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Noriyoshi Fukaya

Publications and source records attributed to Noriyoshi Fukaya.

13 recordsLinked to original sources
arXiv2026

Standing waves for defocusing nonlinear Schrödinger equations with point interaction

We consider standing waves of the nonlinear Schrödinger equation $i\partial_t u = -Δ_αu + |u|^{p-1}u$ in the defocusing case in dimensions $N=2$ and $N=3$. Here, $-Δ_α$ denotes the Laplacian with a point interaction. This operator is bounded from below by a negative constant; consequently, unlike in the free case, the associated energy functional admits non-trivial minimizers. We establish existence and uniqueness of standing waves, and prove further qualitative properties, including radial symmetry, positivity, and stability. Moreover, we build an appropriate functional space for the zero-mass case and establish sharp decay estimates in this case.

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arXiv2025

Stability of standing waves for all frequencies to nonlinear Schrödinger equations with potentials in one dimension

In this paper, we study the orbital stability of standing waves for one-dimensional nonlinear Schrödinger equations with potentials. We show that the standing waves are orbitally stable for all frequencies in the $L^{2}$- subcritical and critical cases. Since the presence of potentials breaks the scale invariance of the equations, it is a delicate problem to apply the abstract theory of Grillakis, Shatah, and Strauss (1987) directly without a perturbative argument. For this reason, little is known about the orbital stability of standing waves for \textit{all} frequencies in the non-scale-invariant setting. We overcome this difficulty by employing the approach of Noris, Tavares, and Verzini (2014).

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arXiv2025

Instability of stationary solutions for double power nonlinear Schrödinger equations in one dimension

We consider a double power nonlinear Schrödinger equation which possesses the algebraically decaying stationary solution $ϕ_0$ as well as exponentially decaying standing waves $e^{iωt}ϕ_ω(x)$ with $ω>0$. It is well-known from the general theory that stability properties of standing waves are determined by the derivative of $ω\mapsto M(ω):=\frac{1}{2}\|ϕ_ω\|_{L^2}^2$; namely $e^{iωt}ϕ_ω$ with $ω>0$ is stable if $M'(ω)>0$ and unstable if $M'(ω)<0$. However, the stability/instability of stationary solutions is outside the general theory from the viewpoint of spectral properties of linearized operators. In this paper we prove the instability of the stationary solution $ϕ_0$ in one dimension under the condition $M'(0):=\lim_{ω\downarrow 0}M'(ω)\in[-\infty, 0)$. The key in the proof is the construction of the one-sided derivative of $ω\mapstoϕ_ω$ at $ω=0$, which is effectively used to construct the unstable direction of $ϕ_0$.

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arXiv2024

Uniqueness and nondegeneracy of ground states for 2d-nonlinear scalar field equations with point interaction

We study uniqueness and nondegeneracy of ground states for nonlinear scalar field equations in two dimensions with a point interaction at the origin. It is known that the all ground states are radial, positive, and decreasing functions. In this paper we prove the uniqueness of positive radial solutions by a method of Pohožaev identities. As a corollary, we obtain the uniqueness of ground states. Moreover, by a variational and ODE technique, we show that the ground state is a nondegenerate critical point of the action in the energy space.

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arXiv2022

Traveling waves for a nonlinear Schrödinger system with quadratic interaction

We study traveling wave solutions for a nonlinear Schrödinger 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.

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arXiv2021

On stability and instability of standing waves for 2d-nonlinear Schrödinger equations with point interaction

We study existence and stability properties of ground-state standing waves for two-dimensional nonlinear Schrödinger equation with a point interaction and a focusing power nonlinearity. The Schrödinger operator with a point interaction describes a one-parameter family of self-adjoint realizations of the Laplacian with delta-like perturbation. The perturbed Laplace operator always has a unique simple negative eigenvalue. We prove that if the frequency of the standing wave is close to the negative eigenvalue, it is stable. Moreover, if the frequency is sufficiently large, we have the stability in the $L^2$-subcritical or critical case, while the instability in the $L^2$-supercritical case.

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arXiv2021

Instability of degenerate solitons for nonlinear Schrödinger equations with derivative

We consider the following nonlinear Schrödinger equation with derivative: \begin{equation} iu_t =-u_{xx} -i |u|^{2}u_x -b|u|^4u , \quad (t,x) \in \mathbb{R}\times\mathbb{R}, \ b \in\mathbb{R}. \end{equation} If $b=0$, this equation is a gauge equivalent form of the well-known derivative nonlinear Schrödinger (DNLS) equation. The soliton profile of DNLS satisfies a certain double power elliptic equation with cubic-quintic nonlinearities. The quintic nonlinearity in our equation only affects the coefficient in front of the quintic term in the elliptic equation, so in this sense the additional nonlinearity is natural as a perturbation preserving soliton profiles of DNLS. When $b\ge 0$, the equation has degenerate solitons whose momentum and energy are zero, and if $b=0$, they are algebraic solitons. Inspired from the works on instability theory of the $L^2$-critical generalized KdV equation, we study the instability of degenerate solitons in a qualitative way, and when $b>0$, we obtain a large set of initial data yielding the instability. The arguments except one step in our proof work for the case $b=0$ in exactly the same way, which is a small step towards understanding the dynamics around algebraic solitons of the DNLS equation.

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arXiv2020

Uniqueness and nondegeneracy of ground states for nonlinear Schrödinger equations with attractive inverse-power potential

We study uniqueness and nondegeneracy of ground states for stationary nonlinear Schrödinger equations with a focusing power-type nonlinearity and an attractive inverse-power potential. We refine the results of Shioji and Watanabe (2016) and apply it to prove the uniqueness and nondegeneracy of ground states for our equations. We also discuss the orbital instability of ground state-standing waves.

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arXiv2020

Instability of algebraic standing waves for nonlinear Schrödinger equations with double power nonlinearities

We consider a nonlinear Schrödinger equation with double power nonlinearity \begin{align*} i\partial_t u+Δu-|u|^{p-1}u+|u|^{q-1}u=0,\quad (t,x)\in\mathbb{R}\times\mathbb{R}^N, \end{align*} where $1<p<q<1+4/(N-2)_+$. Due to the defocusing effect from the lower power order nonlinearity, the equation has algebraically decaying standing waves with zero frequency, which we call algebraic standing waves, as well as usual standing waves decaying exponentially with positive frequency. In this paper we study stability properties of two types of standing waves. We prove strong instability for all frequencies when $q\ge 1+4/N$ and instability for small frequencies when $q<1+4/N$, which especially give the first results on stability properties of algebraic standing waves. The instability result for small positive frequency when $q<1+4/N$ not only improves previous results in one-dimensional case but also gives a first result on instability in higher-dimensional case. The key point in our approach is to take advantage of algebraic standing waves.

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arXiv2018

Strong instability of standing waves with negative energy for double power nonlinear Schrödinger equations

We study the strong instability of ground-state standing waves $e^{iωt}ϕ_ω(x)$ for $N$-dimensional nonlinear Schrödinger equations with double power nonlinearity. One is $L^2$-subcritical, and the other is $L^2$-supercritical. The strong instability of standing waves with positive energy was proven by Ohta and Yamaguchi (2015). In this paper, we improve the previous result, that is, we prove that if $\partial_λ^2S_ω(ϕ_ω^λ)|_{λ=1}\le0$, the standing wave is strongly unstable, where $S_ω$ is the action, and $ϕ_ω^λ(x)\mathrel{\mathop:}=λ^{N/2}ϕ_ω(λx)$ is the $L^2$-invariant scaling.

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arXiv2018

Strong instability of standing waves for nonlinear Schrödinger equations with attractive inverse power potential

We study the strong instability of standing waves $e^{iωt}ϕ_ω(x)$ for nonlinear Schrödinger equations with an $L^2$-supercritical nonlinearity and an attractive inverse power potential, where $ω\in\mathbb{R}$ is a frequency, and $ϕ_ω\in H^1(\mathbb{R}^N)$ is a ground state of the corresponding stationary equation. Recently, for nonlinear Schrödinger equations with a harmonic potential, Ohta (2018) proved that if $\partial_λ^2S_ω(ϕ_ω^λ)|_{λ=1}\le0$, then the standing wave is strongly unstable, where $S_ω$ is the action, and $ϕ_ω^λ(x)\mathrel{\mathop:}=λ^{N/2}ϕ_ω(λx)$ is the scaling, which does not change the $L^2$-norm. In this paper, we prove the strong instability under the same assumption as the above-mentioned in inverse power potential case. Our proof is applicable to nonlinear Schrödinger equations with other potentials such as an attractive Dirac delta potential.

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arXiv2017

A sufficient condition for global existence of solutions to a generalized derivative nonlinear Schrödinger equation

We give a sufficient condition for global existence of the solutions to a generalized derivative nonlinear Schrödinger equation (gDNLS) by a variational argument. The variational argument is applicable to a cubic derivative nonlinear Schrödinger equation (DNLS). For (DNLS), Wu proved that the solution with the initial data $u_0$ is global if $\left\Vert u_0 \right\Vert_{L^2}^2<4π$ by the sharp Gagliardo--Nirenberg inequality in the paper "Global well-posedness on the derivative nonlinear Schrödinger equation", Analysis & PDE 8 (2015), no. 5, 1101--1112. The variational argument gives us another proof of the global existence for (DNLS). Moreover, by the variational argument, we can show that the solution to (DNLS) is global if the initial data $u_0$ satisfies that $\left\Vert u_0 \right\Vert_{L^2}^2=4π$ and the momentum $P(u_0)$ is negative.

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