Generalized quantum theory for accessing nonlinear systems: the case of Li\'{e}nard and Levinson-Smith equations

TL;DR

This paper links a generalized quantum mechanics framework to nonlinear systems like Lie9nard and Levinson-Smith equations, revealing solutions and connections to position-dependent mass systems and solitons.

Contribution

It establishes a novel connection between generalized quantum theory and specific classes of nonlinear differential equations, providing analytical solutions and physical insights.

Findings

Closed form solutions for Lie9nard equations in Abel form

Relevance of Levinson-Smith equations to position-dependent mass systems

Emergence of solitonic solutions from level surface conditions

Abstract

We show that a recently introduced generalized scheme of quantum mechanics has connections to Li\'{e}nard and Levinson-Smith classes of nonlinear systems. For the Li\'{e}nard type, which has coefficients of odd and odd symmetry, we demonstrate that closed form solutions exist on conversion to the Abel form. For the Levinson-Smith equations, we find their relevance to position-dependent mass systems, with an interesting off-shoot that solitonic-like solutions emerge from the condition of the level surface in the system.