Dynamics of solitons for nonlinear quantum walks

Abstract

We present some numerical results for nonlinear quantum walks (NLQWs) studied by the authors analytically (Maeda et al 2018 Discrete Contin. Dyn. Syst. 38 3687-3703; Maeda et al 2018 Quantum Inf.Process. 17 215). 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 et al 2018 Discrete Contin. Dyn. Syst. 38 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.

A remark on uniform Strichartz estimate for quantum walks on 1D lattice

“…The purpose of this note is to combine the the study of QWs as dispersive equation [6,7,8] and the study of the continuous limit of QWs, see [9] and reference therein. In particular, we study the Strichartz estimate of QWs in the continuous limit setting and obtain "uniform" Strichartz estimates, where "uniform" means that the inequality is independent of lattice size δ > 0.…”

Asymptotic stability of small bound state of nonlinear quantum walks

“…The importance of solitons can also be seen by soliton resolution conjecture [58], which claims generic solutions decouple into scattering waves and solitons, as in RAGE theorem for the linear case [20,51]. Indeed, many papers studying nonlinear QWs numerically observe solitonic behavior of the solution and focus on the study of its dynamics [8,9,17,34,37,45,63]. For the stability analysis of bound states, related to the study of topological phases [3,4,10,11,28,29,40,60,61,62], Gerasimenko, Tarasinski, and Beenakker [22], followed by Mochizuki, Kawakami and Obuse [44] studied the linear stability of bound states bifurcating from linear bound states.…”

Continuous limits of linear and nonlinear quantum walks

“…From then, several models of NLQWs have been proposed motivated by simulating nonlinear Dirac equations (NLD) [23,33] and studying the nonlinear effect to the topological insulators [15]. See also [24,25,26] for the study of scattering phenomena, weak limit theorem and soliton-like behavior for NLQWs. We note that for continuous time QWs, which are substantially described by discrete Schrödinger equations, nonlinear models are also attracting interest because it can speed up the quantum search [29].…”