Quasi-rational solutions of the NLS equation and rogue waves

Pierre Gaillard (1)

1 IMB - Institut de Mathématiques de Bourgogne [Dijon]

We degenerate solutions of the NLS equation from the general formulation in terms of theta functions to get quasi-rational solutions of NLS equations. For this we establish a link between Fredholm determinants and Wronskians. We give solutions of the NLS equation as a quotient of two wronskian determinants. In the limit when some parameter goes to $0$, we recover Akhmediev's solutions given recently It gives a new approach to get the well known rogue waves.

Mots clés: NLS equation; Riemann theta function; Wronskians; Fredholm determinants; NLS equation.

Domaines: Equations aux dérivées partielles [math.AP]; Etudes classiques

Pierre Gaillard. Quasi-rational solutions of the NLS equation and rogue waves. 2010. ⟨halshs-00536287v6⟩