{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T14:43:15Z","timestamp":1787236995884,"version":"3.56.0"},"reference-count":64,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Appl. Dyn. Syst."],"published-print":{"date-parts":[[2024,9,30]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>In this article we present a general method to rigorously prove existence of strong solutions to a large class of autonomous semilinear PDEs in a Hilbert space [Formula: see text] ([Formula: see text]) via computer-assisted proofs. Our approach is fully spectral and uses Fourier series to approximate functions in [Formula: see text] as well as bounded linear operators from [Formula: see text] to [Formula: see text]. In particular, we construct approximate inverses of differential operators via Fourier series approximations. Combining this construction with a Newton\u2013Kantorovich approach, we develop a numerical method to prove existence of strong solutions. To do so, we introduce a finite-dimensional trace theorem from which we build smooth functions with support on a hypercube. The method is then generalized to systems of PDEs with extra equations\/parameters, such as eigenvalue problems. As an application, we prove the existence of a traveling wave (soliton) in the Kawahara equation in [Formula: see text] as well as eigenpairs of the linearization about the soliton. These results allow us to prove the stability of the aforementioned traveling wave.<\/jats:p>","DOI":"10.1137\/23m1607507","type":"journal-article","created":{"date-parts":[[2024,7,22]],"date-time":"2024-07-22T04:01:15Z","timestamp":1721620875000},"page":"1966-2017","source":"Crossref","is-referenced-by-count":13,"title":["Rigorous Computation of Solutions of Semilinear PDEs on Unbounded Domains via Spectral Methods"],"prefix":"10.1137","volume":"23","author":[{"given":"Matthieu","family":"Cadiot","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, McGill University, 805 Sherbrooke Street West, Montreal, QC, H3A 0B9, Canada."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jean-Philippe","family":"Lessard","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, McGill University, 805 Sherbrooke Street West, Montreal, QC, H3A 0B9, Canada."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1613-4380","authenticated-orcid":true,"given":"Jean-Christophe","family":"Nave","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, McGill University, 805 Sherbrooke Street West, Montreal, QC, H3A 0B9, Canada."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2024,7,22]]},"reference":[{"key":"ref1","volume-title":"Sobolev Spaces","author":"Adams R. A.","year":"2003"},{"key":"ref2","doi-asserted-by":"publisher","DOI":"10.1080\/03605309208820831"},{"key":"ref3","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-16830-7"},{"key":"ref4","doi-asserted-by":"publisher","DOI":"10.1098\/rspa.2000.0635"},{"key":"ref5","unstructured":"C. Bernardi, M. Dauge, and Y. Maday, Polynomials in the Sobolev World, preprint Hal-00153795v2, 2007."},{"key":"ref6","doi-asserted-by":"publisher","DOI":"10.1137\/141000671"},{"key":"ref7","doi-asserted-by":"publisher","DOI":"10.1016\/j.cnsns.2022.106292"},{"key":"ref8","doi-asserted-by":"publisher","DOI":"10.3934\/dcds.2015.35.4765"},{"key":"ref9","unstructured":"T. Buckmaster, G. Cao-Labora, and J. G\u00f3mez-Serrano, Smooth Imploding Solutions for 3D Compressible Fluids, preprint, https:\/\/arxiv.org\/abs\/2208.09445, 2022."},{"key":"ref10","doi-asserted-by":"publisher","DOI":"10.1007\/s10569-018-9879-8"},{"key":"ref11","doi-asserted-by":"publisher","DOI":"10.1512\/iumj.2003.52.2245"},{"key":"ref12","doi-asserted-by":"publisher","DOI":"10.1512\/iumj.2003.52.2407"},{"key":"ref13","doi-asserted-by":"publisher","DOI":"10.1016\/j.jde.2004.12.003"},{"key":"ref14","author":"Cadiot M.","year":"2023","journal-title":"Proofkawahara.jl"},{"key":"ref15","doi-asserted-by":"publisher","DOI":"10.1088\/1361-6544\/abcb08"},{"key":"ref16","unstructured":"J. Chen and T. Y. Hou, Stable Nearly Self-Similar Blowup of the 2D Boussinesq and 3D Euler Equations with Smooth Data I: Analysis, preprint, https:\/\/arxiv.org\/abs\/arXiv:2210.07191, 2022."},{"key":"ref17","doi-asserted-by":"publisher","DOI":"10.1016\/j.cnsns.2022.106304"},{"key":"ref18","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmaa.2018.05.010"},{"key":"ref19","doi-asserted-by":"publisher","DOI":"10.1090\/gsm\/019"},{"key":"ref20","doi-asserted-by":"publisher","DOI":"10.1016\/j.jde.2010.07.002"},{"key":"ref21","doi-asserted-by":"publisher","DOI":"10.1137\/16M1073789"},{"key":"ref22","doi-asserted-by":"publisher","DOI":"10.1007\/s40324-019-00186-x"},{"key":"ref23","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511623745"},{"key":"ref24","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-66282-9"},{"key":"ref25","series-title":"Tata Institute of Fundamental Research Lectures on Mathematics and Physics 79","volume-title":"Lectures on Numerical Methods in Bifurcation Problems","author":"Keller H. B.","year":"1987"},{"key":"ref26","doi-asserted-by":"publisher","DOI":"10.1137\/S0036144595284180"},{"key":"ref27","doi-asserted-by":"publisher","DOI":"10.1090\/S0273-0979-1982-15008-X"},{"key":"ref28","doi-asserted-by":"crossref","unstructured":"J.P. Lessard and J. D. Mireles James, Computer assisted Fourier analysis in sequence spaces of varying regularity, SIAM J. Math. Anal., 49 (2017), pp. 530\u2013561, https:\/\/doi.org\/10.1137\/16M1056006.","DOI":"10.1137\/16M1056006"},{"key":"ref29","doi-asserted-by":"publisher","DOI":"10.1016\/S0167-2789(98)00245-0"},{"key":"ref30","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2015.03.048"},{"key":"ref31","volume-title":"Strongly Elliptic Systems and Boundary Integral Equations","author":"McLean W.","year":"2000"},{"key":"ref32","doi-asserted-by":"publisher","DOI":"10.1090\/S0273-0979-1995-00558-6"},{"key":"ref33","volume-title":"Interval Analysis","author":"Moore R. E.","year":"1966"},{"key":"ref34","doi-asserted-by":"publisher","DOI":"10.1007\/BF03167855"},{"key":"ref35","doi-asserted-by":"publisher","DOI":"10.1081\/NFA-100105107"},{"key":"ref36","doi-asserted-by":"publisher","DOI":"10.1007\/s00607-004-0111-1"},{"key":"ref37","doi-asserted-by":"publisher","DOI":"10.1007\/978-981-13-7669-6"},{"key":"ref38","doi-asserted-by":"publisher","DOI":"10.1201\/9781315218915"},{"key":"ref39","volume-title":"Linear and Semilinear Partial Differential Equations","author":"Precup R.","year":"2013"},{"key":"ref40","unstructured":"M. Reed and B. Simon, Methods of Modern Mathematical Physics. Fourier Analysis, Self-Adjointness, Academic Press [Harcourt Brace Jovanovich, Publishers], 1975."},{"key":"ref41","doi-asserted-by":"publisher","DOI":"10.1007\/s10543-010-0294-0"},{"key":"ref42","doi-asserted-by":"crossref","first-page":"603","DOI":"10.3934\/dcdsb.2020260","volume":"26","author":"Sander E.","year":"2021","journal-title":"Discrete Contin. Dyn. Syst. - B"},{"key":"ref43","doi-asserted-by":"publisher","DOI":"10.1007\/s00211-020-01155-7"},{"key":"ref44","doi-asserted-by":"publisher","DOI":"10.1016\/S0960-0779(03)00102-4"},{"key":"ref45","doi-asserted-by":"publisher","DOI":"10.1137\/16M1082196"},{"key":"ref46","doi-asserted-by":"publisher","DOI":"10.1007\/s13160-014-0156-2"},{"key":"ref47","doi-asserted-by":"publisher","DOI":"10.1016\/S0764-4442(99)80439-X"},{"key":"ref48","doi-asserted-by":"publisher","DOI":"10.1007\/s002080010018"},{"key":"ref49","doi-asserted-by":"publisher","DOI":"10.1016\/j.jde.2017.11.011"},{"key":"ref50","doi-asserted-by":"publisher","DOI":"10.1007\/s00332-021-09695-4"},{"key":"ref51","doi-asserted-by":"publisher","DOI":"10.1088\/1361-6544\/ad0278"},{"key":"ref52","doi-asserted-by":"publisher","DOI":"10.1016\/j.jde.2018.02.018"},{"key":"ref53","doi-asserted-by":"publisher","DOI":"10.1090\/noti1276"},{"key":"ref54","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-10-02325-2"},{"key":"ref55","doi-asserted-by":"publisher","DOI":"10.1137\/100812008"},{"key":"ref56","doi-asserted-by":"publisher","DOI":"10.1016\/j.indag.2020.01.007"},{"key":"ref57","doi-asserted-by":"publisher","DOI":"10.1137\/17M1155624"},{"key":"ref58","doi-asserted-by":"publisher","DOI":"10.1007\/s13160-019-00344-8"},{"key":"ref59","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2006.06.002"},{"key":"ref60","doi-asserted-by":"publisher","DOI":"10.1080\/03605308708820522"},{"key":"ref61","volume-title":"Linear and Nonlinear Waves","author":"Whitham G. B.","year":"2011"},{"key":"ref62","doi-asserted-by":"publisher","DOI":"10.1016\/j.jde.2020.06.020"},{"key":"ref63","unstructured":"J. M. Wunderlich, Computer-assisted Existence Proofs for Navier-Stokes Equations on an Unbounded Strip with Obstacle, Ph.D. thesis, Karlsruher Institut f\u00fcr Technologie (KIT), 2022, https:\/\/doi.org\/10.5445\/IR\/1000150609."},{"key":"ref64","doi-asserted-by":"publisher","DOI":"10.1016\/j.chaos.2006.10.012"}],"container-title":["SIAM Journal on Applied Dynamical Systems"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/23M1607507","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T14:19:40Z","timestamp":1787235580000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/23M1607507"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,7,22]]},"references-count":64,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2024,9,30]]}},"alternative-id":["10.1137\/23M1607507"],"URL":"https:\/\/doi.org\/10.1137\/23m1607507","relation":{},"ISSN":["1536-0040"],"issn-type":[{"value":"1536-0040","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,7,22]]}}}