The initial-boundary value problem (ibvp) for the mth order Korteweg–de Vries (KdVm) equation on the half-line is studied by extending a novel approach recently developed for the well-posedness of KdV on the half-line, which is based on the solution formula produced via the Fokas unified transform method for the associated forced linear ibvp. Replacing in this formula the forcing by the nonlinearity and using data in Sobolev spaces suggested by the space–time regularity of the Cauchy problem of the linear KdVm equation give an iteration map for the ibvp, which is shown to be a contraction in an appropriately chosen solution space. The proof relies on key linear estimates and a bilinear estimate similar to the one used for the KdV equation on the line by Kenig, Ponce, and Vega [J. Am. Math. Soc. 4, 323–347 (1991)].

中文翻译：

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半线上较高色散 KdV 方程的适定性

通过扩展最近为 KdV 在半线上的适定性而开发的新方法，研究了半线上 m 阶 Korteweg-de Vries (KdVm) 方程的初始边界值问题 (ibvp)，即基于通过 Fokas 统一变换方法为相关联的强制线性 ibvp 生成的解公式。Replacing in this formula the forcing by the nonlinearity and using data in Sobolev spaces suggested by the space–time regularity of the Cauchy problem of the linear KdVm equation give an iteration map for the ibvp, which is shown to be a contraction in an appropriately chosen解空间。证明依赖于关键的线性估计和类似于 Kenig、Ponce 和 Vega 在线使用的 KdV 方程的双线性估计 [J. 是。数学。社会。4, 323–347 (1991)]。