The local well-posedness of the Schrödinger–KdV system in H s 1 × H s 2 , I

In this paper, we study the local well-posedness theory of the Schrödinger–KdV system in Sobolev spaces H s 1 ( R ) × H s 2 ( R ) . We obtain the local well-posedness when s 1 ⩾ 0 , max { − 3 / 4 , s 1 − 3 } ⩽ s 2 ⩽ min { 4 s 1 , s 1 + 2 } . The result is sharp in some sense and improves previous one by Corcho–Linares (2007 Trans. Am. Math. Soc. 359 4089–106), Wu (2010 Differ. Int. Equ. 23 569–600). The endpoint case ( s 1 , s 2 ) = ( 0 , − 3 / 4 ) had been solved in Guo–Wang and Wang–Cui (2010 J. Differ. Equ. 249 2500–20; 2011 J. Differ. Equ. 250 3559–83). We show the necessary and sufficient conditions for related estimates in Bourgain spaces. To solve the borderline cases, we use the U p - V p spaces introduced by Koch–Tataru (2005 Commun. Pure Appl. Math. 58 217–84) and function spaces constructed by Guo–Wang (2010 J. Differ. Equ. 249 2500–20). We also use the normal form argument to control the nonresonant interaction.