Solitary waves, periodic and elliptic solutions to the Benjamin, Bona & Mahony (BBM) equation modified by viscosity

In this paper, we use a traveling wave reduction or a so-called spatial approximation to comprehensively investigate periodic and solitary wave solutions of the modified Benjamin, Bona & Mahony equation (BBM) to include both dissipative and dispersive effects of viscous boundary layers. Under certain circumstances that depend on the traveling wave velocity, classes of periodic and solitary wave like solutions are obtained in terms of Jacobi elliptic functions. An ad-hoc theory based on the dissipative term is presented, in which we have found a set of solutions in terms of an implicit function. Using dynamical systems theory we prove that the solutions of \eqref{BBMv} experience a transcritical bifurcation for a certain velocity of the traveling wave. Finally, we present qualitative numerical results.Comment: 14 pages, 11 figure

Visco-elastic Cosmology for a Sparkling Universe?

We show the analogy between a generalization of the Rayleigh-Plesset equation of bubble dynamics including surface tension, elasticity and viscosity effects with a reformulation of the Friedmann-Lemaître set of equations describing the expansion of space in cosmology assuming a homogeneous and isotropic universe. By comparing both fluid and cosmic equations, we propose a bold generalization of the newly-derived cosmic equation mapping three continuum mechanics contributions. Conversely, the addition of a cosmological constant-like term in the fluid equation would lead also to a new phenomenology. Our work is purely speculative and does not rely on any observations or theoretical derivations from first principles

Formation of Three-Dimensional Surface Waves on Deep-Water Using Elliptic Solutions of Nonlinear Schrödinger Equation

A review of three-dimensional waves on deep-water is presented. Three forms of three-dimensionality, namely oblique, forced and spontaneous types, are identified. An alternative formulation for these three-dimensional waves is given through cubic nonlinear Schrödinger equation. The periodic solutions of the cubic nonlinear Schrödinger equation are found using Weierstrass elliptic ℘ functions. It is shown that the classification of solutions depends on the boundary conditions, wavenumber and frequency. For certain parameters, Weierstrass ℘ functions are reduced to periodic, hyperbolic or Jacobi elliptic functions. It is demonstrated that some of these solutions do not have any physical significance. An analytical solution of cubic nonlinear Schrödinger equation with wind forcing is also obtained which results in how groups of waves are generated on the surface of deep-water in the ocean. In this case, the dependency on the energy-transfer parameter, from wind to waves, makes either the groups of wave to grow initially and eventually dissipate, or simply decay or grow in time

Micro Cavitation Bubbles on the Movement of an Experimental Submarine: Theory and Experiments

To understand the nature of movement of submarine, micro cavitation bubbles were systematically diffused around the exterior of a test body (tube) fully submerged in a water tank. The primary purpose was to assess the feasibility of applying micro cavitation as a means of depth control for underwater vehicles, mainly but not limited to submarines. Ideally, the results would indicate the use of micro cavitation as a more efficient alternative to underwater vehicle depth control than the conventional ballast tank method. The current approach utilizes the Archimedes\u27 principle of buoyancy to alter the density of the object affected, making it less than, or greater than the density of the surrounding fluid. However, this process is too slow for underwater vehicles to react to sudden obstacles inherent in their environment. Rather than altering its internal density, this experiment aimed to investigate the response that would occur if the density of its environment is manipulated instead. In theory, and in a hydrostatic fluid, diffusing micro air bubbles from the top surface of the submarine would dilute the column of water above it with air cavities, thus lowering the density of the water. The resulting pressure differential would then cause the submarine to gain buoyancy

Factorization conditions for nonlinear second-order differential equations

For the case of nonlinear second-order differential equations with a constant coefficient of the first derivative term and polynomial nonlinearities, the factorization conditions of Rosu and Cornejo-Perez are approached in two ways: (i) by commuting the subindices of the factorization functions in the two factorization conditions and (ii) by leaving invariant only the first factorization condition achieved by using monomials or polynomial sequences. For the first case the factorization brackets commute and the generated equations are only equations of Ermakov-Pinney type. The second modification is non commuting, leading to nonlinear equations with different nonlinear force terms, but the same first-order part as the initially factored equation. It is illustrated for monomials with the examples of the generalized Fisher and FitzHugh-Nagumo initial equations. A polynomial sequence example is also included.Comment: 12 pages, 6 figures, 17 references, for NMMP-2022 proceeding

Formation of Three-Dimensional Surface Waves on Deep-Water Using Elliptic Solutions of Nonlinear Schrödinger Equation

A review of three-dimensional waves on deep-water is presented. Three forms of three dimensionality, namely oblique, forced and spontaneous type, are identified. An alternative formulation for these three-dimensional waves is given through cubic nonlinear Schr¨odinger equation. The periodic solutions of the cubic nonlinear Schr¨odinger equation are found using Weierstrass elliptic ℘ functions. It is shown that the classification of solutions depends on the boundary conditions, wavenumber and frequency. For certain parameters, Weierstrass ℘ functions are reduced to periodic, hyperbolic or Jacobi elliptic functions. It is demonstrated that some of these solutions do not have any physical significance. An analytical solution of cubic nonlinear Schr¨odinger equation with wind forcing is also obtained which results in how groups of waves are generated on the surface of deep water in the ocean. In this case the dependency on the energy-transfer parameter, from wind to waves, make either the groups of wave to grow initially and eventually dissipate, or simply decay or grow in time

Formation of Three-Dimensional Surface Waves on Deep-Water Using Elliptic Solutions of Nonlinear Schrödinger Equation

A review of three-dimensional waves on deep-water is presented. Three forms of three-dimensionality, namely oblique, forced and spontaneous types, are identified. An alternative formulation for these three-dimensional waves is given through cubic nonlinear Schrödinger equation. The periodic solutions of the cubic nonlinear Schrödinger equation are found using Weierstrass elliptic ℘ functions. It is shown that the classification of solutions depends on the boundary conditions, wavenumber and frequency. For certain parameters, Weierstrass ℘ functions are reduced to periodic, hyperbolic or Jacobi elliptic functions. It is demonstrated that some of these solutions do not have any physical significance. An analytical solution of cubic nonlinear Schrödinger equation with wind forcing is also obtained which results in how groups of waves are generated on the surface of deep-water in the ocean. In this case, the dependency on the energy-transfer parameter, from wind to waves, makes either the groups of wave to grow initially and eventually dissipate, or simply decay or grow in time

Interactions and Focusing of Nonlinear Water Waves

A theoretical and computational study is undertaken for the modulational instabilities of a pair of nonlinearly interacting two-dimensional waves in deep water. It has been shown that the full dynamics of these interacting waves gives rise to localized large-amplitude wavepackets (wave focusing). The coupled cubic nonlinear Schr¨odinger (CNLS) equations are used to derive a nonlinear dispersion equation which give rise to new class of modulational instabilities and demonstrates the dependence of obliqueness of the interacting waves. The computations, due to nonlinear wave-wave interactions, waves that are separately modulationally stable can give rise to the formation of largeamplitude coherent wave packets with amplitudes several times that of the initial waves. In the case for the original Benjamin-Feir instability, in contrast, waves disintegrate into a wide spectrum

Solitary Waves, Periodic and Elliptic Solutions to the Benjamin, Bona and Mahony Equation Modified by Viscosity

In this paper, we use a traveling wave reduction or a so-called spatial approximation to comprehensively investigate periodic and solitary wave solutions of the modified Benjamin. Bona and Mahony equation (BBM) to include both dissipative and dispersive effects of viscous boundary layers. Under certain circumstances that depend on the traveling wave velocity, classes of periodic and solitary wave like solutions are obtained in terms of Jacobi elliptic functions. An ad-hoc theory based on the dissipative term is presented, in which we have found a set of solutions in terms of an implicit function. Using dynamical systems theory we prove that the solutions experience a transcritical bifurcation for a certain velocity of the traveling wave. Finally, we present qualitative numerical results

Micro Cavitation Bubbles on the Movement of an Experimental Submarine: Theory and Experiments

To understand the nature of movement of submarine, micro cavitation bubbles were systematically diffused around the exterior of a test body (tube) fully submerged in a water tank. The primary purpose was to assess the feasibility of applying micro cavitation as a means of depth control for underwater vehicles, mainly but not limited to submarines. Ideally, the results would indicate the use of micro cavitation as a more efficient alternative to underwater vehicle depth control than the conventional ballast tank method. The current approach utilizes the Archimedes\u27 principle of buoyancy to alter the density of the object affected, making it less than, or greater than the density of the surrounding fluid. However, this process is too slow for underwater vehicles to react to sudden obstacles inherent in their environment. Rather than altering its internal density, this experiment aimed to investigate the response that would occur if the density of its environment is manipulated instead. In theory, and in a hydrostatic fluid, diffusing micro air bubbles from the top surface of the submarine would dilute the column of water above it with air cavities, thus lowering the density of the water. The resulting pressure differential would then cause the submarine to gain buoyancy