{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,3]],"date-time":"2025-11-03T04:57:40Z","timestamp":1762145860688,"version":"build-2065373602"},"reference-count":52,"publisher":"Springer Science and Business Media LLC","issue":"6","license":[{"start":{"date-parts":[[2025,9,25]],"date-time":"2025-09-25T00:00:00Z","timestamp":1758758400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,9,25]],"date-time":"2025-09-25T00:00:00Z","timestamp":1758758400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Nonlinear Sci"],"published-print":{"date-parts":[[2025,12]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    In the work Cho et al. (Jpn J Ind Appl Math 33:145\u2013166, 2016) the authors conjecture that the quadratic nonlinear Schr\u00f6dinger equation (NLS)\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$ i u_t = u_{xx} + u^2 $$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>i<\/mml:mi>\n                            <mml:msub>\n                              <mml:mi>u<\/mml:mi>\n                              <mml:mi>t<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>u<\/mml:mi>\n                              <mml:mrow>\n                                <mml:mi>xx<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>u<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msup>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    for\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$ x \\in \\mathbb {T}$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo>\u2208<\/mml:mo>\n                            <mml:mi>T<\/mml:mi>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    is globally well-posed for real initial data. We identify initial data whose numerical solution blows up in contradiction of this conjecture. The solution exhibits self-similar blowup and potentially nontrivial self-similar dynamics, however the proper scaling ansatz remains elusive. Furthermore, the set of real initial data which blows up under the NLS dynamics appears to occur on a codimension-1 manifold, and we conjecture that it is precisely the stable manifold of the zero equilibrium for the nonlinear heat equation\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$ u_t = u_{xx} + u^2 $$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>u<\/mml:mi>\n                              <mml:mi>t<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>u<\/mml:mi>\n                              <mml:mrow>\n                                <mml:mi>xx<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>u<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msup>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    . We apply the parameterization method to study the internal dynamics of this manifold, offering a heuristic argument in support of our conjecture.\n                  <\/jats:p>","DOI":"10.1007\/s00332-025-10212-0","type":"journal-article","created":{"date-parts":[[2025,9,25]],"date-time":"2025-09-25T18:20:56Z","timestamp":1758824456000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Mechanisms of Unstable Blowup in a Quadratic Nonlinear Schr\u00f6dinger Equation"],"prefix":"10.1007","volume":"35","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8380-3148","authenticated-orcid":false,"given":"Jonathan","family":"Jaquette","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,9,25]]},"reference":[{"issue":"1","key":"10212_CR1","doi-asserted-by":"crossref","first-page":"228","DOI":"10.1016\/j.jfa.2005.08.004","volume":"233","author":"I Bejenaru","year":"2006","unstructured":"Bejenaru, I., Tao, T.: Sharp well-posedness and ill-posedness results for a quadratic non-linear Schr\u00f6dinger equation. 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