Nonclassical symmetry reductions for the dispersive wave equations in shallow water

1992 ◽  
Vol 33 (12) ◽  
pp. 4300-4305 ◽  
Author(s):  
Sen‐yue Lou
Keyword(s):  
Shallow Water ◽  
Wave Equations ◽  
Dispersive Wave ◽  
Symmetry Reductions ◽  
Nonclassical Symmetry

New integrable (2+1)- and (3+1)-dimensional shallow water wave equations: multiple soliton solutions and lump solutions

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Abdul-Majid Wazwaz
Keyword(s):  
Shallow Water ◽  
Water Wave ◽  
Wave Equations ◽  
Soliton Solutions ◽  
Lump Solutions ◽  
Content Type ◽  
Shallow Water Wave ◽  
Multiple Soliton Solutions ◽  
Simplified Hirota’S Method ◽  
Shallow Water Wave Equations

Purpose This study aims to develop two integrable shallow water wave equations, of higher-dimensions, and with constant and time-dependent coefficients, respectively. The author derives multiple soliton solutions and a class of lump solutions which are rationally localized in all directions in space. Design/methodology/approach The author uses the simplified Hirota’s method and lump technique for determining multiple soliton solutions and lump solutions as well. The author shows that the developed (2+1)- and (3+1)-dimensional models are completely integrable in in the Painlené sense. Findings The paper reports new Painlevé-integrable extended equations which belong to the shallow water wave medium. Research limitations/implications The author addresses the integrability features of this model via using the Painlevé analysis. The author reports multiple soliton solutions for this equation by using the simplified Hirota’s method. Practical implications The obtained lump solutions include free parameters; some parameters are related to the translation invariance and the other parameters satisfy a non-zero determinant condition. Social implications The work presents useful algorithms for constructing new integrable equations and for the determination of lump solutions. Originality/value The paper presents an original work with newly developed integrable equations and shows useful findings of solitary waves and lump solutions.

A Coupled-Mode Technique for the Run-Up of Non-Breaking Dispersive Waves on Plane Beaches

2006 ◽  
Author(s):  
K. A. Belibassakis ◽  
G. A. Athanassoulis
Keyword(s):  
Shallow Water ◽  
Water Waves ◽  
Wave Equations ◽  
Neumann Boundary ◽  
Breaking Waves ◽  
Numerical Results ◽  
Coupled Mode Theory ◽  
Mode Theory ◽  
Coupled Mode ◽  
Run Up

A coupled-mode model is developed and applied to the transformation and run-up of dispersive water waves on plane beaches. The present work is based on the consistent coupled-mode theory for the propagation of water waves in variable bathymetry regions, developed by Athanassoulis & Belibassakis (1999) and extended to 3D by Belibassakis et al (2001), which is suitably modified to apply to a uniform plane beach. The key feature of the coupled-mode theory is a complete modal-type expansion of the wave potential, containing both propagating and evanescent modes, being able to consistently satisfy the Neumann boundary condition on the sloping bottom. Thus, the present approach extends previous works based on the modified mild-slope equation in conjunction with analytical solution of the linearised shallow water equations, see, e.g., Massel & Pelinovsky (2001). Numerical results concerning non-breaking waves on plane beaches are presented and compared with exact analytical solutions; see, e.g., Wehausen & Laitone (1960, Sec. 18). Also, numerical results are presented concerning the run-up of non-breaking solitary waves on plane beaches and compared with the ones obtained by the solution of the shallow-water wave equations, Synolakis (1987), Li & Raichlen (2002), and experimental data, Synolakis (1987).

scholarly journals A conservative fully-discrete numerical method for the regularised shallow water wave equations

2021 ◽  
Author(s):  
Dimitrios Mitsotakis ◽  
Hendrik Ranocha ◽  
David I Ketcheson ◽  
Endre Süli
Keyword(s):  
Finite Element ◽  
Shallow Water ◽  
Numerical Method ◽  
Solitary Waves ◽  
Water Waves ◽  
Wave Equations ◽  
Mixed Finite Element ◽  
Boussinesq System ◽  
Fully Discrete ◽  
Long Time

The paper proposes a new, conservative fully-discrete scheme for the numerical solution of the regularised shallow water Boussinesq system of equations in the cases of periodic and reflective boundary conditions. The particular system is one of a class of equations derived recently and can be used in practical simulations to describe the propagation of weakly nonlinear and weakly dispersive long water waves, such as tsunamis. Studies of small-amplitude long waves usually require long-time simulations in order to investigate scenarios such as the overtaking collision of two solitary waves or the propagation of transoceanic tsunamis. For long-time simulations of non-dissipative waves such as solitary waves, the preservation of the total energy by the numerical method can be crucial in the quality of the approximation. The new conservative fully-discrete method consists of a Galerkin finite element method for spatial semidiscretisation and an explicit relaxation Runge--Kutta scheme for integration in time. The Galerkin method is expressed and implemented in the framework of mixed finite element methods. The paper provides an extended experimental study of the accuracy and convergence properties of the new numerical method. The experiments reveal a new convergence pattern compared to standard Galerkin methods.

Nonlinear Fourier Analysis for Shallow Water Waves

2021 ◽  
Author(s):  
Alfred R. Osborne
Keyword(s):  
Fourier Series ◽  
Fourier Analysis ◽  
Shallow Water ◽  
Nonlinear Wave ◽  
Water Waves ◽  
Wave Equations ◽  
Superposition Principle ◽  
Broad Band ◽  
Kp Equation ◽  
Linear Superposition

Abstract I consider nonlinear wave motion in shallow water as governed by the KP equation plus perturbations. I have previously shown that broad band, multiply periodic solutions of the KP equation are governed by quasiperiodic Fourier series [Osborne, OMAE 2020]. In the present paper I give a new procedure for extending this analysis to the KP equation plus shallow water Hamiltonian perturbations. We therefore have the remarkable result that a complex class of nonlinear shallow water wave equations has solutions governed by quasiperiodic Fourier series that are a linear superposition of sine waves. Such a formulation is important because it was previously thought that solving nonlinear wave equations by a linear superposition principle was impossible. The construction of these linear superpositions in shallow water in an engineering context is the goal of this paper. Furthermore, I address the nonlinear Fourier analysis of experimental data described by shallow water physics. The wave fields dealt with here are fully two-dimensional and essentially consist of the linear superposition of generalized cnoidal waves, which nonlinearly interact with one another. This includes the class of soliton solutions and their associated Mach stems, both of which are important for engineering applications. The newly discovered phenomenon of “fossil breathers” is also characterized in the formulation. I also discuss the exact construction of Morison equation forces on cylindrical piles in terms of quasiperiodic Fourier series.

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