{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,11]],"date-time":"2026-10-11T02:36:57Z","timestamp":1791686217009,"version":"4.3.4"},"reference-count":33,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2024,1,2]],"date-time":"2024-01-02T00:00:00Z","timestamp":1704153600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100006769","name":"Russian Science Foundation","doi-asserted-by":"publisher","award":["22-11-00196"],"award-info":[{"award-number":["22-11-00196"]}],"id":[{"id":"10.13039\/501100006769","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>We propose a new hierarchy of the vector derivative nonlinear Schr\u00f6dinger equations and consider the simplest multiphase solutions of this hierarchy. The study of the simplest solutions of these equations led to the following results. First, the three-leaf spectral curves \u0393={(\u03bc,\u03bb)} of the simplest multiphase solutions have a quite simple symmetry. They are invariant with respect to holomorphic involution \u03c4. The type of this involution depends on the genus of the spectral curve. Or the involution has the form \u03c4:(\u03bc,\u03bb)\u2192(\u03bc,\u2212\u03bb), or \u03c4:(\u03bc,\u03bb)\u2192(\u2212\u03bc,\u2212\u03bb). The presence of symmetry leads to the fact that the dynamics of the solution is determined not by the entire spectral curve \u0393, but by its factor \u0393\/\u03c4, which has a smaller genus. Secondly, it turned out that the dynamics of the two-component vector p=(p1,p2)t is determined, first of all, by the dynamics of its length |p|. Independent equations determine the dependence of the direction of the vector p from its length. In cases where the direction of the vector p is fixed, the corresponding spectral curve splits into separate components. In conclusion, we note that, as in the case of the Manakov system, the equation of the spectral curve is invariant with respect to the orthogonal transformation of the vector solutions. I.e., the solution can be found from the spectral curve up to the orthogonal transformation. This fact indicates that the spectral curve does not depend on the individual components of the solution, but on their symmetric functions. Thus, the spectral data of multiphase solutions have two symmetries. These symmetries make it difficult to reconstruct signals from their spectral data. The work contains examples illustrating these statements.<\/jats:p>","DOI":"10.3390\/sym16010060","type":"journal-article","created":{"date-parts":[[2024,1,2]],"date-time":"2024-01-02T06:04:09Z","timestamp":1704175449000},"page":"60","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["On a Hierarchy of Vector Derivative Nonlinear Schr\u00f6dinger Equations"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6781-2105","authenticated-orcid":false,"given":"Aleksandr O.","family":"Smirnov","sequence":"first","affiliation":[{"name":"Department of Advanced Mathematics and Mechanics, St.-Petersburg State Universty of Aerospace Instrumentation, Bolshaya Morskaya Str., 67A, Sankt-Petersburg 190000, Russia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0000-6266-306X","authenticated-orcid":false,"given":"Eugene A.","family":"Frolov","sequence":"additional","affiliation":[{"name":"Department of Advanced Mathematics and Mechanics, St.-Petersburg State Universty of Aerospace Instrumentation, Bolshaya Morskaya Str., 67A, Sankt-Petersburg 190000, Russia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Lada L.","family":"Dmitrieva","sequence":"additional","affiliation":[{"name":"Higher School of Applied Mathematics and Computational Physics, Peter the Great St.-Petersburg Polytechnic University, Polytechnicheskaya Str., 29, Sankt-Petersburg 195251, Russia"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2024,1,2]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"561","DOI":"10.1140\/epjp\/s13360-020-00588-1","article-title":"From generalized Fourier transforms to spectral curves for the Manakov hierarchy. 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