Multidimensional analogue of Virasoro algebra and quantum higher-order water wave equation

1993 ◽  
Vol 32 (2) ◽  
pp. 347-351
Author(s):  
P. Guha ◽  
A. Roy Chowdhury
Keyword(s):  
Wave Equation ◽  
Water Wave ◽  
Virasoro Algebra ◽  
Higher Order ◽  
Multidimensional Analogue

Exact Explode-Decay Soliton Solutions of a Cylindrical Higher-Order Water-Wave Equation

1985 ◽  
Vol 54 (6) ◽  
pp. 2105-2109 ◽  
Author(s):  
Akira Nakamura
Keyword(s):  
Wave Equation ◽  
Water Wave ◽  
Soliton Solutions ◽  
Higher Order

scholarly journals A Higher-Order Water-Wave Equation and the Method for Solving It

1975 ◽  
Vol 54 (2) ◽  
pp. 396-408 ◽  
Author(s):  
D. J. Kaup
Keyword(s):  
Wave Equation ◽  
Water Wave ◽  
Higher Order

scholarly journals Approximate asymptotic integration of a higher order water-wave equation using the inverse scattering transform

1997 ◽  
Vol 4 (1) ◽  
pp. 29-53 ◽  
Author(s):  
A. R. Osborne
Keyword(s):  
Wave Equation ◽  
Inverse Scattering ◽  
Water Wave ◽  
Wave Equations ◽  
Solitary Wave Solution ◽  
Laboratory Data ◽  
Inverse Scattering Transform ◽  
Higher Order ◽  
Linear Dispersion ◽  
Scattering Transform

Abstract. The complete mathematical and physical characterization of nonlinear water wave dynamics has been an important goal since the fundamental partial differential equations were discovered by Euler over 200 years ago. Here I study a subset of the full solutions by considering the irrotational, unidirectional multiscale expansion of these equations in shallow-water. I seek to integrate the first higher-order wave equation, beyond the order of the Korteweg- deVries equation, using the inverse scattering transform. While I am unable to integrate this equation directly, I am instead able to integrate an analogous equation in a closely related hierarchy. This new integrable wave equation is tested for physical validity by comparing its linear dispersion relation and solitary wave solution with those of the full water wave equations and with laboratory data. The comparison is remarkably close and thus supports the physical applicability of the new equation. These results are surprising because the inverse scattering transform, long thought to be useful for solving only very special, low-order nonlinear wave equations, can now be thought of as a useful tool for approximately integrating a wide variety of physical systems to higher order. I give a simple scenario for adapting these results to the nonlinear Fourier analysis of experimentally measured wave trains.

On the integrable models of the higher order water wave equation

1993 ◽  
Vol 174 (3) ◽  
pp. 237-240 ◽  
Author(s):  
M Daniel ◽  
K Porsezian ◽  
M Lakshmanan
Keyword(s):  
Wave Equation ◽  
Water Wave ◽  
Higher Order ◽  
Integrable Models

On a direct bilinearization method: Kaup's higher-order water wave equation as a modified nonlocal Boussinesq equation

1994 ◽  
Vol 27 (15) ◽  
pp. 5325-5334 ◽  
Author(s):  
F Lambert ◽  
I Loris ◽  
J Springael ◽  
R Willer
Keyword(s):  
Wave Equation ◽  
Water Wave ◽  
Boussinesq Equation ◽  
Higher Order

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.

scholarly journals Exact Solutions and Conserved Vectors of the Two-Dimensional Generalized Shallow Water Wave Equation

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1439
Author(s):  
Chaudry Masood Khalique ◽  
Karabo Plaatjie
Keyword(s):  
Differential Equation ◽  
Wave Equation ◽  
Shallow Water ◽  
Water Wave ◽  
Elliptic Integral ◽  
Momentum Conservation ◽  
Two Dimensional ◽  
Order Ordinary Differential Equation ◽  
Closed Form Solutions ◽  
Shallow Water Wave

In this article, we investigate a two-dimensional generalized shallow water wave equation. Lie symmetries of the equation are computed first and then used to perform symmetry reductions. By utilizing the three translation symmetries of the equation, a fourth-order ordinary differential equation is obtained and solved in terms of an incomplete elliptic integral. Moreover, with the aid of Kudryashov’s approach, more closed-form solutions are constructed. In addition, energy and linear momentum conservation laws for the underlying equation are computed by engaging the multiplier approach as well as Noether’s theorem.

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