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

TL;DR

This work addresses the second harmonic generation (SHG) equation under phase-matching with nonzero boundary conditions by formulating and solving a Riemann-Hilbert problem. Leveraging a Lax pair and a uniformization variable $z$, it constructs and analyzes direct scattering, then solves RH problems with single and double poles to reconstruct the potentials $q_1$ and $q_2$, yielding explicit soliton and breather solutions. It provides one- and two-breather interactions, including determinant representations in the reflectionless case, and conducts thorough asymptotic analysis to obtain exact breather displacement formulas on a nonzero background, revealing velocity-dependent region decompositions in the $(x,t)$-plane. The results advance analytic control of breather dynamics in SHG and demonstrate the utility of RH/IST methods for nonlinear optics models, with future work aimed at rigorous long-time asymptotics via the Deift-Zhou method and matrix-double-pole scenarios.

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.

Figures (7)

• Figure 1: Two-soliton solutions with $\tilde{q}_{2,-}=1,\ \zeta_1=2i+\frac{1}{2}$. (a) The three-dimension plot of $q_1$, (b) the three-dimension plot of $q_2$, (c) the density map of $q_1$, (d) the density map of $q_2$.
• Figure 2: Breather solutions with $\tilde{q}_{2,-}=1,\ \zeta_{1}=2\sqrt{2}i$. (a) The three-dimension plot of $q_1$, (b) the three-dimension plot of $q_2$, (c) the density map of $q_1$, (d) the density map of $q_2$.
• Figure 3: Two-breather solutions with $\tilde{q}_{2,-}=1,\ \zeta_1=\frac{1}{4}-\frac{\sqrt{3}}{4}i,\zeta_2=-\frac{1}{4}+\frac{\sqrt{3}}{4}i$. (a) The three-dimension plot of $q_1$, (b) the three-dimension plot of $q_2$, (c) the density map of $q_1$, (d) the density map of $q_2$.
• Figure 4: Multi-soliton solutions with $\tilde{q}_{2,-}=1,\ \zeta _1=e^{-\frac{\pi}{12}i},\zeta _2=e^{\frac{13\pi}{12}i}$. (a) The three-dimension plot of $q_1$, (b) the three-dimension plot of $q_2$, (c) the density map of $q_1$, (d) the density map of $q_2$.
• Figure 5: Two-soliton and breather-soliton solution with $q_{2,-}=1,\ \alpha = 0.5,\ \zeta_1=e^{-\frac{1}{48}\pi i}$. (a) The three-dimension plot of $q_1$, (b) the three-dimension plot of $q_2$, (c) the density map of $q_1$, (d) the density map of $q_2$.