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Satoshi Masaki

Publications and source records attributed to Satoshi Masaki.

At least 19 recordsLinked to original sources
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.

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arXiv2026

High-Velocity Inverse Scattering for Nonlinear Schr\"odinger Equations with Spatially Dependent Nonlinearities

We study a high-velocity inverse scattering problem for nonlinear Schr\"odinger equations with spatially dependent nonlinearities in dimensions $d\ge3$. We consider the whole mass-supercritical and energy-subcritical range, including the endpoint cases. By introducing a moving frame adapted to highly boosted initial data, we construct the scattering operator for a class of large incoming states generated by Galilean boosts. The key observation is that, although the boosted data become large in Sobolev norms, the nonlinear interaction becomes effectively weak at high velocity due to rapid spatial separation. Using the resulting high-velocity asymptotics, we derive a reconstruction formula for the X-ray transform of the coefficient. As a consequence, we prove that the scattering operator uniquely determines both the nonlinearity exponent and the spatial coefficient. Our results extend previous work of Watanabe to all dimensions $d \ge 3$, include the endpoint nonlinearities, and replace the repulsiveness and radial monotonicity assumptions on the coefficient by suitable decay conditions.

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arXiv2026

Modified scattering type asymptotic behavior for a quadratic nonlinear Schr\"odinger system under the mass-resonance condition in two dimensions

We study a system of nonlinear Schrodinger equations under the mass-resonance condition and provide a complete description of the asymptotic behavior of small solutions in two space dimensions. Our analysis is based on a detailed study of an associated system of ordinary differential equations governing the asymptotic profile. We establish a phase-amplitude representation for this ODE with arbitrary initial data, where the amplitude is expressed explicitly in terms of Jacobi elliptic functions and the phase is given by elliptic integrals of the third kind. As a consequence, we obtain a fully explicit characterization of the asymptotic profile for the original PDE. In particular, the long-time behavior is described by a modified-scattering-type profile whose profile evolves according to the above integrable structure. While elliptic-function-type asymptotics were previously constructed for special solutions in final-state problems, the present work provides the first complete characterization, in two space dimensions, of asymptotic dynamics associated with arbitrarily small initial data through elliptic-function-type profiles.

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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.

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arXiv2025

Non-polynomial conserved quantities for ODE systems and its application to the long-time behavior of solutions to cubic NLS systems

In this paper, we investigate the asymptotic behavior of small solutions to the initial value problem for a system of cubic nonlinear Schrodinger equations (NLS) in one spatial dimension. We identify a new class of NLS systems for which the global boundedness and asymptotics of small solutions can be established, even in the absence of any effective conserved quantity. The key to this analysis lies in utilizing conserved quantities for the reduced ordinary differential equation (ODE) systems derived from the original NLS systems. In a previous study, the first author investigated conserved quantities expressed as quartic polynomials. In contrast, the conserved quantities considered in the present paper are of a different type and are not necessarily polynomial.

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arXiv2024

Global existence and large-time behavior of solutions to cubic nonlinear Schrödinger systems without coercive conserved quantity

In this article, we investigate the large-time behavior of small solutions to a system of one-dimensional cubic nonlinear Schrödinger equations with two components. In previous studies, a structural condition on the nonlinearity has been employed to guarantee the existence of a coercive, mass-type conserved quantity. We identify a new class of systems that do not satisfy such a condition and thus lack a coercive conserved quantity. Nonetheless, we establish the global existence and describe the large-time behavior for small solutions in this class. In this setting, the asymptotic profile is described in terms of solutions to the corresponding system of ordinary differential equations (ODEs). A key element of our analysis is the use of a quartic conserved quantity associated with the ODE system. Moreover, for a specific example within this class, we solve the ODE system explicitly, showing that the asymptotic behavior is expressed using Jacobi elliptic functions.

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arXiv2024

Scattering problem for the generalized Korteweg-de Vries equation

In this paper we study the scattering problem for the initial value problem of the generalized Korteweg-de Vries (gKdV) equation. The purpose of this paper is to achieve two primary goals. Firstly, we show small data scattering for (gKdV) in the weighted Sobolev space, ensuring the initial and the asymptotic states belong to the same class. Secondly, we introduce two equivalent characterizations of scattering in the weighted Sobolev space. In particular, this involves the so-called conditional scattering in the weighted Sobolev space. A key ingredient is incorporation of the scattering criterion for (gKdV) in the Fourier-Lebesgue space by the authors into the the scattering problem in the weighted Sobolev space.

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arXiv2023

Partial classification of the large-time behavior of solutions to cubic nonlinear Schrödinger systems

In this paper, we study the large-time behavior of small solutions to the standard form of the systems of 1D cubic nonlinear Schrödinger equations consisting of two components and possessing a coercive mass-like conserved quantity. The cubic nonlinearity is known to be critical in one space dimension in view of the large-time behavior. By employing the result by Katayama and Sakoda, one can obtain the large-time behavior of the solution if we can integrate the corresponding ODE system. We introduce an integration scheme suited to the system. The key idea is to rewrite the ODE system, which is cubic, as a quadratic system of quadratic quantities of the original unknown. By using this technique, we described the large-time behavior of solutions in terms of elementary functions and the Jacobi elliptic functions for several examples of standard systems.

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arXiv2023

Scattering for the focusing, $L^2$-supercritical fourth-order NLS in one dimension

In this paper, we consider the fourth-order Schrödinger equations with focusing, $L^2$-supercritical nonlinearity in one dimension. We prove the global existence and scattering of solutions below the ground state threshold under the evenness assumption. This results extends to the result by Guo and Dinh in higher dimensions under the radial assumption.

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arXiv2023

Scattering below ground states for a class of systems of nonlinear Schrodinger equations

In this paper, we consider the scattering problem for a class of $N$-coupled systems of the cubic nonlinear Schrödinger equations in three space dimensions. We prove the scattering of solutions that have a mass-energy quantity less than that for the ground states. This result is previously obtained by Duyckaerts-Holmer-Roudenko for the single cubic nonlinear Schrödinger equation in three space dimensions. It turns out that the result can be extended to a wide class of $N$-coupled systems.

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arXiv2023

On scalar-type standing-wave solutions to systems of nonlinear Schrödinger equations

In this article, we study the standing-wave solutions to a class of systems of nonlinear Schrödinger equations. Our target is all the standard forms of the NLS systems, with two unknowns, that have a common linear part and cubic gauge-invariant nonlinearities and that yield a Hamiltonian with a coercive kinetic-energy part. We give a necessary and sufficient condition on the existence of the ground state. Further, we give a characterization of the shape of the ground state. It will turn out that the ground states are scalar-type, i.e., multiples of a constant vector and a scalar function. We further give a sufficient condition on the existence of excited states of the same form. The stability and the instability of the ground states are also studied. To this end, we introduce an abstract treatment on the study of scalar-type standing-wave solution that applies to a wide class of NLS systems with homogeneous energy-subcritical nonlinearity. By the argument, some previous results are reproduced.

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arXiv2022

Scattering for the mass-critical nonlinear Klein-Gordon equations in three and higher dimensions

In this paper we consider the real-valued mass-critical nonlinear Klein-Gordon equations in three and higher dimensions. We prove the dichotomy between scattering and blow-up below the ground state energy in the focusing case, and the energy scattering in the defocusing case. We use the concentration-compactness/rigidity method as R. Killip, B. Stovall, and M. Visan [Trans. Amer. Math. Soc. 364 (2012)]. The main new novelty is to approximate the large scale (low-frequency) profile by the solution of the mass-critical nonlinear Schrödinger equation when the nonlinearity is not algebraic.

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arXiv2022

Polynomial deceleration for a system of cubic nonlinear Schrödinger equations in one space dimension

In this paper, we consider the initial value problem of a specific system of cubic nonlinear Schrödinger equations. Our aim of this research is to specify the asymptotic profile of the solution in $L^{\infty}$ as $t \to \infty$. It is then revealed that the solution decays slower than a linear solution does. Further, the difference of the decay rate is a polynomial order. This deceleration of the decay is due to an amplification effect by the nonlinearity. This nonlinear amplification phenomena was previously known for several specific systems, however the deceleration of the decay in these results was by a logarithmic order. As far as we know, the system studied in this paper is the first model in that the deceleration in a polynomial order is justified.

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arXiv2022

Global dynamics below excited solitons for the non-radial NLS with potential

We consider the global dynamics of solutions to the $3d$ cubic nonlinear Schrödinger equation in the presence of an external potential, in the setting in which the equation admits both ground state solitons and excited solitons at small mass. We prove that small mass solutions with energy below that of the excited solitons either scatter to the ground states or grow their $H^1$-norm in time. In particular, we give an extension of the result of Nakanishi [19] from the radial to the non-radial setting.

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arXiv2021

Asymptotic behavior in time of solution to system of cubic nonlinear Schr"odinger equations in one space dimension

In this paper, we consider the large time asymptotic behavior of solutions to systems of two cubic nonlinear Schr"odinger equations in one space dimension. It turns out that for a system there exists a small solution of which asymptotic profile is a sum of two parts oscillating in a different way. This kind of behavior seems new. Further, several examples of systems which admit solution with several types of behavior such as modified scattering, nonlinear amplification, and nonlinear dissipation, are given. We also extend our previous classification result of nonlinear cubic systems.

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arXiv2021

On asymptotic behavior of solutions to cubic nonlinear Klein-Gordon systems in one space dimension

In this paper, we consider the large time asymptotic behavior of solutions to systems of two cubic nonlinear Klein-Gordon equations in one space dimension. We classify the systems by studying the quotient set of a suitable subset of systems by the equivalence relation naturally induced by the linear transformation of the unknowns. It is revealed that the equivalence relation is well described by an identification with a matrix. In particular, we characterize some known systems in terms of the matrix and specify all systems equivalent to them. An explicit reduction procedure from a given system in the suitable subset to a model system, i.e., to a representative, is also established. The classification also draws our attention to some model systems which admit solutions with a new kind of asymptotic behavior. Especially, we find new systems which admit a solution of which decay rate is worse than that of a solution to the linear Klein-Gordon equation by logarithmic order.

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arXiv2020

Asymptotic stability of solitary waves for the $1d$ NLS with an attractive delta potential

We consider the one-dimensional nonlinear Schrödinger equation with an attractive delta potential and mass-supercritical nonlinearity. This equation admits a one-parameter family of solitary wave solutions in both the focusing and defocusing cases. We establish asymptotic stability for all solitary waves satisfying a suitable spectral condition, namely, that the linearized operator around the solitary wave has a two-dimensional generalized kernel and no other eigenvalues or resonances. In particular, we extend our previous result beyond the regime of small solitary waves and extend the results of Fukuizumi-Ohta-Ozawa and Kaminaga-Ohta from orbital to asymptotic stability for a suitable family of solitary waves.

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