arXiv subjects

Yoshio Tsutsumi

Publications and source records attributed to Yoshio Tsutsumi.

8 recordsLinked to original sources

arXiv2023

Gauge transformation for the kinetic derivative nonlinear Schrödinger equation on the torus

We consider the kinetic derivative nonlinear Schrödinger equation, which is a one-dimensional nonlinear Schrödinger equation with a cubic derivative nonlinear term containing the Hilbert transformation. In our previous work, we proved small-data global well-posedness of the Cauchy problem on the torus in Sobolev space $H^s$ for $s>1/2$ by combining the Fourier restriction norm method with the parabolic smoothing effect, which is available in the periodic setting. In this article, we improve the regularity range to $s>1/4$ for the global well-posedness by constructing an effective gauge transformation. Moreover, we remove the smallness assumption by making use of the dissipative nature of the equation.

math.AP↗

arXiv2023

Low regularity a priori estimate for KDNLS via the short-time Fourier restriction method

In this article, we consider the kinetic derivative nonlinear Schrödinger equation (KDNLS), which is a one-dimensional nonlinear Schrödinger equation with a cubic derivative nonlinear term containing the Hilbert transformation. For the Cauchy problem both on the real line and on the circle, we apply the short-time Fourier restriction method to establish a priori estimate for small and smooth solutions in Sobolev spaces $H^s$ with $s>1/4$.

math.AP↗

arXiv2022

Boundedness of the conformal hyperboloidal energy for a wave-Klein-Gordon model

We consider the global evolution problem for a model which couples together a nonlinear wave equation and a nonlinear Klein-Gordon equation, and was independently introduced by LeFloch and Y. Ma and by Q. Wang. By revisiting the Hyperboloidal Foliation Method, we establish that a weighted energy of the solutions remains (almost) bounded for all times. The new ingredient in the proof is a hierarchy of fractional Morawetz energy estimates (for the wave component of the system) which is defined from two conformal transformations. The optimal case for these energy estimates corresponds to using the scaling vector field as a multiplier for the wave component.

math.AP↗

arXiv2021

Well-posedness of the Cauchy Problem for the Kinetic DNLS on $\mathbf{T}$

We consider the Cauchy problem for the kinetic derivative nonlinear Schrödinger equation on the torus: \[ \partial_t u - i \partial_x^2 u = α\partial_x \big( |u|^2 u \big) + β\partial_x \big[ H \big( |u|^2 \big) u \big] , \quad (t, x) \in [0,T] \times \mathbf{T}, \] where the constants $α,β$ are such that $α\in \mathbf{R}$ and $β<0$, and $H$ denotes the Hilbert transform. This equation has dissipative nature, and the energy method is applicable to prove local well-posedness of the Cauchy problem in Sobolev spaces $H^s$ for $s>3/2$. However, the gauge transform technique, which is useful for dealing with the derivative loss in the nonlinearity when $β=0$, cannot be directly adapted due to the presence of the Hilbert transform. In particular, there has been no result on local well-posedness in low regularity spaces or global solvability of the Cauchy problem. In this article, we shall prove local and global well-posedness of the Cauchy problem for small initial data in $H^s(\mathbf{T})$, $s>1/2$. To this end, we make use of the parabolic-type smoothing effect arising from the resonant part of the nonlocal nonlinear term $β\partial_x [H(|u|^2)u]$, in addition to the usual dispersive-type smoothing effect for nonlinear Schrödinger equations with cubic nonlinearities. As by-products of the proof, we also obtain smoothing effect and backward-in-time ill-posedness results.

math.AP↗

arXiv2020

Quasi-Invariance of Gaussian Measures Transported by the Cubic NLS with Third-Order Dispersion on $\mathbf{T}$

We consider the Nonlinear Schrödinger (NLS) equation and prove that the Gaussian measure with covariance $(1-\partial_x^2)^{-α}$ on $L^2(\mathbf T)$ is quasi-invariant for the associated flow for $α>1/2$. This is sharp and improves a previous result obtained in \cite{OTT} where the values $α>3/4$ were obtained. Also, our method is completely different and simpler, it is based on an explicit formula for the Radon-Nikodym derivative. We obtain an explicit formula for this latter in the same spirit as in \cite{Cruz1} and \cite{Cruz2}. The arguments are general and can be used to other Hamiltonian equations.

math.AP↗

arXiv2019

Quasi-invariant Gaussian measures for the cubic nonlinear Schrödinger equation with third order dispersion

In this paper, we consider the cubic nonlinear Schrödinger equation with third order dispersion on the circle. In the non-resonant case, we prove that the mean-zero Gaussian measures on Sobolev spaces $H^s(\mathbb{T})$, $s > \frac 34$, are quasi-invariant under the flow. In establishing the result, we apply gauge transformations to remove the resonant part of the dynamics and use invariance of the Gaussian measures under these gauge transformations.

math.AP↗

arXiv2018

Ill-Posedness of the Third Order NLS Equation with Raman Scattering Term

We consider the ill-posedness and well-posedness of the Cauchy problem for the third order NLS equation with Raman scattering term on the one dimensional torus. It is regarded as a mathematical model for the photonic crystal fiber oscillator. Regarding the ill-posedness, we show the nonexistence of solutions in the Sobolev space and the norm inflation of the data-solution map under slightly different conditions, respectively. We also prove the local unique existence of solutions in the analytic function space.

math.AP↗

arXiv2010

1D quintic nonlinear Schrödinger equation with white noise dispersion

In this article, we improve the Strichartz estimates obtained in [12] for the Schrödinger equation with white noise dispersion in one dimension. This allows us to prove global well posedness when a quintic critical nonlinearity is added to the equation. We finally show that the white noise dispersion is the limit of smooth random dispersion.

math.AP↗