Solitonic Fixed Point Attractors in the Complex Ginzburg–Landau Equation for Associative Memories

Alexey N Pyrkov,Tim Byrnes,Valentin V Cherny

doi:10.3390/sym12010024

Abstract

It was recently shown that the nonlinear Schrodinger equation with a simplified dissipative perturbation features a zero-velocity solitonic solution of non-zero amplitude which can be used in analogy to attractors of Hopfield’s associative memory. In this work, we consider a more complex dissipative perturbation adding the effect of two-photon absorption and the quintic gain/loss effects that yields the complex Ginzburg–Landau equation (CGLE). We construct a perturbation theory for the CGLE with a small dissipative perturbation, define the behavior of the solitonic solutions with parameters of the system and compare the solution with numerical simulations of the CGLE. We show, in a similar way to the nonlinear Schrodinger equation with a simplified dissipation term, a zero-velocity solitonic solution of non-zero amplitude appears as an attractor for the CGLE. In this case, the amplitude and velocity of the solitonic fixed point attractor does not depend on the quintic gain/loss effects. Furthermore, the effect of two-photon absorption leads to an increase in the strength of the solitonic fixed point attractor.

Highlights

Neuromorphic computing—the study of information processing using articficial systems mimicking neuro-biological architectures—has attracted a huge amount of interest in modern information science [1,2,3,4,5]
The nonlinear Schrodinger equation (NLSE), which can be realized in Bose–Einstein condensates (BECs), with a simplified dissipative perturbation which creates a frictional force acting on soliton [25,52,53,54,55,56] was considered in an application to associative memory and pattern recognition [57]
We show that to the simplified model, a zero-velocity solitonic solution of non-zero amplitude appears in the complex Ginzburg–Landau equation (CGLE), and we investigate the behavior of the solitonic solution on various parameter choices

Summary

Introduction

Neuromorphic computing—the study of information processing using articficial systems mimicking neuro-biological architectures—has attracted a huge amount of interest in modern information science [1,2,3,4,5]. The nonlinear Schrodinger equation (NLSE), which can be realized in BEC, with a simplified dissipative perturbation which creates a frictional force acting on soliton [25,52,53,54,55,56] was considered in an application to associative memory and pattern recognition [57]. The perturbation makes the zero-velocity solitonic solution of non-zero amplitude into an attractor for all evolution trajectories whose initial conditions are moving solitons This paves the way to store information in a large dimensional dynamical system using principles which are completely analogous to that of Hopfield’s associative memory. We show that to the simplified model, a zero-velocity solitonic solution of non-zero amplitude appears in the CGLE, and we investigate the behavior of the solitonic solution on various parameter choices

The Complex Ginzburg–Landau Equation

Solitonic Fixed Point Attractors

Conclusions