{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T12:31:30Z","timestamp":1787229090265,"version":"build-2736575974"},"reference-count":99,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","funder":[{"DOI":"10.13039\/501100000923","name":"Australian Research Council","doi-asserted-by":"publisher","award":["DP190101190"],"award-info":[{"award-number":["DP190101190"]}],"id":[{"id":"10.13039\/501100000923","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Appl. Dyn. Syst."],"published-print":{"date-parts":[[2021,1]]},"abstract":"<jats:p>We study \u201cnanoptera,\u201d which are nonlocalized solitary waves with exponentially small but nondecaying oscillations, in two singularly perturbed Hertzian chains with precompression. These two systems are woodpile chains (which we model as systems of Hertzian particles and springs) and diatomic Hertzian chains with alternating masses. We demonstrate that nanoptera arise from the Stokes phenomenon and appear as special curves (called Stokes curves) are crossed in the complex plane. We use techniques from exponential asymptotics to obtain approximations of the oscillation amplitudes. Our analysis demonstrates that traveling-wave solutions in a singularly perturbed woodpile chain have a single Stokes curve, which generates oscillations behind the wave front. Comparing these asymptotic predictions with numerical simulations reveals that our asymptotic approximation accurately describes the nondecaying oscillatory behavior in a woodpile chain. We perform a similar analysis of a diatomic Hertzian chain, and we show that each nanopteron solution has two distinct exponentially small oscillatory contributions. We demonstrate that there exists a set of mass ratios for which these two contributions cancel to produce localized solitary waves. This result builds on prior experimental and numerical observations that there exist mass ratios that support localized solitary waves in diatomic Hertzian chains without precompression. Comparing our asymptotic and numerical results for a diatomic Hertzian chain with precompression reveals that our exponential asymptotic approach accurately predicts the oscillation amplitude for a wide range of system parameters, but it fails to identify several values of the mass ratio that correspond to localized solitary-wave solutions.<\/jats:p>","DOI":"10.1137\/21m1398410","type":"journal-article","created":{"date-parts":[[2021,11,22]],"date-time":"2021-11-22T10:50:02Z","timestamp":1637578202000},"page":"2412-2449","source":"Crossref","is-referenced-by-count":10,"title":["Nanoptera in Weakly Nonlinear Woodpile Chains and Diatomic Granular Chains"],"prefix":"10.1137","volume":"20","author":[{"given":"Guo","family":"Deng","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9504-277X","authenticated-orcid":true,"given":"Christopher J.","family":"Lustri","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5166-0717","authenticated-orcid":true,"given":"Mason A.","family":"Porter","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2021,11,22]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1088\/1361-6544\/ab1294"},{"key":"atypb2","unstructured":"M. P. Allen and D. J. Tildesley,\n                      Computer Simulation of Liquids\n                      , Clarendon Press, Oxford, UK, 1987."},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.79.046607"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.89.053202"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.84.046610"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1016\/0167-2789(93)90091-E"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1007\/BF02698550"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1098\/rspa.1989.0018"},{"key":"atypb9","first-page":"1","author":"Berry M. V.","year":"1991","journal-title":"The Netherlands"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1098\/rspa.1990.0111"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.104.244302"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1088\/0951-7715\/3\/1\/010"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1016\/0167-2789(91)90056-F"},{"key":"atypb14","doi-asserted-by":"crossref","unstructured":"J. P. Boyd,\n                      Weakly Nonlocal Solitary Waves and Beyond-All-Orders Asymptotics: Generalized Solitons and Hyperasymptotic Perturbation Theory\n                      , Math. Appl. 442, Kluwer Academic Publishers, Amsterdam, The Netherlands, 1998.","DOI":"10.1007\/978-1-4615-5825-5"},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1023\/A:1006145903624"},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1137\/S003614450444436X"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1098\/rspa.1998.0278"},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1088\/1361-648X\/aa7672"},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.56.6104"},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.96.058002"},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.73.026610"},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1063\/1.5121427"},{"key":"atypb23","doi-asserted-by":"publisher","DOI":"10.1140\/epjp\/s13360-020-00587-2"},{"key":"atypb24","unstructured":"R. B. Dingle,\n                      Asymptotic Expansions: Their Derivation and Interpretation\n                      , Academic Press, New York, NY, USA, 1973."},{"key":"atypb25","doi-asserted-by":"publisher","DOI":"10.1007\/s100510050119"},{"key":"atypb26","doi-asserted-by":"publisher","DOI":"10.1090\/qam\/1548"},{"key":"atypb27","doi-asserted-by":"publisher","DOI":"10.1016\/j.padiff.2021.100128"},{"key":"atypb28","doi-asserted-by":"publisher","DOI":"10.1063\/1.4902071"},{"key":"atypb29","doi-asserted-by":"publisher","DOI":"10.1088\/0951-7715\/15\/4\/317"},{"key":"atypb30","doi-asserted-by":"publisher","DOI":"10.1007\/BF02099784"},{"key":"atypb31","doi-asserted-by":"publisher","DOI":"10.3934\/mine.2019.3.419"},{"key":"atypb32","first-page":"105","volume":"430","author":"Goddard J. D.","year":"1990","journal-title":"Proc. Roy. Soc. A"},{"key":"atypb33","doi-asserted-by":"publisher","DOI":"10.1137\/S0036139993243825"},{"key":"atypb34","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.82.011306"},{"key":"atypb35","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.110.144101"},{"key":"atypb36","doi-asserted-by":"publisher","DOI":"10.1007\/s00707-009-0163-6"},{"key":"atypb37","doi-asserted-by":"crossref","first-page":"156","DOI":"10.1515\/crll.1882.92.156","volume":"92","author":"Hertz H.","year":"1882","journal-title":"J. Reine Angew. Math."},{"key":"atypb38","unstructured":"E. J. Hinch,\n                      Perturbation Methods\n                      , Cambridge Texts Appl. Math., Cambridge University Press, Cambridge, UK, 1991."},{"key":"atypb39","doi-asserted-by":"publisher","DOI":"10.1098\/rspa.1999.0447"},{"key":"atypb40","doi-asserted-by":"publisher","DOI":"10.1016\/j.physd.2017.07.004"},{"key":"atypb41","doi-asserted-by":"publisher","DOI":"10.1209\/0295-5075\/101\/44003"},{"key":"atypb42","doi-asserted-by":"publisher","DOI":"10.1016\/0167-2789(88)90054-1"},{"key":"atypb43","doi-asserted-by":"publisher","DOI":"10.1007\/s002200050821"},{"key":"atypb44","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.83.036606"},{"key":"atypb45","doi-asserted-by":"publisher","DOI":"10.1007\/s00332-012-9155-0"},{"key":"atypb46","doi-asserted-by":"publisher","DOI":"10.1063\/1.3216805"},{"key":"atypb47","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.94.178002"},{"key":"atypb48","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.92.062201"},{"key":"atypb49","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.114.118002"},{"key":"atypb50","doi-asserted-by":"publisher","DOI":"10.1038\/s41467-018-03015-3"},{"key":"atypb51","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmps.2014.06.012"},{"key":"atypb52","first-page":"39","volume":"5","author":"Korteweg D. J.","year":"1895","journal-title":"London Edinburgh Dublin Philos. Mag. J. Sci. Ser."},{"key":"atypb53","doi-asserted-by":"publisher","DOI":"10.1007\/BF00910379"},{"key":"atypb54","doi-asserted-by":"publisher","DOI":"10.1016\/j.physd.2016.05.007"},{"key":"atypb55","doi-asserted-by":"publisher","DOI":"10.1088\/0951-7715\/29\/11\/3496"},{"key":"atypb56","doi-asserted-by":"publisher","DOI":"10.1016\/j.physd.2019.132239"},{"key":"atypb57","doi-asserted-by":"publisher","DOI":"10.1137\/16M108639X"},{"key":"atypb58","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.66.016616"},{"key":"atypb59","doi-asserted-by":"publisher","DOI":"10.1016\/S0167-2789(01)00302-5"},{"key":"atypb60","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.85.031308"},{"key":"atypb61","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.90.032209"},{"key":"atypb62","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.93.022902"},{"key":"atypb63","doi-asserted-by":"publisher","DOI":"10.1098\/rsta.2017.0139"},{"key":"atypb64","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.93.052224"},{"key":"atypb65","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.80.056602"},{"key":"atypb66","doi-asserted-by":"publisher","DOI":"10.1007\/BF00905892"},{"key":"atypb67","unstructured":"V. F. Nesterenko,\n                      High Rate Deformation of Heterogeneous Materials\n                      , Nauka, Novosibirsk, Russia, 1992 (in Russian)."},{"key":"atypb68","first-page":"121","volume":"28","author":"Nesterenko V. F.","year":"1992","journal-title":"Fiz. Goreniya Vzryva"},{"key":"atypb69","first-page":"132","volume":"29","author":"Nesterenko V. F.","year":"1993","journal-title":"Fiz. Goreniya Vzryva"},{"key":"atypb70","first-page":"134","volume":"29","author":"Nesterenko V. F.","year":"1993","journal-title":"Fiz. Goreniya Vzryva"},{"key":"atypb71","doi-asserted-by":"crossref","unstructured":"V. F. Nesterenko,\n                      Dynamics of Heterogeneous Materials\n                      , Springer-Verlag, Heidelberg, Germany, 2001.","DOI":"10.1007\/978-1-4757-3524-6"},{"key":"atypb72","doi-asserted-by":"publisher","DOI":"10.1098\/rsta.2017.0130"},{"key":"atypb73","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.95.158702"},{"key":"atypb74","doi-asserted-by":"publisher","DOI":"10.1143\/JPSJ.59.2647"},{"key":"atypb75","doi-asserted-by":"publisher","DOI":"10.1137\/S0036139994261769"},{"key":"atypb76","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.82.021301"},{"key":"atypb77","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.77.015601"},{"key":"atypb78","doi-asserted-by":"publisher","DOI":"10.1016\/j.physd.2008.12.010"},{"key":"atypb79","doi-asserted-by":"publisher","DOI":"10.1063\/PT.3.2981"},{"key":"atypb80","doi-asserted-by":"publisher","DOI":"10.1007\/s11340-012-9673-6"},{"key":"atypb81","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.91.042207"},{"key":"atypb82","doi-asserted-by":"publisher","DOI":"10.1088\/1742-5468\/aa9a62"},{"key":"atypb83","doi-asserted-by":"publisher","DOI":"10.1016\/j.physrep.2007.10.007"},{"key":"atypb84","doi-asserted-by":"publisher","DOI":"10.1016\/j.physa.2004.04.092"},{"key":"atypb85","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.64.056605"},{"key":"atypb86","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.57.2386"},{"key":"atypb87","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.90.022905"},{"key":"atypb88","first-page":"55","volume":"305","author":"Spence D. A.","year":"1968","journal-title":"Proc. Roy. Soc. A"},{"key":"atypb89","first-page":"106","volume":"10","author":"Stokes G. G.","year":"1864","journal-title":"Trans. Cam. Phil. Soc."},{"key":"atypb90","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.83.066605"},{"key":"atypb91","doi-asserted-by":"publisher","DOI":"10.1143\/JPSJ.65.3689"},{"key":"atypb92","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.82.056604"},{"key":"atypb93","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.93.042210"},{"key":"atypb94","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.95.108002"},{"key":"atypb95","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.73.066623"},{"key":"atypb96","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRev.159.98"},{"key":"atypb97","doi-asserted-by":"publisher","DOI":"10.1088\/0022-3727\/44\/4\/045402"},{"key":"atypb98","doi-asserted-by":"publisher","DOI":"10.1088\/1751-8113\/48\/19\/195204"},{"key":"atypb99","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.95.062216"}],"container-title":["SIAM Journal on Applied Dynamical Systems"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/21M1398410","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T12:21:08Z","timestamp":1787228468000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/21M1398410"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,1]]},"references-count":99,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2021,1]]}},"alternative-id":["10.1137\/21M1398410"],"URL":"https:\/\/doi.org\/10.1137\/21m1398410","relation":{},"ISSN":["1536-0040"],"issn-type":[{"value":"1536-0040","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,1]]}}}