On some integrability aspects of the two-component Camassa–Holm equation - Deng-Shan Wang (Partial Differential Equations in Applied Mathematics, 2022)
The integrability of the two-component Camassa–Holm shallow water equation with the gravity acceleration points downwards and upwards is investigated by prolongation structure theory and the geometric approach of pseudo-spherical surfaces. It is shown that this equation is completely integrable in sense of having Lax pair and describing the pseudo-spherical surface. The reciprocal transformation is proposed to connect its solution with the solution of a conservative equation.