Well-posedness of Cauchy problems for Korteweg-de Vries-Benjamin-Ono equation and Hirota equation

Abstract

Abstract The well-posedness of the Cauchy problems to the Korteweg-de Vries-Benjamin-Ono equation and Hirota equation is considered. For the Korteweg-de Vries-Benjamin-Ono equation, local result is established for data in Hs (R)(s⩾−1/8). Moreover, the global well-posedness for data in L 2(R) can be obtained. For Hirota equation, local result is established for initial data in Hs (s⩾1/4). In addition, the local solution is proved to be global in Hs (s⩾1)if the initial data are in Hs (s⩾1) by energy inequality and the generalization of the trilinear estimates associated with the Fourier restriction norm method.