arXiv subjects
Publications and source records attributed to Masaya Maeda.
In this short note, we prove the non-existence of slow and fast small nontrivial compact solutions for the Euler-Poisson system in $1$D. The proof is based on the virial estimate which provides local in space average decay of bounded small solutions.
We apply the idea of using a combination of virial inequalities and Kato smoothing, previously applied to NLS and generalized KdV pure power equations to Euler-Poisson: we assume that a solution remains very close for all times to a soliton in an appropriate space and then we prove an asymptotic convergence to a soliton for $t\to +\infty$.
Assuming as hypotheses the results proved numerically by Chang et al. \cite{Chang} for the exponent $p\in (3,5)$, we prove that some of the ground states of the nonlinear Schrödinger equation (NLS) with pure power nonlinearity of exponent $p$ in the line are asymptotically stable for a certain set of values of the exponent $p$ where the FGR occurs by means of a discrete mode 3rd or 4th order power interaction with the continuous mode. For the 3rd the result is true for generic $p$ while for the 4th order case we assume that there are $p$'s satisfying Fermi Golden rule and the non-resonance condition of the threshold of the continuous spectrum of the linearization. The argument is similar to our recent result valid for $p$ near 3 contained in \cite{CM24D1}.
We apply our idea, which previously we used in the analysis of the pure power NLS, consisting in spitting the virial inequality method into a large energy inequality combined with Kato smoothing, to the case of generalized Korteweg--De Vries pure power equations. We assume that a solution remains for all positive times very close to a soliton and then we prove an asymptotic stability result for $t\to +\infty$.
For exponents $p$ satisfying $0<|p-3|\ll 1$ and only in the context of spatially even solutions we prove that the ground states of the nonlinear Schrödinger equation (NLS) with pure power nonlinearity of exponent $p$ in the line are asymptotically stable. The proof is similar to a related result of Martel, preprint arXiv:2312.11016, for a cubic quintic NLS. Here we modify the second part of Martel's argument, replacing the second virial inequality for a transformed problem with a smoothing estimate on the initial problem, appropriately tamed by multiplying the initial variables and equations by a cutoff.
We continue our series devoted, after references \cite{CM24D1} and \cite{CM243}, at proving the asymptotic stability of ground states of the pure power Nonlinear Schrödinger equation on the line. Here we assume some results on the spectrum of the linearization obtained computationally by Chang et al. \cite{Chang} and then we explore the equation for exponents $p\le 2$ sufficiently close to 2. The ensuing loss of regularity of the nonlinearity requires new arguments.
We prove in dimension $d=1$ a result similar to Soffer and Weinstein Jour. Diff. Eq. 98 (1992) capturing for pure power nonlinearities the whole range of exponents $p>1$. The proof is based on the virial inequality of Kowalczyk \textit{et al.} J. Eur. Math. Soc. (JEMS) 24 (2022) with smoothing estimates like in Mizumachi J. Math. Kyoto Univ. 48 (2008).
We extend the result M. Kowalczyk, Y. Martel, C. Muñoz, JEMS 2022, on the existence, in the context of spatially even solutions, of asymptotic stability on a center hypersurface at the soliton of the defocusing power Nonlinear Klein Gordon Equation with $p>3$, to the case $2\ge p> \frac{5}{3}$.
We provide a detailed proof that the Nonlinear Fermi Golden Rule coefficient that appears in our recent proof of the asymptotic stability of ground states for the pure power Nonlinear Schrödinger equations in $\mathbb{R}$ with exponent $0<|p-3|\ll 1$ is nonzero.
Combining virial inequalities by Kowalczyk, Martel and Munoz and Kowalczyk, Martel, Munoz and Van Den Bosch with our theory on how to derive nonlinear induced dissipation on discrete modes, and in particular the notion of Refined Profile, we show how to extend the theory by Kowalczyk, Martel, Munoz and Van Den Bosch to the case when there is a large number of discrete modes in the cubic NLS with a trapping potential which is associate to a repulsive potential by a series of Darboux transformations. This a simpler model than the kink stability for wave equations, but is still a classical one and retains some of the main difficulties.
We give a partial extension to dimension 1 of the result proved by Bambusi and Cuccagna on the absence of small energy real valued periodic solutions for the NLKG in dimension 3. We combine the framework in Kowalczyk and Martel with the notion of "refined profile".
We study the dynamics of solutions of nonlinear Schrödinger equation near unstable ground states. The existence of the local center stable manifold around ground states and the asymptotic stability for the solutions on the manifold is proved. The novelty of our result is that we allow the existence of internal modes.
We give a sufficient condition, in the spirit of Kowalczyk-Martel-Munoz-Van Den Bosch \cite{KMMvdB21AnnPDE}, for the local asymptotic stability of kinks under odd perturbations. In particular, we allow the existence of quite general configuration of internal modes. The extension of our result to moving kinks remains an open problem.
In this paper we give a simple and short proof of asymptotic stability of soliton for discrete nonlinear Schrödinger equation near anti-continuous limit. Our novel insight is that the analysis of linearized operator, usually non-symmetric, can be reduced to a study of simple self-adjoint operator almost like the free discrete Laplacian restricted on odd functions.
In this short note, we study quantum walks (QWs) on one dimensional lattice $δ\mathbb{Z}$. Following Hong-Yang, we prove Strichartz estimates for QWs independent of the lattice width $δ$.
In this paper, we study the long time behavior of nonlinear quantum walks when the initial data is small in $l^2$. In particular, we study the case where the linear part of the quantum walk evolution operator has exactly two eigenvalues and show that the solution decomposed into nonlinear bound states bifurcating from the eigenvalues and scattering waves.
In this paper, we give an alternative proof for the asymptotic stability of solitons for nonlinear Schrödinger equations with internal modes. The novel idea is to use "refined profiles" developed by the authors for the analysis of small bound states. By this new strategy, we able to avoid the normal forms. Further, we can track the functions appearing in the Fermi Golden Rule hypothesis.
This paper is a continuation of the paper \cite{W} by the third author, which studied quantum walks with special long-range perturbations of the coin operator. In this paper, we consider general long-range perturbations of the coin operator and prove the non-existence of a singular continuous spectrum and embedded eigenvalues. The proof relies on the construction of generalized eigenfunctions (Jost solutions) which was studied in the short-range case in \cite{MSSSSdis}.