Water Wave Model Research Articles

Invariance analysis, optimal system, closed-form solutions and dynamical wave structures of a (2+1)-dimensional dissipative long wave system

By employing the one-parameter Lie group of transformations method, abundant exact invariant solutions are obtained for a (2+1)-dimensional dissipative long wave (DLW) system, which describes the water wave model of hydrodynamics with wide channels or open seas of finite depth and can also be used to illustrate nonlinear wave propagation in the dissipative medium. Initially, we derived the Lie symmetries, geometric vector fields, and commutative table, and adjoint relations of the examined vectors for the system. A one-dimensional optimal system of symmetry subalgebras is well-structured. It leads to the reduction of independent variables of the considered DLW system and leaves the system invariant. The (2+1)-dimensional DLW system is reduced into nonlinear ordinary differential equations (NLODEs) for the various subalgebras obtained through two phases of similarity reductions. Some novel closed-form solutions are obtained, including arbitrary independent functional parameters, and have never been reported in the literature. Moreover, we depict the dynamical behaviors of obtained closed-form invariant solutions like multi-wave solitons, single solitons, breather-waves, rogue waves, bell-shaped structures, Kink-waves, and Lump-waves, wave-wave interactions, and annihilation of Kink-wave profiles through three-dimensional graphics. Eventually, a comparison with other results is discussed.

Global bifurcation of solitary waves for the Whitham equation

The Whitham equation is a nonlocal shallow water-wave model which combines the quadratic nonlinearity of the KdV equation with the linear dispersion of the full water wave problem. Whitham conjectured the existence of a highest, cusped, traveling-wave solution, and his conjecture was recently verified in the periodic case by Ehrnström and Wahlén. In the present paper we prove it for solitary waves. Like in the periodic case, the proof is based on global bifurcation theory but with several new challenges. In particular, the small-amplitude limit is singular and cannot be handled using regular bifurcation theory. Instead we use an approach based on a nonlocal version of the center manifold theorem. In the large-amplitude theory a new challenge is a possible loss of compactness, which we rule out using qualitative properties of the equation. The highest wave is found as a limit point of the global bifurcation curve.

Wilton Ripples in Weakly Nonlinear Models of Water Waves: Existence and Computation

Wilton Ripples in Weakly Nonlinear Models of Water Waves: Existence and Computation

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.

Explicit wave phenomena to the couple type fractional order nonlinear evolution equations

We utilize the fractional modified Riemann-Liouville derivative in the sense to develop careful arrangements of space–time fractional coupled Boussinesq equation which emerges in genuine applications, for instance, nonlinear framework waves iron sound waves in plasma and in vibrations in nonlinear string and space–time fractional-coupled Boussinesq Burger equation that emerges in the investigation of liquids stream in a dynamic framework and depicts engendering of shallow-water waves. A decent comprehension of its solutions is exceptionally useful for beachfront and engineers to apply the nonlinear water wave model to the harbor and seaside plans. A summed-up partial complex transformation is correctly used to change this equation to a standard differential equation thus, many precise logical arrangements are acquired with all the free parameters. At this point, the traveling wave arrangements are articulated by hyperbolic functions, trigonometric functions, and rational functions, if these free parameters are considered as specific values. We obtain kink wave solution, periodic solutions, singular kink type solution, and anti-kink type solutions which are shown in 3D and contour plots. The presentation of the method is dependable and important and gives even more new broad accurate arrangements.

On the wave solutions of two-mode equations

In this work, we consider that the two-mode equations, especially two-mode Korteweg-de Vries equation and two-mode Sharma-Tasso-Olver equations, which are used for modelling of shallow water waves, electromagnetism, electrical and electronics engineering, signal analysis, quantum mechanics etc. and their analytical solutions are obtained via the well-known Bernoulli equation method through the symbolic computation.

Conservation laws, classical symmetries and exact solutions of a (1 + 1)-dimensional fifth-order integrable equation

In this paper, we present a study of a fifth-order nonlinear partial differential equation, which was recently introduced in the literature. This equation can be used as a model for bidirectional water waves propagating in a shallow medium. Using elements of an optimal system of one-dimensional subalgebras, we perform similarity reductions culminating in analytic solutions. Rational, hyperbolic, power series and elliptic solutions are obtained. Furthermore, by using the multiple exponential function method we obtain one and two soliton solutions. Finally, local and low-order conserved quantities are derived by enlisting the multiplier approach.

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.

A comparison of smoothed particle hydrodynamics simulation with exact results from a nonlinear water wave model

The aim of this paper is to verify the velocity profile and the pressure variation inside the fluid domain over one wavelength obtained from a numerically simulated Smoothed Particle Hydrodynamics model with some exact qualitative results (i.e., increasing/decreasing trend or constant value of a flow field) from a fully nonlinear Euler equation for water wave model. A numerical wave flume has been modeled and a regular wave train is created by the horizontal displacement of a wave paddle on one side of the flume. A passive beach is used to dissipate the energy of the wave on the other side. The extracted numerical results are compared with some recently available exact results from a nonlinear steady water wave model based on the Euler equations for irrotational flow. The flow properties under wave crests, wave troughs, and along the distance from the wave crest to the wave trough over one wavelength are investigated. The horizontal and vertical velocity components and the pressure in the fluid domain agree well with the analytical results.

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.

Coupled 3D numerical model for a landslide-induced impulse water wave: A case study of the Fuquan landslide

Large impulse waves are generated when a highly mobilized landslide impacts a water body along the runout path. On August 27, 2014, a massive landslide occurred in Fuquan, Guizhou, China and induced an impulse wave upon contact with a quarry lake. The landslide and its associated impulse wave were a multiphase and multicomponent mixture that collectively destroyed two villages. The present study provides insights into multiphase dynamic analysis by employing an existing 3D coupled discrete element modeling and computational fluid dynamics approach to model a case study of the Fuquan landslide and its impulse wave. EDEM and Fluent were two-way coupled by an application programming interface, which transferred data between the models to simulate subaerial movement and impulse wave generation. The simulated results indicate a maximum velocity of 31.7 m/s achieved by the subaerial movement and a runout duration of 39 s. The detached mass entered the quarry lake at a velocity of 24.3 m/s, at t = 7.1 s after the landslide initiation. Energy was transferred between the sliding mass and lake water, and the generated water wave rushed toward the opposite slope. The generated wave had a run-up of 28.4 m and traveled approximately 90 m in the lateral direction, with a maximum dynamic pressure of 3.03 MPa. Qualitative validation of the present model was performed using Fritz's experimental setup for wave generation, and the velocity evolution was physically verified. This study provides calibrated parameters for the forward simulation of similar cases of landslide-induced impulse water waves and related hazard zonation.

Whitham modulation theory for generalized Whitham equations and a general criterion for modulational instability

AbstractThe Whitham equation was proposed as a model for surface water waves that combines the quadratic flux nonlinearity of the Korteweg–de Vries equation and the full linear dispersion relation of unidirectional gravity water waves in suitably scaled variables. This paper proposes and analyzes a generalization of Whitham's model to unidirectional nonlinear wave equations consisting of a general nonlinear flux function and a general linear dispersion relation . Assuming the existence of periodic traveling wave solutions to this generalized Whitham equation, their slow modulations are studied in the context of Whitham modulation theory. A multiple scales calculation yields the modulation equations, a system of three conservation laws that describe the slow evolution of the periodic traveling wave's wavenumber, amplitude, and mean. In the weakly nonlinear limit, explicit, simple criteria in terms of general and establishing the strict hyperbolicity and genuine nonlinearity of the modulation equations are determined. This result is interpreted as a generalized Lighthill–Whitham criterion for modulational instability.

Propagation of wave solutions of nonlinear Heisenberg ferromagnetic spin chain and Vakhnenko dynamical equations arising in nonlinear water wave models

Abstract Manuscript purpose is to find analytical solutions of the (2 + 1)-dimensional Heisenberg Ferromagnetic Spin Chain and Vakhnenko dynamical equations by using the generalized direct algebraic and simple equation analytical methods. Numerous exact traveling results are obtained in exponential, trigonometric and hyperbolic wave solutions. To understand the physical behavior of the models, the two and three dimensions graphs along with contour are plotted after assigning the particular values to the parameters. Obtained structures provide a rich platform for solving nonlinear wave problems with many applications.

On High Order ADER Discontinuous Galerkin Schemes for First Order Hyperbolic Reformulations of Nonlinear Dispersive Systems

This paper is on arbitrary high order fully discrete one-step ADER discontinuous Galerkin schemes with subcell finite volume limiters applied to a new class of first order hyperbolic reformulations of nonlinear dispersive systems based on an extended Lagrangian approach introduced by Dhaouadi et al. (Stud Appl Math 207:1–20, 2018), Favrie and Gavrilyuk (Nonlinearity 30:2718–2736, 2017). We consider the hyperbolic reformulations of two different nonlinear dispersive systems, namely the Serre–Green–Naghdi model of dispersive water waves and the defocusing nonlinear Schrödinger equation. The first order hyperbolic reformulation of the Schrödinger equation is endowed with a curl involution constraint that needs to be properly accounted for in multiple space dimensions. We show that the original model proposed in Dhaouadi et al. (2018) is only weakly hyperbolic in the multi-dimensional case and that strong hyperbolicity can be restored at the aid of a novel thermodynamically compatible GLM curl cleaning approach that accounts for the curl involution constraint in the PDE system. We show one and two-dimensional numerical results applied to both systems and compare them with available exact, numerical and experimental reference solutions whenever possible.

Bound Coherent Structures Propagating on the Free Surface of Deep Water

This article presents a study of bound periodically oscillating coherent structures arising on the free surface of deep water. Such structures resemble the well known bi-soliton solution of the nonlinear Schrödinger equation. The research was carried out in the super-compact Dyachenko-Zakharov equation model for unidirectional deep water waves and the full system of nonlinear equations for potential flows of an ideal incompressible fluid written in conformal variables. The special numerical algorithm that includes a damping procedure of radiation and velocity adjusting was used for obtaining such bound structures. The results showed that in both nonlinear models for deep water waves after the damping is turned off, a periodically oscillating bound structure remains on the fluid surface and propagates stably over hundreds of thousands of characteristic wave periods without losing energy.

Solitary wave solitons to one model in the shallow water waves

The current study utilizes the generalized $$\tan (K(\rho )/2)$$ -expansion method, the generalized $$\tanh $$ - $$\coth $$ method and He’s semi-inverse variational method in constructing various soliton and other solutions to the (2+1)-dimensional coupled variant Boussinesq equations which describes the elevation of water wave surface for slowly modulated shallow water waves in lakes and ocean beaches. The system represents collision of a nonlinear wave propagating along y-axis and long wave along x-axis. The integration mechanism that was adopted is modified direct algebraic method, which extracts different solitons (dark and singular) and combo (dark-singular) solitons for different values of parameters. The existence criteria for solitons production is also established for both Kerr and power nonlinearities. Additionally, some other solutions known as singular periodic and rational are also emerged in the process of derivation

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.

Well-Posedness for a Coupled System of Kawahara/KdV Type Equations

We consider the initial value problem (IVP) associated to a coupled system of Kawahara/KdV type equations. We prove the well-posedness results for given data in a Gevrey spaces. 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 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. The techniques presented in this work were based in Grujic 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, whose the initial data belongs to Gevrey spaces.

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.

Variational theory for a kind of non-linear model for water waves

The Whitham-Broer-Kaup equation exists widely in shallow water waves, but unsmooth boundary seriously affects the properties of solitary waves and has certain deviations in scientific research. The aim of this paper is to introduce its modification with fractal derivatives in a fractal space and to establish a fractal variational formulation by the semi-inverse method. The obtained fractal variational principle shows conservation laws in an energy form in the fractal space and also hints its possible solution structure.