[{"date_updated":"2026-06-30T07:20:56Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","article_processing_charge":"No","oa_version":"Published Version","publication_status":"published","fulldoi":"https://doi.org/10.2140/apde.2023.16.1245","external_id":{"arxiv":["2101.12274"]},"issue":"5","abstract":[{"lang":"eng","text":"We consider the derivative nonlinear Schrödinger equation in one space dimension, posed both on the line and on the circle. This model is known to be completely integrable and L^2-critical with respect to scaling. We first discuss whether ensembles of orbits with L^2-equicontinuous initial data remain equicontinuous under evolution. We prove that this is true under the restriction \r\nM(q)=∫∣∣q∣∣2<4π. We conjecture that this restriction is unnecessary. Further, we prove that the problem is globally well posed for initial data in H1∕6 under the same restriction on M. Moreover, we show that this restriction would be removed by a successful resolution of our equicontinuity conjecture."}],"tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","short":"CC BY (4.0)"},"status":"public","year":"2023","publication_identifier":{"issn":["2157-5045"],"eissn":["1948-206X"]},"doi":"10.2140/apde.2023.16.1245","month":"08","citation":{"mla":"Killip, Rowan, et al. “On the Well-Posedness Problem for the Derivativenonlinear Schrödinger Equation.” <i>Analysis &#38; PDE</i>, vol. 16, no. 5, Mathematical Sciences Publishers, 2023, pp. 1245–70, doi:<a href=\"https://doi.org/10.2140/apde.2023.16.1245\">10.2140/apde.2023.16.1245</a>.","chicago":"Killip, Rowan, Maria Ntekoume, and Monica Vişan. “On the Well-Posedness Problem for the Derivativenonlinear Schrödinger Equation.” <i>Analysis &#38; PDE</i>. Mathematical Sciences Publishers, 2023. <a href=\"https://doi.org/10.2140/apde.2023.16.1245\">https://doi.org/10.2140/apde.2023.16.1245</a>.","short":"R. Killip, M. Ntekoume, M. Vişan, Analysis &#38; PDE 16 (2023) 1245–1270.","ista":"Killip R, Ntekoume M, Vişan M. 2023. On the well-posedness problem for the derivativenonlinear Schrödinger equation. Analysis &#38; PDE. 16(5), 1245–1270.","ama":"Killip R, Ntekoume M, Vişan M. On the well-posedness problem for the derivativenonlinear Schrödinger equation. <i>Analysis &#38; PDE</i>. 2023;16(5):1245-1270. doi:<a href=\"https://doi.org/10.2140/apde.2023.16.1245\">10.2140/apde.2023.16.1245</a>","ieee":"R. Killip, M. Ntekoume, and M. Vişan, “On the well-posedness problem for the derivativenonlinear Schrödinger equation,” <i>Analysis &#38; PDE</i>, vol. 16, no. 5. Mathematical Sciences Publishers, pp. 1245–1270, 2023.","apa":"Killip, R., Ntekoume, M., &#38; Vişan, M. (2023). On the well-posedness problem for the derivativenonlinear Schrödinger equation. <i>Analysis &#38; PDE</i>. Mathematical Sciences Publishers. <a href=\"https://doi.org/10.2140/apde.2023.16.1245\">https://doi.org/10.2140/apde.2023.16.1245</a>"},"day":"12","type":"journal_article","main_file_link":[{"open_access":"1","url":"https://doi.org/10.2140/apde.2023.16.1245"}],"language":[{"iso":"eng"}],"arxiv":1,"date_published":"2023-08-12T00:00:00Z","quality_controlled":"1","mathsc":["35Q55"],"volume":16,"OA_type":"diamond","OA_place":"publisher","title":"On the well-posedness problem for the derivativenonlinear Schrödinger equation","author":[{"full_name":"Killip, Rowan","first_name":"Rowan","last_name":"Killip"},{"first_name":"Maria","last_name":"Ntekoume","full_name":"Ntekoume, Maria"},{"last_name":"Visan","first_name":"Monica","id":"056daca0-b8d1-11f0-964f-f91054abf8ca","full_name":"Visan, Monica"}],"date_created":"2026-06-19T08:15:32Z","oa":1,"has_accepted_license":"1","article_type":"original","page":"1245-1270","ddc":["500"],"das_tickbox":"1","_id":"22067","extern":"1","publication":"Analysis & PDE","scopus_import":"1","intvolume":"        16","publisher":"Mathematical Sciences Publishers"},{"abstract":[{"lang":"eng","text":"We prove an adiabatic theorem for the Landau–Pekar equations. This allows us to derive new results on the accuracy of their use as effective equations for the time evolution generated by the Fröhlich Hamiltonian with large coupling constant α. In particular, we show that the time evolution of Pekar product states with coherent phonon field and the electron being trapped by the phonons is well approximated by the Landau–Pekar equations until times short compared to α2."}],"issue":"7","year":"2021","acknowledgement":"N. L. and R. S. gratefully acknowledge financial support by the European Research Council\r\n(ERC) under the European Union’s Horizon 2020 research and innovation programme (grant\r\nagreement No 694227). B. S. acknowledges support from the Swiss National Science Foundation (grant 200020_172623) and from the NCCR SwissMAP. N. L. would like to thank\r\nAndreas Deuchert and David Mitrouskas for interesting discussions. B. S. and R. S. would\r\nlike to thank Rupert Frank for stimulating discussions about the time-evolution of a polaron.\r\n","status":"public","publication_identifier":{"eissn":["1948-206X"],"issn":["2157-5045"]},"doi":"10.2140/APDE.2021.14.2079","month":"11","citation":{"mla":"Leopold, Nikolai K., et al. “The Landau–Pekar Equations: Adiabatic Theorem and Accuracy.” <i>Analysis and PDE</i>, vol. 14, no. 7, Mathematical Sciences Publishers, 2021, pp. 2079–100, doi:<a href=\"https://doi.org/10.2140/APDE.2021.14.2079\">10.2140/APDE.2021.14.2079</a>.","chicago":"Leopold, Nikolai K, Simone Anna Elvira Rademacher, Benjamin Schlein, and Robert Seiringer. “The Landau–Pekar Equations: Adiabatic Theorem and Accuracy.” <i>Analysis and PDE</i>. Mathematical Sciences Publishers, 2021. <a href=\"https://doi.org/10.2140/APDE.2021.14.2079\">https://doi.org/10.2140/APDE.2021.14.2079</a>.","short":"N.K. Leopold, S.A.E. Rademacher, B. Schlein, R. Seiringer, Analysis and PDE 14 (2021) 2079–2100.","ama":"Leopold NK, Rademacher SAE, Schlein B, Seiringer R. The Landau–Pekar equations: Adiabatic theorem and accuracy. <i>Analysis and PDE</i>. 2021;14(7):2079-2100. doi:<a href=\"https://doi.org/10.2140/APDE.2021.14.2079\">10.2140/APDE.2021.14.2079</a>","apa":"Leopold, N. K., Rademacher, S. A. E., Schlein, B., &#38; Seiringer, R. (2021). The Landau–Pekar equations: Adiabatic theorem and accuracy. <i>Analysis and PDE</i>. Mathematical Sciences Publishers. <a href=\"https://doi.org/10.2140/APDE.2021.14.2079\">https://doi.org/10.2140/APDE.2021.14.2079</a>","ista":"Leopold NK, Rademacher SAE, Schlein B, Seiringer R. 2021. The Landau–Pekar equations: Adiabatic theorem and accuracy. Analysis and PDE. 14(7), 2079–2100.","ieee":"N. K. Leopold, S. A. E. Rademacher, B. Schlein, and R. Seiringer, “The Landau–Pekar equations: Adiabatic theorem and accuracy,” <i>Analysis and PDE</i>, vol. 14, no. 7. Mathematical Sciences Publishers, pp. 2079–2100, 2021."},"type":"journal_article","day":"10","ec_funded":1,"main_file_link":[{"url":"https://arxiv.org/abs/1904.12532","open_access":"1"}],"date_updated":"2026-08-12T08:46:52Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","oa_version":"Preprint","article_processing_charge":"No","publication_status":"published","department":[{"_id":"RoSe"}],"fulldoi":"https://doi.org/10.2140/APDE.2021.14.2079","external_id":{"arxiv":["1904.12532"],"isi":["000733976600004"]},"oa":1,"date_created":"2022-02-06T23:01:33Z","project":[{"name":"Analysis of quantum many-body systems","call_identifier":"H2020","grant_number":"694227","_id":"25C6DC12-B435-11E9-9278-68D0E5697425"}],"article_type":"original","_id":"10738","page":"2079-2100","scopus_import":"1","intvolume":"        14","publication":"Analysis and PDE","isi":1,"publisher":"Mathematical Sciences Publishers","arxiv":1,"language":[{"iso":"eng"}],"corr_author":"1","date_published":"2021-11-10T00:00:00Z","volume":14,"quality_controlled":"1","author":[{"orcid":"0000-0002-0495-6822","full_name":"Leopold, Nikolai K","id":"4BC40BEC-F248-11E8-B48F-1D18A9856A87","last_name":"Leopold","first_name":"Nikolai K"},{"full_name":"Rademacher, Simone Anna Elvira","orcid":"0000-0001-5059-4466","last_name":"Rademacher","first_name":"Simone Anna Elvira","id":"856966FE-A408-11E9-977E-802DE6697425"},{"last_name":"Schlein","first_name":"Benjamin","full_name":"Schlein, Benjamin"},{"full_name":"Seiringer, Robert","orcid":"0000-0002-6781-0521","first_name":"Robert","last_name":"Seiringer","id":"4AFD0470-F248-11E8-B48F-1D18A9856A87"}],"title":"The Landau–Pekar equations: Adiabatic theorem and accuracy"},{"date_published":"2016-11-07T00:00:00Z","language":[{"iso":"eng"}],"arxiv":1,"title":"The final-state problem for the cubic-quintic NLS with nonvanishing boundary conditions","author":[{"last_name":"Killip","first_name":"Rowan","full_name":"Killip, Rowan"},{"last_name":"Murphy","first_name":"Jason","full_name":"Murphy, Jason"},{"full_name":"Visan, Monica","first_name":"Monica","last_name":"Visan","id":"056daca0-b8d1-11f0-964f-f91054abf8ca"}],"quality_controlled":"1","mathsc":["35Q55"],"volume":9,"OA_type":"green","OA_place":"repository","page":"1523-1574","das_tickbox":"1","_id":"22051","publication":"Analysis & PDE","extern":"1","scopus_import":"1","intvolume":"         9","oa":1,"date_created":"2026-06-19T07:54:01Z","article_type":"original","publisher":"Mathematical Sciences Publishers","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","article_processing_charge":"No","oa_version":"Preprint","date_updated":"2026-06-25T08:23:10Z","fulldoi":"https://doi.org/10.2140/apde.2016.9.1523","external_id":{"arxiv":["1506.06151"]},"publication_status":"published","status":"public","year":"2016","doi":"10.2140/apde.2016.9.1523","publication_identifier":{"issn":["2157-5045"],"eissn":["1948-206X"]},"issue":"7","abstract":[{"lang":"eng","text":"We construct solutions with prescribed scattering state to the cubic-quintic NLS (mathematical formular)in three spatial dimensions in the class of solutions with (mathematical formular). This models disturbances in an infinite expanse of (quantum) fluid in its quiescent state— the limiting modulus c corresponds to a local minimum in the energy density.\r\nOur arguments build on work of Gustafson, Nakanishi, and Tsai on the (defocusing) Gross–Pitaevskii equation. The presence of an energy-critical nonlinearity and changes in the geometry of the energy\r\nfunctional add several new complexities. One new ingredient in our argument is a demonstration that\r\nsolutions of such (perturbed) energy-critical equations exhibit continuous dependence on the initial data\r\nwith respect to the weak topology on H1/x."}],"type":"journal_article","day":"07","keyword":["final-state problem","wave operators","cubic-quintic NLS","nonvanishing boundary conditions"],"main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.1506.06151"}],"month":"11","citation":{"short":"R. Killip, J. Murphy, M. Vişan, Analysis &#38; PDE 9 (2016) 1523–1574.","chicago":"Killip, Rowan, Jason Murphy, and Monica Vişan. “The Final-State Problem for the Cubic-Quintic NLS with Nonvanishing Boundary Conditions.” <i>Analysis &#38; PDE</i>. Mathematical Sciences Publishers, 2016. <a href=\"https://doi.org/10.2140/apde.2016.9.1523\">https://doi.org/10.2140/apde.2016.9.1523</a>.","mla":"Killip, Rowan, et al. “The Final-State Problem for the Cubic-Quintic NLS with Nonvanishing Boundary Conditions.” <i>Analysis &#38; PDE</i>, vol. 9, no. 7, Mathematical Sciences Publishers, 2016, pp. 1523–74, doi:<a href=\"https://doi.org/10.2140/apde.2016.9.1523\">10.2140/apde.2016.9.1523</a>.","ieee":"R. Killip, J. Murphy, and M. Vişan, “The final-state problem for the cubic-quintic NLS with nonvanishing boundary conditions,” <i>Analysis &#38; PDE</i>, vol. 9, no. 7. Mathematical Sciences Publishers, pp. 1523–1574, 2016.","ista":"Killip R, Murphy J, Vişan M. 2016. The final-state problem for the cubic-quintic NLS with nonvanishing boundary conditions. Analysis &#38; PDE. 9(7), 1523–1574.","apa":"Killip, R., Murphy, J., &#38; Vişan, M. (2016). The final-state problem for the cubic-quintic NLS with nonvanishing boundary conditions. <i>Analysis &#38; PDE</i>. Mathematical Sciences Publishers. <a href=\"https://doi.org/10.2140/apde.2016.9.1523\">https://doi.org/10.2140/apde.2016.9.1523</a>","ama":"Killip R, Murphy J, Vişan M. The final-state problem for the cubic-quintic NLS with nonvanishing boundary conditions. <i>Analysis &#38; PDE</i>. 2016;9(7):1523-1574. doi:<a href=\"https://doi.org/10.2140/apde.2016.9.1523\">10.2140/apde.2016.9.1523</a>"}},{"scopus_import":"1","intvolume":"         5","extern":"1","publication":"Analysis & PDE","_id":"22024","das_tickbox":"1","page":"855-885","article_type":"original","date_created":"2026-06-19T07:32:05Z","oa":1,"publisher":"Mathematical Sciences Publishers","date_published":"2012-11-27T00:00:00Z","arxiv":1,"language":[{"iso":"eng"}],"author":[{"full_name":"Killip, Rowan","first_name":"Rowan","last_name":"Killip"},{"full_name":"Visan, Monica","id":"056daca0-b8d1-11f0-964f-f91054abf8ca","last_name":"Visan","first_name":"Monica"}],"title":"Global well-posedness and scattering for the defocusing quintic NLS in three dimensions","OA_place":"repository","OA_type":"green","volume":5,"quality_controlled":"1","publication_identifier":{"eissn":["1948-206X"],"issn":["2157-5045"]},"doi":"10.2140/apde.2012.5.855","year":"2012","status":"public","abstract":[{"text":"We revisit the proof of global well-posedness and scattering for the defocusing energy-critical NLS in three space dimensions in light of recent developments. This result was obtained previously by Colliander, Keel, Staffilani, Takaoka, and Tao.","lang":"eng"}],"issue":"4","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.1102.1192"}],"keyword":["energy critical","nonlinear Schrödinger"],"day":"27","type":"journal_article","citation":{"short":"R. Killip, M. Vişan, Analysis &#38; PDE 5 (2012) 855–885.","mla":"Killip, Rowan, and Monica Vişan. “Global Well-Posedness and Scattering for the Defocusing Quintic NLS in Three Dimensions.” <i>Analysis &#38; PDE</i>, vol. 5, no. 4, Mathematical Sciences Publishers, 2012, pp. 855–85, doi:<a href=\"https://doi.org/10.2140/apde.2012.5.855\">10.2140/apde.2012.5.855</a>.","chicago":"Killip, Rowan, and Monica Vişan. “Global Well-Posedness and Scattering for the Defocusing Quintic NLS in Three Dimensions.” <i>Analysis &#38; PDE</i>. Mathematical Sciences Publishers, 2012. <a href=\"https://doi.org/10.2140/apde.2012.5.855\">https://doi.org/10.2140/apde.2012.5.855</a>.","ista":"Killip R, Vişan M. 2012. Global well-posedness and scattering for the defocusing quintic NLS in three dimensions. Analysis &#38; PDE. 5(4), 855–885.","apa":"Killip, R., &#38; Vişan, M. (2012). Global well-posedness and scattering for the defocusing quintic NLS in three dimensions. <i>Analysis &#38; PDE</i>. Mathematical Sciences Publishers. <a href=\"https://doi.org/10.2140/apde.2012.5.855\">https://doi.org/10.2140/apde.2012.5.855</a>","ieee":"R. Killip and M. Vişan, “Global well-posedness and scattering for the defocusing quintic NLS in three dimensions,” <i>Analysis &#38; PDE</i>, vol. 5, no. 4. Mathematical Sciences Publishers, pp. 855–885, 2012.","ama":"Killip R, Vişan M. Global well-posedness and scattering for the defocusing quintic NLS in three dimensions. <i>Analysis &#38; PDE</i>. 2012;5(4):855-885. doi:<a href=\"https://doi.org/10.2140/apde.2012.5.855\">10.2140/apde.2012.5.855</a>"},"month":"11","article_processing_charge":"No","oa_version":"Preprint","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_updated":"2026-06-22T10:20:04Z","external_id":{"arxiv":["1102.1192"]},"fulldoi":"https://doi.org/10.2140/apde.2012.5.855","publication_status":"published"},{"publication_status":"published","fulldoi":"https://doi.org/10.2140/apde.2008.1.229","external_id":{"arxiv":["0708.0849"]},"date_updated":"2026-07-01T07:15:15Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","article_processing_charge":"No","oa_version":"Preprint","month":"06","citation":{"ieee":"R. Killip, M. Vişan, and X. Zhang, “The mass-critical nonlinear Schrödinger equation with radial data in dimensions three and higher,” <i>Analysis &#38; PDE</i>, vol. 1, no. 2. Mathematical Sciences Publishers, pp. 229–266, 2008.","ama":"Killip R, Vişan M, Zhang X. The mass-critical nonlinear Schrödinger equation with radial data in dimensions three and higher. <i>Analysis &#38; PDE</i>. 2008;1(2):229-266. doi:<a href=\"https://doi.org/10.2140/apde.2008.1.229\">10.2140/apde.2008.1.229</a>","ista":"Killip R, Vişan M, Zhang X. 2008. The mass-critical nonlinear Schrödinger equation with radial data in dimensions three and higher. Analysis &#38; PDE. 1(2), 229–266.","apa":"Killip, R., Vişan, M., &#38; Zhang, X. (2008). The mass-critical nonlinear Schrödinger equation with radial data in dimensions three and higher. <i>Analysis &#38; PDE</i>. Mathematical Sciences Publishers. <a href=\"https://doi.org/10.2140/apde.2008.1.229\">https://doi.org/10.2140/apde.2008.1.229</a>","short":"R. Killip, M. Vişan, X. Zhang, Analysis &#38; PDE 1 (2008) 229–266.","chicago":"Killip, Rowan, Monica Vişan, and Xiaoyi Zhang. “The Mass-Critical Nonlinear Schrödinger Equation with Radial Data in Dimensions Three and Higher.” <i>Analysis &#38; PDE</i>. Mathematical Sciences Publishers, 2008. <a href=\"https://doi.org/10.2140/apde.2008.1.229\">https://doi.org/10.2140/apde.2008.1.229</a>.","mla":"Killip, Rowan, et al. “The Mass-Critical Nonlinear Schrödinger Equation with Radial Data in Dimensions Three and Higher.” <i>Analysis &#38; PDE</i>, vol. 1, no. 2, Mathematical Sciences Publishers, 2008, pp. 229–66, doi:<a href=\"https://doi.org/10.2140/apde.2008.1.229\">10.2140/apde.2008.1.229</a>."},"type":"journal_article","day":"01","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.0708.0849","open_access":"1"}],"abstract":[{"lang":"eng","text":"We establish global well-posedness and scattering for solutions to the mass-critical nonlinear Schrödinger equation (mathematical formular)  for large spherically symmetric L2/x(R^d) initial data in dimensions d >= 3.\r\nIn the focusing case we require that the mass is strictly less than that of the ground state. As a consequence, we obtain that in the focusing case, any spherically symmetric blowup solution must concentrate at least the mass of the ground state at the blowup time."}],"issue":"2","year":"2008","status":"public","doi":"10.2140/apde.2008.1.229","publication_identifier":{"issn":["2157-5045"],"eissn":["1948-206X"]},"volume":1,"OA_type":"green","quality_controlled":"1","OA_place":"repository","title":"The mass-critical nonlinear Schrödinger equation with radial data in dimensions three and higher","author":[{"full_name":"Killip, Rowan","last_name":"Killip","first_name":"Rowan"},{"full_name":"Visan, Monica","first_name":"Monica","last_name":"Visan","id":"056daca0-b8d1-11f0-964f-f91054abf8ca"},{"last_name":"Zhang","first_name":"Xiaoyi","full_name":"Zhang, Xiaoyi"}],"arxiv":1,"language":[{"iso":"eng"}],"date_published":"2008-06-01T00:00:00Z","publisher":"Mathematical Sciences Publishers","oa":1,"date_created":"2026-06-19T08:27:33Z","article_type":"original","_id":"22082","page":"229-266","das_tickbox":"1","intvolume":"         1","scopus_import":"1","extern":"1","publication":"Analysis & PDE"}]