arXiv subjects

Hayato Miyazaki

Publications and source records attributed to Hayato Miyazaki.

At least 19 recordsLinked to original sources

arXiv2026

Threshold for the existence of scattering states for inhomogeneous nonlinear Schr\"odinger equations without gauge invariance

We consider the asymptotic behavior of solutions to inhomogeneous nonlinear Schr\"odinger equations with non-gauge-invariant nonlinearities. The spatial coefficient in the nonlinear term may have different orders of singularity at the origin and decay at infinity. Under a coercivity condition involving the coefficient and the nonlinearity, we show that no scattering states exist for the equation below a Strauss-type threshold determined by the decay at infinity. Our class includes nonlinearities with a dominant non-oscillatory component. We also prove a complementary small-data scattering result above this threshold, using non-admissible Strichartz estimates in Lorentz spaces adapted to the singular coefficient.

math.AP↗

arXiv2026

Weak Ferromagnetism in NiS$_2$ under Nanocrystallization

Structurally well-ordered NiS$_2$ nanocrystals with an average diameter of $27.0 \pm 6.5$ nm retain the bulk-like two-step antiferromagnetic transitions, as shown by magnetization and heat-capacity measurements. Below the lower transition, the nanocrystals exhibit a hysteretic ferromagnetic response with large coercivity, exchange bias, and a vertical loop shift after field cooling, whereas the $M$-$H$ response just above the transition is nearly linear. These features are best explained by uncompensated surface moments generated where the low-temperature antiferromagnetic order terminates at the nanocrystal surface. The absence of a clear additional bulk-like weak-ferromagnetic component constrains homogeneous-canting models and indirectly favors a domain-wall scenario for the weak ferromagnetism of bulk NiS$_2$.

cond-mat.str-el↗

arXiv2026

Correction to the article "Global well-posedness and scattering in weighted space for nonlinear Schr\"{o}dinger equations below the Strauss exponent without gauge-invariance"

This article resolves some errors in the paper ``Global well-posedness and scattering in weighted space for nonlinear Schr\"{o}dinger equations below the Strauss exponent without gauge-invariance. Math. Ann. 392, 1051-1097 (2025)''. The errors are in the proof of contraction of a map associated with our equation in two and three dimensions.

math.AP↗

arXiv2026

Large-data $L^2$-decay for attractive-dissipative nonlinear Schr\"odinger equations without the strong dissipative condition

We prove a large-data $L^2$-decay estimate for nonlinear dissipative Schr\"odinger equations with attractive-dissipative power nonlinearity. The main difficulty is the lack of sign definiteness of the standard energy when $\Re\lambda<0$, which prevents the usual energy argument from directly yielding a uniform gradient bound. We introduce an augmented energy, obtained by adding a suitable multiple of the decreasing $L^2$-norm to the standard energy. This produces an additional dissipative term and gives a direct uniform-in-time $H^1$ bound without the iteration argument used in previous works. Consequently, for arbitrary initial data in the weighted energy space $\Sigma = H^1 \cap \mathcal{F}H^1$, we obtain the decay rate previously known under the strong dissipative condition throughout the sharp decay range $1<p\le 1+2/d$. This removes the remaining restriction $p\le 1+4/(3d)$ in the attractive-dissipative case.

math.AP↗

arXiv2026

A Unified Integral Equation Approach to Conservation Laws for Nonlinear Schr\"odinger Equations

We present a unified framework for the rigorous derivation of conservation laws and related identities for nonlinear Schr\"odinger equations with power-type nonlinearities. This approach treats the equation in its Duhamel form and uses the space-time integrability provided by Strichartz estimates, without relying on smooth approximations or regularization procedures. It was first introduced by the third author in [20] and subsequently developed in [7, 13]. In this paper, we establish a single integral identity from which all of the laws and identities considered here follow systematically. These include the conservation of charge (mass), energy, and momentum, the pseudo-conformal conservation law, and virial-type identities.

math.AP↗

arXiv2025

Remarks on the derivation of the virial identity for nonlinear Schrödinger equations

We revisit the derivation of the virial identity for nonlinear Schrödinger equations. In \cite{O06, FM17}, several conservation laws, such as for the charge and the energy, were derived without constructing a sequence of approximate solutions. Their approach involves additional properties of solutions due to Strichartz' estimate. In this paper, we derive the virial identity without constructing the sequence of approximate solutions or employing a regularizing argument for weights, by exploiting the properties of solutions.

math.AP↗

arXiv2025

Threshold for the existence of scattering states for nonlinear Schrödinger equations without gauge invariance

This paper is concerned with a threshold phenomenon for the existence of scattering states for nonlinear Schrödinger equations. The nonlinearity includes a non-oscillatory term of the order lower than the Strauss exponent. We show that no scattering states exist for the equation in a weighted Sobolev space. It is emphasized that our method admits initial data with good properties, such as compactly supported smooth functions. The result indicates that the Strauss exponent acts as a threshold for the power of the nonlinearity that determines whether solutions scatter or not in the weighted space.

math.AP↗

arXiv2025

Global well-posedness and scattering in weighted space for nonlinear Schrödinger equations below the Strauss exponent without gauge-invariance

In this paper, we consider the nonlinear Schrödinger equation (NLS) with a general homogeneous nonlinearity in dimensions up to three. We assume that the degree (i.e., power) of the nonlinearity is such that the equation is mass-subcritical and short-range. We establish global well-posedness (GWP) and scattering for small data in the standard weighted space for a class of homogeneous nonlinearities, including non-gauge-invariant ones. Additionally, we include the case where the degree is less than or equal to the Strauss exponent. When the nonlinearity is not gauge-invariant, the standard Duhamel formulation fails to work effectively in the weighted Sobolev space; for instance, the Duhamel term may not be well-defined as a Bochner integral. To address this issue, we introduce an alternative formulation that allows us to establish GWP and scattering, even in the presence of poor time continuity of the Duhamel term.

math.AP↗

arXiv2025

Refinement of the $L^{2}$-decay estimate of solutions to nonlinear Schrödinger equations with attractive-dissipative nonlinearity

This paper is concerned with the $L^{2}$-decay estimate of solutions to nonlinear dissipative Schrödinger equations with power-type nonlinearity of the order $p$. It is known that the sign of the real part of the dissipation coefficient affects the long-time behavior of solutions, when neither size restriction on the initial data nor strong dissipative condition is imposed. In that case, if the sign is negative, then Gerelmaa, the first and third author [7] obtained the $L^{2}$-decay estimate under the restriction $p \le 1+1/d$. In this paper, we relax the restriction to $p \le 1+4/(3d)$ by refining an energy-type estimate. Furthermore, when $p < 1+ 4/(3d)$, using an iteration argument, the best available decay rate is established, as given by Hayashi, Li and Naumkin [11].

math.AP↗

arXiv2024

Modified scattering operator for nonlinear Schrödinger equations with time-decaying harmonic potentials

This paper is concerned with nonlinear Schrödinger equations with a time-decaying harmonic potential. The nonlinearity is gauge-invariant of the long-range critical order. In [24] and [22], it is proved that the equation admits a nontrivial solution that behaves like a free solution with a logarithmic phase correction in the frameworks of both the final state problem and the initial value problem. Furthermore, a modified scattering operator has been established in the case without the potential in [15]. In this paper, we construct a modified scattering operator for our equation by utilizing a generator of the Galilean transformation. Moreover, we remove a restriction for the coefficient of the potential which is required in [22].

math.AP↗

arXiv2022

Long-range scattering for a critical homogeneous type nonlinear Schrödinger equation with time-decaying harmonic potentials

This paper is concerned with the final state problem for the homogeneous type nonlinear Schrödinger equation with time-decaying harmonic potential. The nonlinearity has the critical order and is not necessarily the form of a polynomial. In the case of the gauge-invariant power-type nonlinearity, the first author proves that the equation admits a nontrivial solution that behaves like a free solution with a logarithmic phase correction in [22]. In this paper, we extend his result into the case with the general homogeneous nonlinearity by the technique due to the Fourier series expansion introduced by Masaki and the second author [26]. To adapt the argument in the aforementioned paper, we develop a factorization identity for the propagator and require a little stronger decay condition for the Fourier coefficients arising from the harmonic potential. Moreover, in two or three dimensions, we improve the regularity condition of the final data in [26, 28].

math.AP↗

arXiv2021

Local well-posedness for the higher-order generalized KdV type equation with low-degree of nonlinearity

This paper is concerned with the local well-posedness for the higher-order generalized KdV type equation with low-degree of nonlinearity. The equation arises as a non-integrable and lower nonlinearity version of the higher-order KdV equation. As for the lower nonlinearity model of the KdV equation, Linares, the author and Ponce [11] prove the local well-posedness under a non-degenerate condition introduced by Cazenave and Naumkin [1]. In this paper, it turns out that the well-posedness result can be extended into the higher-order equation. We also give a lower bound for the lifespan of the solution. The lifespan depends on two quantities determined by the initial data.

math.AP↗

arXiv2021

Modified scattering for inhomogeneous nonlinear Schrödinger equations with and without inverse-square potential

We consider the final state problem for the inhomogeneous nonlinear Schrödinger equation with a critical long-range nonlinearity. Given a prescribed asymptotic profile, which has a logarithmic phase correction compared with the free evolution, we construct a unique global solution which converges to the profile. As a consequence, the existence of modified wave operators for localized small scattering data is obtained. We also study the same problem for the case with the critical inverse-square potential under the radial symmetry. In particular, we construct the modified wave operators for the long-range nonlinear Schrödinger equation with the critical inverse-square potential in three space dimensions, under the radial symmetry.

math.AP↗

arXiv2020

Asymptotic behavior for the long-range nonlinear Schrödinger equation on star graph with the Kirchhoff boundary condition

We consider the cubic nonlinear Schrödinger equation on the star graph with the Kirchhoff boundary condition. We prove modified scattering for the final state problem and the initial value problem. Moreover, we also consider the failure of scattering for the Schrödinger equation with power-type long-range nonlinearities. These results are extension of the results for NLS on the one dimensional Euclidean space.

math.AP↗

arXiv2020

Lifespan of solutions to nonlinear Schrödinger equations with general homogeneous nonlinearity of the critical order

This paper is concerned with the upper bound of the lifespan of solutions to nonlinear Schrödinger equations with general homogeneous nonlinearity of the critical order. In [8], Masaki and the first author obtain the upper bound of the lifespan of solutions to our equation via a test function method introduced by [16, 17]. Their nonlinearity contains a non-oscillating term $|u|^{1+2/d}$ which causes difficultly for constructing an even small data global solution. The non-oscillating term corresponds to the $L^1$-scaling critical. In this paper, it turns out that the upper bound can be refined by employing an unified test function by Ikeda and the second author [5].

math.AP↗

arXiv2019

Strong blow-up instability for standing wave solutions to the system of the quadratic nonlinear Klein-Gordon equations

This paper is concerned with strong blow-up instability (Definition 1.3) for standing wave solutions to the system of the quadratic nonlinear Klein-Gordon equations. In the single case, namely the nonlinear Klein-Gordon equation with power type nonlinearity, stability and instability for standing wave solutions have been extensively studied. On the other hand, in the case of our system, there are no results concerning the stability and instability as far as we know. In this paper, we prove strong blow-up instability for the standing wave to our system. The proof is based on the techniques in Ohta and Todorova [25]. It turns out that we need the mass resonance condition in two or three space dimensions whose cases are the mass-subcritical case.

math.AP↗

arXiv2018

The derivation of the conservation law for defocusing nonlinear Schroedinger equations with non-vanishing initial data at infinity

For nonlinear Schrödinger equations in less than or equal to four dimension, with non-vanishing initial data at infinity, a new approach to derive the conservation law is obtained. Since this approach does not contain approximating procedure, the argument is simplified and some of technical assumption of the nonlinearity to derive the conservation law and time global solutions, is removed.

math.AP↗

arXiv2018

On a class of solutions to the generalized KdV type equation

We consider the IVP associated to the generalized KdV equation with low degree of non-linearity \begin{equation*} \partial_t u + \partial_x^3 u \pm |u|^α\partial_x u = 0,\; x,t \in \mathbb{R},\;α\in (0,1). \end{equation*} By using an argument similar to that introduced by Cazenave and Naumkin [2] we establish the local well-posedness for a class of data in an appropriate weighted Sobolev space. Also, we show that the solutions obtained satisfy the propagation of regularity principle proven in [3] in solutions of the $k$-generalized KdV equation.

math.AP↗