A Few New 2+1-Dimensional Nonlinear Dynamics and the Representation of Riemann Curvature Tensors

We first introduced a linear stationary equation with a quadratic operator in ∂xand ∂y, then a linear evolution equation is given byN-order polynomials of eigenfunctions. As applications, by takingN=2, we derived a (2+1)-dimensional generalized linear heat equation with two constant parameters associative with a symmetric space. When takingN=3, a pair of generalized Kadomtsev-Petviashvili equations with the same eigenvalues with the case ofN=2 are generated. Similarly, a second-order flow associative with a homogeneous space is derived from the integrability condition of the two linear equations, which is a (2+1)-dimensional hyperbolic equation. WhenN=3, the third second flow associative with the homogeneous space is generated, which is a pair of new generalized Kadomtsev-Petviashvili equations. Finally, as an application of a Hermitian symmetric space, we established a pair of spectral problems to obtain a new (2+1)-dimensional generalized Schrödinger equation, which is expressed by the Riemann curvature tensors.