Nonsensitive nonlinear homotopy approach

TL;DR

This paper introduces a novel nonlinear homotopy method that combines perturbative and nonperturbative techniques, using artificial parameters and minimal sensitivity principles to solve complex quantum and physical problems.

Contribution

A new general framework that integrates perturbation and nonperturbation approaches via nonlinear homotopy and minimal sensitivity, applicable to various physical systems.

Findings

Successfully applied to quantum anharmonic oscillators

Effectively handled non-Hermitian parity-time symmetric systems

Potential for broad applications in physics fields

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. In this Letter, a new theory that combines perturbation and nonperturbation is constructed. An artificial nonlinear homotopy parameter plays the role of a perturbation parameter, while other artificial nonlinear parameters, of which the original problems are independent, introduced in the nonlinear homotopy models are nonperturbatively determined by means of a principle 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 that 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.