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articleFree Access
On the $N$ $N$ -waves hierarchy with constant boundary conditions. Spectral properties
- Vladimir S. Gerdjikov and
- Georgi G. Grahovski
International Journal of Geometric Methods in Modern Physics13 Apr 2024
Preview Abstract
This paper is devoted to $N$ $N$ -wave equations with constant boundary conditions related to symplectic Lie algebras. We study the spectral properties of a class of Lax operators $L$ $L$ , whose potentials $Q (x, t)$ $Q (x, t)$ tend to constants $Q_{\pm}$ $Q_{\pm}$ for $x \to \pm \infty$ $x \to \pm \infty$ . For special choices of $Q_{\pm}$ $Q_{\pm}$ , we outline the spectral properties of $L$ $L$ , the direct scattering transform and construct its fundamental analytic solutions. We generalize Wronskian relations for the case of CBC — this allows us to analyze the mapping between the scattering data and the $x$ $x$ -derivative of the potential $Q_{x}$ $Q_{x}$ . Next, using the Wronskian relations, we derive the dispersion laws for the $N$ $N$ -wave hierarchy and describe the NLEE related to the given Lax operator.

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