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Takahisa Inui

Publications and source records attributed to Takahisa Inui.

At least 19 recordsLinked to original sources
arXiv2026

Asymptotic behavior for the damped Schr\"{o}dinger equation with nonlinear dissipation

We consider large time asymptotics of solutions to the damped Schr\"{o}dinger equation with the nonlinear dissipation in the mass-subcritical case. We prove that the optimal $L^2$-decay rate of the nonlinear solution coincides with that of the corresponding linear solution even in the presence of nonlinear dissipation. Moreover, we give the optimal convergence rate for scattering for any power in the mass-subcritical regime.

math.AP↗
arXiv2026

Existence of small semi-vortex solutions for the cubic nonlinear Schr\"{o}dinger system with Rashba type Spin-Orbit coupling on $\mathbb{R}^2$

We consider the cubic nonlinear Schr\"{o}dinger system with Rashba type Spin-Orbit Coupling (SOC) on $\mathbb{R}^2$. The system describes SO-coupled spinor BEC in physics. In the literature of physics, the small semi-vortex solutions, small ground state, and the so-called mixed mode, which are mixture of semi-vortex solutions, are investigated. The semi-vortex solutions cause from the resonance on the essential spectrum of the linear operator. In the present paper, we give mathematical proofs of the existence of the semi-vortex and the ground state by finding minimizers of the energy under small mass constraint based on concentration compactness argument. Moreover, we also discuss the mixed modes in the case where all the coefficients of the nonlinear terms are equal.

math.AP↗
arXiv2025

Existence of solutions to the semilinear damped wave equation with non-$L^2$ slowly decaying data : polynomial nonlinearity case

We study the Cauchy problem of the semilinear damped wave equation with polynomial nonlinearity, and establish the local and global existence of the solution for slowly decaying initial data not belonging to $L^2(\mathbb{R}^n)$ in general. Our approach is based on the $L^p$-$L^q$ estimates of linear solutions and the fractional Leibniz rule in suitable homogeneous Besov spaces.

math.AP↗
arXiv2023

Scattering and Blow-up for threshold even solutions to the nonlinear Schrödinger equation with repulsive delta potential at low frequencies

We consider the $L^2$-supercritical nonlinear Schrödinger equation with a repulsive Dirac delta potential in one dimensional space. In a previous work, we clarified the global dynamics of even solutions with the same action as the high-frequency ground state standing wave solutions. In that case, there are obvious non-scattering global solutions, i.e., the standing waves. In this paper, we show a scattering and blow-up dichotomy for threshold even solutions in the low-frequency case. We emphasize that this dichotomy still holds at the critical frequency between high and low.

math.AP↗
arXiv2023

Multi-solitons for the nonlinear Schrödinger equation with repulsive Dirac delta potential

We prove the existence of multi-soliton solutions for the nonlinear Schrödinger equation with repulsive Dirac delta potential and $L^2$-supercritical focusing nonlinear term. Our main contribution is to treat the unmoving part of the multi-solitons, which is the ground state of the equation. The linearized operator around it has two unstable eigenvalues. This is the main difference from NLS without potential, whose existence of multi-solitons is investigated by Côte, Martel, and Merle (2011).

math.AP↗
arXiv2023

Two-solitons with logarithmic separation for 1D NLS with repulsive delta potential

We consider the one-dimensional nonlinear Schrödinger equation with focusing, power nonlinearity, and a repulsive delta potential. We show that if the potential is not too strong, the construction by Nguy\~{ê}n (2019) of solutions converging strongly at time infinity to a pair of logarithmically separating solitons can be adapted to accommodate the effect of the potential. On the other hand, we show that if the potential is stronger, no such solutions exist.

math.AP↗
arXiv2022

Global dynamics below a threshold for the nonlinear Schrödinger equations with the Kirchhoff boundary and the repulsive Dirac delta boundary on a star graph

We consider the nonlinear Schrödinger equations on the star graph with the Kirchhoff boundary and the repulsive Dirac delta boundary at the origin. In the present paper, we show the scattering-blowup dichotomy result below the mass-energy of the ground state on the real line. The proof of the scattering part is based on a concentration compactness and rigidity argument. Our main contribution is to give a linear profile decomposition on the star graph by using a symmetrical decomposition.

math.AP↗
arXiv2022

Traveling waves for a nonlinear Schr\"odinger system with quadratic interaction

We study traveling wave solutions for a nonlinear Schr\"odinger system with quadratic interaction. For the non mass resonance case, the system has no Galilean symmetry, which is of particular interest in this paper. We construct traveling wave solutions by variational methods and see that for the non mass resonance case there exist specific traveling wave solutions which correspond to the solutions for ``zero mass" case in nonlinear elliptic equations. We also establish the new global existence result for oscillating data as an application. Both of our results essentially come from the lack of Galilean invariance in the system.

math.AP↗
arXiv2022

Blow-up or Grow-up for the threshold solutions to the nonlinear Schrödinger equation

We consider the nonlinear Schrödinger equation with $L^{2}$-supercritical and $H^{1}$-subcritical power type nonlinearity. Duyckaerts and Roudenko and Campos, Farah, and Roudenko studied the global dynamics of the solutions with same mass and energy as that of the ground state. In these papers, finite variance is assumed to show the finite time blow-up. In the present paper, we remove the finite-variance assumption and prove a blow-up or grow-up result.

math.AP↗
arXiv2022

Non relativistic and ultra relativistic limits in 2d stochastic nonlinear damped Klein-Gordon equation

We study the non relativistic and ultra relativistic limits in the two-dimensional nonlinear damped Klein-Gordon equation driven by a space-time white noise on the torus. In order to take the limits, it is crucial to clarify the parameter dependence in the estimates of solution. In this paper we present two methods to confirm this parameter dependence. One is the classical, simple energy method. Another is the method via Strichartz estimates.

math.AP↗
arXiv2022

Threshold odd solutions to the nonlinear Schrödinger equation in one dimension

We consider odd solutions to the Schrödinger equation with the $L^2$-supercritical power type nonlinearity in one dimensional Euclidean space. It is known that the odd solution scatters or blows up if its action is less than twice as that of the ground state. In the present paper, we show that the odd solutions with the action as twice as that of the ground state scatter or blow up.

math.AP↗
arXiv2021

Threshold scattering for the focusing NLS with a repulsive Dirac delta potential

We establish the scattering of solutions to the focusing mass supercritical nonlinear Schrödinger equation with a repulsive Dirac delta potential \[ i\partial_{t}u+\partial^{2}_{x}u+γδ(x)u+|u|^{p-1}u=0, \quad (t,x)\in {\mathbb R}\times{\mathbb R}, \] at the mass-energy threshold, namely, when $E_γ(u_{0})[M(u_{0})]^σ=E_{0}(Q)[M(Q)]^σ$ where $u_{0}\in H^{1}({\mathbb R})$ is the initial data, $Q$ is the ground state of the free NLS on the real line ${\mathbb R}$, $E_γ$ is the energy, $M$ is the mass and $σ=(p+3)/(p-5)$. We also prove failure of the uniform space-time bounds at the mass-energy threshold.

math.AP↗
arXiv2021

Non-delay limit in the energy space from the nonlinear damped wave equation to the nonlinear heat equation

We consider a singular limit problem from the damped wave equation with a power type nonlinearity to the corresponding heat equation. We call our singular limit problem non-delay limit. Our proofs are based on the argument for non-relativistic limit from the nonlinear Klein-Gordon equation to the nonlinear Schrödinger equation by the second author, Nakanishi, and Ozawa (2002), Nakanishi (2002), and Masmoudi and Nakanishi (2002). We can obtain better results for the non-delay limit problem than that for the non-relativistic limit problem due to the dissipation property. More precisely, we get the better convergence rate of the $L^2$-norm and we also obtain the global-in-time uniform convergence of the non-delay limit in the $L^2$-supercritical case.

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

Remarks on asymptotic order for the linear wave equation with the scale-invariant damping and mass with $L^r$-data

In the present paper, we consider the linear wave equation with the scale-invariant damping and mass. It is known that the global behavior of the solution depends on the size of the coefficients in front of the damping and mass at initial time $t=0$. Indeed, the solution satisfies the similar decay estimate to that of the corresponding heat equation if it is large and to that of the modified wave equation if it is small. In our previous paper, we obtain the scattering result and its asymptotic order for the data in the energy space $H^1\times L^2$ when the coefficients are in the wave regime. In fact, the threshold of the coefficients relies on the spatial decay of the initial data. Namely, it varies depending on $r$ when the initial data is in $L^r$ ($1\leq r < 2$). In the present paper, we will show the scattering result and the asymptotic order in the wave regime for $L^r$-data, which is wider than the wave regime for the data in the energy space. Moreover, we give an improvement of the asymptotic order obtained in our previous paper for the data in the energy space.

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