Analysis of nonlinear water wave interaction solutions and energy exchange between different wave modes

Abstract

In this study, we consider the ideal fluid model of an inviscid fluid, assuming that the fluid motion is adiabatic; the flow is irrotational, that is, the individual fluid particles do not rotate; vorticity ω̃=0; and the flow is incompressible, in which the density of fluid particles does not vary significantly with fluid motion and can be considered constant throughout the fluid volume and throughout the motion. We start with equations representing continuity, conservation of momentum, conservation of entropy, and streamline equations, respectively. It is then reduced to a standard system of equations describing motion in two dimensions, defined by the Laplace equation with appropriate kinematic and dynamic boundary conditions, in terms of velocity potential and surface elevation. Finally, the one-dimensional nonlinear Korteweg–De Vries (KdV) equation is derived. Then, we further investigate the interaction of multiple periodic waves using the KdV equation and explain the interaction wave energy transfer procedure between the primary and higher order harmonics, and the Phillips [“On the dynamics of unsteady gravity waves of finite amplitude. I. The elementary interactions,” J. Fluid Mech. 9, 193–217 (1960)] wave resonance criterion is employed for capturing the periodic wave interaction whose energy conversion is analyzed via Fourier spectra. It is also found that for solitons, multiple collisions of different solitons eventually regain their original shape and that higher-energy solitons have faster velocities than lower-energy solitons, which, to the best of our knowledge, is overlooked.

On the analytical soliton approximations to fractional forced Korteweg–de Vries equation arising in fluids and plasmas using two novel techniques

“…By using equation (46) and the Adomian polynomials for nonlinear term as given in equation (25). Hence, solution…”

“…One of the PDEs that describes a broad spectrum of physical nonlinear phenomena as well as engineering materials is the well-known Korteweg-de Vries (KdV) equation [21][22][23][24][25][26]. This equation and its family, whether it contains third-order dispersion [27][28][29] or fifth-order dispersion [30][31][32] or other physical effects such as collision and viscosity forces, have effectively elucidated numerous nonlinear structures that emerge and propagate in diverse physical and engineering systems, including fluid physics and plasma physics.…”

Explorations of certain nonlinear waves of the Boussinesq and Camassa–Holm equations using physics-informed neural networks

“…As we know, the wave-packet behaviours of both equations cannot be fully determined via the analytical methods, except with certain initial and boundary conditions [3,4,[7][8][9][10][11][12]. Therefore, various conventional meshbased numerical methods have been introduced to explore the Boussinesq and Camassa-Holm equations, such as the finite-element method, the finite volume method and the spectral method [20][21][22][23][24][25][26][27]. It has been revealed that for the Boussinesq equation in the 'good' case, those numerical methods can provide satisfactory results, while for the Boussinesq equation in the 'bad' case, most numerical methods fail due to the instability of the equation, even for the simulation of the simplest bell-shaped solitons [20,21].…”