Comment on “Integrability, modulational instability and mixed localized wave solutions for the generalized nonlinear Schrödinger equation”

Abstract

In a recent paper Li et al. (Z Angew Math Phys 73:52, 2022. https://doi.org/10.1007/s00033-022-01681-4) have considered a generalized nonlinear Schrödinger equation which has extensive applications in various fields of physics and engineering. After proving Liouville integrability of this equation, they investigated the phenomenon of the modulational instability for the possible reason of the formation of the rogue waves. By means of the generalized (\(2,N-2\))-fold Darboux transformation, authors presented several mixed localized wave solutions, such as breathers, rogue waves and semi-rational solitons for their model equation, and accurately analyzed a number of important physical quantities. It is the aim of this Comment to point out that (i) the baseband modulation instability was developed in a wrong way and (ii) one of the two different types of Taylor series expansions for solution of Lax pair used in that article for building analytical solutions, especially the one obtained with \(\xi _{j}=Z\) does not correspond to any solution of the spectral problem (2.1) when using \( u_{0}\left( x,t\right) \) as the seed solution. Consequently, all mixed localized solutions that involve the mentioned Taylor series are invalid.