[{"extern":"1","article_type":"original","oa":1,"issue":"3","doi":"10.2140/paa.2025.7.615","language":[{"iso":"eng"}],"date_published":"2025-06-18T00:00:00Z","publisher":"Mathematical Sciences Publishers","year":"2025","quality_controlled":"1","external_id":{"arxiv":["2411.05300"]},"das_tickbox":"1","page":"615-637","_id":"22021","arxiv":1,"scopus_import":"1","month":"06","citation":{"ieee":"S. Haque, R. Killip, M. Vişan, and Y. Zhang, “ Global well-posedness and equicontinuity for modified Korteweg–de Vries equations in modulation spaces,” <i>Pure and Applied Analysis</i>, vol. 7, no. 3. Mathematical Sciences Publishers, pp. 615–637, 2025.","ama":"Haque S, Killip R, Vişan M, Zhang Y.  Global well-posedness and equicontinuity for modified Korteweg–de Vries equations in modulation spaces. <i>Pure and Applied Analysis</i>. 2025;7(3):615-637. doi:<a href=\"https://doi.org/10.2140/paa.2025.7.615\">10.2140/paa.2025.7.615</a>","apa":"Haque, S., Killip, R., Vişan, M., &#38; Zhang, Y. (2025).  Global well-posedness and equicontinuity for modified Korteweg–de Vries equations in modulation spaces. <i>Pure and Applied Analysis</i>. Mathematical Sciences Publishers. <a href=\"https://doi.org/10.2140/paa.2025.7.615\">https://doi.org/10.2140/paa.2025.7.615</a>","short":"S. Haque, R. Killip, M. Vişan, Y. Zhang, Pure and Applied Analysis 7 (2025) 615–637.","ista":"Haque S, Killip R, Vişan M, Zhang Y. 2025.  Global well-posedness and equicontinuity for modified Korteweg–de Vries equations in modulation spaces. Pure and Applied Analysis. 7(3), 615–637.","chicago":"Haque, Saikatul, Rowan Killip, Monica Vişan, and Yunfeng Zhang. “ Global Well-Posedness and Equicontinuity for Modified Korteweg–de Vries Equations in Modulation Spaces.” <i>Pure and Applied Analysis</i>. Mathematical Sciences Publishers, 2025. <a href=\"https://doi.org/10.2140/paa.2025.7.615\">https://doi.org/10.2140/paa.2025.7.615</a>.","mla":"Haque, Saikatul, et al. “ Global Well-Posedness and Equicontinuity for Modified Korteweg–de Vries Equations in Modulation Spaces.” <i>Pure and Applied Analysis</i>, vol. 7, no. 3, Mathematical Sciences Publishers, 2025, pp. 615–37, doi:<a href=\"https://doi.org/10.2140/paa.2025.7.615\">10.2140/paa.2025.7.615</a>."},"publication_status":"published","type":"journal_article","volume":7,"OA_place":"repository","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","article_processing_charge":"No","status":"public","date_created":"2026-06-19T07:30:23Z","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2411.05300"}],"publication":"Pure and Applied Analysis","mathsc":["35Q53","35Q55","37K10"],"publication_identifier":{"eissn":["2578-5885"],"issn":["2578-5893"]},"OA_type":"green","title":" Global well-posedness and equicontinuity for modified Korteweg–de Vries equations in modulation spaces","day":"18","intvolume":"         7","date_updated":"2026-06-24T13:22:40Z","oa_version":"Preprint","abstract":[{"lang":"eng","text":"We establish global well-posedness for both the defocusing and\r\nfocusing complex-valued modified Korteweg–de Vries equations on the real line\r\nin modulation spaces Ms,2p (R), for all 1 \u0014 p < 1 and 0 \u0014 s < 3/2 − 1/p. We\r\nwill also show that such solutions admit global-in-time bounds in these spaces\r\nand that equicontinuous sets of initial data lead to equicontinuous ensembles\r\nof orbits. Indeed, such information forms a crucial part of our well-posedness\r\nargument."}],"author":[{"first_name":"Saikatul","last_name":"Haque","full_name":"Haque, Saikatul"},{"last_name":"Killip","full_name":"Killip, Rowan","first_name":"Rowan"},{"first_name":"Monica","id":"056daca0-b8d1-11f0-964f-f91054abf8ca","full_name":"Visan, Monica","last_name":"Visan"},{"last_name":"Zhang","full_name":"Zhang, Yunfeng","first_name":"Yunfeng"}]},{"status":"public","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2404.02366","open_access":"1"}],"date_created":"2026-06-19T08:20:28Z","article_processing_charge":"No","mathsc":["37J70","37K10","37K60","35Q53","35Q55"],"publication_identifier":{"issn":["1078-0947"],"eissn":["1553-5231"]},"publication":"Discrete and Continuous Dynamical Systems","title":"The modified Korteweg–de Vries limit of the Ablowitz–Ladik system","day":"01","ddc":["500"],"OA_type":"green","author":[{"last_name":"Killip","full_name":"Killip, Rowan","first_name":"Rowan"},{"full_name":"Ouyang, Zhimeng","last_name":"Ouyang","first_name":"Zhimeng"},{"last_name":"Visan","full_name":"Visan, Monica","id":"056daca0-b8d1-11f0-964f-f91054abf8ca","first_name":"Monica"},{"first_name":"Lei","last_name":"Wu","full_name":"Wu, Lei"}],"date_updated":"2026-06-30T07:34:20Z","intvolume":"        45","oa_version":"Preprint","abstract":[{"lang":"eng","text":"For slowly-varying initial data, solutions to the Ablowitz–Ladik system have been proven to converge to solutions of the cubic Schrödinger equation. In this paper we show that in the continuum limit, solutions to the Ablowitz–Ladik system with H^1 initial data may also converge to solutions of the modified Korteweg–de Vries equation. To exhibit this new limiting behavior, it suffices that the initial data is supported near the inflection points of the dispersion relation associated with the Ablowitz–Ladik system.\r\n\r\nOur arguments employ harmonic analysis tools, Strichartz estimates, and the conservation of mass and energy. Correspondingly, they are applicable beyond the completely integrable models of greatest interest to us."}],"oa":1,"issue":"3","language":[{"iso":"eng"}],"doi":"10.3934/dcds.2024114","publisher":"American Institute of Mathematical Sciences","date_published":"2025-03-01T00:00:00Z","extern":"1","article_type":"original","das_tickbox":"1","external_id":{"arxiv":["2404.02366"]},"page":"821-846","year":"2025","quality_controlled":"1","arxiv":1,"month":"03","scopus_import":"1","_id":"22069","OA_place":"repository","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","citation":{"short":"R. Killip, Z. Ouyang, M. Vişan, L. Wu, Discrete and Continuous Dynamical Systems 45 (2025) 821–846.","ista":"Killip R, Ouyang Z, Vişan M, Wu L. 2025. The modified Korteweg–de Vries limit of the Ablowitz–Ladik system. Discrete and Continuous Dynamical Systems. 45(3), 821–846.","chicago":"Killip, Rowan, Zhimeng Ouyang, Monica Vişan, and Lei Wu. “The Modified Korteweg–de Vries Limit of the Ablowitz–Ladik System.” <i>Discrete and Continuous Dynamical Systems</i>. American Institute of Mathematical Sciences, 2025. <a href=\"https://doi.org/10.3934/dcds.2024114\">https://doi.org/10.3934/dcds.2024114</a>.","mla":"Killip, Rowan, et al. “The Modified Korteweg–de Vries Limit of the Ablowitz–Ladik System.” <i>Discrete and Continuous Dynamical Systems</i>, vol. 45, no. 3, American Institute of Mathematical Sciences, 2025, pp. 821–46, doi:<a href=\"https://doi.org/10.3934/dcds.2024114\">10.3934/dcds.2024114</a>.","ieee":"R. Killip, Z. Ouyang, M. Vişan, and L. Wu, “The modified Korteweg–de Vries limit of the Ablowitz–Ladik system,” <i>Discrete and Continuous Dynamical Systems</i>, vol. 45, no. 3. American Institute of Mathematical Sciences, pp. 821–846, 2025.","apa":"Killip, R., Ouyang, Z., Vişan, M., &#38; Wu, L. (2025). The modified Korteweg–de Vries limit of the Ablowitz–Ladik system. <i>Discrete and Continuous Dynamical Systems</i>. American Institute of Mathematical Sciences. <a href=\"https://doi.org/10.3934/dcds.2024114\">https://doi.org/10.3934/dcds.2024114</a>","ama":"Killip R, Ouyang Z, Vişan M, Wu L. The modified Korteweg–de Vries limit of the Ablowitz–Ladik system. <i>Discrete and Continuous Dynamical Systems</i>. 2025;45(3):821-846. doi:<a href=\"https://doi.org/10.3934/dcds.2024114\">10.3934/dcds.2024114</a>"},"publication_status":"published","volume":45,"type":"journal_article"},{"citation":{"ieee":"B. Harrop-Griffiths, R. Killip, and M. Vişan, “Sharp well-posedness for the cubic NLS and mKdV in H^s(R),” <i>Forum of Mathematics, Pi</i>, vol. 12. Cambridge University Press, 2024.","apa":"Harrop-Griffiths, B., Killip, R., &#38; Vişan, M. (2024). Sharp well-posedness for the cubic NLS and mKdV in H^s(R). <i>Forum of Mathematics, Pi</i>. Cambridge University Press. <a href=\"https://doi.org/10.1017/fmp.2024.4\">https://doi.org/10.1017/fmp.2024.4</a>","ama":"Harrop-Griffiths B, Killip R, Vişan M. Sharp well-posedness for the cubic NLS and mKdV in H^s(R). <i>Forum of Mathematics, Pi</i>. 2024;12. doi:<a href=\"https://doi.org/10.1017/fmp.2024.4\">10.1017/fmp.2024.4</a>","short":"B. Harrop-Griffiths, R. Killip, M. Vişan, Forum of Mathematics, Pi 12 (2024).","mla":"Harrop-Griffiths, Benjamin, et al. “Sharp Well-Posedness for the Cubic NLS and MKdV in H^s(R).” <i>Forum of Mathematics, Pi</i>, vol. 12, e6, Cambridge University Press, 2024, doi:<a href=\"https://doi.org/10.1017/fmp.2024.4\">10.1017/fmp.2024.4</a>.","ista":"Harrop-Griffiths B, Killip R, Vişan M. 2024. Sharp well-posedness for the cubic NLS and mKdV in H^s(R). Forum of Mathematics, Pi. 12, e6.","chicago":"Harrop-Griffiths, Benjamin, Rowan Killip, and Monica Vişan. “Sharp Well-Posedness for the Cubic NLS and MKdV in H^s(R).” <i>Forum of Mathematics, Pi</i>. Cambridge University Press, 2024. <a href=\"https://doi.org/10.1017/fmp.2024.4\">https://doi.org/10.1017/fmp.2024.4</a>."},"publication_status":"published","volume":12,"type":"journal_article","OA_place":"publisher","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","_id":"22079","tmp":{"short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png"},"arxiv":1,"scopus_import":"1","month":"04","year":"2024","quality_controlled":"1","das_tickbox":"1","external_id":{"arxiv":["2003.05011"]},"extern":"1","article_type":"original","oa":1,"language":[{"iso":"eng"}],"doi":"10.1017/fmp.2024.4","date_published":"2024-04-02T00:00:00Z","publisher":"Cambridge University Press","date_updated":"2026-06-30T12:16:50Z","intvolume":"        12","article_number":"e6","oa_version":"Published Version","abstract":[{"lang":"eng","text":"We prove that the cubic nonlinear Schrödinger equation (both focusing and defocusing) is globally well-posed in H^s(R) for any regularity s > −1/2. Well-posedness has long been known for s ≥ 0, see [55], but not previously for any s < 0. The scaling-critical value s = −1/2 is necessarily excluded here, since instantaneous norm inflation is known to occur [11, 40, 48]. We also prove (in a parallel fashion) well-posedness of the real- and complex-valued modified Korteweg–de Vries equations in H^s(R) for any s > −1/2. The best regularity achieved previously was s ≥ 1/4 (see [15, 24, 33, 39]). To overcome the failure of uniform continuity of the data-to-solution map, we employ the method of commuting flows introduced in [37]. In stark contrast with our arguments in [37], an essential ingredient in this paper is the demonstration of a local smoothing effect for both equations. Despite the nonperturbative nature of the well-posedness, the gain of derivatives matches that of the underlying linear equation. To compensate for the local nature of the smoothing estimates, we also demonstrate tightness of orbits. The proofs of both local smoothing and tightness rely on our discovery of a new one-parameter family of coercive microscopic conservation laws that remain meaningful at this low regularity. "}],"author":[{"first_name":"Benjamin","full_name":"Harrop-Griffiths, Benjamin","last_name":"Harrop-Griffiths"},{"first_name":"Rowan","full_name":"Killip, Rowan","last_name":"Killip"},{"id":"056daca0-b8d1-11f0-964f-f91054abf8ca","first_name":"Monica","last_name":"Visan","full_name":"Visan, Monica"}],"OA_type":"gold","has_accepted_license":"1","title":"Sharp well-posedness for the cubic NLS and mKdV in H^s(R)","day":"02","ddc":["500"],"publication":"Forum of Mathematics, Pi","DOAJ_listed":"1","mathsc":["35Q55","35Q53","37K10"],"publication_identifier":{"eissn":["2050-5086"]},"article_processing_charge":"No","PlanS_conform":"1","status":"public","date_created":"2026-06-19T08:26:10Z","main_file_link":[{"url":"https://doi.org/10.1017/fmp.2024.4","open_access":"1"}]},{"arxiv":1,"scopus_import":"1","month":"01","_id":"22056","OA_place":"repository","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","citation":{"ieee":"R. Killip, S. Kwon, S. Shao, and M. Vişan, “On the mass-critical generalized KdV equation,” <i>Discrete and Continuous Dynamical Systems</i>, vol. 32, no. 1. American Institute of Mathematical Sciences, pp. 191–221, 2012.","ama":"Killip R, Kwon S, Shao S, Vişan M. On the mass-critical generalized KdV equation. <i>Discrete and Continuous Dynamical Systems</i>. 2012;32(1):191-221. doi:<a href=\"https://doi.org/10.3934/dcds.2012.32.191\">10.3934/dcds.2012.32.191</a>","apa":"Killip, R., Kwon, S., Shao, S., &#38; Vişan, M. (2012). On the mass-critical generalized KdV equation. <i>Discrete and Continuous Dynamical Systems</i>. American Institute of Mathematical Sciences. <a href=\"https://doi.org/10.3934/dcds.2012.32.191\">https://doi.org/10.3934/dcds.2012.32.191</a>","chicago":"Killip, Rowan, Soonsik Kwon, Shuanglin Shao, and Monica Vişan. “On the Mass-Critical Generalized KdV Equation.” <i>Discrete and Continuous Dynamical Systems</i>. American Institute of Mathematical Sciences, 2012. <a href=\"https://doi.org/10.3934/dcds.2012.32.191\">https://doi.org/10.3934/dcds.2012.32.191</a>.","mla":"Killip, Rowan, et al. “On the Mass-Critical Generalized KdV Equation.” <i>Discrete and Continuous Dynamical Systems</i>, vol. 32, no. 1, American Institute of Mathematical Sciences, 2012, pp. 191–221, doi:<a href=\"https://doi.org/10.3934/dcds.2012.32.191\">10.3934/dcds.2012.32.191</a>.","ista":"Killip R, Kwon S, Shao S, Vişan M. 2012. On the mass-critical generalized KdV equation. Discrete and Continuous Dynamical Systems. 32(1), 191–221.","short":"R. Killip, S. Kwon, S. Shao, M. Vişan, Discrete and Continuous Dynamical Systems 32 (2012) 191–221."},"publication_status":"published","type":"journal_article","volume":32,"oa":1,"issue":"1","date_published":"2012-01-01T00:00:00Z","language":[{"iso":"eng"}],"doi":"10.3934/dcds.2012.32.191","publisher":"American Institute of Mathematical Sciences","extern":"1","article_type":"original","external_id":{"arxiv":["0907.5412"]},"das_tickbox":"1","page":"191-221","year":"2012","quality_controlled":"1","title":"On the mass-critical generalized KdV equation","day":"01","OA_type":"green","author":[{"full_name":"Killip, Rowan","last_name":"Killip","first_name":"Rowan"},{"first_name":"Soonsik","last_name":"Kwon","full_name":"Kwon, Soonsik"},{"last_name":"Shao","full_name":"Shao, Shuanglin","first_name":"Shuanglin"},{"last_name":"Visan","full_name":"Visan, Monica","id":"056daca0-b8d1-11f0-964f-f91054abf8ca","first_name":"Monica"}],"date_updated":"2026-06-29T09:59:46Z","intvolume":"        32","oa_version":"Preprint","abstract":[{"lang":"eng","text":"We consider the mass-critical generalized Korteweg{de Vries equation (∂t + ∂xxx)u = ±∂ x(u 5) for real-valued functions u(t; x). We prove that if the global well-posedness and scattering conjecture for this equation failed, then, conditional on a positive answer to the global well-posedness and scattering conjecture for the masscritical nonlinear Schrffodinger equation (-i∂ t + ∂xx)u = ±(|u| 4u), there exists a minimal-mass blowup solution to the mass-critical generalized KdV equation which is almost periodic modulo the symmetries of the equation. Moreover, we can guarantee that this minimal-mass blowup solution is either a self-similar solution, a soliton-like solution, or a double high-to-low frequency cascade solution."}],"status":"public","date_created":"2026-06-19T07:57:09Z","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.0907.5412"}],"article_processing_charge":"No","publication_identifier":{"issn":["1078-0947"],"eissn":["1553-5231"]},"mathsc":["35Q53"],"publication":"Discrete and Continuous Dynamical Systems"}]