When a reasonable long-range Coulomb interaction is introduced into the model Hamiltonian,two-dimensional modulations or the checkerboard patterns may appear for the underdoped samples,which is consistent with the scanning-tunneling-microscopy experiments.

The studyresultindicates thatthe Coulomb force,Van Der Waals force,Ross-linking force,electro-statically induced force and electric field force have obviˉouslyeffect on mechanical behaviors.

Study on the Stability of a Nonlinear Dynamic System --dynamic system effected by coulomb attractive force and linear spring force;