Shallow Water Waves Research Articles

From granular collapses to shallow water waves: A predictive model for tsunami generation

Sudden large-scale geophysical flows such as cliff collapses, rockfalls, or massive landslides are known to be tsunamigenic and constitute a significant hazard for coastal populations and infrastructures. A key challenge is to be able to predict a priori the amplitude of the tsunami wave that would be engendered by a given cliff collapse. In this study, we introduce a comprehensive model of shallow water waves generated by the subaerial collapse of a granular column. The proposed model is able to capture the role of the geometry of the collapse as well as the water depth and is successfully compared to a large dataset of experiments.

RETRACTED: Lie symmetry analysis, optimal system and exact solutions for variable-coefficients (2 + 1)-dimensional dissipative long-wave system

In this article, the symmetry, optimal system, exact solutions and conservation laws of (2+1)-dimensional dissipative long-wave system with variable-coefficients (vcDLW) are investigated. These are important for mathematicians and physicists to solve problems about shallow water waves. The infinitesimal generators of independent variables are solved using Lie symmetry analysis method firstly. Based on the linear combination of vector fields, the representative elements of one-dimensional subalgebras optimal system are introduced by Olver's method. Some new exact solutions of the (2+1)-dimensional vcDLW are obtained by using Riccati equation method and tanh expansion method, such as kink solutions and soliton solutions. It is shown that this system has different solutions for different reductions. These solutions are important for describing a number of physical phenomena. The difference in this article is that the coefficients of some terms are functions. The unique feature of this article is that we can obtain many different types of equations when the coefficient functions change. This makes the (2+1)-dimensional vcDLW describe more general physical phenomena. In addition, the dynamical behaviour of the exact solutions of the (2+1)-dimensional vcDLW is analysed. And the evolution of several types of solutions is studied depending on the change in time. This process can be described better with the help of figures. Finally, The conservation laws of the (2+1)-dimensional vcDLW are obtained using the new conservation theorem with two modificatory rules.

A modeling and comparative study of bottom pressure signals generated by waves and ship in shallow water

For the bottom pressure signal generated by an advancing ship in the ocean, the bottom pressure signal generated by waves is a strong background disturbance. Aiming at the differences between their characteristics, the modeling methods for predicting the bottom pressure signals generated by ship and waves in shallow water are presented and validated, meanwhile, their characteristics in time and frequency domains are simulated, compared, and analyzed. The findings clearly show that their characteristics differ in some key parameters, such as the pressure spectrum, the pressure peak, and the negative duration, etc. The modeling and comparative studies proposed in this paper are very beneficial for the design of underwater weapons, and helpful for identifying whether the target is indeed a ship or just an ocean environment interference.

Lie Symmetry Analysis, Particular Solutions and Conservation Laws of a New Extended (3+1)-Dimensional Shallow Water Wave Equation

In this paper, a new extended (3+1)-dimensional shallow water wave equation is discussed via Lie symmetry analysis. Making use of symmetric nodes, we obtain two kinds of symmetrically reduced ODEs. By means of power series, we obtain the two kinds of exact power series solutions. By invoking a new conservation theorem of Ibragimov, the conservation laws are constructed.

Orbital stability of smooth solitary waves for the Degasperis-Procesi equation

The Degasperis-Procesi (DP) equation is an integrable Camassa-Holm-type model which is an asymptotic approximation for the unidirectional propagation of shallow water waves. This work establishes the orbital stability of localized smooth solitary waves to the DP equation on the real line, extending our previous work on their spectral stability [J. Math. Pures Appl. (9) 142 (2020), pp. 298–314]. The main difficulty stems from the fact that the natural energy space is a subspace of L 3 L^3 , but the translation symmetry for the DP equation gives rise to a conserved quantity equivalent to the L 2 L^2 -norm, resulting in L 3 L^3 higher-order nonlinear terms in the augmented Hamiltonian. But the usual coercivity estimate is in terms of L 2 L^2 norm for DP equation, which cannot be used to control the L 3 L^3 higher order term directly. The remedy is to observe that, given a sufficiently smooth initial condition satisfying some mild constraint, the L ∞ L^\infty orbital norm of the perturbation is bounded above by a function of its L 2 L^2 orbital norm, yielding the higher order control and the orbital stability in the L 2 ∩ L ∞ L^2\cap L^\infty space.

A numerical study of [formula omitted]-periodic wave solutions to four integrable equations

Based on the direct method proposed by A. Nakamura, we present a numerical process to derive N-periodic wave solutions of integrable equations. In particular, we investigate numerical 3-periodic wave solutions of the shallow water wave equation, modified generalized Vakhnenko equation, (2+1)-dimensional BKP equation and a (2+1)-dimensional Boussinesq equation. Numerical experiments are conducted using Gauss–Newton and Levenberg–Marquardt method, respectively. The comparison of the two numerical approaches is given.

A deep water and nearshore wave height calibration of the ECOWAVES hindcasting database

Significant wave height (SWH) in shallow waters is assessed by generating two wave hindcasts; the first uses ERA-Interim wind fields and the second one from ERA5 to quantify the improvement of the ERA5 surface winds on the SWH representativeness, both in deep and shallow waters along the Chilean coastline. Additionally, wind field predictions from the Global Forecast System (GFS) were used to assess the representativeness of shallow waters. Oceanographic buoys were used to validate SWH in deep waters, while Acoustic Doppler Current Profiler (ADCPs) was equipped to measure waves in shallow waters. Energy spectrums coupling Wavewatch III and Simulating Waves Nearshore (SWAN) models were transferred to evaluate the performance of shallow water simulations. In general, the SWH from both wave hindcasts showed good performance. Nonetheless, those forced by ERA5 presented a better qualitative comparison of sea state temporal variability, which increased the correlation coefficients (>0.9), coefficients of determination (>0.8), and minor errors (RMSE, MAE, and BIAS) compared to oceanographic buoys and ADCPs. Additionally, in simulations forced by GFS, the temporal variability of the waves in shallow waters was successfully reproduced. Nevertheless, an increase in the RMSE, MAE, and BIAS error was statistically verified compared to ERA-Interim and ERA 5.

Reappraisal of Whitham’s 1967 theory for wave–meanflow interaction in shallow water

The modulation equations for Stokes waves in shallow water coupled to wave-generated meanflow, derived in Whitham (1967), based on an averaged Lagrangian are re-visited. Firstly, it is shown that they can be recast into two coupled classical shallow water equations, with modified gravity having the sign of the Whitham index: sign(ω0′′ω2). Secondly, it is shown that the amplitude of the meanflow and amplitude of the wave are, in general, independent. Thirdly, the implications of the coalescing characteristics, whose unfolding is associated with the Benjamin–Feir instability, are studied.

A Mathematical Study of the (3+1)-D Variable Coefficients Generalized Shallow Water Wave Equation with Its Application in the Interaction between the Lump and Soliton Solutions

In this paper, the Hirota bilinear method, which is an important scheme, is used. The equation of the shallow water wave in oceanography and atmospheric science is extended to (3+1) dimensions, which is a well-known equation. A lot of classes of rational solutions by selecting the interaction between a lump and one- or two-soliton solutions are obtained. The bilinear form is considered in terms of Hirota derivatives. Accordingly, the logarithm algorithm to obtain the exact solutions of a (3+1)-dimensional variable-coefficient (VC) generalized shallow water wave equation is utilized. The analytical treatment of extended homoclinic breather wave solutions is studied and plotted in three forms 3D, 2D, and density plots. Using suitable mathematical assumptions, the established solutions are included in view of a combination of two periodic and two solitons in terms of two trigonometric and two hyperbolic functions for the governing equation. Maple software for computing the complicated calculations of nonlinear algebra equations is used. The effect of the free parameters on the behavior of acquired figures to a few obtained solutions for two nonlinear rational exact cases was also discussed.

A new generalized KdV equation: Its lump-type, complexiton and soliton solutions

A new generalized KdV equation, describing the motions of long waves in shallow water under the gravity field, is considered in this paper. By adopting a series of well-organized methods, the Bäcklund transformation, the bilinear form and diverse wave structures of the governing model are formally extracted. The exact solutions listed in this paper are categorized as lump-type, complexiton, and soliton solutions. To exhibit the physical mechanism of the obtained solutions, several graphical illustrations are given for particular choices of the involved parameters. As a direct consequence, diverse wave structures given in this paper enrich the studies on the KdV-type equations.

The Exact Solutions for Fractional-Stochastic Drinfel’d–Sokolov–Wilson Equations Using a Conformable Operator

The fractional-stochastic Drinfel’d–Sokolov–Wilson equations (FSDSWEs) perturbed by the multiplicative Wiener process are studied. The mapping method is used to obtain rational, hyperbolic, and elliptic stochastic solutions for FSDSWEs. Due to the importance of FSDSWEs in describing the propagation of shallow water waves, the derived solutions are significantly more useful and effective in understanding various important challenging physical phenomena. In addition, we use the MATLAB Package to generate 3D graphs for specific FSDSWE solutions in order to discuss the impact of fractional order and the Wiener process on the solutions of FSDSWEs.

Dynamic mechanism of nonlinear waves for the (3+1)-dimensional generalized variable-coefficient shallow water wave equation

Under investigation in this paper is a (3+1)-dimensional generalized variable-coefficient shallow water wave equation, which can be used to describe the flow below a pressure surface in oceanography and atmospheric science. Employing the Kadomtsev-Petviashvili hierarchy reduction, we obtain the breather and lump solutions in terms of Grammian. We investigate the generation mechanism and conversion of the breathers, lumps and rogue waves. We find that the breather is produced by the superposition of three parts: The soliton part, the periodic wave part and the background part. The angle between the soliton part and the periodic wave part affects the shape of the breather. Considering the influences of the variable coefficients, we observe the breathers propagating on the periodic backgrounds, with double peaks and the breathers propagating periodic with time, respectively. Taking the long-wave limits, we get the rational solutions which describe the lumps. We find that the characteristic lines keep unchanged on the x − y plane, which means that the lump is similar to a part of the breather. Linear rogue waves only appear on the y − z plane.

A One-Step Block Hybrid Integrator for Solving Fifth Order Korteweg-de Vries Equations

Fifth-order Korteweg-de Vries (KdV) equations, arise in modeling waves phenomena such as the propagation of shallow water waves over a flat surface, gravity-capillary waves and sound waves in plasmas. In this work, a one-step block hybrid linear multistep method was derived using the collocation technique, to solve fifth-order KdV models via the Method of Line (MoL). The consistency, stability and convergence of the method were established. The efficiency of the method can be seen from comparison of the exact solutions of problems and other methods cited from literature.

Numerical solution of fractional Kersten–Krasil’shchik coupled KdV–mKdV system arising in shallow water waves

Numerical solution of fractional Kersten–Krasil’shchik coupled KdV–mKdV system arising in shallow water waves

On new analytical solutions of fractional systems in shallow water dynamics

Recently, fractional calculus has got considerable attention from researchers since many problems in natural sciences and engineering are modelled with differential equations having fractional order. The nonlinear coupled time-fractional Boussinesq-Burger (B-B) equation, the nonlinear time-fractional long water wave (ALW) equation, and the nonlinear (2+1)-dimensional space-time fractional generalized Nizhnik-Novikov-Veselov (GNNV) equation are used to express the structure of shallow water waves (SWWs) with different distributions. The analytical solutions of these equations play a substantial role in explaining the properties of complex phenomena in applied sciences. In the current work, we utilize the exponential rational function (ERF) method with the definition of fractional derivative in the conformable sense to achieve new exact traveling wave solutions of these fractional systems. The correctness, validity, and graphics of the new traveling wave solutions are achieved with the aid of Mathematica. Results demonstrate the effectiveness and strength of this technique to solve the system of fractional differential equations (FDEs).

Hamiltonian structure, optimal classification, optimal solutions and conservation laws of the classical Boussinesq–Burgers system

Shallow water waves phenomena have been vastly studied in the literature for many decades. Understanding their intricacies is not only important in predicting the timing and magnitude of their destructive nature (such as tsunami) but also in knowing how to prevent their occurrences. Among the equations used for gaining accurate and deep understanding of shallow water waves for two-layered fluid flow is the classical Boussinesq–Burgers system. Its solutions have been studied in the literature using various methods; however, no study has considered its connection with (some of) the Painlevé transcendents which lead to infinite rational and special function solutions. The present study explore this avenue through the Lie group method — a powerful and versatile method that has been effectively and extensively used for various analyses of nonlinear differential equations. Invariant, rational and special function solutions of the classical Boussinesq–Burgers equations are derived and non-trivial conserved quantities are also constructed.

Sawi Transform Based Homotopy Perturbation Method for Solving Shallow Water Wave Equations in Fuzzy Environment

In this manuscript, a new hybrid technique viz Sawi transform-based homotopy perturbation method is implemented to solve one-dimensional shallow water wave equations. In general, the quantities involved with such equations are commonly assumed to be crisp, but the parameters involved in the actual scenario may be imprecise/uncertain. Therefore, fuzzy uncertainty is introduced as an initial condition. The main focus of this study is to find the approximate solution of one-dimensional shallow water wave equations with crisp, as well as fuzzy, uncertain initial conditions. First, by taking the initial condition as crisp, the approximate series solutions are obtained. Then these solutions are compared graphically with existing solutions, showing the reliability of the present method. Further, by considering uncertain initial conditions in terms of Gaussian fuzzy number, the governing equation leads to fuzzy shallow water wave equations. Finally, the solutions obtained by the proposed method are presented in the form of Gaussian fuzzy number plots.

New Efficient Computations with Symmetrical and Dynamic Analysis for Solving Higher-Order Fractional Partial Differential Equations

Due to the rapid development of theoretical and computational techniques in the recent years, the role of nonlinearity in dynamical systems has attracted increasing interest and has been intensely investigated. A study of nonlinear waves in shallow water is presented in this paper. The classic form of the Korteweg–de Vries (KdV) equation is based on oceanography theory, shallow water waves in the sea, and internal ion-acoustic waves in plasma. A shallow fluid assumption is shown in the framework by a sequence of nonlinear fractional partial differential equations. Indeed, the primary purpose of this study is to use a semi-analytical technique based on Fractional Taylor Series to achieve numerical results for nonlinear fifth-order KdV models of non-integer order. Caputo is the operator used for dealing with fractional derivatives. The generated solutions of nonlinear fifth-order KdV models of non-integer order for modeling turbulence processes in the field of ocean engineering are compared analytically and numerically, to demonstrate the behaviors of several parameters of the current model. We verified the method’s convergence analysis and provided an error estimate by showing 2D and 3D graphs to further confirm its efficacy.

The Analytical Solutions for the Stochastic-Fractional Broer–Kaup Equations

We take into account here the stochastic-fractional Broer–Kaup equations (SFBKEs) perturbed by the multiplicative Wiener process. To get rational, hyperbolic, and elliptic stochastic solutions for SBKEs, we utilize the Jacobi elliptic function method. The derived solutions are significantly more useful and effective in comprehending various important challenging physical phenomena due to the important of SFBKEs in describing the propagation of shallow water waves. Also, we use the MATLAB Package to create 2D and 3D graphs for certain solutions of SFBKEs in order to discuss the impact of fractional order and the Wiener process on the solutions of SFBKEs.

Uniqueness and generic regularity of global weak conservative solutions to the Constantin-Lannes equation

ABSTRACT This paper is devoted to the uniqueness of global-in-time conservative solutions and generic regularity for the shallow water waves of moderate amplitude equation (Constantin-Lannes equation). The Constantin-Lannes equation possible development of singularities in finite time, and beyond the occurrence of wave breaking, it exists global conservative solutions. In the present paper, we will prove the uniqueness of global-in-time conservative solutions for the Constantin-Lannes equation with general initial data by analyzing the evolution of the quantities u and along each characteristic. Moreover, we consider that piecewise smooth solutions with only generic singularities are dense in the whole solution set by Thom's transversality Lemma.