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.

Euler operators and conservation laws of the BBM equation

1979 ◽  
Vol 85 (1) ◽  
pp. 143-160 ◽  
Author(s):  
Peter J. Olver
Keyword(s):  
Wave Equation ◽  
Conservation Laws ◽  
Calculus Of Variations ◽  
Long Wave ◽  
Regularized Long Wave Equation ◽  
Formal Calculus ◽  
Euler Operators ◽  
Bbm Equation

AbstractThe BBM or Regularized Long Wave Equation is shown to possess only three non-trivial independent conservation laws. In order to prove this result, a new theory of Euler-type operators in the formal calculus of variations will be developed in detail.

Solitary waves of the regularised long-wave equation

1990 ◽  
Vol 91 (2) ◽  
pp. 441-459 ◽  
Author(s):  
L.R.T Gardner ◽  
G.A Gardner
Keyword(s):  
Wave Equation ◽  
Solitary Waves ◽  
Long Wave

scholarly journals WEAKLY NONPARAXIAL EFFECTS ON THE PROPAGATION OF (1+1)D SPATIAL SOLITONS IN INHOMOGENEOUS KERR MEDIA

2005 ◽  
Vol 14 (02) ◽  
pp. 203-219 ◽  
Author(s):  
A. SURYANTO ◽  
E. VAN GROESEN ◽  
M. HAMMER
Keyword(s):  
Wave Equation ◽  
Numerical Simulations ◽  
Beam Propagation ◽  
Bound State ◽  
Higher Order ◽  
Spatial Solitons ◽  
Nls Equation ◽  
Soliton Propagation ◽  
Kerr Media ◽  
Higher Order Terms

The widely-used approach to study the beam propagation in Kerr media is based on the slowly varying envelope approximation (SVEA) which is also known as the paraxial approximation. Within this approximation, the beam evolution is described by the nonlinear Schrödinger (NLS) equation. In this paper, we extend the NLS equation by including higher-order terms to study the effects of nonparaxiality on the soliton propagation in inhomogeneous Kerr media. The result is still a one-way wave equation which means that all back-reflections are neglected. The accuracy of this approximation exceeds the standard SVEA. By performing several numerical simulations, we show that the NLS equation produces reasonably good predictions for relatively small degrees of nonparaxiality, as expected. However, in the regions where the envelope beam is changing rapidly as in the breakup of a multisoliton bound state, the nonparaxiality plays an important role.

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