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We argue that near the critical point of gradient catastrophe of the dispersionless system, the solutions to a suitable initial value problem for the perturbed equations are approximately described by particular solutions to the Painlev\u00e9-I (P<jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$_I$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mrow\/>\n                    <mml:mi>I<\/mml:mi>\n                  <\/mml:msub>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>) equation or its fourth-order analogue P<jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$_I^2$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msubsup>\n                    <mml:mrow\/>\n                    <mml:mi>I<\/mml:mi>\n                    <mml:mn>2<\/mml:mn>\n                  <\/mml:msubsup>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>. As concrete examples, we discuss nonlinear Schr\u00f6dinger equations in the semiclassical limit. A numerical study of these cases provides strong evidence in support of the conjecture.<\/jats:p>","DOI":"10.1007\/s00332-015-9236-y","type":"journal-article","created":{"date-parts":[[2015,2,10]],"date-time":"2015-02-10T09:27:30Z","timestamp":1423560450000},"page":"631-707","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":26,"title":["On Critical Behaviour in Systems of Hamiltonian Partial Differential Equations"],"prefix":"10.1007","volume":"25","author":[{"given":"Boris","family":"Dubrovin","sequence":"first","affiliation":[]},{"given":"Tamara","family":"Grava","sequence":"additional","affiliation":[]},{"given":"Christian","family":"Klein","sequence":"additional","affiliation":[]},{"given":"Antonio","family":"Moro","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2015,2,11]]},"reference":[{"key":"9236_CR1","volume-title":"Nonlinear Fiber Optics","author":"GP Agrawal","year":"2006","unstructured":"Agrawal, G.P.: Nonlinear Fiber Optics, 4th edn. 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