Modified energy method and applications for the well-posedness for the higher-order Benjamin–Ono equation and the higher-order intermediate long wave equation

In this paper, the well-posedness of the higher-order Benjamin–Ono equation u t + ℋ ⁢ ( u x ⁢ x ) + u x ⁢ x ⁢ x = u ⁢ u x - ∂ x ⁡ ( u ⁢ ℋ ⁢ ∂ x ⁡ u + ℋ ⁢ ( u ⁢ ∂ x ⁡ u ) ) u_{t}+\mathcal{H}(u_{xx})+u_{xxx}=uu_{x}-\partial_{x}(u\mathcal{H}\partial_{x}% u+\mathcal{H}(u\partial_{x}u)) is considered. The modified energy method is introduced to consider the equation. It is shown that the Cauchy problem of the higher-order Benjamin–Ono equation is locally well-posed in H 3 / 4 {H^{3/4}} without using the gauge transformation. Moreover, the well-posedness of the higher-order intermediate long wave equation u t + 𝒢 δ ⁢ ( u x ⁢ x ) + u x ⁢ x ⁢ x = u ⁢ u x - ∂ x ⁡ ( u ⁢ 𝒢 δ ⁢ ∂ x ⁡ u + 𝒢 δ ⁢ ( u ⁢ ∂ x ⁡ u ) ) , 𝒢 δ = ℱ x - 1 ⁢ i ⁢ ( coth ⁡ ( δ ⁢ ξ ) ) ⁢ ℱ x , u_{t}+\mathcal{G}_{\delta}(u_{xx})+u_{xxx}=uu_{x}-\partial_{x}(u\mathcal{G}_{% \delta}\partial_{x}u+\mathcal{G}_{\delta}(u\partial_{x}u)),\quad\mathcal{G}_{% \delta}=\mathcal{F}_{x}^{-1}i(\coth(\delta\xi))\mathcal{F}_{x}, is considered. It is shown that the Cauchy problem of the higher-order intermediate long wave equation is locally well-posed in H 3 / 4 {H^{3/4}} .