{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:28:30Z","timestamp":1760239710104,"version":"build-2065373602"},"reference-count":21,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2020,12,31]],"date-time":"2020-12-31T00:00:00Z","timestamp":1609372800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The concept of ranked order probability distribution unveils natural probabilistic interpretations for the kink waves (and hence the solitons) solving higher order dispersive Burgers\u2019 type PDEs. Thanks to this underlying structure, it is possible to propose a systematic derivation of exact solutions for PDEs with a quadratic nonlinearity of the Burgers\u2019 type but with arbitrary dispersive orders. As illustrations, we revisit the dissipative Kotrweg de Vries, Kuramoto-Sivashinski, and Kawahara equations (involving third, fourth, and fifth order dispersion dynamics), which in this context appear to be nothing but the simplest special cases of this infinitely rich class of nonlinear evolutions.<\/jats:p>","DOI":"10.3390\/sym13010057","type":"journal-article","created":{"date-parts":[[2020,12,31]],"date-time":"2020-12-31T14:31:49Z","timestamp":1609425109000},"page":"57","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Brownian Swarm Dynamics and Burgers\u2019 Equation with Higher Order Dispersion"],"prefix":"10.3390","volume":"13","author":[{"given":"Max-Olivier","family":"Hongler","sequence":"first","affiliation":[{"name":"Ecole Polytechnique F\u00e9d\u00e9rale de Lausanne (EPFL), EPFL STI IMT-GE\u2014Rue de la Maladi\u00e8re 71b, Case postale 526, CH-2002 Neuch\u00e2tel, Switzerland"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,12,31]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Debnath, L. 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