Higher order KdV-type equations with Robin boundary conditions on $\mathbb{R}^+$

Abstract

We study the initial boundary value problem (IBVP) for the higher order Korteweg--de Vries type equation \[\partial_tu+(-1)^{j+1}\partial_x^{2j+1}u+\frac12\partial_x(u^2)=0,\qquad j\in\mathbb{N},\] on the right half line, subject to the Robin boundary conditions \[(\partial_x+\gamma)\partial_x^{\ell-1}u(t,0)=\varphi_\ell(t),\qquad 1\le\ell\le j,\] where $\gamma\in\mathbb{R}$ is common to all boundary conditions. We prove local well posedness for \[u_0\in H^s(\mathbb{R}^+),\qquad \varphi_\ell\in H^{\frac{s+j-\ell}{2j+1}}(0,T),\qquad -j+\frac14 0$. Combining this representation with linear estimates in modified Fourier restriction spaces and higher order KdV bilinear estimates yields the low regularity well posedness result. The representation also recovers the classical KdV formulas with Robin and Neumann boundary data when $j=1$ and is formally consistent with the higher order Dirichlet representation under the reciprocal Robin limit.