Two-Dimensional Chebyshev Wavelet Method for Camassa-Holm Equation with Riesz Fractional Derivative Describing Propagation of Shallow Water Waves

2017 ◽  
Vol 151 (1-4) ◽  
pp. 77-89
Author(s):  
A.K. Gupta ◽  
S. Saha Ray
Keyword(s):  
Fractional Derivative ◽  
Shallow Water ◽  
Water Waves ◽  
Two Dimensional ◽  
Shallow Water Waves ◽  
Riesz Fractional Derivative ◽  
Wavelet Method ◽  
Holm Equation ◽  
Chebyshev Wavelet

scholarly journals Ermakov Equation and Camssa-Holm Waves

2016 ◽  
Vol 34 (1-2) ◽  
pp. 47-51
Author(s):  
Haret C. Rosu ◽  
Stefan C. Mancas
Keyword(s):  
Shallow Water ◽  
Spectral Problem ◽  
Water Waves ◽  
Shallow Water Waves ◽  
Important Ingredient ◽  
Interesting Issue ◽  
Holm Equation

Since the works of [1] and [2], it is known that the solution of the Ermakov equation is an important ingredient in the spectral problem of the Camassa-Holm equation. Here, we review this interesting issue and consider in addition more features of the Ermakov equation which have an impact on the behavior of the shallow water waves as described by the Camassa-Holm equation.

The string density problem and the Camassa–Holm equation

2007 ◽  
Vol 365 (1858) ◽  
pp. 2299-2312 ◽  
Author(s):  
Richard Beals ◽  
David H Sattinger ◽  
Jacek Szmigielski
Keyword(s):  
Inverse Problem ◽  
Special Reference ◽  
Shallow Water ◽  
Spectral Problem ◽  
Water Waves ◽  
Shallow Water Waves ◽  
Numerical Examples ◽  
Holm Equation ◽  
Density Problem

Recently, the string density problem, considered in the pioneering work of M. G. Krein, has arisen naturally in connection with the Camassa–Holm equation for shallow water waves. In this paper we review the forward and inverse string density problems, with some numerical examples, and relate it to the Camassa–Holm equation, with special reference to multi-peakon/anti-peakon solutions. Under stronger assumptions, the Camassa–Holm spectral problem and the string density problem can be transformed to the Schrödinger spectral problem and its inverse problem. Recent results exploiting this transformation are reviewed briefly.

scholarly journals Comment on “Two-dimensional third-and fifth-order nonlinear evolution equations for shallow water waves with surface tension” [Nonlinear Dyn,  doi:10.1007/s11071-017-3938-7]

2021 ◽  
Author(s):  
Piotr Rozmej ◽  
Anna Karczewska
Keyword(s):  
Surface Tension ◽  
Shallow Water ◽  
Water Waves ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonlinear Evolution Equations ◽  
Two Dimensional ◽  
Shallow Water Waves ◽  
Small Parameters ◽  
Fifth Order

AbstractThe authors of the paper “Two-dimensional third-and fifth-order nonlinear evolution equations for shallow water waves with surface tension” Fokou et al. (Nonlinear Dyn 91:1177–1189, 2018) claim that they derived the equation which generalizes the KdV equation to two space dimensions both in first and second order in small parameters. Moreover, they claim to obtain soliton solution to the derived first-order (2+1)-dimensional equation. The equation has been obtained by applying the perturbation method Burde (J Phys A: Math Theor 46:075501, 2013) for small parameters of the same order. The results, if correct, would be significant. In this comment, it is shown that the derivation presented in Fokou et al. (Nonlinear Dyn 91:1177–1189, 2018) is inconsistent because it violates fundamental properties of the velocity potential. Therefore, the results, particularly the new evolution equation and the dynamics that it describes, bear no relation to the problem under consideration.

Two-dimensional KdV-burgers for shallow-water waves

1984 ◽  
Vol 81 (2) ◽  
pp. 260-272
Author(s):  
M. Bartuccelli ◽  
V. Muto ◽  
P. Carbonaro
Keyword(s):  
Shallow Water ◽  
Water Waves ◽  
Two Dimensional ◽  
Shallow Water Waves

scholarly journals Water Waves and Light: Two Unlikely Partners

2021 ◽  
Author(s):  
Georgios N. Koutsokostas ◽  
Theodoros P. Horikis ◽  
Dimitrios J. Frantzeskakis ◽  
Nalan Antar ◽  
İlkay Bakırtaş
Keyword(s):  
Shallow Water ◽  
Water Waves ◽  
Water Wave ◽  
Soliton Solutions ◽  
Type Equation ◽  
Beam Propagation ◽  
Generic Model ◽  
Two Dimensional ◽  
Shallow Water Waves ◽  
Nonlocal Nonlinear

We study a generic model governing optical beam propagation in media featuring a nonlocal nonlinear response, namely a two-dimensional defocusing nonlocal nonlinear Schrödinger (NLS) model. Using a framework of multiscale expansions, the NLS model is reduced first to a bidirectional model, namely a Boussinesq or a Benney-Luke-type equation, and then to the unidirectional Kadomtsev-Petviashvili (KP) equation – both in Cartesian and cylindrical geometry. All the above models arise in the description of shallow water waves, and their solutions are used for the construction of relevant soliton solutions of the nonlocal NLS. Thus, the connection between water wave and nonlinear optics models suggests that patterns of water may indeed exist in light. We show that the NLS model supports intricate patterns that emerge from interactions between soliton stripes, as well as lump and ring solitons, similarly to the situation occurring in shallow water.

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