A direct method for solving Lax equations

The integrable $(1+1)$-dimensional Lax equations and their various exact solutions, including multisoliton solutions, can be derived by a straightforward algebraic procedure. This method starts with a specific case of the Sylvester equation, eliminating the necessity of introducing an initial value problem. The Lax equation, along with its modified and Schwarzian forms, is constructed using elements present in the Sylvester equation, allowing the exact solutions to be expanded in terms of the solutions of the Sylvester equation. In particular, we obtain the Lax pair for the Lax equation by this direct approach and analyze its soliton solutions asymptotically. Furthermore, we extend the dispersion relations associated with the Lax equation and formulate the $(2+1)$-dimensional C-type Kadomtsev–Petviashvili equation, along with its novel solutions.