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

RESUMO

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 \[u_0\in H^s(\mathbb{R}^+),\qquad \varphi_\ell\in H^{\frac{s+j-\ell}{2j+1}}(0,T),\qquad -j+\frac14<s<\frac32.\] In particular, the regularity range extends the low regularity theory for the higher order Dirichlet IBVP to Robin boundary conditions. The main ingredient is an explicit unified transform representation for the higher order IBVP with Robin boundary conditions. The Robin hierarchy couples adjacent boundary traces and leads, in each spectral sector, to a nontrivial system for the unknown boundary transforms. Rather than computing the inverse of this system, we resolve precisely the linear combination required by the unified transform representation through Lagrange interpolation at the rotated spectral points. This yields an exact factorization in which the dependence on the Robin parameter is concentrated in the single factor $(k-i\gamma)^{-1}$. Consequently, the only possible Robin pole is $k=i\gamma$, which contributes the residue mode \[e^{-\gamma x+\gamma^{2j+1}t}\] exactly when $j$ is odd and $\gamma>0$. 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.