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Hiroaki Kikuchi

Publications and source records attributed to Hiroaki Kikuchi.

At least 19 recordsLinked to original sources
arXiv2026

Transverse Instability and Bifurcation Analysis of the line soliton for the NLS equation on a Fractured Strip

We consider the nonlinear Schrödinger equation on a two-dimensional strip with an attractive $δ$ interaction and power nonlinearity. We investigate the transverse stability and bifurcation of line solitons as the width of the strip varies. We first establish local well-posedness in $H^1$, conservation of mass and energy, and global existence in the $H^1$-subcritical regime. We then identify a critical width $L_*$ at which the line soliton undergoes a transverse instability. More precisely, we prove orbital stability for $L L_*$. At the critical width, a simple eigenvalue of the linearized operator crosses zero, and we construct, via the Lyapunov-Schmidt reduction, a branch of positive nontrivial stationary solutions bifurcating from the line soliton. We determine the direction of this bifurcation by computing the second-order variation of the width along the branch. Finally, we investigate the orbital stability of the bifurcating solitons and obtain a stability criterion which can be evaluated in the regime of sufficiently small interaction strength.

math.AP↗
arXiv2026

Benchmarking the Utility of Privacy-Preserving Cox Regression Under Data-Driven Clipping Bounds: A Multi-Dataset Simulation Study

Differential privacy (DP) is a mathematical framework that guarantees individual privacy; however, systematic evaluation of its impact on statistical utility in survival analyses remains limited. In this study, we systematically evaluated the impact of DP mechanisms (Laplace mechanism and Randomized Response) with data-driven clipping bounds on the Cox proportional hazards model, using 5 clinical datasets ($n = 168$--$6{,}524$), 15 levels of $\varepsilon$ (0.1--1000), and $B = 1{,}000$ Monte Carlo iterations. The data-driven clipping bounds used here are observed min/max and therefore do not provide formal $\varepsilon$-DP guarantees; the results represent an optimistic lower bound on utility degradation under formal DP. We compared three types of input perturbations (covariates only, all inputs, and the discrete-time model) with output perturbations (dfbeta-based sensitivity), using loss of significance rate (LSR), C-index, and coefficient bias as metrics. At standard DP levels ($\varepsilon \leq 1$), approximately 90% (90--94%) of the significant covariates lost significance, even in the largest dataset ($n = 6{,}524$), and the predictive performance approached random levels (test C-index $\approx 0.5$) under many conditions. Among the input perturbation approaches, perturbing only covariates preserved the risk-set structure and achieved the best recovery, whereas output perturbation (dfbeta-based sensitivity) maintained near-baseline performance at $\varepsilon \geq 5$. At $n \approx 3{,}000$, the significance recovered rapidly at $\varepsilon = 3$--10; however, in practice, $\varepsilon \geq 10$ (for predictive performance) to $\varepsilon \geq 30$--60 (for significance preservation) is required. In the moderate-to-high $\varepsilon$ range, false-positive rates increased for variables whose baseline $p$-values were near the significance threshold.

cs.CR↗
arXiv2025

Singular solutions and bifurcation diagram of semilinear elliptic equations with general nonlinearity in two dimensions

In this paper, we investigate semilinear elliptic equations with general exponential-type nonlinearities in two dimensions. For such nonlinearities, we establish two main results. The first is the construction of a singular solution. Recently, Fujishima, Ioku, Ruf, and Terraneo [10] proved the existence of singular solutions under certain assumptions for nonlinearities. We succeed in relaxing these conditions by providing the precise asymptotic form of a singular solution. Our second result concerns the bifurcation diagram of regular solutions. While the bifurcation structure has been extensively studied in three or higher dimensions, comparatively little was known in two dimensions until recently. In [18], the second author proved that the bifurcation curve possesses infinitely many turning points for supercritical analytic nonlinearities. In the present work, we refine this analysis by showing the bifurcation curve oscillates infinitely many times around some point, without assuming analyticity of the nonlinearities. The novelty of our approach lies in the introduction of a generalized Emden-type transformation.

math.AP↗
arXiv2025

Classification and qualitative properties of positive solutions to double-power nonlinear stationary Schrödinger equations

In this paper, we investigate positive radial solutions to double-power nonlinear stationary Schrodinger equations in three space dimensions. It is now known that the non-uniqueness of H^{1}-positive solutions can occur in three dimensions when the frequency is sufficiently small. Under suitable conditions, in addition to the ground state solution (whose L^{\infty} norm vanishes as the frequency tends to zero), there exists another positive solution that minimizes a different constrained variational problem, with an L^{\infty} norm diverging as the frequency tends to zero (see Theorem 1.4). We classify all positive solutions with small frequency into two categories: the ground state and the Aubin-Talenti type solution. As a consequence, we establish the multiplicity of positive solutions. Finally, we also examine the non-degeneracy and Morse index of each positive solution.

math.AP↗
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).

math.AP↗
arXiv2025

Data Poisoning Attacks to Locally Differentially Private Range Query Protocols

Local Differential Privacy (LDP) has been widely adopted to protect user privacy in decentralized data collection. However, recent studies have revealed that LDP protocols are vulnerable to data poisoning attacks, where malicious users manipulate their reported data to distort aggregated results. In this work, we present the first study on data poisoning attacks targeting LDP range query protocols, focusing on both tree-based and grid-based approaches. We identify three key challenges in executing such attacks, including crafting consistent and effective fake data, maintaining data consistency across levels or grids, and preventing server detection. To address the first two challenges, we propose novel attack methods that are provably optimal, including a tree-based attack and a grid-based attack, designed to manipulate range query results with high effectiveness. \textbf{Our key finding is that the common post-processing procedure, Norm-Sub, in LDP range query protocols can help the attacker massively amplify their attack effectiveness.} In addition, we study a potential countermeasure, but also propose an adaptive attack capable of evading this defense to address the third challenge. We evaluate our methods through theoretical analysis and extensive experiments on synthetic and real-world datasets. Our results show that the proposed attacks can significantly amplify estimations for arbitrary range queries by manipulating a small fraction of users, providing 5-10x more influence than a normal user to the estimation.

cs.CR↗
arXiv2022

Castell: Scalable Joint Probability Estimation of Multi-dimensional Data Randomized with Local Differential Privacy

Performing randomized response (RR) over multi-dimensional data is subject to the curse of dimensionality. As the number of attributes increases, the exponential growth in the number of attribute-value combinations greatly impacts the computational cost and the accuracy of the RR estimates. In this paper, we propose a new multi-dimensional RR scheme that randomizes all attributes independently, and then aggregates these randomization matrices into a single aggregated matrix. The multi-dimensional joint probability distributions are then estimated. The inverse matrix of the aggregated randomization matrix can be computed efficiently at a lightweight computation cost (i.e., linear with respect to dimensionality) and with manageable storage requirements. To overcome the limitation of accuracy, we propose two extensions to the baseline protocol, called {\em hybrid} and {\em truncated} schemes. Finally, we have conducted experiments using synthetic and major open-source datasets for various numbers of attributes, domain sizes, and numbers of respondents. The results using UCI Adult dataset give average distances between the estimated and the real (2 through 6-way) joint probability are $0.0099$ for {\em truncated} and $0.0155$ for {\em hybrid} schemes, whereas they are $0.03$ and $0.04$ for LoPub, which is the state-of-the-art multi-dimensional LDP scheme.

cs.CR↗
arXiv2022

Threshold solutions for the 3D focusing cubic-quintic nonlinear Schrodinger equation at low frequencies

This paper addresses the focusing cubic-quintic nonlinear Schrodinger equation in three space dimensions. Especially, we study the global dynamics of solutions whose energy and mass equal to those of the ground state in the sprits of Duyckaerts and Merle (2009). When we try to obtain the corresponding result, we meet several difficulties due to the cubic-quintic nonlinearity. We overcome them by using the one-pass theorem (no return theorem) developed by Nakanishi and Schlag (2012).

math.AP↗
arXiv2022

Designing a Location Trace Anonymization Contest

For a better understanding of anonymization methods for location traces, we have designed and held a location trace anonymization contest that deals with a long trace (400 events per user) and fine-grained locations (1024 regions). In our contest, each team anonymizes her original traces, and then the other teams perform privacy attacks against the anonymized traces. In other words, both defense and attack compete together, which is close to what happens in real life. Prior to our contest, we show that re-identification alone is insufficient as a privacy risk and that trace inference should be added as an additional risk. Specifically, we show an example of anonymization that is perfectly secure against re-identification and is not secure against trace inference. Based on this, our contest evaluates both the re-identification risk and trace inference risk and analyzes their relationship. Through our contest, we show several findings in a situation where both defense and attack compete together. In particular, we show that an anonymization method secure against trace inference is also secure against re-identification under the presence of appropriate pseudonymization. We also report defense and attack algorithms that won first place, and analyze the utility of anonymized traces submitted by teams in various applications such as POI recommendation and geo-data analysis.

cs.CR↗
arXiv2022

Pitchfork bifurcation at line solitons for nonlinear Schrödinger equations on the product space $\mathbb{R} \times \mathbb{T}$

In this paper, we study the bifurcation problem from a line soliton for a stationary nonlinear Schrödinger equation on the product space $\mathbb{R} \times \mathbb{T}$. We extend earlier results to a larger class of the nonlinearity in the equation. The salient point of our analysis relies on a lower bound of solution to the ``auxiliary equation'' and then on the application of the Crandall-Rabinowitz argument

math.AP↗
arXiv2021

Non-existence of ground states and gap of variational problems for combined power-type nonlinear scalar field equations involving the Sobolev critical exponent in three space dimensions

In this paper, we consider minimization problems related to the combined power-type nonlinear scalar field equations involving the Sobolev critical exponent in three space dimensions. In four and higher space dimensions, it is known that for any frequency and any power of the subcritical nonlinearity, there exists a ground state. In contrast to those cases, when the space dimension is three and the subcritical power is three or less, we can show that there exists a threshold frequency, above which no ground state exists, and below which the ground state exists. Furthermore, we prove the difference between two typical variational problems used to characterize the ground states.

math.AP↗
arXiv2021

Transverse stability of line soliton and characterization of ground state for wave guide Schrödinger equations

In this paper, we study the transverse stability of the line Schrödinger soliton under a full wave guide Schrödinger flow on a cylindrical domain $\mathbb R\times\mathbb T$. When the nonlinearity is of power type $|ψ|^{p-1}ψ$ with $p>1$, we show that there exists a critical frequency $ω_{p} >0$ such that the line standing wave is stable for $0<ω< ω_{p}$ and unstable for $ω> ω_{p}$. Furthermore, we characterize the ground state of the wave guide Schrödinger equation. More precisely, we prove that there exists $ω_{*} \in (0, ω_{p}]$ such that the ground states coincide with the line standing waves for $ω\in (0, ω_{*}]$ and are different from the line standing waves for $ω\in (ω_{*}, \infty)$.

math.AP↗
arXiv2019

Uniqueness and nondegeneracy of ground states to nonlinear scalar field equations involving the Sobolev critical exponent in their nonlinearities for high frequencies

The study of the uniqueness and nondegeneracy of ground state solutions to semilinear elliptic equations is of great importance because of the resulting energy landscape and its implications for the various dynamics. In [AIKN3], semilinear elliptic equations with combined power-type nonlinearities involving the Sobolev critical exponent are studied. There, it is shown that if the dimension is four or higher, and the frequency is sufficiently small, then the positive radial ground state is unique and nondegenerate. In this paper, we extend these results to the case of high frequencies when the dimension is five and higher. After suitably rescaling the equation, we demonstrate that the main behavior of the solutions is given by the Sobolev critical part for which the ground states are explicit, and their degeneracy is well characterized. Our result is a key step towards the study of the different dynamics of solutions of the corresponding nonlinear Schrödinger and Klein-Gordon equations with energies above the energy of the ground state. Our restriction on the dimension is mainly due to the existence of resonances in dimension three and four.

math.AP↗
arXiv2019

Non-uniqueness for an energy-critical heat equation on $\mathbb{R}^2$

We construct a singular solution of a stationary nonlinear Schrödinger equation on $\mathbb{R}^2$ with square-exponential nonlinearity having linear behavior around zero. In view of Trudinger-Moser inequality, this type of nonlinearity has an energy-critical growth. We use this singular solution to prove non-uniqueness of strong solutions for the Cauchy problem of the corresponding semilinear heat equation. The proof relies on explicit computation showing a regularizing effect of the heat equation in an appropriate functional space.

math.AP↗
arXiv2018

Linear instability and nondegeneracy of ground state for combined power-type nonlinear scalar field equations with the Sobolev critical exponent and large frequency parameter

We consider combined power-type nonlinear scalar field equations with the Sobolev critical exponent. In \cite{AIKN3}, it was shown that if the frequency parameter is sufficiently small, then the positive ground state is nondegenerate and linearly unstable, together with an application to a study of global dynamics for nonlinear Schrödinger equations. In this paper, we prove the nondegeneracy and linear instability of the ground state frequency for sufficiently large frequency parameters. Moreover, we show that the derivative of the mass of ground state with respect to the frequency is negative.

math.AP↗
arXiv2018

Remarks on solitary waves and Cauchy problem for a Half-wave-Schrödinger equations

In this paper, we study the solitary wave and the Cauchy problem for Half-wave-Schrödinger equations in the plane. First, we show the existence and orbital stability of the ground states. Secondly, we prove that traveling waves exist and converge to zero as the velocity tends to $1$. Finally, we solve the Cauchy problem for initial data in $L^{2}_{x}H^{s}_{y}(\mathbb{R}^{2})$, with $s>\frac{1}{2}$.

math.AP↗
arXiv2011

Existence of a ground state and blow-up problem for a nonlinear Schrodinger equation with critical growth

In this paper we show the existence of ground-state solutions for the energy-critical NLS perturbed with subcritical terms when the space dimension $d\geq4$. However in dimension three, we show that when the perturbation is small enough, then such solution does not exist. For the evolution equation, we show the existence of finite time blow up of solutions with radially symmetric data with energy below the one of the ground state.

math.AP↗