Nonlinear Shallow Water Waves Research Articles

Dynamical control on the homotopy analysis method for solving nonlinear shallow water wave equation

In this paper, the nonlinear shallow water wave equation is illustrated. The famous semi-analytical method, homotopy analysis method (HAM) is applied for solving this equation. The main novelty, of this study is to validate the numerical results using the stochastic arithmetic, the CESTAC method and the CADNA library. Based on this method, we can find the optimal iteration of the HAM, optimal approximation of the shallow water wave equation and optimal error. The main theorem of the CESTAC method is proved. Based on this theorem, we can show that the number of common significant digits for two successive approximations are almost equal to the number of common significant digits for exact and approximate solutions. Thus instead of traditional absolute error to show the accuracy of method we can apply the new termination criterion depends on two successive approximations. In order to find the convergence region of the HAM, several ħ-curves are demonstrated.

Approximate Numerical solutions for the nonlinear dispersive shallow water waves as the Fornberg–Whitham model equations

The nonlinear partial differential equations having travelling or solitary wave solutions is numerically challenging, in which one of the important type is the Fornberg–Whitham model equation. This article aims to solve the Fornberg–Whitham type equations numerically via the variational iteration algorithm-I (MVIA-I). The MVIA-I gives approximate and exact solutions with easily computable terms to linear and nonlinear PDEs without the linearization or discretization, small perturbation and Adomian polynomials. To assess the precision, reliability and compactness of the recommended algorithm, we have compared the obtained results with the traditional variational iteration method (VIM), homotopy analysis method, reproducing kernel Hilbert space method and Adomian’s decomposition method which reveals that the MVIA-I is computationally attractive, exceptionally productive and is more reliable than the others techniques used in the literature.

Mathematical Modeling of Regular and Irregular Shallow Water Waves Using Boussinesq Equation with Improved Dispersion

A mathematical model has been formulated to analyze the behaviour of small amplitude linear and nonlinear shallowwater waves in the coastal region. The coupled Boussinesq equations (BEs) are obtained from the Euler's equation in terms of velocities variable as the velocity measured from arbitrary distance from mean water level. BEs improves the dispersion characteristics and is applicable to variable water depth compared to conventional BEs, which is in terms of the depth-averaged velocity. The solution of the time-dependent BEs with kinematic and dynamic boundary conditions is obtained by utilizing Crank- Nicolson procedure of finite difference method (FDM). Further, the Von Neumann stability analysis for the Crank Nicolson scheme is also conducted for linearized BEs. The numerical simulation of regular and irregular waves propagating over the variable water depth is validated with the previous studies and experimental results. The present numerical model can be utilized to determine the wave characteristics in the nearshore region, including diffraction, refraction, shoaling, reflection, and nonlinear wave interactions. Therefore, the current model provides a competent tool for simulating the water waves in harbour or coastal regions for practical application.

The role of constant vorticity on weakly nonlinear surface gravity waves

This manuscript describes the derivation of systems of equations for weakly nonlinear gravity waves in shallow water in the presence of constant vorticity. The derivation is based on a multi-layer generalization of the traditional columnar Ansatz. A perturbative development in a nonlinear parameter and a dispersive parameter allow us to obtain sets of equations, for the horizontal fluid velocity and the free surface, able to describe propagation of weakly nonlinear and dispersive surface waves moving in water with some prescribed initial constant vorticity. We have shown that vorticity plays a central role on the dispersive properties of the system. When it is weak, it acts as a correction in linear and nonlinear dispersive terms. When stronger, it can also influence the nondispersive behavior of the system. Explicit steady solutions of the system corresponding to zero, weak, normal or strong vorticity are obtained. They correspond to solitary waves. Evolution of the soliton celerity, amplitude and width for these four cases are discussed.

Experimental Study on Spatial Variation of Mass Transport Induced by Surface Waves Generated in a Finite-Depth Laboratory Flume

AbstractThe aim of laboratory experiments conducted in a wave flume is to confirm that wave-induced mass transport varies along the way of propagation of regular mechanically generated waves in water of finite depth. This effect is attributed to the interactions between primary waves generated in the flume and associated free waves. Measurements of wave kinematics in several locations along the flume are performed to evaluate mass transport. Wave gauges and the particle image velocimetry technique are employed, respectively, to provide information on the evolution of the oscillating water surface and the wave velocity field under regular waves. Laboratory data are compared with numerical predictions of the derived nonlinear wavemaker model based on a pseudospectral approach. The experimental data exhibit good agreement with the numerical results with respect to particle trajectories and mass transport for nonlinear transitional and shallow water waves. Numerical and experimental investigations confirm that free-surface oscillatory motion and particle kinematics are affected by second-order free waves, which modify the characteristics of laboratory waves, especially in a strongly nonlinear shallow water regime. Moreover, the present study gives valuable insight into the effects of the interaction between two independent waves on wave-induced mass transport, which highlights the need of including nonlinear energy transfers between higher harmonics for the reliable estimation of wave processes in ocean circulation models.

Wave Height Interval Distribution Method Based on Nonlinear Shallow Water Wave Function

Li, X.; Zhao, Y., and Jin, H., 2020. Wave height interval distribution method based on nonlinear shallow water wave function. In: Yang, Y.; Mi, C.; Zhao, L., and Lam, S. (eds.), Global Topics and New Trends in Coastal Research: Port, Coastal and Ocean Engineering. Journal of Coastal Research, Special Issue No. 103, pp. 843–846. Coconut Creek (Florida), ISSN 0749-0208.In order to improve the ability to analyze and judge the interval distribution of wave height, it is necessary to estimate the interval distribution of wave height. An interval distribution estimation method of wave height based on nonlinear shallow water wave function is proposed. A statistical time series analysis model for estimating the interval distribution of sea wave height is constructed, large data mining and feature extraction for estimating the interval distribution of sea wave height are carried out by adopting a large data feature detection method, ordered clustering of statistical feature sequences of the interval distribution of sea wave height is carried out based on a nonlinear shallow water wave function estimation idea, information clustering and attribute merging in the process of estimating the interval distribution of sea wave height are carried out by combining a fuzzy C-means clustering analysis method. The feature quantity of association rules of statistical time series of wave height interval distribution is extracted, and the association rule set of statistical time series of wave height interval distribution is analyzed by using principal component analysis method, so as to realize accurate estimation of wave height interval distribution quantity in weighted Markov chain. The simulation results show that the method has high accuracy in estimating the interval distribution of wave height and improves the quantitative analysis capability of the interval distribution of wave height.

Invariant Subspace Classification and Exact Explicit Solutions to a Class of Nonlinear Wave Equation

In this paper, the invariant subspace classification of a class of nonlinear shallow water wave equation is given, then some exact explicit solutions to the nonlinear equation are provided by using the invariant subspace method. This method is a dynamical system method by nature, for its key step is to transform a nonlinear partial differential equation (PDE) into ordinary differential equation (ODE) systems, then by solving the ODE systems, the exact solutions to the nonlinear PDE are obtained.

Exact Traveling and Nano-Solitons Wave Solitons of the Ionic Waves Propagating along Microtubules in Living Cells

In this paper, the weakly nonlinear shallow-water wave model is mathematically investigated by applying the modified Riccati-expansion method and Adomian decomposition method. This model is used to describe the propagation of waves in weakly nonlinear and dispersive media. We obtain exact and solitary wave solutions of this model by using the modified Riccati-expansion method then using these solutions to determine the boundary and initial conditions. These conditions are employed to evaluate the semi-analytical wave solutions and calculate the absolute value of error. The values of absolute error show the accuracy of the obtained solutions. Some solutions are sketched to show the perspective view of the solution of this model. Moreover, the novelty of the obtained solutions is illustrated by showing the similarity and differences between our and previous solutions of the model.

Homotopy perturbation method for predicting tsunami wave propagation with crisp and uncertain parameters

PurposeThe purpose of this paper is to find the solution of classical nonlinear shallow-water wave (SWW) equations in particular to the tsunami wave propagation in crisp and interval environment.Design/methodology/approachHomotopy perturbation method (HPM) has been used for handling crisp and uncertain differential equations governing SWW equations.FindingsThe wave height and depth-averaged velocity of a tsunami wave in crisp and interval cases have been obtained.Originality/valuePresent results by HPM are compared with the existing solution (in crisp case), and they are found to be in good agreement. Also, the residual error of the solutions is found for the convergence conformation and reliability of the present results.

Different Groups of Variational Principles for Whitham-Broer-Kaup Equations in Shallow Water

Because the variational theory is the theoretical basis for many kinds of analytical or numerical methods, it is an essential but difficult task to seek explicit functional formulations whose extrema are sought by the nonlinear and complex models. By the semi-inverse method and designing trial-Lagrange functional skillfully, two different groups of variational principles are constructed for the Whitham-Broer-Kaup equations, which can model a lot of nonlinear shallow-water waves. Furthermore, by a combination of different variational formulations, new families of variational principles are established. The obtained variational principles provide conservation laws in an energy form and are proved correct by minimizing the functionals with the calculus of variations. All variational principles are firstly discovered, which can help to study the symmetries and find conserved quantities, and might find lots of applications in numerical simulation. The procedure reveals that the semi-inverse method is highly efficient and powerful, and can be extended to more other nonlinear equations.

Symmetry Reductions, Dynamical Behavior and Exact Explicit Solutions to a Class of Nonlinear Shallow Water Wave Equation

By using Lie symmetry analysis and dynamical systems method for a class of nonlinear shallow water wave equation, the exact solutions based on the Lie group method are provided. Especially, the bifurcations and exact explicit parametric representations of the traveling solutions are given, and the possible solitary wave solutions and many uncountable infinite periodic wave solutions to the nonlinear equation are obtained. To guarantee the existence of the above solutions, all parameter conditions are determined. Furthermore, we give some exact analytic solutions by using the power series method. This result enriches the types of solutions of nonlinear shallow water wave equation and has important physical significance for further study of this kind of equation.

On open boundary conditions for long wave equation in the hodograph plane

We discuss implications of the seaward boundary conditions used in initial-boundary value problem formulation of nonlinear shallow-water wave propagation over a linear slope. We first demonstrate the reflection of wave velocity in the case of Dirichlet condition and that of water elevation in the case of Neumann condition. We then show that linear superposition of the two boundary conditions results in much less reflection at the artificial boundary. We also propose a new boundary condition of mixed type and compare its results with that of the aforementioned conditions.

Energy accumulation during the growth of forced wave induced by a moving atmospheric pressure disturbance

ABSTRACTAtmospheric pressure disturbances of high translational speed can excite significant water waves, which are widely believed to be the main cause of destructive meteotsunamis. During the response of sea surface to adisturbance, energy is transferred from the atmosphere to the ocean, while the forced wave induced by the disturbance gradually grows and finally reaches a quasi-steady state. This study aims at quantifying the time required to excite the forced wave. Based on the nonlinear shallow water wave model, the evolution of the forced wave is numerically investigated. Analyses of the accumulation of wave energy are imported to help understand the evolution of the forced wave in different situations. The characteristics of energy accumulation show a great difference in the cases of different Froude number Fr. A parameter named response time is defined to quantify the time required for the evolution of the forced wave. It can be seen that Fr is the principal parameter that controls the wave evolution, while the effects of nonlinearity and bottom friction are significant only when Fr is close to 1. Simple relationships are proposed to roughly estimate the response time for Fr<0.9 and Fr>1.1, which can also provide a lower limit for Fr≈1.

Parameter dependence of acoustic mode quantities in an idealized model for shallow-water nonlinear internal wave ducts.

Nonlinear internal waves in shallow water have significant acoustic impacts and cause three-dimensional ducting effects, for example, energy trapping in a duct between curved wavefronts that propagates over long distances. A normal mode approach applied to a three-dimensional idealized parametric model [Lin, McMahon, Lynch, and Siegmann, J. Acoust. Soc. Am. 133(1), 37-49 (2013)] determines the dependence of such effects on parameters of the features. Specifically, an extension of mode number conservation leads to convenient analytical formulas for along-duct (angular) acoustic wavenumbers. The radial modes are classified into five types depending on geometric characteristics, resulting in five distinct formulas to obtain wavenumber approximations. Examples of their dependence on wavefront curvature and duct width, along with benchmark comparisons, demonstrate approximation accuracy over a broad range of physical values, even including situations where transitions in mode types occur with parameter changes. Horizontal-mode transmission loss contours found from approximate and numerically exact wavenumbers agree well in structure and location of intensity features. Cross-sectional plots show only small differences between pattern phases and amplitudes of the two calculations. The efficiency and accuracy of acoustic wavenumber and field approximations, in combination with the mode-type classifications, suggest their application to determining parameter sensitivity and also to other feature models.

Numerical approximation of the generalized regularized long wave equation using Petrov–Galerkin finite element method

AbstractThe generalized regularized long wave (GRLW) equation has been developed to model a variety of physical phenomena such as ion‐acoustic and magnetohydrodynamic waves in plasma, nonlinear transverse waves in shallow water and phonon packets in nonlinear crystals. This paper aims to develop and analyze a powerful numerical scheme for the nonlinear GRLW equation by Petrov–Galerkin method in which the element shape functions are cubic and weight functions are quadratic B‐splines. The proposed method is implemented to three reference problems involving propagation of the single solitary wave, interaction of two solitary waves and evolution of solitons with the Maxwellian initial condition. The variational formulation and semi‐discrete Galerkin scheme of the equation are firstly constituted. We estimate rate of convergence of such an approximation. Using Fourier stability analysis of the linearized scheme we show that the scheme is unconditionally stable. To verify practicality and robustness of the new scheme error norms L2, L∞ and three invariants I1, I2, and I3 are calculated. The computed numerical results are compared with other published results and confirmed to be precise and effective.

Interaction behavior associated with a generalized (2+1)-dimensional Hirota bilinear equation for nonlinear waves

• A generalized bilinear equation is proposed and linked with a nonlinear model describing nonlinear wave in fluids and oceans. • Two types of interaction solution (lump-kink and lump-soliton) have been derived, based on which rogue wave may be predicted. • Completely non-elastic interaction between lump and stripe has been presented (the lump is drowned/shallowed by the stripe). • Interaction between lump and soliton has been given (lump moves from one branch to the other branch of the soliton). • These phenomena exhibit the dynamical process of nonlinear waves and are useful to study the nonlinear interaction behavior. In this paper, we focus on the interaction behavior associated with a generalized (2+1)-dimensional Hirota bilinear equation. With symbolic computation, two types of interaction solutions including lump-kink and lump-soliton ones are derived through mixing two positive quadratic functions with an exponential function, or two positive quadratic functions with a hyperbolic cosine function in the bilinear equation. The completely non-elastic interaction between a lump and a stripe is presented, which shows the lump is drowned or shallowed by the stripe. The interaction between lump and soliton is also given, where the lump moves from one branch to the other branch of the soliton. These phenomena exhibit the dynamics of nonlinear waves and the solutions are useful for the study on interaction behavior of nonlinear waves in shallow water, plasma, nonlinear optics and Bose–Einstein condensates.

Coupling Conditions for Water Waves at Forks

We considered the propagation of nonlinear shallow water waves in a narrow channel presenting a fork. We aimed at computing the coupling conditions for a 1D effective model, using 2D simulations and an analysis based on the conservation laws. For small amplitudes, this analysis justifies the well-known Stoker interface conditions, so that the coupling does not depend on the angle of the fork. We also find this in the numerical solution. Large amplitude solutions in a symmetric fork also tend to follow Stoker’s relations, due to the symmetry constraint. For non symmetric forks, 2D effects dominate so that it is necessary to understand the flow inside the fork. However, even then, conservation laws give some insight in the dynamics.

Novel interaction phenomena of localized waves in the generalized (3+1)-dimensional KP equation

Based on Hirota bilinear method, the N-soliton solution of the (3+1)-dimensional generalized Kadomtsev–Petviashvili equation is derived explicitly, from which some localized waves such as soliton, breather, lump and their interactions are obtained by the approach of long wave limit. Especially, by selecting particular parameter constraints in the N-soliton solutions, the one breather or one lump can be obtained from two-soliton; the elastic interaction solutions between one bell-shaped soliton and one breather or between one bell-shaped soliton and one lump can be obtained from three-soliton; the elastic interaction solutions among two bell-shaped solitons and one breather, among two bell-shaped solitons and one lump, between two breathers or between two lumps can be obtained from four-soliton; the elastic interactions solutions among one bell-shaped soliton and two breathers, among one breather and three bell-shaped solitons, among one lump and three bell-shaped solitons, among one bell-shaped soliton and two breathers or among one breather, one lump and one bell-shaped soliton can be obtained from five-soliton. Detailed behaviors of such interaction phenomena are illustrated analytically and graphically. The results obtained in this paper may be helpful for understanding the evolution of nonlinear localized waves in shallow water.

A class of exact nonlinear traveling wave solutions for shallow water with a non-stationary bottom surface

Abstract The GN nonlinear shallow water wave equations developed by Green and Naghdi (1976) [5,6] are valid for a non-stationary, non-uniform bottom surface and a non-uniform pressure on the top surface. In contrast, the S nonlinear shallow water wave equations developed by Serre (1953) [12,13] for uniform depth and later generalized by Seabra-Santos et al. (1987) for non-uniform depth are limited to a stationary bottom surface and a uniform pressure applied to the top surface. This paper develops a class of exact nonlinear traveling wave solutions of the GN equations for a non-stationary, non-uniform bottom surface. Also, the explicit expressions for the pressure acting on the bottom surface and the average pressure through the depth in the GN equations are used to place physical restrictions on the motion which ensure that these pressures remain non-negative preventing cavitation.

Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations

We introduce physics-informed neural networks – neural networks that are trained to solve supervised learning tasks while respecting any given laws of physics described by general nonlinear partial differential equations. In this work, we present our developments in the context of solving two main classes of problems: data-driven solution and data-driven discovery of partial differential equations. Depending on the nature and arrangement of the available data, we devise two distinct types of algorithms, namely continuous time and discrete time models. The first type of models forms a new family of data-efficient spatio-temporal function approximators, while the latter type allows the use of arbitrarily accurate implicit Runge–Kutta time stepping schemes with unlimited number of stages. The effectiveness of the proposed framework is demonstrated through a collection of classical problems in fluids, quantum mechanics, reaction–diffusion systems, and the propagation of nonlinear shallow-water waves.