[{"type":"journal_article","mathsc":["35Q55"],"date_published":"2018-01-01T00:00:00Z","status":"public","publication_status":"published","publisher":"Society for Industrial & Applied Mathematics","article_processing_charge":"No","year":"2018","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"}],"month":"01","fulldoi":"https://doi.org/10.1137/17m1116702","OA_place":"repository","author":[{"full_name":"Killip, Rowan","last_name":"Killip","first_name":"Rowan"},{"last_name":"Murphy","full_name":"Murphy, Jason","first_name":"Jason"},{"last_name":"Visan","full_name":"Visan, Monica","first_name":"Monica","id":"056daca0-b8d1-11f0-964f-f91054abf8ca"}],"article_type":"original","oa_version":"Preprint","publication_identifier":{"issn":["0036-1410","1095-7154"]},"das_tickbox":"1","day":"01","extern":"1","doi":"10.1137/17m1116702","language":[{"iso":"eng"}],"intvolume":"        50","issue":"3","external_id":{"arxiv":["1702.04413"]},"quality_controlled":"1","keyword":["cubic-quintic NLS","nonvanishing boundary conditions","space-time resonances","scattering"],"date_created":"2026-06-19T07:49:03Z","page":"2681-2739","volume":50,"_id":"22045","arxiv":1,"citation":{"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>","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>","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.","short":"R. Killip, J. Murphy, M. Vişan, SIAM Journal on Mathematical Analysis 50 (2018) 2681–2739.","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."},"publication":"SIAM Journal on Mathematical Analysis","date_updated":"2026-06-25T07:49:21Z","scopus_import":"1","OA_type":"green","title":"The initial-value problem for the cubic-quintic NLS with nonvanishing boundary conditions","oa":1,"main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.1702.04413"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87"},{"page":"1523-1574","_id":"22051","volume":9,"arxiv":1,"citation":{"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>","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>","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.","short":"R. Killip, J. Murphy, M. Vişan, Analysis &#38; PDE 9 (2016) 1523–1574."},"publication":"Analysis & PDE","scopus_import":"1","date_updated":"2026-06-25T08:23:10Z","title":"The final-state problem for the cubic-quintic NLS with nonvanishing boundary conditions","oa":1,"OA_type":"green","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.1506.06151","open_access":"1"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","doi":"10.2140/apde.2016.9.1523","language":[{"iso":"eng"}],"intvolume":"         9","issue":"7","external_id":{"arxiv":["1506.06151"]},"quality_controlled":"1","date_created":"2026-06-19T07:54:01Z","keyword":["final-state problem","wave operators","cubic-quintic NLS","nonvanishing boundary conditions"],"fulldoi":"https://doi.org/10.2140/apde.2016.9.1523","OA_place":"repository","author":[{"last_name":"Killip","full_name":"Killip, Rowan","first_name":"Rowan"},{"first_name":"Jason","last_name":"Murphy","full_name":"Murphy, Jason"},{"id":"056daca0-b8d1-11f0-964f-f91054abf8ca","first_name":"Monica","full_name":"Visan, Monica","last_name":"Visan"}],"article_type":"original","oa_version":"Preprint","publication_identifier":{"issn":["2157-5045"],"eissn":["1948-206X"]},"das_tickbox":"1","extern":"1","day":"07","mathsc":["35Q55"],"type":"journal_article","date_published":"2016-11-07T00:00:00Z","status":"public","publication_status":"published","year":"2016","article_processing_charge":"No","publisher":"Mathematical Sciences Publishers","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."}],"month":"11"}]