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Homoclinic orbits and solitary waves within the nondissipative Lorenz model and KdV equation. (English) Zbl 1471.34084

In this study, mathematical similarities for the solutions of homoclinic orbits and solitons within the 3D-NLM, NLS, and KdV equations are presented. Simplification of the 3D-NLM into the so-called error growth model and an improved error growth model is also discussed. As shown, the \(X\) and \(Z\) components of the homoclinic orbit, which are a hyperbolic secant function (sech) and a hyperbolic secant squared function (sech2), respectively, have the same mathematical form as solutions for the solitary wave envelope of the NLS equation and the solitary wave of the KdV equation, respectively. Specifically, the same second-order ODE for the \(Z\) component and the KdV, and the same solitary pattern solutions for both systems are obtained. Additionally, the ODE for the \(X\) 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.
Reviewer: Yingxin Guo (Qufu)

MSC:

34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
35C07 Traveling wave solutions
92D25 Population dynamics (general)

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