arXiv subjects

Toshiki Kondo

Publications and source records attributed to Toshiki Kondo.

5 recordsLinked to original sources

arXiv2026

Nowhere continuity of the flow map of an integrable derivative nonlinear Schrödinger system on the torus

We consider a derivative nonlinear Schrödinger system called the Chen-Lee-Liu type system on the torus. This system is known as a completely integrable system. We prove the flow map fails to be continuous at every point in the Sobolev space $H^s(\mathbb{T}) \times H^s(\mathbb{T})$. Moreover, we establish an additional condition required for the flow map to be continuous. For the discontinuity, we take a sequence converging to the initial data for which the corresponding solutions do not exist.

math.AP↗

arXiv2026

Norm inflation for quadratic derivative fractional nonlinear Schrödinger equations

We consider the Cauchy problem for quadratic derivative fractional nonlinear Schrödinger equations on $\mathbb{R}$ or $\mathbb{T}$. We determine the sharp exponents of the fractional derivatives for which the Cauchy problem is well-posed in the Sobolev space. Thanks to the global well-posedness result established by Nakanishi and Wang (2025), we can expand the solution as a sum of iterated terms. By deriving estimates for each iterated term, we establish norm inflation with infinite loss of regularity, which in particular implies ill-posedness.

math.AP↗

arXiv2025

Well- and Ill-posedness of the Cauchy problem for derivative fractional nonlinear Schrödinger equations on the torus

We consider the Cauchy problem for derivative fractional Schrödinger equations (fNLS) on the torus $\mathbb T$. Recently, the second and third authors established a necessary and sufficient condition on the nonlinearity for well-posedness of semi-linear Schrödinger equations on $\mathbb T$. In this paper, we extend this result to derivative fNLS. More precisely, we prove that the necessary and sufficient condition on the nonlinearity is the same as that for semi-linear Schrödinger equations. However, since we can not employ a gauge transformation for derivative fNLS, we use the modified energy method to prove well-posedness. We need to inductively construct correction terms for the modified energy when the fractional Laplacian is of order between $1$ and $\frac 32$. For the ill-posedness, we prove the non-existence of solutions to the Cauchy problem by exploiting a Cauchy-Riemann-type operator that appears in nonlinear interactions.

math.AP↗

arXiv2025

Well- and ill-posedness of the Cauchy problem for semi-linear Schrödinger equations on the torus

We consider the Cauchy problem for semi-linear Schrödinger equations on the torus $\mathbb T$. We establish a necessary and sufficient condition on the polynomial nonlinearity for the Cauchy problem to be well-posed in the Sobolev space $H^s(\mathbb T)$ for $s>\frac 52$. For the well-posedness, we use the energy estimates and the gauge transformation. For the ill-posedness, we prove the non-existence of solutions to the Cauchy problem.

math.AP↗

arXiv2024

Norm inflation for a higher-order nonlinear Schrödinger equation with a derivative on the circle

We consider a periodic higher-order nonlinear Schrödinger equation with the nonlinearity $u^k \partial_x u$, where $k$ is a natural number. We prove the norm inflation in a subspace of the Sobolev space $H^s(\mathbb{T})$ for any $s \in \mathbb{R}$. In particular, the Cauchy problem is ill-posed in $H^s(\mathbb{T})$ for any $s \in \mathbb{R}$.

math.AP↗