Simple Asymptotics for a Generalized Wave Equation with Degenerating Velocity and Their Applications in the Linear Long Wave Run-Up Problem

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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Novel implicit/explicit local conservative schemes for the regularized long-wave equation and convergence analysis

Journal of Mathematical Analysis and Applications ◽

10.1016/j.jmaa.2016.09.047 ◽

2017 ◽

Vol 447 (1) ◽

pp. 17-31 ◽

Cited By ~ 4

Author(s):

Jiaxiang Cai ◽

Yuezheng Gong ◽

Hua Liang

Keyword(s):

Wave Equation ◽

Convergence Analysis ◽

Conservative Schemes ◽

Long Wave ◽

Regularized Long Wave Equation

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Numerical Investigation of Wave Period Influence on Long Wave Run-Up on Slope With Rigid Vegetation

Volume 7: Ocean Engineering ◽

10.1115/omae2016-54298 ◽

2016 ◽

Author(s):

Jun Tang ◽

Yongming Shen

Keyword(s):

Shallow Water Equations ◽

Water Wave ◽

Wave Period ◽

Coastal Vegetation ◽

Coastal Structures ◽

Rigid Vegetation ◽

Comparison With Experiment ◽

Long Wave ◽

Nonlinear Shallow Water Equations ◽

Run Up

Coastal vegetation can not only provide shade to coastal structures but also reduce wave run-up. Study of long water wave climb on vegetation beach is fundamental to understanding that how wave run-up may be reduced by planted vegetation along coastline. The present study investigates wave period influence on long wave run-up on a partially-vegetated plane slope via numerical simulation. The numerical model is based on an implementation of Morison’s formulation for rigid structures induced inertia and drag stresses in the nonlinear shallow water equations. The numerical scheme is validated by comparison with experiment results. The model is then applied to investigate long wave with diverse periods propagating and run-up on a partially-vegetated 1:20 plane slope, and the sensitivity of run-up to wave period is investigated based on the numerical results.

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Euler operators and conservation laws of the BBM equation

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100055572 ◽

1979 ◽

Vol 85 (1) ◽

pp. 143-160 ◽

Cited By ~ 156

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.

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