{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,11]],"date-time":"2025-09-11T22:35:56Z","timestamp":1757630156477,"version":"3.44.0"},"reference-count":23,"publisher":"Springer Science and Business Media LLC","issue":"5","license":[{"start":{"date-parts":[[2025,8,19]],"date-time":"2025-08-19T00:00:00Z","timestamp":1755561600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,8,19]],"date-time":"2025-08-19T00:00:00Z","timestamp":1755561600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["258734477","258734477"],"award-info":[{"award-number":["258734477","258734477"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100009534","name":"Universit\u00e4t Stuttgart","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100009534","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Nonlinear Sci"],"published-print":{"date-parts":[[2025,10]]},"abstract":"<jats:title>Abstract<\/jats:title>\n          <jats:p>We are interested in improving validity results for the nonlinear Schr\u00f6dinger (NLS) approximation beyond the natural time scale for completely integrable systems. As a first step, we consider this approximation for the Korteweg\u2013de Vries equation with initial conditions for which the scattering data contain no eigenvalues. By performing a linear Schr\u00f6dinger approximation for the scattering data, the error made by this approximation has only to be estimated for a purely linear problem which gives estimates beyond the natural NLS time scale.  The inverse scattering transform allows us to transfer these estimates to the original variables.<\/jats:p>","DOI":"10.1007\/s00332-025-10195-y","type":"journal-article","created":{"date-parts":[[2025,8,19]],"date-time":"2025-08-19T11:52:24Z","timestamp":1755604344000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A Linear Schr\u00f6dinger Approximation for the KdV Equation via IST Beyond the Natural NLS Time Scale"],"prefix":"10.1007","volume":"35","author":[{"given":"Sarah","family":"Hofbauer","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Guido","family":"Schneider","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,8,19]]},"reference":[{"key":"10195_CR1","doi-asserted-by":"publisher","first-page":"13","DOI":"10.1002\/sapm197757113","volume":"57","author":"MJ Ablowitz","year":"1977","unstructured":"Ablowitz, M.J., Segur, H.: Asymptotic solutions of the Korteweg\u2013deVries equation. Stud. Appl. Math. 57, 13\u201344 (1977)","journal-title":"Stud. Appl. Math."},{"issue":"1","key":"10195_CR2","first-page":"109","volume":"2","author":"D Bambusi","year":"2002","unstructured":"Bambusi, D., Carati, A., Ponno, A.: The nonlinear Schr\u00f6dinger equation as a resonant normal form. Discrete Contin. Dyn. Syst. Ser. B 2(1), 109\u2013128 (2002)","journal-title":"Discrete Contin. Dyn. Syst. Ser. B"},{"issue":"15","key":"10195_CR3","doi-asserted-by":"publisher","first-page":"1965","DOI":"10.1002\/mma.884","volume":"30","author":"M Chirilus-Bruckner","year":"2007","unstructured":"Chirilus-Bruckner, M., Schneider, G., Uecker, H.: On the interaction of NLS-described modulating pulses with different carrier waves. Math. Methods Appl. Sci. 30(15), 1965\u20131978 (2007)","journal-title":"Math. Methods Appl. Sci."},{"issue":"7","key":"10195_CR4","doi-asserted-by":"publisher","first-page":"2161","DOI":"10.1016\/j.jde.2012.06.003","volume":"253","author":"M Chirilus-Bruckner","year":"2012","unstructured":"Chirilus-Bruckner, M., Schneider, G.: Detection of standing pulses in periodic media by pulse interaction. J. Differ. Equ. 253(7), 2161\u20132190 (2012)","journal-title":"J. Differ. Equ."},{"issue":"1","key":"10195_CR5","doi-asserted-by":"publisher","first-page":"304","DOI":"10.1016\/j.jmaa.2008.05.101","volume":"347","author":"M Chirilus-Bruckner","year":"2008","unstructured":"Chirilus-Bruckner, M., Chong, C., Schneider, G., Uecker, H.: Separation of internal and interaction dynamics for NLS-described wave packets with different carrier waves. J. Math. Anal. Appl. 347(1), 304\u2013314 (2008)","journal-title":"J. Math. Anal. Appl."},{"key":"10195_CR6","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781139172059","volume-title":"Solitons: An Introduction. Cambridge Texts in Applied Mathematics","author":"PG Drazin","year":"1989","unstructured":"Drazin, P.G., Johnson, R.S.: Solitons: An Introduction. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (1989)"},{"issue":"2","key":"10195_CR7","doi-asserted-by":"publisher","first-page":"831","DOI":"10.1007\/s00205-020-01586-4","volume":"239","author":"W-P D\u00fcll","year":"2021","unstructured":"D\u00fcll, W.-P.: Validity of the nonlinear Schr\u00f6dinger approximation for the two-dimensional water wave problem with and without surface tension in the arc length formulation. Arch. Ration. Mech. Anal. 239(2), 831\u2013914 (2021)","journal-title":"Arch. Ration. Mech. Anal."},{"issue":"1","key":"10195_CR8","doi-asserted-by":"publisher","first-page":"175","DOI":"10.1007\/BF02099741","volume":"165","author":"PA Deift","year":"1994","unstructured":"Deift, P.A., Zhou, X.: Long-time asymptotics for integrable systems. Higher order theory. Commun. Math. Phys. 165(1), 175\u2013191 (1994)","journal-title":"Commun. Math. Phys."},{"key":"10195_CR9","volume-title":"The Inverse Scattering Transformation and the Theory of Solitons: An Introduction","author":"W Eckhaus","year":"1981","unstructured":"Eckhaus, W., van Harten, A.: The Inverse Scattering Transformation and the Theory of Solitons: An Introduction. North-Holland Publ, Company (1981)"},{"key":"10195_CR10","volume-title":"Schr\u00f6dinger Operators: Eigenvalues and Lieb\u2013Thirring Inequalities, volume 200 of Cambridge Studies in Advanced Mathematics","author":"RL Frank","year":"2023","unstructured":"Frank, R.L., Laptev, A., Weidl, T.: Schr\u00f6dinger Operators: Eigenvalues and Lieb\u2013Thirring Inequalities, volume 200 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2023)"},{"issue":"3","key":"10195_CR11","doi-asserted-by":"publisher","first-page":"287","DOI":"10.1007\/s11040-009-9062-2","volume":"12","author":"K Grunert","year":"2009","unstructured":"Grunert, K., Teschl, G.: Long-time asymptotics for the Korteweg\u2013de Vries equation via nonlinear steepest descent. Math. Phys. Anal. Geom. 12(3), 287\u2013324 (2009)","journal-title":"Math. Phys. Anal. Geom."},{"issue":"2","key":"10195_CR12","doi-asserted-by":"publisher","first-page":"457","DOI":"10.1070\/SM1988v060n02ABEH003181","volume":"60","author":"LA Kalyakin","year":"1988","unstructured":"Kalyakin, L.A.: Asymptotic decay of a one-dimensional wavepacket in a nonlinear dispersive medium. Math. USSR Sbornik 60(2), 457\u2013483 (1988)","journal-title":"Math. USSR Sbornik"},{"key":"10195_CR13","volume-title":"KdV & KAM, volume\u00a045 of Ergeb. Math. Grenzgeb. 3. Folge","author":"T Kappeler","year":"2003","unstructured":"Kappeler, T., P\u00f6schel, J.: KdV & KAM, volume\u00a045 of Ergeb. Math. Grenzgeb. 3. Folge. Springer, Berlin (2003)"},{"issue":"1\u20132","key":"10195_CR14","doi-asserted-by":"publisher","first-page":"85","DOI":"10.1017\/S0308210500020989","volume":"122","author":"P Kirrmann","year":"1992","unstructured":"Kirrmann, P., Schneider, G., Mielke, A.: The validity of modulation equations for extended systems with cubic nonlinearities. Proc. R. Soc. Edinb. Sect. A 122(1\u20132), 85\u201391 (1992)","journal-title":"Proc. R. Soc. Edinb. Sect. A"},{"issue":"5","key":"10195_CR15","doi-asserted-by":"publisher","first-page":"769","DOI":"10.1088\/0951-7715\/8\/5\/007","volume":"8","author":"RD Pierce","year":"1995","unstructured":"Pierce, R.D., Wayne, C.E.: On the validity of mean-field amplitude equations for counterpropagating wavetrains. Nonlinearity 8(5), 769\u2013779 (1995)","journal-title":"Nonlinearity"},{"issue":"2","key":"10195_CR16","doi-asserted-by":"publisher","first-page":"354","DOI":"10.1016\/j.jde.2005.04.018","volume":"216","author":"G Schneider","year":"2005","unstructured":"Schneider, G.: Justification and failure of the nonlinear Schr\u00f6dinger equation in case of non-trivial quadratic resonances. J. Differ. Equ. 216(2), 354\u2013386 (2005)","journal-title":"J. Differ. Equ."},{"key":"10195_CR17","doi-asserted-by":"publisher","first-page":"4","DOI":"10.1155\/2011\/854719","volume":"2011","author":"G Schneider","year":"2011","unstructured":"Schneider, G.: Justification of the NLS approximation for the KdV equation using the Miura transformation. Adv. Math. Phys. 2011, 4 (2011)","journal-title":"Adv. Math. Phys."},{"key":"10195_CR18","first-page":"2015","volume":"267\u2013275","author":"G Schneider","year":"2015","unstructured":"Schneider, G., Chirilus-Bruckner, M.: Interaction of oscillatory packets of water waves. Discrete Contin. Dyn. Syst. 267\u2013275, 2015 (2015)","journal-title":"Discrete Contin. Dyn. Syst."},{"key":"10195_CR19","doi-asserted-by":"publisher","first-page":"721","DOI":"10.1017\/S0022112073001813","volume":"59","author":"H Segur","year":"1973","unstructured":"Segur, H.: The Korteweg\u2013de Vries equation and water waves. Solutions of the equation. I. J. Fluid Mech. 59, 721\u2013736 (1973)","journal-title":"J. Fluid Mech."},{"issue":"1","key":"10195_CR20","doi-asserted-by":"publisher","first-page":"83","DOI":"10.1016\/0167-2789(88)90088-7","volume":"32","author":"ER Tracy","year":"1988","unstructured":"Tracy, E.R., Larson, J.W., Osborne, A.R., Bergamasco, L.: On the nonlinear Schr\u00f6dinger limit of the Korteweg\u2013de Vries equation. Physica D 32(1), 83\u2013106 (1988)","journal-title":"Physica D"},{"issue":"3","key":"10195_CR21","doi-asserted-by":"publisher","first-page":"817","DOI":"10.1007\/s00220-012-1422-2","volume":"310","author":"N Totz","year":"2012","unstructured":"Totz, N., Sijue, W.: A rigorous justification of the modulation approximation to the 2D full water wave problem. Commun. Math. Phys. 310(3), 817\u2013883 (2012)","journal-title":"Commun. Math. Phys."},{"key":"10195_CR22","first-page":"190","volume":"4","author":"VE Zakharov","year":"1968","unstructured":"Zakharov, V.E.: Stability of periodic waves of finite amplitude on the surface of a deep fluid. Sov. Phys. J. Appl. Mech. Tech. Phys 4, 190\u2013194 (1968)","journal-title":"Sov. Phys. J. Appl. Mech. Tech. Phys"},{"key":"10195_CR23","doi-asserted-by":"publisher","first-page":"455","DOI":"10.1016\/0167-2789(86)90214-9","volume":"18","author":"VE Zakharov","year":"1986","unstructured":"Zakharov, V.E., Kuznetsov, E.A.: Multi-scale expansions in the theory of systems integrable by the inverse scattering transform. Physica D 18, 455\u2013463 (1986)","journal-title":"Physica D"}],"container-title":["Journal of Nonlinear Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00332-025-10195-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00332-025-10195-y\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00332-025-10195-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,9,10]],"date-time":"2025-09-10T16:58:06Z","timestamp":1757523486000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00332-025-10195-y"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,8,19]]},"references-count":23,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2025,10]]}},"alternative-id":["10195"],"URL":"https:\/\/doi.org\/10.1007\/s00332-025-10195-y","relation":{},"ISSN":["0938-8974","1432-1467"],"issn-type":[{"type":"print","value":"0938-8974"},{"type":"electronic","value":"1432-1467"}],"subject":[],"published":{"date-parts":[[2025,8,19]]},"assertion":[{"value":"1 July 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"17 July 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"19 August 2025","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors have no relevant financial or non-financial interests to disclose. The authors have no Conflict of interest to declare that are relevant to the content of this article. All authors certify that they have no affiliations with or involvement in any organization or entity with any financial interest or non-financial interest in the subject matter or materials discussed in this manuscript. The authors have no financial or proprietary interests in any material discussed in this article.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"103"}}