Nonlinear Water Wave Equations Research Articles

Time-fractional Davey–Stewartson equation: Lie point symmetries, similarity reductions, conservation laws and traveling wave solutions

As a celebrated nonlinear water wave equation, the Davey–Stewartson equation is widely studied by researchers, especially in the field of mathematical physics. On the basis of the Riemann–Liouville fractional derivative, the time-fractional Davey–Stewartson equation is investigated in this paper. By application of the Lie symmetry analysis approach, the Lie point symmetries and symmetry groups are obtained. At the same time, the similarity reductions are derived. Furthermore, the equation is converted to a system of fractional partial differential equations and a system of fractional ordinary differential equations in the sense of Riemann–Liouville fractional derivative. By virtue of the symmetry corresponding to the scalar transformation, the equation is converted to a system of fractional ordinary differential equations in the sense of Erdélyi–Kober fractional integro-differential operators. By using Noether’s theorem and Ibragimov’s new conservation theorem, the conserved vectors and the conservation laws are derived. Finally, the traveling wave solutions are achieved and plotted.

Sensitivity analysis and solitary wave solutions to the (2+1)-dimensional Boussinesq equation in dispersive media

This paper explores the dynamic behavior of the [Formula: see text]-dimensional Boussinesq equation, which is a nonlinear water wave equation used to model wave packets in dispersive media with weak nonlinearity. Specifically, we investigate the equation’s traveling wave solutions using the Riccati equation mapping method. Our results include solitary and soliton solutions, each with their own set of parameter values. To provide a comprehensive understanding of these solutions, we present them in general form and visualize their significance using various graphs, such as 3D, 2D, and contour plots. The computational effort and resulting outcomes highlight the efficacy of our approach, which has the potential to be applied to other nonlinear physical problems in fields such as mathematical physics, engineering, and nonlinear science.

Analytical study of nonlinear water wave equations for their fractional solution structures

This paper examines the three-dimensional nonlinear time-fractional water wave equations for their analytical wave solutions. These are the equations of the names [Formula: see text]-Zakharov–Kuznetsov–Burgers equation and [Formula: see text]-Yu–Toda–Sasa–Fukuyama equation. The obtained wave solutions are in the form of kink, periodic and singular waves by utilizing the [Formula: see text]-expansion approach. The aforesaid solutions are verified and demonstrated graphically via symbolic soft computations.

Specific wave structures of a fifth-order nonlinear water wave equation

Investigated in the present paper is a fifth-order nonlinear evolution (FONLE) equation, known as a nonlinear water wave (NLWW) equation, with applications in the applied sciences. More precisely, a traveling wave hypothesis is firstly applied that reduces the FONLE equation to a 1D domain. The Kudryashov methods (KMs) are then adopted as leading techniques to construct specific wave structures of the governing model which are classified as W-shaped and other solitons. In the end, the effect of changing the coefficients of nonlinear terms on the dynamical features of W-shaped and other solitons is investigated in detail for diverse groups of the involved parameters.

An effective computational method to deal with a time-fractional nonlinear water wave equation in the Caputo sense

The authors’ concern of the present paper is to conduct a systematic study on a time-fractional nonlinear water wave equation which is an evolutionary version of the Boussinesq system. The study goes on by adopting a new analytical method based on the Laplace transform and the homotopy analysis method to the governing model and obtaining its approximate solutions in the presence of the Caputo fractional derivative. To analyze the influence of the Caputo operator on the dynamical behavior of the approximate solutions, some graphical illustrations in two- and three-dimensions are formally presented. Furthermore, several numerical tables are given to support the performance of the new analytical method in handling the time-fractional nonlinear water wave equation.

Spatial evolution of the kurtosis of steep unidirectional random waves

Abstract

A numerical study of the run-up and the force exerted on a vertical wall by a solitary wave propagating over two tandem trenches

The propagation and transformation of water waves over varying bathymetries is a subject of fundamental interest to ocean, coastal and harbor engineers. The specific bathymetry considered in this paper consists of one or two, naturally formed or man-made, trenches. The problem we focus on is the transformation of an incoming solitary wave by the trench(es), and the impact of the resulting wave system on a vertical wall located after the trench(es). The maximum run-up and the maximum force exerted on the wall are calculated for various lengths and heights of the trench(es), and are compared with the corresponding quantities in the absence of them. The calculations have been performed using the fully nonlinear water-wave equations, in the form of the Hamiltonian coupled-mode theory, recently developed by Papoutsellis et al. (Eur J Mech B/Fluids 72:199–224, 2018). Comparisons of the calculated free-surface elevation with existing experimental results indicate that the effect of the vortical flow, inevitably developed within and near the trench(es) but not captured by any potential theory, is not important concerning the frontal wave flow regime. This suggests that the predictions of the run-up and the force on the wall by means of nonlinear potential theory are expected to be nearly realistic. The main conclusion of our investigation is that the presence of two tandem trenches in front of the wall may reduce the run-up from (about) 20 to 45% and the force from 15 to 38%, depending on the trench dimensions and the wave amplitude. The percentage reduction is greater for higher waves. The presence of only one trench leads to reductions 1.4–1.7 times smaller.

Comparison of Numerical Methods for SWW Equations

Abstract: In this paper we consider three methods of approximation for the nonlinear water wave equation. In particular we are interested of KdV equation as a stationary water wave. The first is the method of approximation with a polynomial, the second method is the finite–volume method and the third method is Laplace decomposition method (LDM). A comparison between the methods is mentioned in this article.
 We treat the considered methods comparing the obtained solutions with the exact ones. We give in particular the numerical results compared with the analytical results. We show that the used methods are effective and convenient for solving the water wave equations. We can propose and sure that the method of approximation with a polynomial gives accurate results.

Invariant subspaces, exact solutions and stability analysis of nonlinear water wave equations

The key purpose of the present research is to derive the exact solutions of nonlinear water wave equations (NLWWEs) in oceans through the invariant subspace scheme (ISS). In this respect, the NLWWEs which describe specific nonlinear waves are converted to a number of systems of ordinary differential equations (ODEs) such that the resulting systems can be efficiently handled by computer algebra systems. As an accomplishment, the performance of the well-designed ISS in extracting a group of exact solutions is formally confirmed. In the end, the stability analysis for the NLWWE is investigated through the linear stability scheme.

On the near resonances of collinear steady-state wave groups in finite water depth

In this paper, the role nearly resonant components play in collinear steady-state wave groups in finite water depth is investigated theoretically. Fully nonlinear water wave equations are solved when the resonance conditions are nearly satisfied. Convergent solutions are obtained for steady-state wave groups with multiple near resonances. Spectra analysis shows that, as water depth decreases from deep water to finite water depth, the four-wave nearly resonant interactions play a dominant role in nonlinear steady-state wave groups. Meanwhile the dispersion relation changes with the water depth and some low and high frequency components appear in the spectrum due to the three-wave nearly resonant interactions. Comparison between the linear and nonlinear wave groups are made in finite water depth when the mean rates of mass, momentum and energy fluxes are almost conserved. It is found that wave energy in nonlinear wave groups is redistributed as more components join the resonance when the nonlinearity increases. Besides, both the wave group steepness and magnitude of water particle horizontal velocities near the crests and troughs in nonlinear wave groups increase more rapidly with the nonlinearity. The significance of near resonances in finite water depth for collinear steady-state wave groups is demonstrated.

Finite amplitude steady-state wave groups with multiple near resonances in deep water

In this paper, finite amplitude steady-state wave groups with multiple nearly resonant interactions in deep water are investigated theoretically. The nonlinear water wave equations are solved by the homotopy analysis method (HAM), which imposes no constraint on either the number or the amplitude of the wave components, to resolve the small-divisor problems caused by near resonances. A new kind of auxiliary linear operator in the framework of the HAM is proposed to transform the small divisors associated with the non-trivial nearly resonant components to singularities associated with the exactly resonant ones. Primary components, exactly resonant components together with nearly resonant components are considered as the initial non-trivial components, since all of them are homogeneous solutions to the auxiliary linear operator. For wave groups with weak nonlinearity, the energy transfer between nearby nearly resonant components is remarkable. As the nonlinearity increases, the number of steady-state wave groups increases as more components join the near resonance. This indicates that the probability of existence of steady-state resonant waves increases with the nonlinearity of wave groups. The frequency band broadens and spectral asymmetry becomes more and more pronounced. The amplitude of each component may either increase or decrease with the nonlinearity of wave groups, while the amplitude of the whole wave group increases continuously and finite amplitude wave groups are obtained. This work shows the wide existence of steady-state waves when multiple nearly resonant interactions are considered.

Three-dimensional waves beneath an ice sheet due to a steadily moving pressure

Solutions of the nonlinear water wave equations under an ice sheet are computed using a boundary integral equation method. The ice sheet is modelled as a thin elastic plate and the fluid equations are nonlinear. Depending on the velocity of the moving disturbance generating the flow, different types of responses of the floating ice sheet are discussed.

The inverse spectral problem for the discrete cubic string

Given a measure m on the real line or a finite interval, the cubic string is the third-order ODE − ϕ′′′ = zmϕ where z is a spectral parameter. If equipped with Dirichlet-like boundary conditions this is a non-self-adjoint boundary value problem which has recently been shown to have a connection to the Degasperis–Procesi nonlinear water wave equation. In this paper, we study the spectral and inverse spectral problem for the case of Neumann-like boundary conditions which appear in a high-frequency limit of the Degasperis–Procesi equation. We solve the spectral and inverse spectral problem for the case of m being a finite positive discrete measure. In particular, explicit determinantal formulae for the measure m are given. These formulae generalize Stieltjes' formulae used by Krein in his study of the corresponding second-order ODE −ϕ″ = zmϕ.

Stability of steep gravity–capillary solitary waves in deep water

The stability of steep gravity–capillary solitary waves in deep water is numerically investigated using the full nonlinear water-wave equations with surface tension. Out of the two solution branches that bifurcate at the minimum gravity–capillary phase speed, solitary waves of depression are found to be stable both in the small-amplitude limit when they are in the form of wavepackets and at finite steepness when they consist of a single trough, consistent with observations. The elevation-wave solution branch, on the other hand, is unstable close to the bifurcation point but becomes stable at finite steepness as a limit point is passed and the wave profile features two well-separated troughs. Motivated by the experiments of Longuet-Higgins & Zhang (1997), we also consider the forced problem of a localized pressure distribution applied to the free surface of a stream with speed below the minimum gravity–capillary phase speed. We find that the finite-amplitude forced solitary-wave solution branch computed by Vanden-Broeck & Dias (1992) is unstable but the branch corresponding to Rayleigh’s linearized solution is stable, in agreement also with a weakly nonlinear analysis based on a forced nonlinear Schrödinger equation. The significance of viscous effects is assessed using the approach proposed by Longuet-Higgins (1997): while for free elevation waves the instability predicted on the basis of potential-flow theory is relatively weak compared with viscous damping, the opposite turns out to be the case in the forced problem when the forcing is strong. In this régime, which is relevant to the experiments of Longuet-Higgins & Zhang (1997), the effects of instability can easily dominate viscous effects, and the results of the stability analysis are used to propose a theoretical explanation for the persistent unsteadiness of the forced wave profiles observed in the experiments.

Exact traveling wave solutions of some nonlinear physical models. II. Coupled KdV type equations

New exact traveling wave solutions are derived for two coupled nonlinear water wave equations by using a delicate way of rank analysis two-step ansatz method.

A linearized implicit pseudo-spectral method for certain non-linear water wave equations

An efficient numerical method is developed for the numerical solution of non-linear wave equations typified by the third- and fifth-order Korteweg–de Vries equations and their generalizations. The method developed uses a pseudo-spectral (Fourier transform) treatment of the space dependence together with a linearized implicit scheme in time. An important advantage to be gained from the use of this method over the pseudo-spectral scheme proposed by Fornberg and Whitham (a Fourier transform treatment of the space variable together with a leap-frog scheme in time) which is conditionally stable, is the ability to vary the mesh length, thereby reducing the computational time. Using a linearized stability analysis, it is shown that the proposed method is unconditionally stable. The method presented here is for the Korteweg–de Vries equations and their generalized forms, but it can be implemented to a broad class of non-linear wave equations (equation (1)), with obvious changes in the various formulae. To illustrate the application of this method, numerical results portraying a single soliton solution and the collision of two solitons are reported for the third- and fifth-order Korteweg–de Vries equations. © 1998 John Wiley & Sons, Ltd.

Maximum amplitude of modulated wavetrain

For water wave predictions, it is of practical importance to know the size of the biggest wave that emerges from a given uniform wavetrain when it experiences Benjamin-Feir's modulational instability. As far as we are aware, the series of experiments performed by Su and Green [5] are the only work that has pursued this problem so far. In this work their experimental result is compared with theoretical predictions given by the nonlinear Schrödinger equation as well as with the numerical simulation of the fully nonlinear water wave equation. The result of the fully nonlinear simulation shows that the height of the biggest wave reaches more than 3.5 times the initial amplitude of the wavetrain at the maximum modulation when the initial steepness a 0 k 0 of the wavetrain is 0.125.