Water Wave Equation Research Articles

Sensitivity analysis and solitary wave solutions to the (2+1)-dimensional Boussinesq equation in dispersive media

This paper explores the dynamic behavior of the [Formula: see text]-dimensional Boussinesq equation, which is a nonlinear water wave equation used to model wave packets in dispersive media with weak nonlinearity. Specifically, we investigate the equation’s traveling wave solutions using the Riccati equation mapping method. Our results include solitary and soliton solutions, each with their own set of parameter values. To provide a comprehensive understanding of these solutions, we present them in general form and visualize their significance using various graphs, such as 3D, 2D, and contour plots. The computational effort and resulting outcomes highlight the efficacy of our approach, which has the potential to be applied to other nonlinear physical problems in fields such as mathematical physics, engineering, and nonlinear science.

Residual symmetry, CRE integrability and interaction solutions of two higher-dimensional shallow water wave equations

Two (3+1)-dimensional shallow water wave equations are studied by using residual symmetry and the consistent Riccati expansion (CRE) method. Through localization of residual symmetries, symmetry reduction solutions of the two equations are obtained. The CRE method is applied to the two equations to obtain new Bäcklund transformations from which a type of interesting interaction solution between solitons and periodic waves is generated.

Propagation characteristics of tsunami waves in shear flow based on the Boussinesq model with constant vorticity

Propagation of tsunami wave in linear shear flow with constant vorticity is studied in this paper. The nonlinear and dispersive Boussinesq equations for water waves in shear flow are derived by introducing uniform vorticity into the Euler equations. Generally, the transoceanic propagation of tsunami can be well simulated by the weakly nonlinear Boussinesq model. Thus, the nonlinear and dispersive terms are retained to the first-order and second-order (ϵ,μ2) respectively in the Boussinesq equations with uniform vorticity in the present work, which satisfies the simulation accuracy especially the dispersion effects of the long-distance propagation of tsunamis and does not require huge computational cost for the simulation. The set of equations is solved with the total variation diminishing (TVD) Lax–Wendroff scheme which is second-order both in space and time. This TVD scheme is applicable to handle the dispersion effects of the N-waves and undular bores. Different patterns of tsunami wave in the water with presence of vorticity are simulated by the numerical model. Wave height and wavelength are affected by the current with vorticity. The pattern of solitary wave is significantly changed in upstream and downstream conditions. The wave deformation and propagation speed of N-waves and undular bores in the shear flow are also studied. The vorticity has impacts on the leading crest amplitude and the maximum wave appearance time for N-wave and changes the wave amplitude and the following water level for undular bores.

Dynamics of breather, multi-wave, interaction and other wave solutions to the new (3+1)-dimensional integrable fourth-order equation for shallow water waves

PurposeThe purpose of this paper is to study the new (3 + 1)-dimensional integrable fourth-order nonlinear equation which is used to model the shallow water waves.Design/methodology/approachBy means of the Cole–Hopf transform, the bilinear form of the studied equation is extracted. Then the ansatz function method combined with the symbolic computation is implemented to construct the breather, multiwave and the interaction wave solutions. In addition, the subequation method tis also used to search for the diverse travelling wave solutions.FindingsThe breather, multiwave and the interaction wave solutions and other wave solutions like the singular periodic wave structure and dark wave structure are obtained. To the author’s knowledge, the solutions obtained are all new and have never been reported before.Originality/valueThe solutions obtained in this work have never appeared in other literature and can be regarded as an extension of the solutions for the new (3 + 1)-dimensional integrable fourth-order nonlinear equation.

Reflected wave energy by two collinear submerged wavy breakwaters

Under the assumption of a potential flow, we obtain an asymptotic solution up to the second order in a regular expansion, for the problem of the interaction between linear long waves and two submerged floating breakwaters of wavy surfaces, placed in a collinear manner. The hydrodynamics that results from the interaction between the waves and the structures is modeled with the aid of the well-known shallow water wave equations together with the Laplace equation. For this purpose, the domain perturbation method is used to obtain the solution of the governing equations and assume, as is common, small amplitudes of the breakwater's wavy surfaces. This solution is compared with classical analytical solutions reported in the specialized literature, and they adjust properly. Several geometrical configurations of the breakwaters are analyzed. As the breakwaters are near the free surface elevation, just at one-third of the total water depth, larger values of the reflection coefficients are obtained. The maximum wave reflection occurs for four undulations of the breakwater surfaces. In addition, the wave reflection increases as the amplitude of the surfaces of the breakwaters increases. The results of this study are expected to be used by coastal engineers for preliminary feasibility and desk design of submerged breakwaters with wavy surfaces.

Bilinear form, auto-Bäcklund transformations, Pfaffian, soliton, and breather solutions for a (3 + 1)-dimensional extended shallow water wave equation

Study of the water waves remains central to fluid physics, ocean dynamics, and engineering. In this paper, a (3 + 1)-dimensional extended shallow water wave equation is investigated via symbolic computation. Bilinear form and two kinds of the bilinear auto-Bäcklund transformations with some solutions are given via the Hirota method. The Nth-order Pfaffian solutions are worked out by means of the Pfaffian technique, where N is a positive integer. N-soliton solutions are exported through the Nth-order Pfaffian solutions. By virtue of the asymptotic analysis, elastic and inelastic interactions between the two solitons on some periodic backgrounds are discussed. Interaction among the three solitons is illustrated graphically. The higher-order breather solutions are investigated via the complex parameter relation. Elastic and inelastic interactions between the two breathers on the periodic backgrounds are depicted graphically. Hybrid solutions consisting of the solitons and breathers are obtained. Interaction between the one soliton and one breather on a periodic background is presented.

Gram-type, three-breather and hybrid solutions for a (3＋1)-dimensional generalized variable-coefficient shallow water wave equation

In this paper, a (3＋1)-dimensional generalized variable-coefficient shallow water wave equation is studied, which characterizes the flow below a pressure surface in a fluid. Through the Kadomtsev–Petviashvili hierarchy reduction, we construct some Gram-type solutions, three-breather solutions and three kinds of the hybrid solutions. Via the asymptotic analysis, we choose certain parameters to make the three connected breathers parallel to the x axis and reduce the first-order breather to the first-order homoclinic orbit on the x–z plane. Interaction among the three breathers is displayed and we observe that the one breather splits into the two breathers on the x–y plane, where x, y, z are the scaled spatial coordinates. For the three hybrid solutions, we reduce the V-shaped soliton to the one kink soliton on the y–z plane and show that the one breather converts into the one kink soliton on the y–z plane. Asymptotic analysis indicates that the interaction between the V-shaped soliton and kink soliton leads to the increase/decrease of the constant background of the V-shaped soliton as the constant background of the kink soliton decrease/increase. Furthermore, we present the two interactions among the V-shaped soliton and three breathers.

Numerical Solutions of the Nonlinear Dispersive Shallow Water Wave Equations Based on the Space–Time Coupled Generalized Finite Difference Scheme

This study applies the space–time generalized finite difference scheme to solve nonlinear dispersive shallow water waves described by the modified Camassa–Holm equation, the modified Degasperis–Procesi equation, the Fornberg–Whitham equation, and its modified form. The proposed meshless numerical scheme combines the space–time generalized finite difference method, the two-step Newton’s method, and the time-marching method. The space–time approach treats the temporal derivative as a spatial derivative. This enables the discretization of all partial derivatives using a spatial discretization method and efficiently handles mixed derivatives with the proposed mesh-less numerical scheme. The space–time generalized finite difference method is derived from Taylor series expansion and the moving least-squares method. The numerical discretization process only involves functional data and weighting coefficients on the central and neighboring nodes. This results in a sparse matrix system of nonlinear algebraic equations that can be efficiently solved using the two-step Newton’s method. Additionally, the time-marching method is employed to advance the space–time domain along the time axis. Several numerical examples are presented to validate the effectiveness of the proposed space–time generalized finite difference scheme.

The dynamics of some exact solutions of the (3+1)-dimensional generalized shallow water wave equation

The dynamics of some exact solutions of the (3+1)-dimensional generalized shallow water wave equation

New breather wave and interaction solutions of the generalized ($$2+1$$)-dimensional variable-coefficient shallow water wave equation

New breather wave and interaction solutions of the generalized ($$2+1$$)-dimensional variable-coefficient shallow water wave equation

Auto-Bäcklund transformations and analytic solutions of a generalized (4 + 1)-dimensional variable-coefficient Fokas equation for the shallow water waves

In this paper, we investigate a generalized (4 + 1)-dimensional variable-coefficient Fokas equation for the shallow water waves. Through the truncated Painlevé expansion, we give the auto-Bäcklund transformations. Based on the Hirota method, we get the two-soliton solutions. With different choices of certain variable coefficients, we observe some phenomena of the two solitons. We obtain the ring-type and periodic-type two solitons. In addition, we derive the resonant two solitons with the linear, hyperbolic, and periodic types.

Construction of Analytical Solutions to the Conformable New (3+1)-Dimensional Shallow Water Wave Equation

This study investigates the new (3+1)-dimensional shallow water wave equation. To do so, the definitions of conformable derivatives and their descriptions are given. Using the Riccati equation and modified Kudryashov methods, exact solutions to this problem are discovered. The gathered data's contour plot surfaces and related 3D and 2D surfaces emphasize the result's physical nature. To monitor the problem's physical activity, exact and complete solutions are necessary. The results demonstrate the potential applicability of additional nonlinear physical models from mathematical physics and under-investigation in real-world settings. In order to solve fractional differential equations, it may prove helpful to use these methods in various situations.

Generalized (G’ / G) – expansion method for the loaded shallow water wave equation

This article is devoted to finding solutions for the traveling wave of the loaded wave equation in shallow water. One of the approaches to finding solutions by the expansion method (G / G) is given, which is one of the most effective ways to obtain solutions. When parameters are taken as special values, solitary waves are also derived from traveling waves. Solutions for the traveling wave are expressed by hyperbolic and trigonometric functions. This method is easy to implement using well-known software packages that allow solving complex nonlinear evolutionary equations of mathematical physics.

ON THE THIRD ORDER SOLUTION OF KdV EQUATION BY USING HOMOTOPY PERTURBATION METHOD

In this research we discussed about the solution of the KdV equation using Homotopy Perturbation method. The KdV equation that describing water wave equation solved by using the mixing method between Homotopy and Perturbation method. Homotopy was built with embedding parameter p∈[0,1] which undergoes a deformation process from linear problems to nonlinear problems and the assumed solution of the KdV equation is expressed in the form of a power series p up to the third order. The result show that in each order solution we obtained resonance term. for handling the condition, we used Lindsteadt-Poincare method.the wave number k2 and dispersion relation can be obtained in the second order solution as the effect of using Lindsteadt-Poincare method.

Dynamics of breather waves, lump-kink solutions and interaction solutions for a (3+1)-dimensional generalized shallow water waves equation

PurposeThis paper aims to study the breather, lump-kink and interaction solutions of a (3 + 1)-dimensional generalized shallow water waves (GSWW) equation, which describes water waves propagating in the ocean or is used for simulating weather.Design/methodology/approachHirota bilinear form and the direct method are used to construct breather and lump-kink solutions of the GSWW equation. The “rational-cosh-cos-type” test function is applied to obtain three kinds of interaction solutions.FindingsThe fusion and fission of the interaction solutions between a lump wave and a 1-kink soliton of the GSWW equation are studied. The dynamics of three kinds of interaction solutions between lump, kink and periodic waves are discussed graphically.Originality/valueThis paper studies the breather, lump-kink and interaction solutions of the GSWW equation by using various approaches and provides some phenomena that have not been studied.

Dynamic behaviors of the lump solutions and mixed solutions to a (2+1)-dimensional nonlinear model

In this paper, we propose a combined form of the bilinear Kadomtsev–Petviashvili equation and the bilinear extended (2+1)-dimensional shallow water wave equation, which is linked with a novel (2+1)-dimensional nonlinear model. This model might be applied to describe the evolution of nonlinear waves in the ocean. Under the effect of a novel combination of nonlinearity and dispersion terms, two cases of lump solutions to the (2+1)-dimensional nonlinear model are derived by searching for the quadratic function solutions to the bilinear form. Moreover, the one-lump-multi-stripe solutions are constructed by the test function combining quadratic functions and multiple exponential functions. The one-lump-multi-soliton solutions are derived by the test function combining quadratic functions and multiple hyperbolic cosine functions. Dynamic behaviors of the lump solutions and mixed solutions are analyzed via numerical simulation. The result is of importance to provide efficient expressions to model nonlinear waves and explain some interaction mechanism of nonlinear waves in physics.

Breather and Interaction Solutions for a $$(3+1)$$-Dimensional Generalized Shallow Water Wave Equation

Breather and Interaction Solutions for a $$(3+1)$$-Dimensional Generalized Shallow Water Wave Equation

Dynamic study of bifurcation, chaotic behavior and multi-soliton profiles for the system of shallow water wave equations with their stability

The purpose of this study is to investigate the deeper characteristics of the system of shallow water wave equations that is used to model the turbulence in the atmosphere and oceans from different standpoints. The multi-soliton structures such as 1-soliton, 2-soliton, 3-soliton solutions are successfully generated with the help of simplified Hirota’s method. For the sake of physical demonstration and visual presentation, we graphically illustrate the identified solutions in 3D, 2D, and contour plots using Mathematica software. The Lie symmetry technique is used to identify the Lie group invariant transformations and symmetry reductions of the examined system, which aid in reducing the dimension by one or into an ordinary differential equation. Further, the qualitative behavior of the time-dependent dynamical system is observed using the bifurcation and chaos theory. The phase portraits of bifurcation are observed at the equilibrium point of a planar dynamical system. We discuss various tools to identify chaos (random and unpredictable behavior) in autonomous dynamic systems, such as 3D phase portraits, 2D phase portraits, time series, and Poincaré maps. At various initial conditions, the sensitivity and modulation instability analyses are also presented, and it is discovered that the investigated system is stable, as a small change in the initial conditions does not cause an abrupt change in solutions. The findings of this investigation will contribute in the overall depiction of soliton theory and nonlinear dynamical systems.

An enormous diversity of soliton solutions to the (2+1)-dimensional extended shallow water wave equation using three analytical methods

In this paper, we obtain a variety of analytical wave solutions of a ([Formula: see text])-dimensional extended shallow water wave equation. The applications of the governing equation are enormous in ocean modeling and investigation of moist-convection properties in atmospheric dynamics. Two powerful mathematical approaches, the [Formula: see text]-expansion method (EEM) and the generalized projective Riccati equations method (GPREM), were used to find closed-form traveling wave solutions. In addition, the breather wave solutions were described using the Cole-Hopf transform, derived with the help of Hirota bilinear method (HBM). Eventually, we obtain 46 closed-form solutions in explicit form. The general form of obtained solutions include trigonometric, rational, exponential and hyperbolic function solutions. To illustrate the physical significance of some novel results, we provided contour, two- and three-dimensional graphics under the suitable free choices of unknown parameters. The dynamical shapes of these solutions are one soliton, multiple solitons, kink, anti-kink, lump and periodic solutions. We believe that the applied methods of this work are well-organized, genuine and powerful mathematical tools for solving the nonlinear evolution equations (NLEEs) occurring in the study of ocean science and engineering.

Fractional solitons: New phenomena and exact solutions

The fractional solitons have demonstrated many new phenomena, which cannot be explained by the traditional solitary wave theory. This paper studies some famous fractional wave equations including the fractional KdV–Burgers equation and the fractional approximate long water wave equation by a modified tanh-function method. The solving process is given in details, and new solitons can be rigorously explained by the obtained exact solutions. This paper offers a new window for studying fractional solitons.