Exterior differential expression of the (1 + 1)-dimensional nonlinear evolution equation with Lax integrability

A (1+1)(1+1)-dimensional nonlinear evolution equation with Lax integrability is investigated in this paper with the aid of differential forms and exterior differentials. Equivalent definition of the Lax integrability is given. Relation between the Lax integrability and complete integrability is discussed. Geometric interpretation of the Lax equation is presented. Gauge transformation and Darboux transformation are restated in terms of the differential forms and exterior differentials. If λI−SλI−S is a Darboux transformation between two Lax equations, the system that the matrix S should satisfy is given, where λ is the spectral parameter and I   is the identity matrix. The completely integrable condition of that system is obtained according to the Frobenius theorem, and it is shown that S=HΛH−1S=HΛH−1 satisfies that completely integrable condition with H and Λ as two matrices.