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Masaya Maeda

Publications and source records attributed to Masaya Maeda.

At least 37 records · Page 2Linked to original sources
arXiv2021

A note on small data soliton selection for nonlinear Schrödinger equations with potential

In this note, we give an alternative proof of the theorem on soliton selection for small energy solutions of nonlinear Schrödinger equations (NLS) which we studied in Anal. PDE 8 (2015), 1289-1349 and more recently in Annals of PDE (2021) 7:16. As in in the latter paper we use the notion of Refined Profile, with the difference that here we do not modify the modulation coordinates and we do not search for Darboux coordinates. This shortens considerably the proof.

math.AP↗
arXiv2021

Dispersive estimates for quantum walks on 1D lattice

We consider quantum walks with position dependent coin on 1D lattice $\mathbb{Z}$. The dispersive estimate $\|U^tP_c u_0\|_{l^\infty}\lesssim (1+|t|)^{-1/3} \|u_0\|_{l^1}$ is shown under $l^{1,1}$ perturbation for the generic case and $l^{1,2}$ perturbation for the exceptional case, where $U$ is the evolution operator of a quantum walk and $P_c$ is the projection to the continuous spectrum. This is an analogous result for Schrödinger operators and discrete Schrödinger operators. The proof is based on the estimate of oscillatory integrals expressed by Jost solutions.

math-ph↗
arXiv2020

Coordinates at small energy and refined profiles for the Nonlinear Schrödinger Equation

In this paper we give a new and simplified proof of the theorem on selection of standing waves for small energy solutions of the nonlinear Schrödinger equations (NLS) that we gave in \cite{CM15APDE}. We consider a NLS with a Schrödinger operator with several eigenvalues, with corresponding families of small standing waves, and we show that any small energy solution converges to the orbit of a time periodic solution plus a scattering term. The novel idea is to consider the "refined profile", a quasi--periodic function in time which almost solves the NLS and encodes the discrete modes of a solution. The refined profile, obtained by elementary means, gives us directly an optimal coordinate system, avoiding the normal form arguments in \cite{CM15APDE}, giving us also a better understanding of the Fermi Golden Rule.

math.AP↗
arXiv2019

Stabilization of small solutions of discrete NLS with potential having two eigenvalues

We study the long time behavior of small (in $l^2$) solutions of discrete nonlinear Schrödinger equations with potential. In particular, we are interested in the case that the corresponding discrete Schrödinger operator has exactly two eigenvalues. We show that under the nondegeneracy condition of Fermi Golden Rule, all small solutions decompose into a nonlinear bound state and dispersive wave. We further show the instability of excited states and generalized equipartition property.

math.AP↗
arXiv2019

Continuous limits of linear and nonlinear quantum walks

In this paper, we consider the continuous limit of a nonlinear quantum walk (NLQW) that incorporates a linear quantum walk as a special case. In particular, we rigorously prove that the walker (solution) of the NLQW on a lattice $δ\mathbb Z$ uniformly converges (in Sobolev space $H^s$) to the solution to a nonlinear Dirac equation (NLD) on a fixed time interval as $δ\to 0$. Here, to compare the walker defined on $δ\mathbb Z$ and the solution to the NLD defined on $\mathbb R$, we use Shannon interpolation.

math-ph↗
arXiv2019

On stability of small solitons of the 1--D NLS with a trapping delta potential

We consider a Nonlinear Schrödinger Equation with a very general non linear term and with a trapping $δ$ potential on the line. We then discuss the asymptotic behavior of all its small solutions, generalizing a recent result by Masaki et al. We give also a result of dispersion in the case of defocusing equations with a non--trapping delta potential.

math.AP↗
arXiv2019

Dynamics of solitons for nonlinear quantum walks

We present some numerical results for nonlinear quantum walks (NLQWs) studied by the authors analytically \cite{MSSSS18DCDS, MSSSS18QIP}. It was shown that if the nonlinearity is weak, then the long time behavior of NLQWs are approximated by linear quantum walks. In this paper, we observe the linear decay of NLQWs for range of nonlinearity wider than studied in \cite{MSSSS18DCDS}. In addition, we treat the strong nonlinear regime and show that the solitonic behavior of solutions appears. There are several kinds of soliton solutions and the dynamics becomes complicated. However, we see that there are some special cases so that we can calculate explicit form of solutions. In order to understand the nonlinear dynamics, we systematically study the collision between soliton solutions. We can find a relationship between our model and a nonlinear differential equation.

quant-ph↗
arXiv2018

Long time oscillation of solutions of nonlinear Schrödinger equations near minimal mass ground state

In this paper, we consider the long time dynamics of radially symmetric solutions of nonlinear Schrödinger equations (NLS) having a minimal mass ground state. In particular, we show that there exist solutions with initial data near the minimal mass ground state that oscillate for long time. More precisely, we introduce a coordinate defined near the minimal mass ground state which consists of finite and infinite dimensional part associated to the discrete and continuous part of the linearized operator. Then, we show that the finite dimensional part, two dimensional, approximately obeys Newton's equation of motion for a particle in an anharmonic potential well. Showing that the infinite dimensional part is well separated from the finite dimensional part, we will have long time oscillation.

math.AP↗
arXiv2018

On nonlinear profile decompositions and scattering for a NLS-ODE model

In this paper, we consider a Hamiltonian system combining a nonlinear Schr\" odinger equation (NLS) and an ordinary differential equation (ODE). This system is a simplified model of the NLS around soliton solutions. Following Nakanishi \cite{NakanishiJMSJ}, we show scattering of $L^2$ small $H^1$ radial solutions. The proof is based on Nakanishi's framework and Fermi Golden Rule estimates on $L^4$ in time norms.

math.AP↗
arXiv2018

Scattering and inverse scattering for nonlinear quantum walks

We study large time behavior of quantum walks (QWs) with self-dependent (nonlinear) coin. In particular, we show scattering and derive the reproducing formula for inverse scattering in the weak nonlinear regime. The proof is based on space-time estimate of (linear) QWs such as dispersive estimates and Strichartz estimate. Such argument is standard in the study of nonlinear Schrödinger equations and discrete nonlinear Schrödinger equations but it seems to be the first time to be applied to QW.

math-ph↗
arXiv2018

Weak limit theorem for a nonlinear quantum walk

This paper continues the study of large time behavior of a nonlinear quantum walk begun in arXiv:1801.03214. In this paper, we provide a weak limit theorem for the distribution of the nonlinear quantum walk. The proof is based on the scattering theory of the nonlinear quantum walk and the limit distribution is obtained in terms of its asymptotic state.

math-ph↗
arXiv2017

On nonlinear scattering for quantum walks

We study large time behavior of quantum walks (QW) with self-dependent coin. In particular, we show scattering and derive the reproducing formula for inverse scattering in the weak nonlinear regime. The proof is based on space-time estimate of (linear) QW such as Strichartz estimate. Such argument is standard in the study of nonlinear Schrödinger equations but it seems to be the first time to be applied to QW. We also numerically study the dynamics of QW and observe soliton like solutions.

math-ph↗
arXiv2016

A note on small periodic solutions of discrete nonlinear Klein-Gordon equations

In this note, we consider discrete nonlinear Klein-Gordon equations with potential. By the pioneering work of Sigal, it is known that for the "continuous" nonlinear Klein-Gordon equation, no small time periodic solution exists generically. However, for the discrete nonlinear Klein-Gordon equations, we show that there exist small time periodic solutions.

math.AP↗
arXiv2015

On small energy stabilization in the NLKG with a trapping potential

We consider a nonlinear Klein Gordon equation (NLKG) with short range potential with eigenvalues and show that in the contest of complex valued solutions the small standing waves are attractors for small solutions of the NLKG. This extends the results already known for the nonlinear Schrödinger equation and for the nonlinear Dirac equation. In addition, this extends a result of Bambusi and Cuccagna (which in turn was an extension of a result by Soffer and Weinstein) which considered only real valued solutions of the NLKG.

math.AP↗
arXiv2015

On small energy stabilization in the NLS with a trapping potential

We describe the asymptotic behavior of small energy solutions of an NLS with a trapping potential. In particular we generalize work of Soffer and Weinstein, and of Tsai et. al. The novelty is that we allow generic spectra associated to the potential. This is yet a new application of the idea to interpret the nonlinear Fermi Golden Rule as a consequence of the Hamiltonian structure.

math.AP↗