Block Toeplitz Inner-Bordering method for the Gelfand-Levitan-Marchenko equations associated with the Zakharov-Shabat system

We propose a generalized method for solving the Gelfand-Levitan-Marchenko equation (GLME) based on the block version of the Toeplitz Inner-Bordering (TIB). The method works for the signals containing both the continuous and the discrete spectra. The method allows us to calculate the potential at an arbitrary point and does not require small spectral data. Using this property, we can perform calculations to the right and to the left of the selected starting point. For the discrete spectrum, the procedure of cutting off exponentially growing matrix elements is suggested to avoid the numerical instability and perform calculations for soliton solutions spaced apart in the time domain.

Automated summary

We propose a generalized method based on the block version of the Toeplitz Inner-Bordering (TIB) to solve the Gelfand-Levitan-Marchenko equation (GLME). This method works for signals with both continuous and discrete spectra, allowing calculation of the potential at any point without requiring small spectral data. The method also enables calculations to be performed to the right and left of a selected starting point. For the discrete spectrum, we suggest truncating exponentially growing matrix elements to avoid numerical instability and calculate soliton solutions spaced apart in the time domain.