[{"oa_version":"Preprint","oa":1,"OA_place":"repository","quality_controlled":"1","language":[{"iso":"eng"}],"issue":"3","type":"journal_article","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","month":"06","citation":{"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.","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.","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.","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>.","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>","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>."},"scopus_import":"1","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2411.05300","open_access":"1"}],"_id":"22021","publication_identifier":{"issn":["2578-5893"],"eissn":["2578-5885"]},"das_tickbox":"1","date_created":"2026-06-19T07:30:23Z","doi":"10.2140/paa.2025.7.615","article_processing_charge":"No","page":"615-637","external_id":{"arxiv":["2411.05300"]},"arxiv":1,"title":" Global well-posedness and equicontinuity for modified Korteweg–de Vries equations in modulation spaces","volume":7,"mathsc":["35Q53","35Q55","37K10"],"publication":"Pure and Applied Analysis","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."}],"year":"2025","day":"18","date_updated":"2026-06-24T13:22:40Z","article_type":"original","publisher":"Mathematical Sciences Publishers","publication_status":"published","status":"public","intvolume":"         7","extern":"1","author":[{"first_name":"Saikatul","last_name":"Haque","full_name":"Haque, Saikatul"},{"first_name":"Rowan","last_name":"Killip","full_name":"Killip, Rowan"},{"first_name":"Monica","last_name":"Visan","full_name":"Visan, Monica","id":"056daca0-b8d1-11f0-964f-f91054abf8ca"},{"first_name":"Yunfeng","last_name":"Zhang","full_name":"Zhang, Yunfeng"}],"OA_type":"green","date_published":"2025-06-18T00:00:00Z"},{"date_updated":"2026-06-25T07:54:44Z","article_type":"original","publisher":"IOP Publishing","status":"public","publication_status":"published","year":"2023","day":"09","author":[{"full_name":"Killip, Rowan","last_name":"Killip","first_name":"Rowan"},{"full_name":"Ouyang, Zhimeng","first_name":"Zhimeng","last_name":"Ouyang"},{"last_name":"Visan","first_name":"Monica","id":"056daca0-b8d1-11f0-964f-f91054abf8ca","full_name":"Visan, Monica"},{"full_name":"Wu, Lei","first_name":"Lei","last_name":"Wu"}],"OA_type":"green","date_published":"2023-06-09T00:00:00Z","keyword":["Ablowitz–Ladik","continuum limit","cubic NLS"],"intvolume":"        36","extern":"1","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","month":"06","type":"journal_article","scopus_import":"1","citation":{"apa":"Killip, R., Ouyang, Z., Vişan, M., &#38; Wu, L. (2023). Continuum limit for the Ablowitz–Ladik system. <i>Nonlinearity</i>. IOP Publishing. <a href=\"https://doi.org/10.1088/1361-6544/acd978\">https://doi.org/10.1088/1361-6544/acd978</a>","ista":"Killip R, Ouyang Z, Vişan M, Wu L. 2023. Continuum limit for the Ablowitz–Ladik system. Nonlinearity. 36(7), 3751–3775.","ama":"Killip R, Ouyang Z, Vişan M, Wu L. Continuum limit for the Ablowitz–Ladik system. <i>Nonlinearity</i>. 2023;36(7):3751-3775. doi:<a href=\"https://doi.org/10.1088/1361-6544/acd978\">10.1088/1361-6544/acd978</a>","chicago":"Killip, Rowan, Zhimeng Ouyang, Monica Vişan, and Lei Wu. “Continuum Limit for the Ablowitz–Ladik System.” <i>Nonlinearity</i>. IOP Publishing, 2023. <a href=\"https://doi.org/10.1088/1361-6544/acd978\">https://doi.org/10.1088/1361-6544/acd978</a>.","short":"R. Killip, Z. Ouyang, M. Vişan, L. Wu, Nonlinearity 36 (2023) 3751–3775.","ieee":"R. Killip, Z. Ouyang, M. Vişan, and L. Wu, “Continuum limit for the Ablowitz–Ladik system,” <i>Nonlinearity</i>, vol. 36, no. 7. IOP Publishing, pp. 3751–3775, 2023.","mla":"Killip, Rowan, et al. “Continuum Limit for the Ablowitz–Ladik System.” <i>Nonlinearity</i>, vol. 36, no. 7, IOP Publishing, 2023, pp. 3751–75, doi:<a href=\"https://doi.org/10.1088/1361-6544/acd978\">10.1088/1361-6544/acd978</a>."},"main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2206.02720","open_access":"1"}],"date_created":"2026-06-19T07:49:24Z","doi":"10.1088/1361-6544/acd978","publication_identifier":{"issn":["0951-7715"],"eissn":["1361-6544"]},"_id":"22046","das_tickbox":"1","oa":1,"oa_version":"Preprint","OA_place":"repository","issue":"7","quality_controlled":"1","language":[{"iso":"eng"}],"title":"Continuum limit for the Ablowitz–Ladik system","publication":"Nonlinearity","abstract":[{"lang":"eng","text":"We show that solutions to the Ablowitz–Ladik system converge to solutions of the cubic nonlinear Schrödinger equation for merely L2 initial data. Furthermore, we consider initial data for this lattice model that excites Fourier modes near both critical points of the discrete dispersion relation and demonstrate convergence to a decoupled system of nonlinear Schrödinger equations."}],"volume":36,"mathsc":["35Q55","37K05","37K10"],"article_processing_charge":"No","external_id":{"arxiv":["2206.02720"]},"page":"3751-3775","arxiv":1},{"extern":"1","intvolume":"       288","date_published":"2018-04-01T00:00:00Z","OA_type":"green","author":[{"first_name":"R.","last_name":"Killip","full_name":"Killip, R."},{"full_name":"Miao, C.","last_name":"Miao","first_name":"C."},{"last_name":"Visan","first_name":"Monica","id":"056daca0-b8d1-11f0-964f-f91054abf8ca","full_name":"Visan, Monica"},{"full_name":"Zhang, J.","last_name":"Zhang","first_name":"J."},{"full_name":"Zheng, J.","last_name":"Zheng","first_name":"J."}],"day":"01","year":"2018","status":"public","publication_status":"published","article_type":"original","publisher":"Springer Nature","date_updated":"2026-06-25T07:36:26Z","arxiv":1,"article_processing_charge":"No","external_id":{"arxiv":["1503.02716"]},"page":"1273-1298","abstract":[{"lang":"eng","text":"We study the L p-theory for the Schrödinger operatorLa with inverse-square potential\r\na|x|^−2. Our main result describes when L p-based Sobolev spaces defined in terms of the\r\noperator (La)^s/2 agree with those defined via (−\u0002)^s/2.We consider all regularities 0 < s < 2.\r\nIn order to make the paper self-contained, we also review (with proofs) multiplier theorems,\r\nLittlewood–Paley theory, and Hardy-type inequalities associated to the operator La."}],"publication":"Mathematische Zeitschrift","volume":288,"mathsc":["35P25","35Q55"],"title":"Sobolev spaces adapted to the Schrödinger operator with inverse-square potential","issue":"3-4","language":[{"iso":"eng"}],"quality_controlled":"1","OA_place":"repository","oa":1,"oa_version":"Preprint","doi":"10.1007/s00209-017-1934-8","date_created":"2026-06-19T07:46:14Z","publication_identifier":{"eissn":["1432-1823"],"issn":["0025-5874"]},"_id":"22042","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.1503.02716"}],"scopus_import":"1","citation":{"apa":"Killip, R., Miao, C., Vişan, M., Zhang, J., &#38; Zheng, J. (2018). Sobolev spaces adapted to the Schrödinger operator with inverse-square potential. <i>Mathematische Zeitschrift</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00209-017-1934-8\">https://doi.org/10.1007/s00209-017-1934-8</a>","ista":"Killip R, Miao C, Vişan M, Zhang J, Zheng J. 2018. Sobolev spaces adapted to the Schrödinger operator with inverse-square potential. Mathematische Zeitschrift. 288(3–4), 1273–1298.","ama":"Killip R, Miao C, Vişan M, Zhang J, Zheng J. Sobolev spaces adapted to the Schrödinger operator with inverse-square potential. <i>Mathematische Zeitschrift</i>. 2018;288(3-4):1273-1298. doi:<a href=\"https://doi.org/10.1007/s00209-017-1934-8\">10.1007/s00209-017-1934-8</a>","chicago":"Killip, R., C. Miao, Monica Vişan, J. Zhang, and J. Zheng. “Sobolev Spaces Adapted to the Schrödinger Operator with Inverse-Square Potential.” <i>Mathematische Zeitschrift</i>. Springer Nature, 2018. <a href=\"https://doi.org/10.1007/s00209-017-1934-8\">https://doi.org/10.1007/s00209-017-1934-8</a>.","short":"R. Killip, C. Miao, M. Vişan, J. Zhang, J. Zheng, Mathematische Zeitschrift 288 (2018) 1273–1298.","mla":"Killip, R., et al. “Sobolev Spaces Adapted to the Schrödinger Operator with Inverse-Square Potential.” <i>Mathematische Zeitschrift</i>, vol. 288, no. 3–4, Springer Nature, 2018, pp. 1273–98, doi:<a href=\"https://doi.org/10.1007/s00209-017-1934-8\">10.1007/s00209-017-1934-8</a>.","ieee":"R. Killip, C. Miao, M. Vişan, J. Zhang, and J. Zheng, “Sobolev spaces adapted to the Schrödinger operator with inverse-square potential,” <i>Mathematische Zeitschrift</i>, vol. 288, no. 3–4. Springer Nature, pp. 1273–1298, 2018."},"month":"04","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","type":"journal_article"},{"author":[{"first_name":"Rowan","last_name":"Killip","full_name":"Killip, Rowan"},{"first_name":"Jason","last_name":"Murphy","full_name":"Murphy, Jason"},{"last_name":"Visan","first_name":"Monica","id":"056daca0-b8d1-11f0-964f-f91054abf8ca","full_name":"Visan, Monica"}],"OA_type":"green","date_published":"2018-01-01T00:00:00Z","keyword":["cubic-quintic NLS","nonvanishing boundary conditions","space-time resonances","scattering"],"intvolume":"        50","extern":"1","date_updated":"2026-06-25T07:49:21Z","publisher":"Society for Industrial & Applied Mathematics","article_type":"original","publication_status":"published","status":"public","year":"2018","day":"01","title":"The initial-value problem for the cubic-quintic NLS with nonvanishing boundary conditions","mathsc":["35Q55"],"volume":50,"abstract":[{"text":"We consider the initial-value problem for the cubic-quintic nonlinear Schrödinger equation (𝑖𝜕𝑡+Δ)⁢𝜓 =𝛼1⁢𝜓 −𝛼3⁢|𝜓|2⁢𝜓 +𝛼5⁢|𝜓|4⁢𝜓 in three spatial dimensions in the class of solutions with |𝜓⁡(𝑥)| →𝑐 >0 as |𝑥| →∞. Here 𝛼1, 𝛼3, 𝛼5, and 𝑐 are such that 𝜓⁡(𝑥) ≡𝑐 is an energetically stable equilibrium solution to this equation. Normalizing the boundary condition to 𝜓⁡(𝑥) →1 as |𝑥| →∞, we study the associated initial-value problem for 𝑢 =𝜓 −1 and prove a scattering result for small initial data in a weighted Sobolev space.","lang":"eng"}],"publication":"SIAM Journal on Mathematical Analysis","page":"2681-2739","external_id":{"arxiv":["1702.04413"]},"article_processing_charge":"No","arxiv":1,"type":"journal_article","month":"01","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","citation":{"mla":"Killip, Rowan, et al. “The Initial-Value Problem for the Cubic-Quintic NLS with Nonvanishing Boundary Conditions.” <i>SIAM Journal on Mathematical Analysis</i>, vol. 50, no. 3, Society for Industrial &#38; Applied Mathematics, 2018, pp. 2681–739, doi:<a href=\"https://doi.org/10.1137/17m1116702\">10.1137/17m1116702</a>.","ieee":"R. Killip, J. Murphy, and M. Vişan, “The initial-value problem for the cubic-quintic NLS with nonvanishing boundary conditions,” <i>SIAM Journal on Mathematical Analysis</i>, vol. 50, no. 3. Society for Industrial &#38; Applied Mathematics, pp. 2681–2739, 2018.","short":"R. Killip, J. Murphy, M. Vişan, SIAM Journal on Mathematical Analysis 50 (2018) 2681–2739.","chicago":"Killip, Rowan, Jason Murphy, and Monica Vişan. “The Initial-Value Problem for the Cubic-Quintic NLS with Nonvanishing Boundary Conditions.” <i>SIAM Journal on Mathematical Analysis</i>. Society for Industrial &#38; Applied Mathematics, 2018. <a href=\"https://doi.org/10.1137/17m1116702\">https://doi.org/10.1137/17m1116702</a>.","ama":"Killip R, Murphy J, Vişan M. The initial-value problem for the cubic-quintic NLS with nonvanishing boundary conditions. <i>SIAM Journal on Mathematical Analysis</i>. 2018;50(3):2681-2739. doi:<a href=\"https://doi.org/10.1137/17m1116702\">10.1137/17m1116702</a>","ista":"Killip R, Murphy J, Vişan M. 2018. The initial-value problem for the cubic-quintic NLS with nonvanishing boundary conditions. SIAM Journal on Mathematical Analysis. 50(3), 2681–2739.","apa":"Killip, R., Murphy, J., &#38; Vişan, M. (2018). The initial-value problem for the cubic-quintic NLS with nonvanishing boundary conditions. <i>SIAM Journal on Mathematical Analysis</i>. Society for Industrial &#38; Applied Mathematics. <a href=\"https://doi.org/10.1137/17m1116702\">https://doi.org/10.1137/17m1116702</a>"},"scopus_import":"1","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.1702.04413","open_access":"1"}],"_id":"22045","publication_identifier":{"issn":["0036-1410","1095-7154"]},"das_tickbox":"1","doi":"10.1137/17m1116702","date_created":"2026-06-19T07:49:03Z","oa":1,"oa_version":"Preprint","OA_place":"repository","language":[{"iso":"eng"}],"quality_controlled":"1","issue":"3"},{"publication":"Analysis & PDE","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."}],"mathsc":["35Q55"],"volume":9,"title":"The final-state problem for the cubic-quintic NLS with nonvanishing boundary conditions","arxiv":1,"page":"1523-1574","article_processing_charge":"No","external_id":{"arxiv":["1506.06151"]},"date_created":"2026-06-19T07:54:01Z","doi":"10.2140/apde.2016.9.1523","das_tickbox":"1","_id":"22051","publication_identifier":{"issn":["2157-5045"],"eissn":["1948-206X"]},"main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.1506.06151"}],"scopus_import":"1","citation":{"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.","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>.","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>","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>"},"month":"11","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","type":"journal_article","issue":"7","quality_controlled":"1","language":[{"iso":"eng"}],"OA_place":"repository","oa":1,"oa_version":"Preprint","date_published":"2016-11-07T00:00:00Z","OA_type":"green","author":[{"full_name":"Killip, Rowan","last_name":"Killip","first_name":"Rowan"},{"full_name":"Murphy, Jason","last_name":"Murphy","first_name":"Jason"},{"full_name":"Visan, Monica","id":"056daca0-b8d1-11f0-964f-f91054abf8ca","first_name":"Monica","last_name":"Visan"}],"extern":"1","intvolume":"         9","keyword":["final-state problem","wave operators","cubic-quintic NLS","nonvanishing boundary conditions"],"status":"public","publication_status":"published","article_type":"original","publisher":"Mathematical Sciences Publishers","date_updated":"2026-06-25T08:23:10Z","day":"07","year":"2016"},{"keyword":["NLS","Gross–Pitaevskii equation","non-vanishing boundary condition"],"intvolume":"        19","extern":"1","author":[{"full_name":"Killip, Rowan","last_name":"Killip","first_name":"Rowan"},{"last_name":"Oh","first_name":"Tadahiro","full_name":"Oh, Tadahiro"},{"full_name":"Pocovnicu, Oana","last_name":"Pocovnicu","first_name":"Oana"},{"id":"056daca0-b8d1-11f0-964f-f91054abf8ca","full_name":"Visan, Monica","last_name":"Visan","first_name":"Monica"}],"OA_type":"green","date_published":"2013-03-15T00:00:00Z","year":"2013","day":"15","date_updated":"2026-06-25T08:33:18Z","article_type":"original","publisher":"International Press of Boston","status":"public","publication_status":"published","page":"969-986","article_processing_charge":"No","external_id":{"arxiv":["1112.1354"]},"arxiv":1,"title":"Global well-posedness of the Gross–Pitaevskii and cubic-quintic nonlinear Schrödinger equations with non-vanishing boundary conditions","abstract":[{"lang":"eng","text":"We consider the Gross–Pitaevskii equation on R^4 and the cubic-quintic nonlinear Schrödinger equation (NLS) on R^3 with non-vanishing boundary conditions at spatial infinity. By viewing these equations as perturbations to the energy-critical NLS, we prove that they are globally well-posed in their energy spaces. In particular, we prove unconditional uniqueness in the energy spaces for these equations."}],"publication":"Mathematical Research Letters","volume":19,"mathsc":["35Q55"],"oa":1,"oa_version":"Preprint","OA_place":"repository","issue":"5","quality_controlled":"1","language":[{"iso":"eng"}],"month":"03","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","type":"journal_article","scopus_import":"1","citation":{"apa":"Killip, R., Oh, T., Pocovnicu, O., &#38; Vişan, M. (2013). Global well-posedness of the Gross–Pitaevskii and cubic-quintic nonlinear Schrödinger equations with non-vanishing boundary conditions. <i>Mathematical Research Letters</i>. International Press of Boston. <a href=\"https://doi.org/10.4310/mrl.2012.v19.n5.a1\">https://doi.org/10.4310/mrl.2012.v19.n5.a1</a>","ista":"Killip R, Oh T, Pocovnicu O, Vişan M. 2013. Global well-posedness of the Gross–Pitaevskii and cubic-quintic nonlinear Schrödinger equations with non-vanishing boundary conditions. Mathematical Research Letters. 19(5), 969–986.","ama":"Killip R, Oh T, Pocovnicu O, Vişan M. Global well-posedness of the Gross–Pitaevskii and cubic-quintic nonlinear Schrödinger equations with non-vanishing boundary conditions. <i>Mathematical Research Letters</i>. 2013;19(5):969-986. doi:<a href=\"https://doi.org/10.4310/mrl.2012.v19.n5.a1\">10.4310/mrl.2012.v19.n5.a1</a>","chicago":"Killip, Rowan, Tadahiro Oh, Oana Pocovnicu, and Monica Vişan. “Global Well-Posedness of the Gross–Pitaevskii and Cubic-Quintic Nonlinear Schrödinger Equations with Non-Vanishing Boundary Conditions.” <i>Mathematical Research Letters</i>. International Press of Boston, 2013. <a href=\"https://doi.org/10.4310/mrl.2012.v19.n5.a1\">https://doi.org/10.4310/mrl.2012.v19.n5.a1</a>.","short":"R. Killip, T. Oh, O. Pocovnicu, M. Vişan, Mathematical Research Letters 19 (2013) 969–986.","ieee":"R. Killip, T. Oh, O. Pocovnicu, and M. Vişan, “Global well-posedness of the Gross–Pitaevskii and cubic-quintic nonlinear Schrödinger equations with non-vanishing boundary conditions,” <i>Mathematical Research Letters</i>, vol. 19, no. 5. International Press of Boston, pp. 969–986, 2013.","mla":"Killip, Rowan, et al. “Global Well-Posedness of the Gross–Pitaevskii and Cubic-Quintic Nonlinear Schrödinger Equations with Non-Vanishing Boundary Conditions.” <i>Mathematical Research Letters</i>, vol. 19, no. 5, International Press of Boston, 2013, pp. 969–86, doi:<a href=\"https://doi.org/10.4310/mrl.2012.v19.n5.a1\">10.4310/mrl.2012.v19.n5.a1</a>."},"main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.1112.1354"}],"doi":"10.4310/mrl.2012.v19.n5.a1","date_created":"2026-06-19T07:54:49Z","publication_identifier":{"eissn":["1945-001X"],"issn":["1073-2780"]},"_id":"22053","das_tickbox":"1"},{"_id":"22049","publication_identifier":{"eissn":["1435-5337"],"issn":["0933-7741"]},"das_tickbox":"1","doi":"10.1515/forum.2008.042","date_created":"2026-06-19T07:53:12Z","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.math/0609690","open_access":"1"}],"citation":{"ieee":"T. Tao, M. Vişan, and X. Zhang, “Minimal-mass blowup solutions of the mass-critical NLS,” <i>Forum Mathematicum</i>, vol. 20, no. 5. De Gruyter, pp. 881–919, 2008.","mla":"Tao, Terence, et al. “Minimal-Mass Blowup Solutions of the Mass-Critical NLS.” <i>Forum Mathematicum</i>, vol. 20, no. 5, De Gruyter, 2008, pp. 881–919, doi:<a href=\"https://doi.org/10.1515/forum.2008.042\">10.1515/forum.2008.042</a>.","short":"T. Tao, M. Vişan, X. Zhang, Forum Mathematicum 20 (2008) 881–919.","chicago":"Tao, Terence, Monica Vişan, and Xiaoyi Zhang. “Minimal-Mass Blowup Solutions of the Mass-Critical NLS.” <i>Forum Mathematicum</i>. De Gruyter, 2008. <a href=\"https://doi.org/10.1515/forum.2008.042\">https://doi.org/10.1515/forum.2008.042</a>.","ama":"Tao T, Vişan M, Zhang X. Minimal-mass blowup solutions of the mass-critical NLS. <i>Forum Mathematicum</i>. 2008;20(5):881-919. doi:<a href=\"https://doi.org/10.1515/forum.2008.042\">10.1515/forum.2008.042</a>","ista":"Tao T, Vişan M, Zhang X. 2008. Minimal-mass blowup solutions of the mass-critical NLS. Forum Mathematicum. 20(5), 881–919.","apa":"Tao, T., Vişan, M., &#38; Zhang, X. (2008). Minimal-mass blowup solutions of the mass-critical NLS. <i>Forum Mathematicum</i>. De Gruyter. <a href=\"https://doi.org/10.1515/forum.2008.042\">https://doi.org/10.1515/forum.2008.042</a>"},"scopus_import":"1","type":"journal_article","month":"11","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","language":[{"iso":"eng"}],"quality_controlled":"1","issue":"5","OA_place":"repository","oa_version":"Preprint","oa":1,"volume":20,"mathsc":["35Q55"],"abstract":[{"lang":"eng","text":"We consider the minimal mass m0 required for solutions to the mass-critical nonlinear Schrödinger (NLS) equation iut + Δu = μ|u|^4/d u to blow up. If m0 is finite, we show that there exists a minimal-mass solution blowing up (in the sense of an infinite spacetime norm) in both time directions, whose orbit in  is compact after quotienting out by the symmetries of the equation. A similar result is obtained for spherically symmetric solutions. Similar results were previously obtained by Keraani, [Keraani S.: On the blow-up phenomenon of the critical nonlinear Schrödinger equation. J. Funct. Anal. 235 (2006), 171–192], in dimensions 1, 2 and Begout and Vargas, [Begout P., Vargas A.: Mass concentration phenomena for the L2-critical nonlinear Schrödinger equation, preprint], in dimensions d ≥ 3 for the mass-critical NLS and by Kenig and Merle, [Kenig C., Merle F.: Global well-posedness, scattering, and blowup for the energy-critical, focusing, non-linear Schrödinger equation in the radial case, preprint], in the energy-critical case. In a subsequent paper we shall use this compactness result to establish global existence and scattering in  for the defocusing NLS in three and higher dimensions with spherically symmetric data."}],"publication":"Forum Mathematicum","title":"Minimal-mass blowup solutions of the mass-critical NLS","arxiv":1,"article_processing_charge":"No","page":"881-919","external_id":{"arxiv":["math/0609690"]},"publication_status":"published","status":"public","article_type":"original","publisher":"De Gruyter","date_updated":"2026-06-25T08:15:22Z","day":"03","year":"2008","date_published":"2008-11-03T00:00:00Z","OA_type":"green","author":[{"last_name":"Tao","first_name":"Terence","full_name":"Tao, Terence"},{"full_name":"Visan, Monica","id":"056daca0-b8d1-11f0-964f-f91054abf8ca","first_name":"Monica","last_name":"Visan"},{"full_name":"Zhang, Xiaoyi","first_name":"Xiaoyi","last_name":"Zhang"}],"extern":"1","intvolume":"        20"},{"type":"journal_article","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","month":"08","citation":{"chicago":"Tao, Terence, Monica Vişan, and Xiaoyi Zhang. “The Nonlinear Schrödinger Equation with Combined Power-Type Nonlinearities.” <i>Communications in Partial Differential Equations</i>. Informa UK Limited, 2007. <a href=\"https://doi.org/10.1080/03605300701588805\">https://doi.org/10.1080/03605300701588805</a>.","ama":"Tao T, Vişan M, Zhang X. The nonlinear Schrödinger equation with combined power-type nonlinearities. <i>Communications in Partial Differential Equations</i>. 2007;32(8):1281-1343. doi:<a href=\"https://doi.org/10.1080/03605300701588805\">10.1080/03605300701588805</a>","ieee":"T. Tao, M. Vişan, and X. Zhang, “The nonlinear Schrödinger equation with combined power-type nonlinearities,” <i>Communications in Partial Differential Equations</i>, vol. 32, no. 8. Informa UK Limited, pp. 1281–1343, 2007.","mla":"Tao, Terence, et al. “The Nonlinear Schrödinger Equation with Combined Power-Type Nonlinearities.” <i>Communications in Partial Differential Equations</i>, vol. 32, no. 8, Informa UK Limited, 2007, pp. 1281–343, doi:<a href=\"https://doi.org/10.1080/03605300701588805\">10.1080/03605300701588805</a>.","short":"T. Tao, M. Vişan, X. Zhang, Communications in Partial Differential Equations 32 (2007) 1281–1343.","apa":"Tao, T., Vişan, M., &#38; Zhang, X. (2007). The nonlinear Schrödinger equation with combined power-type nonlinearities. <i>Communications in Partial Differential Equations</i>. Informa UK Limited. <a href=\"https://doi.org/10.1080/03605300701588805\">https://doi.org/10.1080/03605300701588805</a>","ista":"Tao T, Vişan M, Zhang X. 2007. The nonlinear Schrödinger equation with combined power-type nonlinearities. Communications in Partial Differential Equations. 32(8), 1281–1343."},"scopus_import":"1","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.math/0511070"}],"publication_identifier":{"issn":["0360-5302"],"eissn":["1532-4133"]},"_id":"22047","das_tickbox":"1","date_created":"2026-06-19T07:49:46Z","doi":"10.1080/03605300701588805","oa":1,"oa_version":"Preprint","OA_place":"repository","language":[{"iso":"eng"}],"quality_controlled":"1","issue":"8","title":"The nonlinear Schrödinger equation with combined power-type nonlinearities","volume":32,"mathsc":["35Q55"],"publication":"Communications in Partial Differential Equations","abstract":[{"text":"We undertake a comprehensive study of the nonlinear Schrödinger equation (mathematical formular) where u(t, x) is a complex-valued function in spacetime R, xRn/x, λ1 and λ2 are nonzero real constants, and (mathematical formular). We address questions related to local and global well-posedness, finite time blowup, and asymptotic behaviour. Scattering is considered both in the energy space H^1(ℝ n ) and in the pseudoconformal space Σ := {f ∈ H^1(ℝ^n); xf ∈ L^2(ℝ^n)}. Of particular interest is the case when both nonlinearities are defocusing and correspond to the L2/x-critical, respectively H1/x-critical NLS, that is, λ1, λ2 > 0 and (mathematical formular) . The results at the endpoint p1= 4/n are conditional on a conjectured global existence and spacetime estimate for the L2/x-critical nonlinear Schrödinger equation, which has been verified in dimensions n ≥ 2 for radial data in Tao et al. (Tao et al. to appear a,b) and Killip et al. (preprint).\r\nAs an off-shoot of our analysis, we also obtain a new, simpler proof of scattering in H1/x for solutions to the nonlinear Schrödinger equation (mathematical formular) with 4/n < p < 4/n-2, which was first obtained by Ginibre and Velo (Citation1985).","lang":"eng"}],"external_id":{"arxiv":["math/0511070"]},"page":"1281-1343","article_processing_charge":"No","arxiv":1,"date_updated":"2026-06-25T08:04:20Z","article_type":"original","publisher":"Informa UK Limited","publication_status":"published","status":"public","year":"2007","day":"29","author":[{"first_name":"Terence","last_name":"Tao","full_name":"Tao, Terence"},{"id":"056daca0-b8d1-11f0-964f-f91054abf8ca","full_name":"Visan, Monica","last_name":"Visan","first_name":"Monica"},{"first_name":"Xiaoyi","last_name":"Zhang","full_name":"Zhang, Xiaoyi"}],"OA_type":"green","date_published":"2007-08-29T00:00:00Z","keyword":["Energy-critical","Mass-critical","Nonlinear Schrödinger equation","Wellposedness"],"intvolume":"        32","extern":"1"},{"type":"journal_article","month":"06","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","citation":{"ama":"Vişan M. The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions. <i>Duke Mathematical Journal</i>. 2007;138(2):281-374. doi:<a href=\"https://doi.org/10.1215/s0012-7094-07-13825-0\">10.1215/s0012-7094-07-13825-0</a>","chicago":"Vişan, Monica. “The Defocusing Energy-Critical Nonlinear Schrödinger Equation in Higher Dimensions.” <i>Duke Mathematical Journal</i>. Duke University Press, 2007. <a href=\"https://doi.org/10.1215/s0012-7094-07-13825-0\">https://doi.org/10.1215/s0012-7094-07-13825-0</a>.","short":"M. Vişan, Duke Mathematical Journal 138 (2007) 281–374.","ieee":"M. Vişan, “The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions,” <i>Duke Mathematical Journal</i>, vol. 138, no. 2. Duke University Press, pp. 281–374, 2007.","mla":"Vişan, Monica. “The Defocusing Energy-Critical Nonlinear Schrödinger Equation in Higher Dimensions.” <i>Duke Mathematical Journal</i>, vol. 138, no. 2, Duke University Press, 2007, pp. 281–374, doi:<a href=\"https://doi.org/10.1215/s0012-7094-07-13825-0\">10.1215/s0012-7094-07-13825-0</a>.","apa":"Vişan, M. (2007). The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions. <i>Duke Mathematical Journal</i>. Duke University Press. <a href=\"https://doi.org/10.1215/s0012-7094-07-13825-0\">https://doi.org/10.1215/s0012-7094-07-13825-0</a>","ista":"Vişan M. 2007. The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions. Duke Mathematical Journal. 138(2), 281–374."},"scopus_import":"1","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.math/0508298"}],"_id":"22050","das_tickbox":"1","publication_identifier":{"issn":["0012-7094"]},"doi":"10.1215/s0012-7094-07-13825-0","date_created":"2026-06-19T07:53:37Z","oa":1,"oa_version":"Preprint","OA_place":"repository","language":[{"iso":"eng"}],"quality_controlled":"1","issue":"2","title":"The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions","volume":138,"mathsc":["35Q55"],"abstract":[{"text":"We obtain global well-posedness, scattering, and global L2(n+2)/(n−2)/t,x space-time\r\nbounds for energy-space solutions to the energy-critical nonlinear Schrodinger (NLS) ¨\r\nequation in Rt × Rn/x , n ≥ 5.","lang":"eng"}],"publication":"Duke Mathematical Journal","article_processing_charge":"No","page":"281-374","external_id":{"arxiv":["math/0508298"]},"arxiv":1,"date_updated":"2026-06-25T08:18:44Z","article_type":"original","publisher":"Duke University Press","publication_status":"published","status":"public","year":"2007","day":"01","author":[{"id":"056daca0-b8d1-11f0-964f-f91054abf8ca","full_name":"Visan, Monica","last_name":"Visan","first_name":"Monica"}],"OA_type":"green","date_published":"2007-06-01T00:00:00Z","intvolume":"       138","extern":"1"}]