Higher-Order Approximations for Symmetrical Regular Long Wave Equation

2006 ◽  
Vol 61 (12) ◽  
pp. 607-614
Author(s):  
Xian-Jing Lai ◽  
Jie-Fang Zhang
Keyword(s):  
Wave Equation ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Order Term ◽  
Perturbation Expansion ◽  
Higher Order ◽  
Reductive Perturbation Method ◽  
Third Order ◽  
Long Wave ◽  
Tangent Method

In this work, we extend the application of “the modified reductive perturbation method” to the symmetrical regular long wave equation and try to obtain the contribution of higher-order terms in the perturbation expansion. It is shown that the lowest-order term in the expansion is governed by the nonlinear Schrödinger equation while the second- and third-order terms are governed by the linear Schrödinger equation. By employing the hyperbolic tangent method, progressive wave type solutions are obtained for the first-, second- and third-order terms in the perturbation expansion. - PACS numbers: 02.30.Jr, 42.25.Bs, 42.81.Dp, 43.35.Kp

scholarly journals An Analytic Evolutionary Algorithm To Compute Higher-Order Solutions For Model Equation

2021 ◽  
Author(s):  
Bang-Qing Li ◽  
Yu-Lan Ma
Keyword(s):  
Schrödinger Equation ◽  
Evolutionary Algorithm ◽  
Lower Order ◽  
Schrodinger Equation ◽  
Model Equation ◽  
Evolutionary Computing ◽  
Higher Order ◽  
Third Order ◽  
Generalized Schrödinger Equation ◽  
Basic Features

Abstract In this work, by introducing Darboux operator in evolutionary computing frame, we propose a novel analytic evolutionary algorithm to obtain exact higher-order iteration solutions for model equation. We construct the first-, second- and third-order solutions of a generalized Schrödinger equation by applying this algorithm. The higher-order solutions not only retain the basic features of the lower-order cases, but also become more abundant than the lower-order cases.

scholarly journals On Galilean Invariant and Energy Preserving BBM-Type Equations

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 878
Author(s):  
Alexei Cheviakov ◽  
Denys Dutykh ◽  
Aidar Assylbekuly
Keyword(s):  
Wave Equation ◽  
Conservation Laws ◽  
Numerical Simulations ◽  
Solitary Waves ◽  
Local Symmetry ◽  
Symmetry And Conservation Laws ◽  
Higher Order ◽  
Special Equation ◽  
Long Wave ◽  
Local Symmetries

We investigate a family of higher-order Benjamin–Bona–Mahony-type equations, which appeared in the course of study towards finding a Galilei-invariant, energy-preserving long wave equation. We perform local symmetry and conservation laws classification for this family of Partial Differential Equations (PDEs). The analysis reveals that this family includes a special equation which admits additional, higher-order local symmetries and conservation laws. We compute its solitary waves and simulate their collisions. The numerical simulations show that their collision is elastic, which is an indication of its S−integrability. This particular PDE turns out to be a rescaled version of the celebrated Camassa–Holm equation, which confirms its integrability.

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