{"id":"https://openalex.org/W1971610215","doi":"https://doi.org/10.1137/140965089","title":"The Integrable Nature of Modulational Instability","display_name":"The Integrable Nature of Modulational Instability","publication_year":2015,"publication_date":"2015-01-01","ids":{"openalex":"https://openalex.org/W1971610215","doi":"https://doi.org/10.1137/140965089","mag":"1971610215"},"language":"en","primary_location":{"id":"doi:10.1137/140965089","is_oa":false,"landing_page_url":"https://doi.org/10.1137/140965089","pdf_url":null,"source":{"id":"https://openalex.org/S77200987","display_name":"SIAM Journal on Applied Mathematics","issn_l":"0036-1399","issn":["0036-1399","1095-712X"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Applied Mathematics","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5021223423","display_name":"Gino Biondini","orcid":"https://orcid.org/0000-0003-3835-1343"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Gino Biondini","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5083652944","display_name":"Emily Fagerstrom","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Emily Fagerstrom","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":5.9623,"has_fulltext":false,"cited_by_count":52,"citation_normalized_percentile":{"value":0.96529882,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":94,"max":99},"biblio":{"volume":"75","issue":"1","first_page":"136","last_page":"163"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10248","display_name":"Nonlinear Waves and Solitons","score":0.9998000264167786,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10248","display_name":"Nonlinear Waves and Solitons","score":0.9998000264167786,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11575","display_name":"Nonlinear Photonic Systems","score":0.9998000264167786,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10988","display_name":"Advanced Fiber Laser Technologies","score":0.9988999962806702,"subfield":{"id":"https://openalex.org/subfields/3107","display_name":"Atomic and Molecular Physics, and Optics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/modulational-instability","display_name":"Modulational instability","score":0.763178288936615},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.640140175819397},{"id":"https://openalex.org/keywords/korteweg\u2013de-vries-equation","display_name":"Korteweg\u2013de Vries equation","score":0.5963248014450073},{"id":"https://openalex.org/keywords/inverse-scattering-problem","display_name":"Inverse scattering problem","score":0.5805273652076721},{"id":"https://openalex.org/keywords/instability","display_name":"Instability","score":0.5694975256919861},{"id":"https://openalex.org/keywords/nonlinear-schr\u00f6dinger-equation","display_name":"Nonlinear Schr\u00f6dinger equation","score":0.5618534684181213},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.5589162707328796},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5475101470947266},{"id":"https://openalex.org/keywords/piecewise","display_name":"Piecewise","score":0.5425293445587158},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.5302664041519165},{"id":"https://openalex.org/keywords/constant","display_name":"Constant (computer programming)","score":0.5201135873794556},{"id":"https://openalex.org/keywords/mathematical-physics","display_name":"Mathematical physics","score":0.489362895488739},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.4853760004043579},{"id":"https://openalex.org/keywords/linearization","display_name":"Linearization","score":0.47211992740631104},{"id":"https://openalex.org/keywords/lax-pair","display_name":"Lax pair","score":0.45829662680625916},{"id":"https://openalex.org/keywords/conjecture","display_name":"Conjecture","score":0.4260629713535309},{"id":"https://openalex.org/keywords/boundary-value-problem","display_name":"Boundary value problem","score":0.42293494939804077},{"id":"https://openalex.org/keywords/initial-value-problem","display_name":"Initial value problem","score":0.42278993129730225},{"id":"https://openalex.org/keywords/integrable-system","display_name":"Integrable system","score":0.4185565114021301},{"id":"https://openalex.org/keywords/inverse-problem","display_name":"Inverse problem","score":0.35623013973236084},{"id":"https://openalex.org/keywords/schr\u00f6dinger-equation","display_name":"Schr\u00f6dinger equation","score":0.3549574017524719},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.2675776481628418},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.1841544806957245}],"concepts":[{"id":"https://openalex.org/C125755039","wikidata":"https://www.wikidata.org/wiki/Q17104082","display_name":"Modulational instability","level":3,"score":0.763178288936615},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.640140175819397},{"id":"https://openalex.org/C146630112","wikidata":"https://www.wikidata.org/wiki/Q601796","display_name":"Korteweg\u2013de Vries equation","level":3,"score":0.5963248014450073},{"id":"https://openalex.org/C174558057","wikidata":"https://www.wikidata.org/wiki/Q11639409","display_name":"Inverse scattering problem","level":3,"score":0.5805273652076721},{"id":"https://openalex.org/C207821765","wikidata":"https://www.wikidata.org/wiki/Q405372","display_name":"Instability","level":2,"score":0.5694975256919861},{"id":"https://openalex.org/C83774755","wikidata":"https://www.wikidata.org/wiki/Q2579914","display_name":"Nonlinear Schr\u00f6dinger equation","level":3,"score":0.5618534684181213},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.5589162707328796},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5475101470947266},{"id":"https://openalex.org/C164660894","wikidata":"https://www.wikidata.org/wiki/Q2037833","display_name":"Piecewise","level":2,"score":0.5425293445587158},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.5302664041519165},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.5201135873794556},{"id":"https://openalex.org/C37914503","wikidata":"https://www.wikidata.org/wiki/Q156495","display_name":"Mathematical physics","level":1,"score":0.489362895488739},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.4853760004043579},{"id":"https://openalex.org/C11210021","wikidata":"https://www.wikidata.org/wiki/Q1520713","display_name":"Linearization","level":3,"score":0.47211992740631104},{"id":"https://openalex.org/C172829509","wikidata":"https://www.wikidata.org/wiki/Q4115796","display_name":"Lax pair","level":3,"score":0.45829662680625916},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.4260629713535309},{"id":"https://openalex.org/C182310444","wikidata":"https://www.wikidata.org/wiki/Q1332643","display_name":"Boundary value problem","level":2,"score":0.42293494939804077},{"id":"https://openalex.org/C26955809","wikidata":"https://www.wikidata.org/wiki/Q10167591","display_name":"Initial value problem","level":2,"score":0.42278993129730225},{"id":"https://openalex.org/C200741047","wikidata":"https://www.wikidata.org/wiki/Q1957758","display_name":"Integrable system","level":2,"score":0.4185565114021301},{"id":"https://openalex.org/C135252773","wikidata":"https://www.wikidata.org/wiki/Q1567213","display_name":"Inverse problem","level":2,"score":0.35623013973236084},{"id":"https://openalex.org/C63036615","wikidata":"https://www.wikidata.org/wiki/Q165498","display_name":"Schr\u00f6dinger equation","level":2,"score":0.3549574017524719},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.2675776481628418},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.1841544806957245},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1137/140965089","is_oa":false,"landing_page_url":"https://doi.org/10.1137/140965089","pdf_url":null,"source":{"id":"https://openalex.org/S77200987","display_name":"SIAM Journal on Applied Mathematics","issn_l":"0036-1399","issn":["0036-1399","1095-712X"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Applied Mathematics","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":55,"referenced_works":["https://openalex.org/W96933547","https://openalex.org/W101248039","https://openalex.org/W171105080","https://openalex.org/W204347044","https://openalex.org/W1488636064","https://openalex.org/W1501956976","https://openalex.org/W1509169316","https://openalex.org/W1510817630","https://openalex.org/W1545160759","https://openalex.org/W1552987673","https://openalex.org/W1561675722","https://openalex.org/W1582129286","https://openalex.org/W1582823736","https://openalex.org/W1631114555","https://openalex.org/W1631607997","https://openalex.org/W1676457984","https://openalex.org/W1708785370","https://openalex.org/W1980663754","https://openalex.org/W1983957961","https://openalex.org/W1986788497","https://openalex.org/W2004508559","https://openalex.org/W2017277106","https://openalex.org/W2026236086","https://openalex.org/W2027151088","https://openalex.org/W2028777092","https://openalex.org/W2029064797","https://openalex.org/W2032514295","https://openalex.org/W2034326727","https://openalex.org/W2045239004","https://openalex.org/W2046521424","https://openalex.org/W2049915610","https://openalex.org/W2052638644","https://openalex.org/W2056413318","https://openalex.org/W2062565740","https://openalex.org/W2066716989","https://openalex.org/W2071030514","https://openalex.org/W2074366694","https://openalex.org/W2083045785","https://openalex.org/W2093598519","https://openalex.org/W2099884912","https://openalex.org/W2102582588","https://openalex.org/W2104369108","https://openalex.org/W2111796192","https://openalex.org/W2118575947","https://openalex.org/W2140334330","https://openalex.org/W2141237926","https://openalex.org/W2153786792","https://openalex.org/W2169773650","https://openalex.org/W2795721869","https://openalex.org/W2955491277","https://openalex.org/W3009283957","https://openalex.org/W3038282287","https://openalex.org/W3105088137","https://openalex.org/W3118670128","https://openalex.org/W4243708910"],"related_works":["https://openalex.org/W2329242664","https://openalex.org/W4283373761","https://openalex.org/W2088259254","https://openalex.org/W1999843133","https://openalex.org/W1542159970","https://openalex.org/W2120820547","https://openalex.org/W1977444113","https://openalex.org/W3209339817","https://openalex.org/W3118776131","https://openalex.org/W2041753758"],"abstract_inverted_index":{"We":[0,31,92,175,253],"investigate":[1],"the":[2,6,13,18,35,46,49,57,65,67,70,85,89,96,100,104,108,112,135,140,144,148,151,164,189,207,239,243,246,262,276,280,287,291,304,308,313,317,323,332,336,344,351,354],"nonlinear":[3,20,86,305],"stage":[4,87],"of":[5,48,69,88,107,126,143,153,156,162,166,169,188,220,245,267,279,290,307,322,331,353],"modulational":[7,90,355],"(or":[8],"Benjamin--Feir)":[9],"instability":[10],"by":[11,94],"characterizing":[12],"initial":[14,118,145],"value":[15],"problem":[16,98],"for":[17,42,74,99,111,192,206,222,242,320,350],"focusing":[19],"Schr\u00f6dinger":[21],"(NLS)":[22],"equation":[23,51,209,293],"with":[24,115,210,257,286],"nonzero":[25],"boundary":[26],"conditions":[27,119],"(NZBC)":[28],"at":[29],"infinity.":[30],"do":[32],"so":[33],"using":[34],"recently":[36],"formulated":[37],"inverse":[38,149],"scattering":[39,136,324],"transform":[40],"(IST)":[41],"this":[43,231,339],"problem.":[44],"While":[45],"linearization":[47,289],"NLS":[50,113,208,292],"ceases":[52],"to":[53,64,83,124,249,301,327],"be":[54,238],"valid":[55,73],"when":[56],"perturbations":[58,168,187,221],"have":[59],"grown":[60],"sufficiently":[61],"large":[62],"compared":[63],"background,":[66,297],"results":[68,256],"IST":[71,345],"remain":[72],"all":[75],"times":[76],"and":[77,130,265,282],"therefore":[78],"provide":[79],"a":[80,122,127,131,159,170,218,250,258,295,328],"convenient":[81],"way":[82],"study":[84,152,260],"instability.":[91,356],"begin":[93],"studying":[95],"spectral":[97],"Dirac":[101],"operator":[102],"(i.e.,":[103],"first":[105],"half":[106],"Lax":[109],"pair":[110],"equation)":[114],"piecewise":[116],"constant":[117,171,190,296],"which":[120,193,198,223,298],"are":[121,184,195,227,316],"generalization":[123],"NZBC":[125],"potential":[128,132],"well":[129],"barrier.":[133],"Since":[134],"data":[137],"uniquely":[138],"determine":[139],"time":[141],"evolution":[142],"condition":[146],"via":[147,173],"problem,":[150],"these":[154,255],"kinds":[155],"potentials":[157],"provides":[158],"simple":[160],"means":[161],"investigating":[163],"growth":[165],"small":[167,186],"background":[172,191],"IST.":[174,314],"obtain":[176],"several":[177],"results.":[178],"First,":[179],"we":[180,213,274,283],"prove":[181,214],"that":[182,200,215,235,343],"there":[183,194,216],"arbitrarily":[185],"discrete":[196,225,268],"eigenvalues,":[197],"shows":[199,234,342],"no":[201,224],"area":[202],"theorem":[203],"is":[204,217],"possible":[205],"NZBC.":[211],"Second,":[212],"class":[219],"eigenvalues":[226,269],"present.":[228],"In":[229],"particular,":[230],"latter":[232],"result":[233,341],"solitons":[236],"cannot":[237],"primary":[240],"vehicle":[241],"manifestation":[244],"instability,":[247],"contrary":[248],"recent":[251],"conjecture.":[252],"supplement":[254],"numerical":[259],"about":[261],"existence,":[263],"number,":[264],"location":[266],"in":[270],"other":[271],"situations.":[272],"Finally,":[273],"compute":[275],"small-deviation":[277],"limit":[278],"IST,":[281],"compare":[284],"it":[285],"direct":[288],"around":[294,335],"allows":[299],"us":[300],"precisely":[302],"identify":[303],"analogue":[306],"unstable":[309],"Fourier":[310],"modes":[311],"within":[312],"These":[315],"Jost":[318],"eigenfunctions":[319],"values":[321],"parameter":[325],"belonging":[326],"finite":[329],"interval":[330],"imaginary":[333],"axis":[334],"origin.":[337],"Importantly,":[338],"last":[340],"contains":[346],"an":[347],"automatic":[348],"mechanism":[349],"saturation":[352]},"counts_by_year":[{"year":2026,"cited_by_count":3},{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":4},{"year":2023,"cited_by_count":3},{"year":2022,"cited_by_count":3},{"year":2021,"cited_by_count":5},{"year":2020,"cited_by_count":3},{"year":2019,"cited_by_count":4},{"year":2018,"cited_by_count":11},{"year":2017,"cited_by_count":2},{"year":2016,"cited_by_count":8},{"year":2015,"cited_by_count":4}],"updated_date":"2026-06-17T08:01:34.144755","created_date":"2025-10-10T00:00:00"}