Shallow Water Wave Model Research Articles

Supershear shock front contribution to the tsunami from the 2018 Mw 7.5 Palu, Indonesia earthquake

SUMMARY Hazardous tsunamis are known to be generated predominantly at subduction zones. However, the 2018 Mw 7.5 Palu (Indonesia) earthquake on a strike-slip fault generated a tsunami that devastated the city of Palu. The mechanism by which this tsunami originated from such an earthquake is being debated. Here we present near-field ground motion (GPS) data confirming that the earthquake attained supershear speed, i.e. a rupture speed greater than the shear wave speed of the host medium. We subsequently study the effect of this supershear rupture on tsunami generation by coupling the ground motion to a 1-D non-linear shallow-water wave model accounting for both time-dependent bathymetric displacement and velocity. With the local bathymetric profile of Palu bay around a tidal station, our simulations reproduce the tsunami arrival and motions observed by CCTV cameras. We conclude that Mach (shock) fronts, generated by the supershear speed, interacted with the bathymetry and contributed to the tsunami.

Novel interaction phenomena of the new (2 + 1)-dimensional extended shallow water wave equation

Based on the Hirota bilinear method, we study some kinds of novel interaction solutions of the new integrable [Formula: see text]-dimensional extended shallow water wave equation. Using the symbolic computations, we obtain a rich variety of exact solutions such as the lump solutions, lump-kink solutions, breather solutions and two-kink solutions and these exact solutions are shown analytically and graphically by selecting the proper parameters. These new exact solutions contribute us to our understanding of the physical phenomena of the integrable nonlinear shallow water wave model.

Lie Symmetries, Closed-Form Solutions, and Various Dynamical Profiles of Solitons for the Variable Coefficient (2+1)-Dimensional KP Equations

This investigation focuses on two novel Kadomtsev–Petviashvili (KP) equations with time-dependent variable coefficients that describe the nonlinear wave propagation of small-amplitude surface waves in narrow channels or large straits with slowly varying width and depth and non-vanishing vorticity. These two variable coefficients, Kadomtsev–Petviashvili (VCKP) equations in (2+1)-dimensions, are the main extensions of the KP equation. Applying the Lie symmetry technique, we carry out infinitesimal generators, potential vector fields, and various similarity reductions of the considered VCKP equations. These VCKP equations are converted into nonlinear ODEs via two similarity reductions. The closed-form analytic solutions are achieved, including in the shape of distinct complex wave structures of solitons, dark and bright soliton shapes, double W-shaped soliton shapes, multi-peakon shapes, curved-shaped multi-wave solitons, and novel solitary wave solitons. All the obtained solutions are verified and validated by using back substitution to the original equation through Wolfram Mathematica. We analyze the dynamical behaviors of these obtained solutions with some three-dimensional graphics via numerical simulation. The obtained variable coefficient solutions are more relevant and useful for understanding the dynamical structures of nonlinear KP equations and shallow water wave models.

The Influence of Noise on the Solutions of Fractional Stochastic Bogoyavlenskii Equation

We look at the stochastic fractional-space Bogoyavlenskii equation in the Stratonovich sense, which is driven by multiplicative noise. Our aim is to acquire analytical fractional stochastic solutions to this stochastic fractional-space Bogoyavlenskii equation via two different methods such as the exp(−Φ(η))-expansion method and sine–cosine method. Since this equation is used to explain the hydrodynamic model of shallow-water waves, the wave of leading fluid flow, and plasma physics, scientists will be able to characterize a wide variety of fascinating physical phenomena with these solutions. Furthermore, we evaluate the influence of noise on the behavior of the acquired solutions using 2D and 3D graphical representations.

A comparative study about the propagation of water waves with fractional operators

In the current research article, the truncated M-fractional derivative and Atangana–Baleanu derivative in Riemann–Liouville sense is practiced in the fractional modeling of the weakly non-linear shallow water wave equation. This paper provides some traveling wave transformations for the conversion of the partial fractional differential equation into an ordinary differential equation concerning fractional operators. In order to portray the wave propagation in weakly non-linear and dispersive media, the weakly non-linear shallow water wave model has been utilized. The proficient approach new extended direct algebraic scheme is used to find the solitonic structures. It is observed that the acquired solutions are purely new and have full novelty. The obtained results are carrying dark, dark singular, bright singular, and bright-dark singular solutions, which are obtained via Mathematica. The graphical comparison is also depicted in the perspective of operating fractional differential definitions with the selection of appropriate parametric values.

Exact solitary wave solution for fractal shallow water wave model by He’s variational method

In this paper, the shallow water wave with unsmooth boundaries is depicted by fractal calculus, and its exact fractal solitary wave solution is successfully found by He’s variational method. The numerical examples demonstrate that the method is simple, efficient and convenient. Finally, we elaborate the physical properties of the fractal solitary solution by some graphs. The novel scheme clears up the road to the exact fractal soliton, and it sheds a new light on the fractal soliton theory.

The patterns of traveling wave on shallow water modeled by Green-Naghdi equation

The Green-Naghdi equations are a shallow water waves model which play important roles in nonlinear wave fields. By using the trial equation method and the Complete discrimination system for the polynomial we obtained the classification of travelling wave patterns. Among those patterns, new singular patterns and double periodic patterns are obtained in the first time. And we draw the graphs which help us to understand the dynamics behaviors of the Green-Naghdi model intuitionally.

Diverse wave propagation in shallow water waves with the Kadomtsev–Petviashvili–Benjamin–Bona–Mahony and Benney–Luke integrable models

Abstract The shallow water wave model is one of the completely integrable models illustrating many physical problems. In this article, we investigate new exact wave structures to Kadomtsev–Petviashvili–Benjamin–Bona–Mahony and the Benney–Luke equations which explain the behavior of waves in shallow water. The exact structures are expressed in the shapes of hyperbolic, singular periodic, rational as well as solitary, singular, shock, shock-singular solutions. An efficient computational strategy namely modified direct algebraic method is employed to construct the different shapes of wave structures. Moreover, by fixing parameters, the graphical representations of some solutions are plotted in terms of three-dimensional, two-dimensional and contour plots, which explain the physical movement of the attained results. The accomplished results show that the applied computational technique is valid, proficient, concise and can be applied in more complicated phenomena.

Nonlinear dynamical study to time fractional Dullian–Gottwald–Holm model of shallow water waves

This paper studies the exact traveling wave solutions to the nonlinear Dullin–Gottwald–Holm model which has the application in shallow-water waves in which the fractional derivative is considered in the sense of conformable derivative. Diverse exact solutions in hyperbolic, trigonometric and plane wave forms are obtained using two integration norms. For this purpose [Formula: see text]-expansion method and reccati mapping techniques are used. The 3D plots and their corresponding contour graphs are also depicted. Being concise and straightforward, the calculations demonstrate the effectiveness and convenience of the method for solving other nonlinear partial differential equations.

New Solitary and Periodic Wave Solutions of (n + 1)-Dimensional Fractional Order Equations Modeling Fluid Dynamics

In this study, first, fractional derivative definitions in the literature are examined and their disadvantages are explained in detail. Then, it seems appropriate to apply the (G′G)-expansion method under Atangana’s definition of β-conformable fractional derivative to obtain the exact solutions of the space–time fractional differential equations, which have attracted the attention of many researchers recently. The method is applied to different versions of (n+1)-dimensional Kadomtsev–Petviashvili equations and new exact solutions of these equations depending on the β parameter are acquired. If the parameter values in the new solutions obtained are selected appropriately, 2D and 3D graphs are plotted. Thus, the decay and symmetry properties of solitary wave solutions in a nonlocal shallow water wave model are investigated. It is also shown that all such solitary wave solutions are symmetrical on both sides of the apex. In addition, a close relationship is established between symmetric and propagated wave solutions.

A (2+1)-dimensional generalized Hirota–Satsuma–Ito equations: Lie symmetry analysis, invariant solutions and dynamics of soliton solutions

This paper investigates the exact invariant solutions and the dynamics of soliton solutions to the (2+1)-dimensional generalized Hirota–Satsuma–Ito (g-HSI) equations. By applying the Lie symmetry technique, infinitesimal vectors, the commutation relations, and various similarity reductions are derived from the g-HSI equations. Using the two stages of Lie symmetry reductions, the equation is transformed into various nonlinear ordinary differential equations (NLODEs). After that, by solving the various resulting ODEs, we obtain abundant explicit exact solutions in terms of the involved functional parameters. These closed-form invariant solutions are successfully presented in the form of distinct complex wave-structures of solutions like combo-form solitons, dark-bright solitons, W-shaped solitons, the interaction between multiple solitons, parabolic wave solitons, multi-wave structures, and curved-shaped parabolic solitons. Furthermore, using computerized symbolic computation and numerical simulation, the physical behaviors of some obtained solutions are displayed in three-dimensional graphics. The resulting solutions are found to be useful for understanding the dynamics of the exact closed-form solutions of this model and show the authenticity as well as the effectiveness of the proposed method. Therefore, the gained solutions and their dynamical wave structures are quite significant for understanding the propagation of the excitation waves in shallow water wave models. Furthermore, using the resulting symmetries, the conservation laws of g-HSI equations have been obtained by applying Ibragimov’s theorem.

A two-layer model for landslide generated impulse wave: Simulation of the 1958 Lituya bay landslide impact wave from generation to long–duration transport

An updated one-dimensional two-layer coupled mathematical model is developed for landslide generated impulse wave (LGIW) from wave formation to long-duration travel. The coupled model is composed of a landslide model in a bed-fitted coordinate system and a shallow-water wave model in a global coordinate system. The coupling is realized through a reconstruction and interpolation algorithm. We successfully reproduced the experimental reproduction of 1958 Lituya LGIW for a total duration of 250 s at prototype scale, and well captured the three runups on the right bank and three runups on the left bank, together with the six wave crests and troughs in the channel. This seems to be the first reproduction of the Lituya LGIW for such a long duration. The predicted free surface elevation and runups are comparable with experimental results though with some deviations. However, the level of reproduction of the Lituya LGIW from wave formation to long-duration traveling achieved by numerical simulation is greatly enhanced as compared with the existing interface-capture models and mesh-free/particle-based models. This proposed one-dimensional two-layer coupled model could provide a unified framework for LGIWs from generation to long-duration propagation.

A new approach to mathematical models of Drinfeld-Sokolov-Wilson and coupled viscous Burgers’ equations in water flow

In this paper, it is the first time that we implement conformable Laplace decomposition method (CLDM) to time fractional systems of Drinfeld-Sokolov-Wilson equation (DSWE) and coupled viscous Burgers’ equation (CVBE). DSWE and CVBE have an important place for cceanic, coastal sea research and they are considered as a mathematical model for shallow water waves and hydrodynmic turbulence respectively. At the end, the obtained solutions are compared with the exact solutions by the aid of tables and figures. The obtained results show that,conformable Laplace decomposition method (CLDM) is efficient, reliable, easy to apply and it gives researchers a new perspective for solving a wide variety of nonlinear fractional partial differential equations in physics.

ON THE PHYSICAL BEHAVIOURS OF THE CONFORMABLE FRACTIONAL MODIFIED CAMASSA– HOLM EQUATION USING TWO EFFICIENT METHODS

In recent years, many authors have researched about fractional partial differential equations. Physical phenomena, which arise in engineering and applied science, can be defined more accurately by using FPDEs. Thus, obtaining exact solutions of the FPDEs equations have become more important to understand physical problems. In this article, we have reached the new traveling wave solutions of the conformable fractional modified Camassa – Holm equation via two efficient methods such as first integral method and the functional variable method. The wave transformation and conformable fractional derivative have been used to convert FPDE to the ordinary differential equation. The Camassa – Holm equation is physical model of shallow water waves with non-hydrostatic pressure. Thanks to these powerful methods, some comparisons, such as type of solutions and physical behaviours, have been made. Additionally, mathematica program have been used with the aim of checking of solutions. Investigating results of the fractional differential equations can help understanding complex phenomena in applied mathematics and physics.

M-lump and interaction solutions of a ($$2+1$$)-dimensional extended shallow water wave equation

In this paper, a more extended ( $$2+1$$ )-dimensional shallow water wave equation is considered, which is integrable in terms of the Hirota’s bilinear method. With the aid of the bilinear method, the N-soliton solutions are obtained. By applying the long-wave limit method to the N-soliton solutions, the multiple lump solutions are gained. The dynamical characteristics of the single lumps among the M-lump wave are investigated. From the point of view of mathematical mechanism, the propagation orbits, velocities and the interactions among the lumps are analyzed systematically. Furthermore, various types of elastic and inelastic interaction solutions among the lumps, breathers and solitons are constructed, which develop the types of the solutions of the ( $$2+1$$ )-dimensional extended shallow water wave model. The method presented in this paper can be applied to more nonlinear integrable equations to investigate the localized wave solutions and interaction solutions. The obtained results may provide some useful information to analyze the theory of the shallow water waves and solitons.

Low-dimensional offshore wave input for extreme event quantification

In offshore engineering design, nonlinear wave models are often used to propagate stochastic waves from an input boundary to the location of an offshore structure. Each wave realization is typically characterized by a high-dimensional input time-series, and a reliable determination of the extreme events is associated with substantial computational effort. As the sea depth decreases, extreme events become more difficult to evaluate. We here construct a low-dimensional characterization of the candidate input time series to circumvent the search for extreme wave events in a high-dimensional input probability space. Each wave input is represented by a unique low-dimensional set of parameters for which standard surrogate approximations, such as Gaussian processes, can estimate the short-term exceedance probability efficiently and accurately. We demonstrate the advantages of the new approach with a simple shallow-water wave model based on the Korteweg–de Vries equation for which we can provide an accurate reference solution based on the simple Monte Carlo method. We furthermore apply the method to a fully nonlinear wave model for wave propagation over a sloping seabed. The results demonstrate that the Gaussian process can learn accurately the tail of the heavy-tailed distribution of the maximum wave crest elevation based on only $$1.7\%$$ of the required Monte Carlo evaluations.

Diverse novel analytical and semi-analytical wave solutions of the generalized (2+1)-dimensional shallow water waves model

This article studies the generalized (2 + 1)-dimensional shallow water equation by applying two recent analytical schemes (the extended simplest equation method and the modified Kudryashov method) for constructing abundant novel solitary wave solutions. These solutions describe the bidirectional propagating water wave surface. Some obtained solutions are sketched in two- and three-dimensional and contour plots for demonstrating the dynamical behavior of these waves along shallow water. The accuracy of the obtained solutions and employed analytical schemes is investigated using the evaluated solutions to calculate the initial condition, and then the well-known variational iterational (VI) method is applied. The VI method is one of the most accurate semi-analytical solutions, and it can be applied for high derivative order. The used schemes’ performance shows their effectiveness and power and their ability to handle many nonlinear evolution equations.

Bifurcations and Exact Traveling Wave Solutions of Two Shallow Water Two-Component Systems

This paper studies two two-component shallow water wave models. From the dynamical systems approach and using the singular traveling wave theory developed by Li and Chen [2007], all possible bounded solutions (solitary wave solutions, pseudo-peakons, periodic peakons, as well as smooth periodic wave solutions) are obtained under different parameter conditions. More than six explicit exact parametric representations are derived. More interestingly, it was found that, for the two-component Camassa–Holm equations with constant vorticity, its [Formula: see text]-traveling wave system has a pseudo-peakon wave solution. In addition, its [Formula: see text]-traveling wave system has four families of uncountably infinitely many solitary wave solutions. The new results complete a recent study of Dutykh and Ionescu-Kruse [2019].

Multiple soliton, fusion, breather, lump, mixed kink-lump and periodic solutions to the extended shallow water wave model in (2+1)-dimensions

In this paper, we consider the shallow water wave model in the (2+1)-dimensions. The Hirota simple method is applied to construct the new dynamics one-, two-, three-, [Formula: see text]-soliton solutions, complex multi-soliton, fusion, and breather solutions. By using the quadratic function, the one-lump, mixed kink-lump and periodic lump solutions to the model are obtained. The Hirota bilinear form variable of this model is derived at first via logarithmic variable transform. The physical phenomena to this model are explored. The obtained results verify the proposed model.

Rational Localized Waves and Their Absorb-Emit Interactions in the (2 + 1)-Dimensional Hirota–Satsuma–Ito Equation

In this paper, we investigate the (2 + 1)-dimensional Hirota–Satsuma–Ito (HSI) shallow water wave model. By introducing a small perturbation parameter ϵ, an extended (2 + 1)-dimensional HSI equation is derived. Further, based on the Hirota bilinear form and the Hermitian quadratic form, we construct the rational localized wave solution and discuss its dynamical properties. It is shown that the oblique and skew characteristics of rational localized wave motion depend closely on the translation parameter ϵ. Finally, we discuss two different interactions between a rational localized wave and a line soliton through theoretic analysis and numerical simulation: one is an absorb-emit interaction, and the other one is an emit-absorb interaction. The results show that the delay effect between the encountering and parting time of two localized waves leads to two different kinds of interactions.