Breather interactions in the integrable discrete Manakov system and trigonometric Yang-Baxter maps

TL;DR

The paper analyzes the integrable discrete Manakov (IDM) system, a vector generalization of the Ablowitz–Ladik model, to completely characterize soliton and breather interactions. It employs the inverse scattering transform to derive explicit transmission coefficients for fundamental solitons, fundamental breathers, and composite breathers, and uses long-time asymptotics to obtain polarization maps that describe how solitons and breathers transform after collisions. A key finding is that a fundamental soliton becomes a fundamental breather after interacting with a fundamental breather, while two fundamental breathers typically exchange polarization and may produce a soliton–breather pair; composite breathers interact trivially with others. These interactions are interpreted as refactorization problems that give rise to parametric Yang–Baxter maps of trigonometric type on rank-one projectors, highlighting a novel algebraic structure in discrete multicomponent integrable systems.

Abstract

The goal of this work is to obtain a complete characterization of soliton and breather interactions in the integrable discrete Manakov (IDM) system, a vector generalization of the Ablowitz-Ladik model. The IDM system, which in the continuous limit reduces to the Manakov system (i.e., a 2-component vector nonlinear Schrodinger equation), was shown to admit a variety of discrete vector soliton solutions: fundamental solitons, fundamental breathers, and composite breathers. While the interaction of fundamental solitons was studied early on, no results are presently available for other types of soliton-breather and breather-breather interactions. Our study reveals that upon interacting with a fundamental breather, a fundamental soliton becomes a fundamental breather. Conversely, the interaction of two fundamental breathers generically yields two fundamental breathers with polarization shifts, but may also result in a fundamental soliton and a fundamental breather. Composite breathers interact trivially both with each other and with a fundamental soliton or breather. Explicit formulas for the scattering coefficients that characterize fundamental and composite breathers are given. This allows us to interpret the interactions in terms of a refactorization problem and derive the associated Yang-Baxter maps describing the effect of interactions on the polarizations. These give the first examples of parametric Yang-Baxter maps of trigonometric type.

Figures (7)

• Figure 1: Single soliton solutions corresponding to the same discrete eigenvalue $z_{1}=\exp(0.2-i\pi/8)$, with $|Q_n^{(1)}(\tau)|$ in the top panels, and $|Q_n^{(2)}(\tau)|$ in the bottom panels. (a) Fundamental soliton with ${\mathbf{C}}_{1}=0.300.10$. (b) Fundamental breather with ${\mathbf{C}}_{1}=0.30.30.10.1$. (c) Composite breather with ${\mathbf{C}}_{1}=0.30.30.10.2$.
• Figure 2: (a) Fundamental soliton-fundamental breather interaction with $z_{1}=\exp(0.15+i\pi/8)$, $\boldsymbol\gamma_{1}=(1,2)^{T}$, $\boldsymbol\delta_{1}=(1,0)^{T}$ and $z_{2}=\exp(0.1-i\pi/8)$, $\boldsymbol\gamma_{2}=(1,1)^{T}$, $\boldsymbol\delta_{2}=(0.1,0.1)^{T}$. (b) Reverse view, from which one can see more clearly that the fundamental soliton becomes a fundamental breather after the interaction.
• Figure 3: Interaction between two fundamental breathers with $z_{1}=\exp(0.1+i\pi/3)$, $\boldsymbol\gamma_{1}=(2,3)^{T}$, $\boldsymbol\delta_{1}=(0.04,0.04)^{T}$ and $z_{2}=\exp(0.12)$, $\boldsymbol\gamma_{2}=(0.2,0.2)^{T}$, $\boldsymbol\delta_{2}=(1,0.5)^{T}$.
• Figure 4: The same soliton-breather interaction as in Fig. \ref{['f:2soliton']}, with the predicted asymptotic breathers subtracted in each direction. In particular; in (a) the soliton with polarization vectors ${\mathbf{u}}_{1}^{-},{\mathbf{v}}_{1}^{-}$ is subtracted, in (b) the breather with polarization vectors ${\mathbf{u}}_{1}^{+},{\mathbf{v}}_{1}^{+}$ is subtracted, in (c) the breather with polarization vectors ${\mathbf{u}}_{2}^{-},{\mathbf{v}}_{2}^{-}$ is subtracted, in (d) the breather with polarization vectors ${\mathbf{u}}_{2}^{+},{\mathbf{v}}_{2}^{+}$ is subtracted.
• Figure 5: The same breather-breather interaction as in Fig. \ref{['f:2breather']}, with the predicted asymptotic breathers subtracted in each direction. In particular; in (a) the breather with polarization vectors ${\mathbf{u}}_{1}^{-},{\mathbf{v}}_{1}^{-}$ is subtracted, in (b) the breather with polarization vectors ${\mathbf{u}}_{1}^{+},{\mathbf{v}}_{1}^{+}$ is subtracted, in (c) the breather with polarization vectors ${\mathbf{u}}_{2}^{-},{\mathbf{v}}_{2}^{-}$ is subtracted, in (d) the breather with polarization vectors ${\mathbf{u}}_{2}^{+},{\mathbf{v}}_{2}^{+}$ is subtracted.

Theorems & Definitions (2)

• Proposition 1
• Proposition 2