Solvability of concatenated Runge-Kutta equations for second-order nonlinear PDEs

In this paper, we consider the solvability of concatenated Runge-Kutta (CRK) equations, which are the nonlinear algebraic equations arising from CRK methods applied to some kind of second-order nonlinear partial differential equations. To guarantee the existence and uniqueness of the solution of CRK equations, temporal and spatial step-sizes of CRK methods should be chosen appropriately. Under different assumptions, with the homotopy continuation method, we choose different step-sizes to ensure the solvability of CRK equations.