Local well‐posedness for a type of periodic fifth‐order Korteweg–de Vries equations without nonlinear dispersive term

Abstract

We consider the Cauchy problem of the fifth‐order Korteweg–de Vries (KdV) equations without nonlinear dispersive term ∂tu−∂x5u+b0u∂xu+b1∂xfalse(∂xufalse)2=0,0.30emfalse(t,xfalse)∈ℝ×𝕋. Recently, Kappeler‐Molnar (2018) proved that the fifth‐order KdV equation with nonlinear dispersive term and Hamiltonian structure is globally well‐posed in Hsfalse(𝕋false) with s ≥ 0. Without the nonlinear dispersive term, Equation () is not integrable, and Kappeler–Molnar's approach is not valid. Using the idea of modifying Bourgain space, we prove that Equation () is locally well‐posed in Hsfalse(𝕋false) with s≥58.