On Asymptotic Dynamical Regimes of Manakov $N$-soliton Trains in Adiabatic Approximation

TL;DR

This paper investigates the long-term behavior of N-soliton trains in the Manakov model using the complex Toda chain, identifying conditions for bound and free regimes and validating predictions with numerical simulations.

Contribution

It introduces a novel analysis of asymptotic regimes of Manakov N-solitons via the complex Toda chain, including explicit symmetric configurations and regime classification.

Findings

Explicit conditions for bound state and free regimes.

Excellent agreement between Toda chain predictions and numerical simulations.

Identification of symmetric configurations ensuring specific asymptotic behaviors.

Abstract

We analyze the dynamical behavior of the $N$-soliton train in the adiabatic approximation of the Manakov model. %perturbed by gain/loss effects and also by several types of external potentials. The evolution of Manakov $N$-soliton trains is described by the complex Toda chain (CTC) which is a completely integrable dynamical model. Calculating the eigenvalues of its Lax matrix allows us to determine the asymptotic velocity of each soliton. So we describe sets of soliton parameters that ensure one of the two main types of asymptotic regimes: the bound state regime (BSR) and the free asymptotic regime (FAR). In particular we find explicit description of special symmetric configurations of $N$ solitons that ensure BSR and FAR. We find excellent matches between the trajectories of the solitons predicted by CTC with the ones calculated numerically from the Manakov system for wide classes of soliton parameters. This confirms the validity of our model.