Deformation of surfaces in S~2(R),S~3(R)and affine geometries are studied in Chapter 2,which is derived by endowing with an extra space variable y to the formulae of the curve motions in these geometries,and induces a 2+1-dimensional generalization to the sine-Gordon equation,a 2+1-dimensional complex mKdV equation,a 2+1-dimensional breaking soliton equation and a 2+1-dimensional nonlinear evolution equation to be regarded as a generalization to the 1+1-dimensional KdV equation,respectively.

这种曲面形变是通过对相应几何中的曲线运动公式赋予一个额外的空间变量y得到的,并分别导出了2+1维广义的sine-Gordon方程,2+1维的复mKdV方程,2+1维的破裂孤立子方程和2+1维的非线性演化方程,此方程可以被认为是1+1维的KdV方程在2+1维的推广。