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Generalized quantum theory for accessing nonlinear systems: the case of Levinson-Smith equations

Bijan Bagchi, Anindya Ghose-Choudhury

TL;DR

This work establishes a bridge between generalized nonlinear quantum mechanics (NLQM) and classical nonlinear ODE classes, specifically Levinson-Smith and Liénard forms. By formulating a two-ket NLQM model and focusing on the S = 0 sector, the authors derive coupled dynamics for variables y and x, reveal their reduction to a Levinson-Smith–type equation, and apply Abel and Jacobi Last Multiplier techniques to obtain first integrals and a position-dependent mass framework. They demonstrate exact solutions, including hyperbolic and Jacobi elliptic forms, and reveal soliton-like profiles for certain parameter choices, illustrating rich dynamical behavior and consistent Lagrangian structure. The findings illuminate how NLQM can accommodate PD mass effects and link quantum evolution to solvable nonlinear oscillator dynamics, with potential relevance to optics, shallow water, and non-Hermitian quantum systems. Overall, the paper provides a concrete methodology for translating NLQM dynamics into well-studied nonlinear equations, enabling explicit solutions and new physical interpretations.

Abstract

Motivated by a recently developed generalized scheme of quantum mechanics, we touch upon connections with Levinson-Smith classes of nonlinear systems that contain as a particular case the Liénard family of differential equations. The latter, which has coefficients of odd and odd symmetry, admits a closed form solution when converted to the Abel form. Analysis of the governing condition shows that one of the nontrivial equilibrium points is stable in character. Other classes of differential equations that we encounter speak of solutions involving Jacobi elliptic functions for a certain combination of underlying parameters, while, for a different set, relevance to position-dependent mass systems is shown. In addition, an interesting off-shoot of our results is the emergence of solitonic-like solutions from the condition of the level surface in the system.

Generalized quantum theory for accessing nonlinear systems: the case of Levinson-Smith equations

TL;DR

This work establishes a bridge between generalized nonlinear quantum mechanics (NLQM) and classical nonlinear ODE classes, specifically Levinson-Smith and Liénard forms. By formulating a two-ket NLQM model and focusing on the S = 0 sector, the authors derive coupled dynamics for variables y and x, reveal their reduction to a Levinson-Smith–type equation, and apply Abel and Jacobi Last Multiplier techniques to obtain first integrals and a position-dependent mass framework. They demonstrate exact solutions, including hyperbolic and Jacobi elliptic forms, and reveal soliton-like profiles for certain parameter choices, illustrating rich dynamical behavior and consistent Lagrangian structure. The findings illuminate how NLQM can accommodate PD mass effects and link quantum evolution to solvable nonlinear oscillator dynamics, with potential relevance to optics, shallow water, and non-Hermitian quantum systems. Overall, the paper provides a concrete methodology for translating NLQM dynamics into well-studied nonlinear equations, enabling explicit solutions and new physical interpretations.

Abstract

Motivated by a recently developed generalized scheme of quantum mechanics, we touch upon connections with Levinson-Smith classes of nonlinear systems that contain as a particular case the Liénard family of differential equations. The latter, which has coefficients of odd and odd symmetry, admits a closed form solution when converted to the Abel form. Analysis of the governing condition shows that one of the nontrivial equilibrium points is stable in character. Other classes of differential equations that we encounter speak of solutions involving Jacobi elliptic functions for a certain combination of underlying parameters, while, for a different set, relevance to position-dependent mass systems is shown. In addition, an interesting off-shoot of our results is the emergence of solitonic-like solutions from the condition of the level surface in the system.
Paper Structure (4 sections, 26 equations, 2 figures)

This paper contains 4 sections, 26 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Formulation of a generalized NLQM
  3. Connecting to Levinson-Smith equations
  4. Summary

Figures (2)

  • Figure 1: (a) We plot the variation of $y$ with respect to the variable $z$ as shown by equation (19); (b) We plot the variation of $w$ with respect to $y$ at $a=\frac{2(2\mu+b)}{3}$.
  • Figure 2: Variation of $x(t)$ vs $t$ for the three equations (28), (30), and (31) for positive values of $t$ only. We have taken $E=10$, $N=1$, $\mu=1$ and $b=-2$ in this plot.