, together with an almost-conservation argument controlling the movement of $L^2$ mass in frequency space, rules out the possibility of energy concentration.

math.AP↗
arXiv2002

Polynomial upper bounds for the orbital instability of the 1D cubic NLS below the energy norm

We study the long-time behaviour of the focusing cubic NLS on $\R$ in the Sobolev norms $H^s$ for $0 < s < 1$. We obtain polynomial growth-type upper bounds on the $H^s$ norms, and also limit any orbital $H^s$ instability of the ground state to polynomial growth at worst; this is a partial analogue of the $H^1$ orbital stability result of Weinstein. In the sequel to this paper we generalize this result to other nonlinear Schrödinger equations. Our arguments are based on the ``$I$-method'' from our earlier papers, which pushes down from the energy norm, as well as an ``upside-down $I$-method'' which pushes up from the $L^2$ norm.

math.AP↗