Nonsensitive Nonlinear Homotopy Approach

Abstract

Generally, natural scientific problems are so complicated that one has to establish some effective perturbation or nonperturbation theories with respect to some associated ideal models. We construct a new theory that combines perturbation and nonperturbation. An artificial nonlinear homotopy parameter plays the role of a perturbation parameter, while other artificial nonlinear parameters, which are independent of the original problems, introduced in the nonlinear homotopy models are nonperturbatively determined by means of the principle of minimal sensitivity. The method is demonstrated through several quantum anharmonic oscillators and a non-hermitian parity-time symmetric Hamiltonian system. In fact, the framework of the theory is rather general and can be applied to a broad range of natural phenomena. Possible applications to condensed matter physics, matter wave systems, and nonlinear optics are briefly discussed.