On Differential Equations Describing 3-Dimensional Hyperbolic Spaces

In this paper, we introducethe notion of a (2+1)-dimensional differential equation describingthree-dimensional hyperbolic spaces (3-h.s.). The (2+1)-dimensional coupled nonlinear Schrödinger equation and itssister equation, the (2+1)-dimensional coupled derivative nonlinear Schrödinger equation, are shown to describe 3-h.s. The (2+1)-dimensional generalized HF model:S_t={(1/2i)[S,S_y]+2iσS}_x,σ_x=-(1/4i)tr(SS_xS_y), in which S∈[GL_C(2)]/[GL_C(1)×GL_C(1)], provides another example of (2+1)-dimensional differential equations describing 3-h.s. As a direct consequence, the geometric construction of an infinitenumber of conservation laws of such equations is illustrated.Furthermore we display a new infinite number of conservation lawsof the (2+1)-dimensional nonlinear Schrödinger equation and the(2+1)-dimensional derivative nonlinear Schrödinger equationby a geometric way.