Breather interactions and limit analysis in the second harmonic generation process via Riemann-Hilbert approach

TL;DR

This paper employs the Riemann-Hilbert approach to derive and analyze soliton and breather solutions in second harmonic generation, revealing detailed interaction dynamics and asymptotic behaviors in nonlinear optics.

Contribution

It introduces a Riemann-Hilbert framework for solving the SHG equation, characterizing breather interactions and computing exact position shifts, advancing theoretical understanding of nonlinear optical phenomena.

Findings

Derived multi-soliton and breather solutions for SHG.

Analyzed breather interactions and calculated position shifts.

Numerically simulated spatiotemporal evolution of solutions.

Abstract

The discovery of second harmonic generation (SHG) heralds the emergence of nonlinear optics. In this paper, we focus on the theoretical analysis of the SHG equation under phase-matching conditions. A rich family of soliton solutions are derived via the Riemann-Hilbert (RH) approach, and we characterize breather interactions corresponding to second harmonic solutions. The construction and solution of the RH problem are discussed firstly, including a detailed analysis of the discrete spectrum in the single-zero and double-zero cases. In such cases two-soliton solutions, breather solutions, two-breather solutions, and soliton-breather solutions are obtained. We numerically simulate and visually illustrate the spatiotemporal evolution of these solutions. Furthermore, through asymptotic analysis of the interaction dynamics, the exact position shift magnitudes resulting from breather-breather interaction within a nonzero background field are calculated. When the velocities are distinct, the interaction of two breathers divides the xt-plane into four asymptotic regions by the characteristic trajectories of breathers, and we show that the asymptotic behavior can be explicitly determined by the relative position between the region and the breathers.