The Dullin-Gottwald-Holm Equation on the Half-Line

We investigate the initial-boundary value problem for the Dullin-Gottwald-Holm equation formulated on the half-line, an integrable model that includes the Camassa-Holm equation and the Korteweg-de Vries equation as special cases. Exploiting its complete integrability, we construct eigenfunctions associated with the Lax pair and analyze their properties, which lead to the definition of spectral functions and the derivation of the global relation involving the initial and boundary data. We then formulate a matrix Riemann-Hilbert problem that characterizes the solution of the initial-boundary value problem and develop the corresponding inverse spectral mappings. Finally, we establish an existence result under the assumption that the associated spectral data satisfy the global relation.