We prove that the Ginzburg–Landau energy of non-constant travelling waves of the Gross–Pitaevskii equation has a lower positive bound, depending only on the dimension, in any dimension larger or equal to three. In particular, we conclude that there are no non-constant travelling waves with small energy.
On démontre que l'énergie de Ginzburg–Landau des ondes progressives non constantes de l'équation de Gross–Pitaevskii est bornée inférieurement par une constante positive qui ne dépend que de la dimension, pour toute dimension supérieure ou égale à trois. En particulier, on en déduit qu'il n'existe pas d'onde progressive non constante d'énergie petite.
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André de Laire 1
@article{CRMATH_2009__347_7-8_375_0, author = {Andr\'e de Laire}, title = {Non-existence for travelling waves with small energy for the {Gross{\textendash}Pitaevskii} equation in dimension $ N\ensuremath{\geqslant}3$}, journal = {Comptes Rendus. Math\'ematique}, pages = {375--380}, publisher = {Elsevier}, volume = {347}, number = {7-8}, year = {2009}, doi = {10.1016/j.crma.2009.02.006}, language = {en}, }
TY - JOUR AU - André de Laire TI - Non-existence for travelling waves with small energy for the Gross–Pitaevskii equation in dimension $ N⩾3$ JO - Comptes Rendus. Mathématique PY - 2009 SP - 375 EP - 380 VL - 347 IS - 7-8 PB - Elsevier DO - 10.1016/j.crma.2009.02.006 LA - en ID - CRMATH_2009__347_7-8_375_0 ER -
André de Laire. Non-existence for travelling waves with small energy for the Gross–Pitaevskii equation in dimension $ N⩾3$. Comptes Rendus. Mathématique, Volume 347 (2009) no. 7-8, pp. 375-380. doi : 10.1016/j.crma.2009.02.006. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.02.006/
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