[{"page":"613-680","scopus_import":"1","month":"04","date_published":"2021-04-01T00:00:00Z","arxiv":1,"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","extern":"1","year":"2021","publication":"American Journal of Mathematics","external_id":{"arxiv":["1606.07738"]},"oa":1,"abstract":[{"lang":"eng","text":"We prove that solutions of the cubic nonlinear Schr\\\"odinger equation on $\\Bbb{R}^2$ can be approximated by a finite-dimensional Hamiltonian system, uniformly on bounded sets of initial data. This is despite the wealth of non-compact symmetries: scaling, translation, and Galilei boosts.\r\n\r\nComplementing this approximation result, we show that all solutions of the finite-dimensional Hamiltonian system we use can be approximated by the full PDE.\r\n\r\nA key ingredient in these results is the development of a general methodology for transfering uniform global space-time bounds to suitable Fourier truncations of dispersive PDE models.\r\n\r\nAs an application, we prove symplectic non-squeezing (in the sense of Gromov) for the cubic NLS on $\\Bbb{R}^2$. This is the first symplectic non-squeezing result for a Hamiltonian PDE in infinite volume. It is also the first unconditional symplectic non-squeezing result in a scaling-critical setting.\r\n\r\nFinally, we discuss implications of non-squeezing on the nature of scattering."}],"language":[{"iso":"eng"}],"main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.1606.07738"}],"citation":{"apa":"Killip, R., Vişan, M., & Zhang, X. (2021). Finite-dimensional approximation and non-squeezing for the cubic nonlinear Schrödinger equation on ℝ2. American Journal of Mathematics. Johns Hopkins University Press. https://doi.org/10.1353/ajm.2021.0014","ieee":"R. Killip, M. Vişan, and X. Zhang, “Finite-dimensional approximation and non-squeezing for the cubic nonlinear Schrödinger equation on ℝ2,” American Journal of Mathematics, vol. 143, no. 2. Johns Hopkins University Press, pp. 613–680, 2021.","short":"R. Killip, M. Vişan, X. Zhang, American Journal of Mathematics 143 (2021) 613–680.","ama":"Killip R, Vişan M, Zhang X. Finite-dimensional approximation and non-squeezing for the cubic nonlinear Schrödinger equation on ℝ2. American Journal of Mathematics. 2021;143(2):613-680. doi:10.1353/ajm.2021.0014","chicago":"Killip, Rowan, Monica Vişan, and Xiaoyi Zhang. “Finite-Dimensional Approximation and Non-Squeezing for the Cubic Nonlinear Schrödinger Equation on ℝ2.” American Journal of Mathematics. Johns Hopkins University Press, 2021. https://doi.org/10.1353/ajm.2021.0014.","mla":"Killip, Rowan, et al. “Finite-Dimensional Approximation and Non-Squeezing for the Cubic Nonlinear Schrödinger Equation on ℝ2.” American Journal of Mathematics, vol. 143, no. 2, Johns Hopkins University Press, 2021, pp. 613–80, doi:10.1353/ajm.2021.0014.","ista":"Killip R, Vişan M, Zhang X. 2021. Finite-dimensional approximation and non-squeezing for the cubic nonlinear Schrödinger equation on ℝ2. American Journal of Mathematics. 143(2), 613–680."},"publisher":"Johns Hopkins University Press","author":[{"last_name":"Killip","first_name":"Rowan","full_name":"Killip, Rowan"},{"last_name":"Visan","first_name":"Monica","full_name":"Visan, Monica","id":"056daca0-b8d1-11f0-964f-f91054abf8ca"},{"full_name":"Zhang, Xiaoyi","last_name":"Zhang","first_name":"Xiaoyi"}],"day":"01","date_created":"2026-06-19T07:43:41Z","title":"Finite-dimensional approximation and non-squeezing for the cubic nonlinear Schrödinger equation on ℝ2","status":"public","date_updated":"2026-06-22T12:56:27Z","volume":143,"oa_version":"Preprint","OA_place":"repository","article_type":"original","fulldoi":"https://doi.org/10.1353/ajm.2021.0014","article_processing_charge":"No","das_tickbox":"1","quality_controlled":"1","publication_identifier":{"eissn":["1080-6377"]},"_id":"22035","issue":"2","intvolume":" 143","OA_type":"green","doi":"10.1353/ajm.2021.0014","type":"journal_article","publication_status":"published"},{"page":"613-680","arxiv":1,"date_published":"2021-04-01T00:00:00Z","month":"04","scopus_import":"1","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","year":"2021","extern":"1","publication":"American Journal of Mathematics","external_id":{"arxiv":["1606.07738"]},"oa":1,"language":[{"iso":"eng"}],"abstract":[{"lang":"eng","text":"We prove that solutions of the cubic nonlinear Schrödinger equation on $\\Bbb{R}^2$ can be approximated by a finite-dimensional Hamiltonian system, uniformly on bounded sets of initial data. This is despite the wealth of non-compact symmetries: scaling, translation, and Galilei boosts.\r\n\r\nComplementing this approximation result, we show that all solutions of the finite-dimensional Hamiltonian system we use can be approximated by the full PDE.\r\n\r\nA key ingredient in these results is the development of a general methodology for transfering uniform global space-time bounds to suitable Fourier truncations of dispersive PDE models.\r\n\r\nAs an application, we prove symplectic non-squeezing (in the sense of Gromov) for the cubic NLS on $\\Bbb{R}^2$. This is the first symplectic non-squeezing result for a Hamiltonian PDE in infinite volume. It is also the first unconditional symplectic non-squeezing result in a scaling-critical setting.\r\n\r\nFinally, we discuss implications of non-squeezing on the nature of scattering."}],"citation":{"apa":"Killip, R., Vişan, M., & Zhang, X. (2021). Finite-dimensional approximation and non-squeezing for the cubic nonlinear Schrödinger equation on ℝ2. American Journal of Mathematics. Johns Hopkins University Press. https://doi.org/10.1353/ajm.2021.0014","ieee":"R. Killip, M. Vişan, and X. Zhang, “Finite-dimensional approximation and non-squeezing for the cubic nonlinear Schrödinger equation on ℝ2,” American Journal of Mathematics, vol. 143, no. 2. Johns Hopkins University Press, pp. 613–680, 2021.","mla":"Killip, Rowan, et al. “Finite-Dimensional Approximation and Non-Squeezing for the Cubic Nonlinear Schrödinger Equation on ℝ2.” American Journal of Mathematics, vol. 143, no. 2, Johns Hopkins University Press, 2021, pp. 613–80, doi:10.1353/ajm.2021.0014.","ista":"Killip R, Vişan M, Zhang X. 2021. Finite-dimensional approximation and non-squeezing for the cubic nonlinear Schrödinger equation on ℝ2. American Journal of Mathematics. 143(2), 613–680.","chicago":"Killip, Rowan, Monica Vişan, and Xiaoyi Zhang. “Finite-Dimensional Approximation and Non-Squeezing for the Cubic Nonlinear Schrödinger Equation on ℝ2.” American Journal of Mathematics. Johns Hopkins University Press, 2021. https://doi.org/10.1353/ajm.2021.0014.","short":"R. Killip, M. Vişan, X. Zhang, American Journal of Mathematics 143 (2021) 613–680.","ama":"Killip R, Vişan M, Zhang X. Finite-dimensional approximation and non-squeezing for the cubic nonlinear Schrödinger equation on ℝ2. American Journal of Mathematics. 2021;143(2):613-680. doi:10.1353/ajm.2021.0014"},"author":[{"last_name":"Killip","first_name":"Rowan","full_name":"Killip, Rowan"},{"id":"056daca0-b8d1-11f0-964f-f91054abf8ca","full_name":"Visan, Monica","first_name":"Monica","last_name":"Visan"},{"full_name":"Zhang, Xiaoyi","last_name":"Zhang","first_name":"Xiaoyi"}],"publisher":"Johns Hopkins University Press","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.1606.07738"}],"date_created":"2026-06-19T08:46:12Z","day":"01","status":"public","title":"Finite-dimensional approximation and non-squeezing for the cubic nonlinear Schrödinger equation on ℝ2","date_updated":"2026-07-01T12:38:11Z","volume":143,"oa_version":"Preprint","OA_place":"repository","fulldoi":"https://doi.org/10.1353/ajm.2021.0014","article_type":"original","das_tickbox":"1","article_processing_charge":"No","quality_controlled":"1","publication_identifier":{"eissn":["1080-6377"]},"issue":"2","_id":"22087","OA_type":"green","intvolume":" 143","doi":"10.1353/ajm.2021.0014","type":"journal_article","publication_status":"published"},{"language":[{"iso":"eng"}],"abstract":[{"lang":"eng","text":"We consider the defocusing energy-critical nonlinear Schr\\\"odinger equation in the exterior of a smooth compact strictly convex obstacle in three dimensions. For the initial-value problem with Dirichlet boundary condition we prove global well-posedness and scattering for all initial data in the energy space."}],"oa":1,"date_created":"2026-06-19T07:45:53Z","day":"01","citation":{"ama":"Killip R, Vişan M, Zhang X. Quintic NLS in the exterior of a strictly convex obstacle. American Journal of Mathematics. 2016;138(5):1193-1346. doi:10.1353/ajm.2016.0039","short":"R. Killip, M. Vişan, X. Zhang, American Journal of Mathematics 138 (2016) 1193–1346.","chicago":"Killip, Rowan, Monica Vişan, and Xiaoyi Zhang. “Quintic NLS in the Exterior of a Strictly Convex Obstacle.” American Journal of Mathematics. Johns Hopkins University Press, 2016. https://doi.org/10.1353/ajm.2016.0039.","mla":"Killip, Rowan, et al. “Quintic NLS in the Exterior of a Strictly Convex Obstacle.” American Journal of Mathematics, vol. 138, no. 5, Johns Hopkins University Press, 2016, pp. 1193–346, doi:10.1353/ajm.2016.0039.","ista":"Killip R, Vişan M, Zhang X. 2016. Quintic NLS in the exterior of a strictly convex obstacle. American Journal of Mathematics. 138(5), 1193–1346.","ieee":"R. Killip, M. Vişan, and X. Zhang, “Quintic NLS in the exterior of a strictly convex obstacle,” American Journal of Mathematics, vol. 138, no. 5. Johns Hopkins University Press, pp. 1193–1346, 2016.","apa":"Killip, R., Vişan, M., & Zhang, X. (2016). Quintic NLS in the exterior of a strictly convex obstacle. American Journal of Mathematics. Johns Hopkins University Press. https://doi.org/10.1353/ajm.2016.0039"},"author":[{"last_name":"Killip","first_name":"Rowan","full_name":"Killip, Rowan"},{"full_name":"Visan, Monica","id":"056daca0-b8d1-11f0-964f-f91054abf8ca","last_name":"Visan","first_name":"Monica"},{"full_name":"Zhang, Xiaoyi","last_name":"Zhang","first_name":"Xiaoyi"}],"publisher":"Johns Hopkins University Press","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.1208.4904"}],"date_updated":"2026-06-22T13:20:45Z","volume":138,"status":"public","title":"Quintic NLS in the exterior of a strictly convex obstacle","OA_place":"repository","oa_version":"Preprint","date_published":"2016-10-01T00:00:00Z","arxiv":1,"month":"10","scopus_import":"1","page":"1193-1346","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","year":"2016","extern":"1","external_id":{"arxiv":["1208.4904"]},"publication":"American Journal of Mathematics","publication_identifier":{"eissn":["1080-6377"]},"OA_type":"green","intvolume":" 138","issue":"5","_id":"22041","type":"journal_article","doi":"10.1353/ajm.2016.0039","publication_status":"published","fulldoi":"https://doi.org/10.1353/ajm.2016.0039","article_type":"original","das_tickbox":"1","article_processing_charge":"No","quality_controlled":"1"},{"oa_version":"Preprint","OA_place":"repository","status":"public","title":"The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher","date_updated":"2026-06-19T10:24:15Z","volume":132,"publisher":"Johns Hopkins University Press","author":[{"last_name":"Killip","first_name":"Rowan","full_name":"Killip, Rowan"},{"full_name":"Visan, Monica","id":"056daca0-b8d1-11f0-964f-f91054abf8ca","first_name":"Monica","last_name":"Visan"}],"citation":{"ama":"Killip R, Vişan M. The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher. American Journal of Mathematics. 2010;132(2):361-424. doi:10.1353/ajm.0.0107","short":"R. Killip, M. Vişan, American Journal of Mathematics 132 (2010) 361–424.","chicago":"Killip, Rowan, and Monica Vişan. “The Focusing Energy-Critical Nonlinear Schrödinger Equation in Dimensions Five and Higher.” American Journal of Mathematics. Johns Hopkins University Press, 2010. https://doi.org/10.1353/ajm.0.0107.","mla":"Killip, Rowan, and Monica Vişan. “The Focusing Energy-Critical Nonlinear Schrödinger Equation in Dimensions Five and Higher.” American Journal of Mathematics, vol. 132, no. 2, Johns Hopkins University Press, 2010, pp. 361–424, doi:10.1353/ajm.0.0107.","ista":"Killip R, Vişan M. 2010. The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher. American Journal of Mathematics. 132(2), 361–424.","ieee":"R. Killip and M. Vişan, “The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher,” American Journal of Mathematics, vol. 132, no. 2. Johns Hopkins University Press, pp. 361–424, 2010.","apa":"Killip, R., & Vişan, M. (2010). The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher. American Journal of Mathematics. Johns Hopkins University Press. https://doi.org/10.1353/ajm.0.0107"},"main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.0804.1018"}],"date_created":"2026-06-19T07:31:39Z","day":"31","oa":1,"language":[{"iso":"eng"}],"abstract":[{"lang":"eng","text":"We consider the focusing energy-critical nonlinear Schrödinger equation iut + ∆u = −|u|\r\n4 d−2 u in dimensions d ≥ 5. We prove that if a maximal-lifespan solution u : I × Rd → C obeys supt∈I k∇u(t)k2 < k∇Wk2, then it is global and scatters both forward and backward in time. Here W denotes the ground state, which is a stationary solution of the equation. In\r\nparticular, if a solution has both energy and kinetic energy less than those\r\nof the ground state W at some point in time, then the solution is global and\r\nscatters. We also show that any solution that blows up with bounded kinetic\r\nenergy must concentrate at least the kinetic energy of the ground state. Similar\r\nresults were obtained by Kenig and Merle in [17, 18] for spherically symmetric\r\ninitial data and dimensions d = 3, 4, 5."}],"publication":"American Journal of Mathematics","external_id":{"arxiv":["0804.1018"]},"year":"2010","extern":"1","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","page":"361-424","month":"03","arxiv":1,"date_published":"2010-03-31T00:00:00Z","scopus_import":"1","publication_status":"published","doi":"10.1353/ajm.0.0107","type":"journal_article","issue":"2","_id":"22023","OA_type":"green","intvolume":" 132","publication_identifier":{"issn":["0002-9327"],"eissn":["1080-6377"]},"quality_controlled":"1","article_processing_charge":"No","fulldoi":"https://doi.org/10.1353/ajm.0.0107","article_type":"original"},{"_id":"22089","issue":"2","intvolume":" 132","OA_type":"green","publication_identifier":{"eissn":["1080-6377"]},"publication_status":"published","doi":"10.1353/ajm.0.0107","type":"journal_article","article_type":"original","fulldoi":"https://doi.org/10.1353/ajm.0.0107","quality_controlled":"1","article_processing_charge":"No","das_tickbox":"1","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.0804.1018"}],"publisher":"Johns Hopkins University Press","citation":{"chicago":"Killip, Rowan, and Monica Vişan. “The Focusing Energy-Critical Nonlinear Schrödinger Equation in Dimensions Five and Higher.” American Journal of Mathematics. Johns Hopkins University Press, 2010. https://doi.org/10.1353/ajm.0.0107.","short":"R. Killip, M. Vişan, American Journal of Mathematics 132 (2010) 361–424.","ama":"Killip R, Vişan M. The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher. American Journal of Mathematics. 2010;132(2):361-424. doi:10.1353/ajm.0.0107","mla":"Killip, Rowan, and Monica Vişan. “The Focusing Energy-Critical Nonlinear Schrödinger Equation in Dimensions Five and Higher.” American Journal of Mathematics, vol. 132, no. 2, Johns Hopkins University Press, 2010, pp. 361–424, doi:10.1353/ajm.0.0107.","ista":"Killip R, Vişan M. 2010. The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher. American Journal of Mathematics. 132(2), 361–424.","ieee":"R. Killip and M. Vişan, “The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher,” American Journal of Mathematics, vol. 132, no. 2. Johns Hopkins University Press, pp. 361–424, 2010.","apa":"Killip, R., & Vişan, M. (2010). The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher. American Journal of Mathematics. Johns Hopkins University Press. https://doi.org/10.1353/ajm.0.0107"},"author":[{"full_name":"Killip, Rowan","first_name":"Rowan","last_name":"Killip"},{"first_name":"Monica","last_name":"Visan","id":"056daca0-b8d1-11f0-964f-f91054abf8ca","full_name":"Visan, Monica"}],"day":"01","date_created":"2026-06-19T08:48:46Z","oa":1,"abstract":[{"lang":"eng","text":"We consider the focusing energy-critical nonlinear Schr\\\"odinger equation \r\n$iu_t+\\Delta u = - |u|^{4\\over{d-2}}u$ in dimensions $d\\geq 5$. We prove \r\nthat if a maximal-lifespan solution $u\\colon \\ I\\times {\\Bbb R}^d\\to {\\Bbb C}$ obeys $\\sup_{t\\in I}\\|\\nabla u(t)\\|_2<\\|\\nabla W\\|_2$, then it is \r\nglobal and scatters both forward and backward in time. Here $W$ denotes \r\nthe ground state, which is a stationary solution of the equation. In \r\nparticular, if a solution has both energy and kinetic energy less than \r\nthose of the ground state $W$ at some point in time, then the solution is \r\nglobal and scatters. We also show that any solution that blows up with \r\nbounded kinetic energy must concentrate at least the kinetic energy of the \r\nground state. Similar results were obtained by Kenig and Merle for \r\nspherically symmetric initial data and dimensions $d=3,4,5$.\r\n"}],"language":[{"iso":"eng"}],"oa_version":"Preprint","OA_place":"repository","title":"The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher","status":"public","volume":132,"date_updated":"2026-07-01T12:52:05Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","page":"361-424","scopus_import":"1","date_published":"2010-04-01T00:00:00Z","month":"04","arxiv":1,"publication":"American Journal of Mathematics","external_id":{"arxiv":["0804.1018"]},"extern":"1","year":"2010"},{"publication":"American Journal of Mathematics","external_id":{"arxiv":["math/0501462"]},"extern":"1","year":"2007","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","page":"1-60","scopus_import":"1","month":"02","arxiv":1,"date_published":"2007-02-01T00:00:00Z","oa_version":"Preprint","OA_place":"repository","status":"public","title":"Global well-posedness and scattering for the defocusing energy-critical nonlinear Schrödinger equation in R 1+4","volume":129,"date_updated":"2026-06-29T10:30:01Z","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.math/0501462","open_access":"1"}],"citation":{"apa":"Ryckman, E., & Vişan, M. (2007). Global well-posedness and scattering for the defocusing energy-critical nonlinear Schrödinger equation in R 1+4. American Journal of Mathematics. Johns Hopkins University Press. https://doi.org/10.1353/ajm.2007.0004","ieee":"E. Ryckman and M. Vişan, “Global well-posedness and scattering for the defocusing energy-critical nonlinear Schrödinger equation in R 1+4,” American Journal of Mathematics, vol. 129, no. 1. Johns Hopkins University Press, pp. 1–60, 2007.","mla":"Ryckman, E., and Monica Vişan. “Global Well-Posedness and Scattering for the Defocusing Energy-Critical Nonlinear Schrödinger Equation in R 1+4.” American Journal of Mathematics, vol. 129, no. 1, Johns Hopkins University Press, 2007, pp. 1–60, doi:10.1353/ajm.2007.0004.","ista":"Ryckman E, Vişan M. 2007. Global well-posedness and scattering for the defocusing energy-critical nonlinear Schrödinger equation in R 1+4. American Journal of Mathematics. 129(1), 1–60.","chicago":"Ryckman, E, and Monica Vişan. “Global Well-Posedness and Scattering for the Defocusing Energy-Critical Nonlinear Schrödinger Equation in R 1+4.” American Journal of Mathematics. Johns Hopkins University Press, 2007. https://doi.org/10.1353/ajm.2007.0004.","ama":"Ryckman E, Vişan M. Global well-posedness and scattering for the defocusing energy-critical nonlinear Schrödinger equation in R 1+4. American Journal of Mathematics. 2007;129(1):1-60. doi:10.1353/ajm.2007.0004","short":"E. Ryckman, M. Vişan, American Journal of Mathematics 129 (2007) 1–60."},"author":[{"full_name":"Ryckman, E","last_name":"Ryckman","first_name":"E"},{"last_name":"Visan","first_name":"Monica","id":"056daca0-b8d1-11f0-964f-f91054abf8ca","full_name":"Visan, Monica"}],"publisher":"Johns Hopkins University Press","day":"01","date_created":"2026-06-19T07:57:32Z","oa":1,"abstract":[{"text":" We obtain global well-posedness, scattering, uniform regularity, and global L6t,x spacetime bounds for energy-space solutions to the defocusing energy-critical nonlinear Schrödinger equation in âÃâ4. Our arguments closely follow those of Colliander, Keel, et al., though our derivation of the frequency-localized interaction Morawetz estimate is somewhat simpler. As a consequence, our method yields a better bound on the L6t,x-norm. ","lang":"eng"}],"language":[{"iso":"eng"}],"quality_controlled":"1","article_processing_charge":"No","das_tickbox":"1","article_type":"original","fulldoi":"https://doi.org/10.1353/ajm.2007.0004","publication_status":"published","doi":"10.1353/ajm.2007.0004","type":"journal_article","_id":"22057","issue":"1","intvolume":" 129","OA_type":"green","publication_identifier":{"eissn":["1080-6377"]}}]