{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:52:36Z","timestamp":1760143956043,"version":"build-2065373602"},"reference-count":24,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2024,3,5]],"date-time":"2024-03-05T00:00:00Z","timestamp":1709596800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100003453","name":"Natural Science Foundation of Guangdong Province of China","doi-asserted-by":"publisher","award":["2021A1515012214"],"award-info":[{"award-number":["2021A1515012214"]}],"id":[{"id":"10.13039\/501100003453","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Two systems of mathematical physics are defined by us, which are the first-order differential system (FODS) and the second-order differential system (SODS). Basing on the conventional Legendre transformation, we obtain a new kind of canonical equations of Hamilton (CEH) with some kind of symmetry. We show that the FODS can only be expressed by the new CEH, but do not by the conventional CEH, while the SODS can be done by both the new and the conventional CEHs, on basis of the same conventional Legendre transformation. As an example, we prove that the nonlinear Schr\u00f6dinger equation can be expressed with the new CEH in a consistent way. Based on the new CEH, the approximate soliton solution of the nonlocal nonlinear Schr\u00f6dinger equation is obtained, and the soliton stability is analysed analytically as well. Furthermore, because the symmetry of a system is closely connected with certain conservation theorem of the system, the new CEH may be useful in some complicated systems when the symmetry considerations are used.<\/jats:p>","DOI":"10.3390\/sym16030305","type":"journal-article","created":{"date-parts":[[2024,3,5]],"date-time":"2024-03-05T05:31:19Z","timestamp":1709616679000},"page":"305","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Canonical Equations of Hamilton with Symmetry and Their Applications"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9897-9306","authenticated-orcid":false,"given":"Guo","family":"Liang","sequence":"first","affiliation":[{"name":"Guangdong Provincial Key Laboratory of Nanophotonic Functional Materials and Devices, South China Normal University, Guangzhou 510631, China"},{"name":"School of Electrical & Electronic Engineering, Shangqiu Normal University, Shangqiu 476000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xiangwei","family":"Chen","sequence":"additional","affiliation":[{"name":"School of Electrical & Electronic Engineering, Shangqiu Normal University, Shangqiu 476000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhanmei","family":"Ren","sequence":"additional","affiliation":[{"name":"Guangdong Provincial Key Laboratory of Nanophotonic Functional Materials and Devices, South China Normal University, Guangzhou 510631, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3314-8306","authenticated-orcid":false,"given":"Qi","family":"Guo","sequence":"additional","affiliation":[{"name":"Guangdong Provincial Key Laboratory of Nanophotonic Functional Materials and Devices, South China Normal University, Guangzhou 510631, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,3,5]]},"reference":[{"key":"ref_1","unstructured":"Goldstein, H., Poole, C., and Safko, J. (2001). Classical Mechanics, Addison-Wesley."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"14256","DOI":"10.1073\/pnas.93.25.14256","article-title":"The role of symmetry in fundamental physics","volume":"93","author":"Gross","year":"1996","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"154","DOI":"10.1016\/0167-2789(93)90011-O","article-title":"Numerical study of hydrodynamics using the nonlinear Schr\u00f6dinger equation","volume":"65","author":"Nore","year":"1993","journal-title":"Phys. D"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Hasegawa, A., and Kodama, Y. (1995). Solitons in Optical Communications, Clarenoon Press.","DOI":"10.1093\/oso\/9780198565079.001.0001"},{"key":"ref_5","unstructured":"Haus, H.A. (1984). Waves and Fields in Optoelectronics, Prentice-Hall."},{"key":"ref_6","unstructured":"Agrawal, G. (2001). Nonlinear Fiber Optics, Academic Press."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"38","DOI":"10.1063\/1.2956239","article-title":"Weak-nonlinear acoustic pulse dynamics in a waveguide channel with longitudinal inhomogeneity","volume":"1022","author":"Bisyarin","year":"2008","journal-title":"AIP Conf. Proc."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"033622","DOI":"10.1103\/PhysRevA.71.033622","article-title":"Nonlinear band structure in Bose-Einstein condensates: Nonlinear Schr\u00f6dinger equation with a Kronig-Penney potential","volume":"71","author":"Seaman","year":"2005","journal-title":"Phys. Rev. A"},{"key":"ref_9","unstructured":"Arfken, G.B., and Weber, H.J. (2005). Mathematical Methods for Physicists, Academic Press."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"3135","DOI":"10.1103\/PhysRevA.27.3135","article-title":"Variational approach to nonlinear pulse propagation in optical fibers","volume":"27","author":"Anderson","year":"1983","journal-title":"Phys. Rev. A"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Chen, X., Zhang, G., Zeng, H., Guo, Q., and She, W. (2015). Advances in Nonlinear Optics, De Gruyter. Chapter 4.","DOI":"10.1515\/9783110304497"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1538","DOI":"10.1126\/science.276.5318.1538","article-title":"Accessible Solitons","volume":"276","author":"Snyder","year":"1997","journal-title":"Science"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"016612","DOI":"10.1103\/PhysRevE.64.016612","article-title":"Modulational instability in nonlocal nonlinear Kerr media","volume":"64","author":"Krolikowski","year":"2001","journal-title":"Phys. Rev. E"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Assanto, G. (2012). Nematicons: Spatial Optical Solitons in Nematic Liquid Crystals, John Wiley & Sons. Chapter 2.","DOI":"10.1002\/9781118414637"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"043803","DOI":"10.1103\/PhysRevA.76.043803","article-title":"Solitons in the midst of chaos","volume":"76","author":"Seghete","year":"2007","journal-title":"Phys. Rev. A"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"233901","DOI":"10.1103\/PhysRevLett.107.233901","article-title":"Incoherent soliton turbulence in nonlocal nonlinear media","volume":"107","author":"Picozzi","year":"2011","journal-title":"Phys. Rev. Lett."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"422","DOI":"10.1016\/j.physleta.2007.02.042","article-title":"Two-dimensional nonlocal multisolitons","volume":"366","author":"Lashkin","year":"2007","journal-title":"Phys. Lett. A"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"196","DOI":"10.1016\/j.optcom.2007.07.006","article-title":"Quasi-stable propagation of vortices and soliton clusters in saturable Kerr media with square-root nonlinearity","volume":"279","author":"Petroski","year":"2007","journal-title":"Opt. Commun."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"783","DOI":"10.1007\/BF01031343","article-title":"Stationary solutions of the wave equation in a medium with nonlinearity saturation","volume":"16","author":"Vakhitov","year":"1973","journal-title":"Radiophys. Quantum Electron."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"2082","DOI":"10.1364\/JOSAB.8.002082","article-title":"Variational approach to collapse of optical pulses","volume":"8","author":"Desaix","year":"1991","journal-title":"J. Opt. Soc. Am. B"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1016\/S0370-1573(97)00092-6","article-title":"Wave collapse in physics: Principles and applications to light and plasma waves","volume":"303","author":"Berge","year":"1998","journal-title":"Phys. Rep."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"203902","DOI":"10.1103\/PhysRevLett.90.203902","article-title":"Self-similar optical wave collapse: Observation of the Townes profile","volume":"90","author":"Moll","year":"2003","journal-title":"Phys. Rev. Lett."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"20676","DOI":"10.1364\/OE.16.020676","article-title":"Peakon profiles and collapse-bounce cycles in self-focusing spatial beams","volume":"16","author":"Sun","year":"2008","journal-title":"Opt. Express"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"046619","DOI":"10.1103\/PhysRevE.66.046619","article-title":"Collapse arrest and soliton stabilization in nonlocal nonlinear media","volume":"66","author":"Bang","year":"2002","journal-title":"Phys. Rev. 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