Plenty of wave solutions to the ill-posed Boussinesq dynamic wave equation under shallow water beneath gravity (original) (raw)
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Physica D: Nonlinear Phenomena, 2020
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Indian Journal of Physics, 2019
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Open Physics
The variant Boussinesq equation has significant application in propagating long waves on the surface of the liquid layer under gravity action. In this article, the improved Bernoulli subequation function (IBSEF) method and the new auxiliary equation (NAE) technique are introduced to establish general solutions, some fundamental soliton solutions accessible in the literature, and some archetypal solitary wave solutions that are extracted from the broad-ranging solution to the variant Boussinesq wave equation. The established soliton solutions are knowledgeable and obtained as a combination of hyperbolic, exponential, rational, and trigonometric functions, and the physical significance of the attained solutions is speculated for the definite values of the included parameters by depicting the 3D profiles and interpreting the physical incidents. The wave profile represents different types of waves associated with the free parameters that are related to the wave number and velocity of th...
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In the present study, the nonlinear Boussinesq type equation describe the bi-directional propagation of small amplitude long capillary-gravity waves on the surface of shallow water. By using the extended auxiliary equation method, we obtained some new soliton like solutions for the two-dimensional fourth-order nonlinear Boussinesq equation with constant coefficient. These solutions include symmetrical, non-symmetrical kink solutions, solitary pattern solutions, Jacobi and Weierstrass elliptic function solutions and triangular function solutions. The stability analysis for these solutions are discussed.
Solutions of Differential Equations in Nonlinear Water Waves
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This book is concerned with the study of nonlinear water waves, which is one of the important observable phenomena in Nature. This study is related to the fluid dynamics, in general, and to the oceans dynamics in particular. The solutions of nonlinear PDEs with constant and variable coefficients, which describe the wave motion of undulant bores in shallow water, are investigated by using various analytical methods to illustrate the relation between solitary and water waves. The important ideas and results for nonlinear dispersive properties and solitons, which originated from the investigations of water waves, are discussed. The stability analysis for the second order system of PDEs is studied by using the phase plane method. In addition, we use perturbation methods to study the water wave problems for an incompressible fluid under the acceleration gravity and surface tension. The conservation laws of some PDEs are established. We illustrate the resulting solutions in several 3D-gra...
Higher-order Boussinesq equations for two-way propagation of shallow water waves
Standard perturbation methods are applied to Euler's equations of motion governing the capillary-gravity shallow water waves to derive a general higher-order Boussinesq equation involving the small-amplitude parameter, α = a/ h 0 , and long-wavelength parameter, β = (h 0 /l) 2 , where a and l are the actual amplitude and wavelength of the surface wave, and h 0 is the height of the undisturbed water surface from the flat bottom topography. This equation is also characterized by the surface tension parameter, namely the Bond number τ = Γ /ρgh 2 0 , where Γ is the surface tension coefficient, ρ is the density of water, and g is the acceleration due to gravity. The general Boussinesq equation involving the above three parameters is used to recover the classical model equations of Boussinesq type under appropriate scaling in two specific cases: (1) | 1 3 − τ | | β, and (2) | 1 3 − τ | = O(β). Case 1 leads to the classical (ill-posed and well-posed) fourth-order Boussinesq equations whose dispersive terms vanish at τ = 1 3. Case 2 leads to a sixth-order Boussinesq equation, which was originally introduced on a heuristic ground by Daripa and Hua [P. Daripa, W. Hua, A numerical method for solving an illposed Boussinesq equation arising in water waves and nonlinear lattices, Appl. Math. Comput. 101 (1999) 159–207] as a dispersive regularization of the ill-posed fourth-order Boussinesq equation. The relationship between the sixth-order Boussinesq equation and fifth-order KdV equation is also established in the limiting cases of the two small parameters α and β.