To solve exactly the problem of plane EM wave interaction with multilayer nonlinear structures a non-traditional theoretical method is applied. Wave equation's solution in each layer of a structure is presented in the special form of a single expression. This form of field presentation does not require the fulfillment of superposition principle, contrary to the traditional approach. Multilayer structures of threefold quarter-wavelength bilayers with alternate linear and nonlinear indices are investigated. Multistable dependences of reflection coefficients on incident electric field amplitude are obtained for negative and positive nonlinearity. The results of investigation not only permits to deeply understand the phenomena of intense electromagnetic wave interaction with multilayer structures but also will be very useful for elaboration of nonlinear devices.