前田 昌也

This paper is a continuation of work 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 gauge transformation and construction of generalized eigenfunctions (Jost solutions) which was studied in the short-range case.

<title>Abstract</title> In this paper, we consider a Hamiltonian system combining a nonlinear Schrödinger equation (NLS) and an ordinary differential equation. This system is a simplified model of the NLS around soliton solutions. Following Nakanishi [33], 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.

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 [Formula: see text] uniformly converges (in Sobolev space [Formula: see text]) to the solution to a nonlinear Dirac equation (NLD) on a fixed time interval as [Formula: see text]. Here, to compare the walker defined on [Formula: see text] and the solution to the NLD defined on [Formula: see text], we use Shannon interpolation.

<title>Abstract</title> We present some numerical results for nonlinear quantum walks (NLQWs) studied by the authors analytically (Maeda <italic>et al</italic> 2018 <italic>Discrete Contin. Dyn. Syst.</italic> <bold>38</bold> 3687–3703; Maeda <italic>et al</italic> 2018 <italic>Quantum Inf. Process.</italic> <bold>17</bold> 215). 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 (Maeda <italic>et al</italic> 2018 <italic>Discrete Contin. Dyn. Syst.</italic> <bold>38</bold> 3687–3703). 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.

We prove the existence of a two-parameter family of small quasi-periodic solutions of the discrete nonlinear Schrodinger equation (DNLS). We further show that all small solutions of DNLS decouple to one of these quasi-periodic solutions and dispersive wave. As a byproduct, we show that all small nonlinear bound states including excited states are stable.

We explain how spectrally stable vortices of the nonlinear Schrodinger equation in the plane can be orbitally unstable. This relates to the nonlinear Fermi golden rule, a mechanism which exploits the nonlinear interaction between discrete and continuous modes of the NLS.

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 Schrodinger 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. (C) 2016 Elsevier Ltd. All rights reserved.

We continue our study initiated in [4] of the interaction of a ground state with a potential considering here a class of trapping potentials. We track the precise asymptotic behavior of the solution if the interaction is weak, either because the ground state moves away from the potential or is very fast.

We consider nonlinear Schrodinger equation with nonlocal nonlinearity which is described by a growing interaction potential. This model contains low-dimensional Schrodinger Poisson system. We briefly survey recent progress on this subject and then show existence of ground state in a specific model.

We describe the asymptotic behavior of small energy solutions of an NLS with a trapping potential, generalizing work of Soffer and Weinstein, and of Tsai and Yau. The novelty is that we allow generic spectra associated to the potential. This is a new application of the idea of interpreting the nonlinear Fermi golden rule as a consequence of the Hamiltonian structure.

We consider the following semilinear elliptic equation: {-epsilon(2)Delta u + u - u(p) = 0, u &gt; 0 in ohm(epsilon), partial derivative/u partial derivative nu = 0 on partial derivative ohm(epsilon). Here, epsilon &gt; 0 and p &gt; 1. Omega(epsilon) is a domain in R-2 with smooth boundary partial derivative Omega(epsilon), and nu denotes the outer unit normal to partial derivative Omega(epsilon). The domain Omega(epsilon) depends on epsilon, which shrinks to a straight line in the plane as epsilon -&gt; 0. In this case, a least-energy solution exists for each epsilon sufficiently small, and it concentrates on a line. Moreover, the concentration line converges to the narrowest place of the domain as epsilon -&gt; 0. (C) 2013 Elsevier Inc. All rights reserved.We consider the following semilinear elliptic equation:

We show that ground states of the NLS moving at nonzero speed are asymptotically stable if they either stay far from the potential, or the potential is small, or the ground state has large speed. We search an effective Hamiltonian using the Birkhoff normal forms argument in [11], treating the potential as a perturbation. The so-called Fermi Golden Rule, which is used to describe the decay to 0 of the internal discrete modes of the ground state, is similar to that in [12]. The continuous modes dispersion requires the theory in [40] on charge transfer models. (C) 2013 Elsevier Inc. All rights reserved.

We consider the Zakharov system in two space dimension with periodic boundary condition: {i partial derivative(t)u = -Delta u+nu, partial derivative(tt)n = Delta n+Delta vertical bar u vertical bar(2), (t, x) is an element of [0, T) x T-2. (Z) We prove the existence of finite time blow-up solutions of (Z). Further, we show there exists no minimal mass blow-up solution. (C) 2012 Elsevier Masson SAS. All rights reserved.

In this article, we consider the nonlinear Schrodinger equation with nonlocal nonlinearity, which is a generalized model of the Schrodinger-Poisson system (Schrodinger-Newton equations) in low dimensions. We prove global well-posedness in a wider space than in previous results and show the stability of standing waves including excited states. It turns out that an example of stable excited states with high Morse index is contained. Several examples of traveling-wave-type solutions are also given.

We consider a Hamiltonian systems which is invariant under a one-parameter unitary group and give a criterion for the stability and instability of bound states for the degenerate case. We apply our theorem to the single power nonlinear Klein-Gordon equation and the double power nonlinear Schrodinger equation. (c) 2012 Elsevier Inc. All rights reserved.

In this paper, we study the fourth order nonlinear Schrodinger type equation (4NLS) which is a generalization of the Fukumoto-Moffatt [5] model that arising in the context of the motion of a vortex filament. Firstly, we mention the existence of standing wave solution and the conserved quantities. We next investigate the case that the equation is completely integrable and show that the standing wave obtained in [20] is orbitally stable in Solbolev spaces H-m with m is an element of N. Further, we show that the completely integrable equation is ill-posed in H-s with s is an element of (-1/2, 1/2) by following Kenig-Ponce-Vega [13].

We investigate the minimizers of the energy functional epsilon(u) = 1/2 integral(RN) vertical bar del u vertical bar(2) dx + 1/2 integral(RN) V vertical bar u vertical bar(2) dx - 1/p + 1 integral(RN) b vertical bar u vertical bar(p+1) dx under the constraint of the L(2)-norm. We show that for the case L(2)-norm is small, the minimizer is unique and for the case L(2)-norm is large, the minimizer concentrate at the maximum point of b and decays exponentially. By this result, we can show that if V and b are radially symmetric but b does not attain its maximum at the origin, then the symmetry breaking occurs as the L(2)-norm increases. Further, we show that for the case b has several maximum points, the minimizer concentrates at a point which minimizes a function which is defined by b, V and the unique positive radial solution of -Delta phi + phi - phi(p) = 0. For the case when V and b are radially symmetric, we show that if the minimizer concentrates at the origin, then the minimizer is radially symmetric. Further, we construct an energy functional such that the minimizer breaks its symmetry once but after that it recovers to be symmetric as the L(2)-norm increases.

We prove that every bound state of the nonlinear Schrodinger equation (NLS) with Morse index equal to two, with d(2)/d omega(2) (E(phi(omega)) + omega Q(phi(omega))) &gt; 0, is orbitally unstable. We apply this result to two particular cases. One is the NLS equation with potential and the other is a system of three coupled NUS equations. In both the cases the linear instability is well known but the orbital instability results are new when the spatial dimension is high. (C) 2009 Elsevier Ltd. All rights reserved.