Water Wave Model Research Articles

Invariance properties, exact and explicit solutions of time-fractional Gear–Grimshaw model

In this study, we consider a time-fractional shallow water wave model which was introduced by Gear and Grimshaw. This model represents the dynamics of two-layered shallow water waves in oceans and rivers. Also it covers many shallow water wave models with special values taken to parameters. By performing the group invariant analysis approach for the governing system with Riemann–Liouville derivative, symmetry generators were found. Similarity reduction was constructed by Erdelyi–Kober sense. In addition, by adapting the Sine-Gordon expansion method, we find some exact traveling wave solutions. Furthermore, explicit solutions in form of power series are examined via the residual power series method. Exact and approximate solutions are compared for some [Formula: see text] values and these are given as tables.

Two Models for 2D Deep Water Waves

In this paper we propose two Hamiltonian models to describe two-dimensional deep water waves propagating on the surface of an ideal incompressible three-dimensional fluid. The idea is based on taking advantage of the Zakharov equation for one-dimensional waves which can be written in the form of so-called compact equations. We generalize these equations to the case of two-dimensional waves. As a test of our models, we perform numerical simulations of the dynamics of standing waves in a channel with smooth vertical walls. The results obtained in the proposed models are comparable, indicating that the models are similar to the original Zakharov equation.

The solitonic solutions of finite depth long water wave models

The nonlinear Fokas and breaking soliton equations are important modeling equations for coastal and ocean waves, such as long water waves and tsunami waves. Moreover, these models have effective applications for simulating surface and internal waves in oceans and rivers, wave packets in finite depth water, and Riemann wave propagation interactions. By exploiting the enhanced modified simple equation approach, in this article, we have established further generic, radical, and standard solitons to the erstwhile nonlinear framework with some parameters. The reputed schematic solitons, such as bell-shaped, periodic, cuspon, compacton, plane-shaped, singular soliton, kink, and other solitons, are devised for varied scores of parameters. The accomplished results might assist in better understanding the dynamics of coastal waves modulated by the formerly acknowledged couple of nonlinear models. We have studied the impacts of physical parameters and propagation rate on wave structure and affirmed their role in changing the waveform. Three- and two-dimensional diagrams are sketched to interpret the context profiles of the obtained solitary wave solutions. The established solutions demonstrate the applicability and competence of the approach to investigating nonlinear models.

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.

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.

Some analytic and series solutions of integrable generalized Broer-Kaup system

In this article, we have updated the results obtained by Zhang et al. (2019) and derived some certain features of a famous dispersion water wave model, namely, generalized Broer-Kaup (BK) system. First, we have shown the integrability of the model. Then, a combination of Lie classical approach with polynomial type assumption is used to find its series solutions. Additionally, some soliton solutions are derived via tanh-expansion method and Kudryashov method. Also, local conservation laws of the governing model are constructed by Ibragimov method. The results of this study are novel and can be helpful in several scientific fields.

Energy-based Modeling and Simulation of Shallow Water Waves in a Tube with Moving Boundary

Modeling and simulation of shallow water waves in a tube with moving boundary and arbitrary cross-section are considered. Based on the a variational formulation of the well known one dimensional Saint-Venant equations in material fixed coordinates, energy-conserving finite-dimensional approximations are directly derived by the principle of least action. A particular discretization scheme is investigated, namely, a piecewise constant approximation of the distributed variables. The approach is validated in simulation and compared with alternative discretization-schemes.

Hamiltonian Model for Water Waves in a Triangular Domain

Hamiltonian Model for Water Waves in a Triangular Domain

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.

Well-posedness for a coupled system of Kawahara/KdV type equations with polynomials nonlinearities

<p style='text-indent:20px;'>We consider the initial value problem (IVP) associated to a coupled system of modified Kawahara/KdV type equations with polynomials nonlinearities. For the model in question, the Cauchy problem is of interest, and is shown to be well-posed for given data in a Gevrey spaces. Our results make use of techniques presented in Grujić and Kalisch, who studied the Gevrey regularity for a class of water-wave models and the well-posedness of a IVP associated to a general equation. The proof relies on estimates in space-time norms adapted to the linear part of the equations. In particular, estimates in Bourgain spaces are proven for the linear and nonlinear terms of the system and the main result is obtained by a contraction principle. The class of system in view generalizes the system of modified Kawahara/KdV type equations studied by Kondo and Pes, which contains a number of systems arising in the modeling of waves in fluids, stability and instability of solitary waves and models for wave propagation in physical systems where both nonlinear and dispersive effects are important.</p>

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 bright soliton solutions for Kadomtsev–Petviashvili–Benjamin–Bona–Mahony equations and bidirectional propagation of water wave surface

The governing equations for fluid flows, i.e. Kadomtsev–Petviashvili–Benjamin–Bona–Mahony (KP-BBM) model equations represent a water wave model. These model equations describe the bidirectional propagating water wave surface. In this paper, an auto-Bäcklund transformation is being generated by utilizing truncated Painlevé expansion method for the considered equation. This paper determines the new bright soliton solutions for [Formula: see text] and [Formula: see text]-dimensional nonlinear KP-BBM equations. The simplified version of Hirota’s technique is utilized to infer new bright soliton solutions. The results are plotted graphically to understand the physical behavior of solutions.

Localized nonlinear waves on spatio-temporally controllable backgrounds for a (3+1)-dimensional Kadomtsev-Petviashvili-Boussinesq model in water waves

Physics of nonlinear waves on variable backgrounds and the relevant mathematical analysis continues to be the challenging aspect of the study. In this work, we consider a (3+1)-dimensional nonlinear model describing the dynamics of water waves and construct nonlinear wave solutions on spatio-temporally controllable backgrounds for the first time by using a simple mathematical tool auto-Bäcklund transformation. Mainly, we unravel physically interesting features to control and manipulate the dynamics of nonlinear waves through the background. Adapting an exponential function and general polynomial of degree two as initial seed solutions, we construct single kink-soliton and rogue wave, respectively. We choose arbitrary periodic, localized and combined wave backgrounds by incorporating Jacobi elliptic functions and investigate the modulation of these two nonlinear waves with a clear analysis and graphical demonstrations. The solutions derived in this work give us sufficient freedom to generate exotic nonlinear coherent structures on variable backgrounds and open up an interesting direction to explore the dynamics of various other nonlinear waves propagating through inhomogeneous media.

Global bifurcation of solitary waves to the Boussinesq abcd system

The Boussinesq abcd system arises in the modeling of long wave small amplitude water waves in a channel, where the four parameters (a,b,c,d) satisfy one constraint. In this paper we focus on the solitary wave solutions to such a system. In particular we work in two parameter regimes where the system does not admit a Hamiltonian structure (corresponding to b≠d). We prove via analytic global bifurcation techniques the existence of solitary waves in such parameter regimes. Some qualitative properties of the solutions are also derived, from which sharp results can be obtained for the global solution curves.Specifically, we first construct solutions bifurcating from the stationary waves, and obtain a global continuous curve of solutions that exhibits a loss of ellipticity in the limit. The second family of solutions bifurcate from the classical Boussinesq supercritical waves. We show that the curve associated to the second class either undergoes a loss of ellipticity in the limit or becomes arbitrarily close to having a stagnation point.

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.

Global well-posedness and decay for viscous water wave models

The motion of the free surface of an incompressible fluid is a very active research area. Most of these works examine the case of an inviscid fluid. However, in several practical applications, there are instances where the viscous damping needs to be considered. In this paper, we derive and study a new asymptotic model for the motion of unidirectional viscous water waves. In particular, we establish the global well-posedness in Sobolev spaces. Furthermore, we also establish the global well-posedness and decay of a fourth order partial differential equation modeling bidirectional water waves with viscosity moving in deep water with or without surface tension effects.

Overcoming excessive numerical dissipation in SPH modeling of water waves

Excessive nonphysical energy dissipation is a problem in Smoothed Particle Hydrodynamics (SPH) when modeling free surface waves, resulting in a significant decrease in wave amplitude within a few wavelengths for progressive waves. This dissipation poses a limitation to the physical scale of SPH applications involving water wave propagation. Some prior solutions to this wave decay problem rely on elaborate schemes, which require a complex, or non-straightforward, implementation. Other approaches demand large smoothing lengths that lead to longer simulation times and potential degradation of the results. In this work we present an approach based on a kernel gradient correction. Our scheme is fully 3D and solves the main known drawbacks of kernel gradient corrections, such as instabilities and lack of momentum conservation. The latter is ensured by adopting an averaged correction matrix, so as to conserve reciprocity during particle interactions. We test our model with a standing wave in a basin and a progressive wave train in a wave tank, and in both cases no nonphysical decay occurs. A comparison to an approach based on large smoothing factors shows advantages both in quality of the results and simulation time.