Homoclinic Orbits and Solitary Waves within the Nondissipative Lorenz Model and KdV Equation

Abstract

Recent studies using the classical Lorenz model and generalized Lorenz models present abundant features of both chaotic and oscillatory solutions that may change our view on the nature of the weather as well as climate. In this study, the mathematical universality of solutions in different physical systems is presented. Specifically, the main goal is to reveal mathematical similarities for solutions of homoclinic orbits and solitary waves within a three-dimensional nondissipative Lorenz model (3D-NLM), the Korteweg–de Vries (KdV) equation, and the Nonlinear Schrodinger (NLS) equation. A homoclinic orbit for the [Formula: see text], [Formula: see text], and [Formula: see text] state variables of the 3D-NLM connects the unstable and stable manifolds of a saddle point. The [Formula: see text] and [Formula: see text] solutions for the homoclinic orbit can be expressed in terms of a hyperbolic secant function ([Formula: see text]) and a hyperbolic secant squared function ([Formula: see text]), respectively. Interestingly, these two solutions have the same mathematical form as solitary solutions for the NLS and KdV equations, respectively. After introducing new independent variables, the same second-order ordinary differential equation (ODE) and solutions for the [Formula: see text] component and the KdV equation were obtained. Additionally, the ODE for the [Formula: see text] component has the same form as the NLS for the solitary wave envelope. Finally, how a logistic equation, also known as the Lorenz error growth model, and an improved error growth model can be derived by simplifying the 3D-NLM is also discussed. Future work will compare the solutions of the 3D-NLM and KdV equation in order to understand the different physical role of nonlinearity in their solutions and the solutions of the error growth model and the 3D-NLM, as well as other Lorenz models, to propose an improved error growth model for better representing error growth at linear and nonlinear stages for both oscillatory and nonoscillatory solutions.