On the $N$-waves hierarchy with constant boundary conditions. Spectral properties

Abstract

The paper is devoted to $N$-wave equations with constant boundary conditions related to symplectic Lie algebras. We study the spectral properties of a class of Lax operators $L$, whose potentials $Q(x,t)$ tend to constants $Q_\pm$ for $x\to \pm \infty$. For special choices of $Q_\pm$ we outline the spectral properties of $L$, the direct scattering transform and construct its fundamental analytic solutions. We generalise Wronskian relations for the case of CBC -- this allows us to analyse the mapping between the scattering data and the $x$-derivative of the potential $Q_x$. Next, using the Wronskian relations we derive the dispersion laws for the $N$-wave hierarchy and describe the NLEE related to the given Lax operator.

Figures (3)

• Figure 1: The elementary wave-decay corresponding to each term $H(\alpha ,\beta ,\gamma )$ in \ref{['eq:decay']}. If we assign to each wave $q_\alpha$ ($\alpha\in \Delta_+$) a wave number $k_{\alpha }$ and frequency $\omega _{\alpha }$, then the resonance condition $\omega (k_{\alpha })=\omega (k_{\beta }) + \omega (k_{\gamma })$ will imply preservation of the wave number: $k_{\alpha } = k_{\beta } + k_{\gamma }$ for each elementary decay $q_\alpha \rightarrow q_\beta + q_\gamma$ (for any triple of roots $\alpha, \beta, \gamma \in \Delta_+$, such that $\beta + \gamma=\alpha$).
• Figure 2: The wave-decay diagram corresponding to the cascade interaction Hamiltonian \ref{['eq:H4']}.
• Figure 3: The spectrum of the asymptotic operators $L_\pm$. On the left panel (a) it is located on the two pairs of cuts on the complex $\lambda$-plane, that determine the Riemannian surfaces $s_1\cup s_2$; on the right panel (b) we show it on the complex $z_1$-plane, where $z_1$ is the uniformizing variable for the first Riemannian surface $s_1$; see also Remark \ref{['rem:1']} below.

Theorems & Definitions (1)

• Remark 1