{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,19]],"date-time":"2026-02-19T08:25:17Z","timestamp":1771489517887,"version":"3.50.1"},"reference-count":137,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2023,10,18]],"date-time":"2023-10-18T00:00:00Z","timestamp":1697587200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100003336","name":"Bulgarian National Science Foundation","doi-asserted-by":"publisher","award":["KP-06N42-2"],"award-info":[{"award-number":["KP-06N42-2"]}],"id":[{"id":"10.13039\/501100003336","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The standard approach to integrable nonlinear evolution equations (NLEE) usually uses the following steps: (1) Lax representation [L,M]=0; (2) construction of fundamental analytic solutions (FAS); (3) reducing the inverse scattering problem (ISP) to a Riemann-Hilbert problem (RHP) \u03be+(x,t,\u03bb)=\u03be\u2212(x,t,\u03bb)G(x,t\u03bb) on a contour \u0393 with sewing function G(x,t,\u03bb); (4) soliton solutions and possible applications. Step 1 involves several assumptions: the choice of the Lie algebra g underlying L, as well as its dependence on the spectral parameter, typically linear or quadratic in \u03bb. In the present paper, we propose another approach that substantially extends the classes of integrable NLEE. Its first advantage is that one can effectively use any polynomial dependence in both L and M. We use the following steps: (A) Start with canonically normalized RHP with predefined contour \u0393; (B) Specify the x and t dependence of the sewing function defined on \u0393; (C) Introduce convenient parametrization for the solutions \u03be\u00b1(x,t,\u03bb) of the RHP and formulate the Lax pair and the nonlinear evolution equations (NLEE); (D) use Zakharov\u2013Shabat dressing method to derive their soliton solutions. This requires correctly taking into account the symmetries of the RHP. (E) Define the resolvent of the Lax operator and use it to analyze its spectral properties.<\/jats:p>","DOI":"10.3390\/sym15101933","type":"journal-article","created":{"date-parts":[[2023,10,18]],"date-time":"2023-10-18T09:55:06Z","timestamp":1697622906000},"page":"1933","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Riemann\u2013Hilbert Problems, Polynomial Lax Pairs, Integrable Equations and Their Soliton Solutions"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1058-6565","authenticated-orcid":false,"given":"Vladimir Stefanov","family":"Gerdjikov","sequence":"first","affiliation":[{"name":"Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. Georgi Bonchev Str., Block 8, 1113 Sofia, Bulgaria"},{"name":"Institute for Advanced Physical Studies, 111 Tsarigradsko Chaussee, 1784 Sofia, Bulgaria"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6614-3464","authenticated-orcid":false,"given":"Aleksander Aleksiev","family":"Stefanov","sequence":"additional","affiliation":[{"name":"Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. Georgi Bonchev Str., Block 8, 1113 Sofia, Bulgaria"},{"name":"Faculty of Mathematics and Informatics, Sofia University \u201cSt. Kliment Ohridski\u201d, 5 James Bourchier Blvd., 1164 Sofia, Bulgaria"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,10,18]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"280","DOI":"10.1007\/BF01086739","article-title":"Korteweg-de Vries equation: A completely integrable Hamiltonian system","volume":"5","author":"Zakharov","year":"1971","journal-title":"Funct. Anal. 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