Benney-Luke Equation Research Articles

Lump, Breather, Ma-Breather, Kuznetsov–Ma-Breather, Periodic Cross-Kink and Multi-Waves Soliton Solutions for Benney–Luke Equation

The goal of this research is to utilize some ansatz forms of solutions to obtain novel forms of soliton solutions for the Benney–Luke equation. It is a mathematically valid approximation that describes the propagation of two-way water waves in the presence of surface tension. By using ansatz forms of solutions, with an appropriate set of parameters, the lump soliton, periodic cross-kink waves, multi-waves, breather waves, Ma-breather, Kuznetsov–Ma-breather, periodic waves and rogue waves solutions can be obtained. Breather waves are confined, periodic, nonlinear wave solutions that preserve their amplitude and shape despite alternating between compression and expansion. For some integrable nonlinear partial differential equations, a lump soliton is a confined, stable solitary wave solution. Rogue waves are unusually powerful and sharp ocean surface waves that deviate significantly from the surrounding wave pattern. They pose a threat to maritime safety. They typically show up in solitary, seemingly random circumstances. Periodic cross-kink waves are a particular type of wave pattern that has frequent bends or oscillations that cross at right angles. These waves provide insights into complicated wave dynamics and arise spontaneously in a variety of settings. In order to predict the wave dynamics, certain 2D, 3D and contour profiles are also analyzed. Since these recently discovered solutions contain certain arbitrary constants, they can be used to describe the variation in the qualitative characteristics of wave phenomena.

The Cauchy Problem for the Sixth Order $p$-Generalized Benney-Luke Equation

The Cauchy Problem for the Sixth Order $p$-Generalized Benney-Luke Equation

New Solutions of Benney-Luke Equation Using The (G’/G,1/G) Method

The Benney-Luke equation has contributed to studying the propagation of the water wave surfaces. This paper illustrates the (G’/G,1/G)-method to obtain the solutions of the Benney-Luke equation and an extension of the Benney-Luke equation. The new types of solutions are also constructed to gather the performance and visualization in three dimensions for observing the behaviors. The solutions are found in the expressions of hyperbolic functions giving the general performance by selecting arbitrary constants.

The bilinear neural network method for solving Benney–Luke equation

Benney–Luke equation, the estimation of water wave propagation on the water’s surface, is significantly important in studying the tension of water waves in physics. This paper focuses on the Bilinear Neural Network method (BNNM) to find the solutions to the Benney–Luke equation. Using the Hirota bilinear operator, the Benney–Luke equation is transformed into a bilinear expression and establishes models of neurons. The method construction is straightforward, well-organized, and effective for finding various exact analytic solutions of the Benny–Luke equation consisting of one-soliton solutions, two-soliton solutions, one-soliton hyperbolic wave solutions, two-soliton hyperbolic wave solutions, mixed-soliton hyperbolic solutions, and mixed-multi hyperbolic solutions by choosing the appropriate test functions. The solutions are rigorously depicted in three dimensions to observe the mutable behavior.

New analytical technique for prototype closed form solutions of certain nonlinear partial differential equations

This work proposes a novel modified (G’/G2)-expansion method to construct the exact travelling wave solutions of three famous nonlinear partial differential equations namely, nanoionic currents along microtubules equation, Benney-Luke equation and the generalized Hirota-Satsuma coupled KdV system. In the result, these solutions can be divided into different types i.e. trigonometric, rational and exponential solutions. The computational tool Maple is used to arrive at these solutions. Applying the novel modified (G’/G2)-expansion approach yields results that are trustworthy, easy to understand, and useful in a variety of disciplines including physics, biology, finance, engineering, statistics, and more. Software is used to create the two- and three-dimensional graphs. The comparison of the obtained solutions is also given in the form of Tables 1, 2 and 3, to show the novelty of our obtained solutions.

DF-ParPINN: parallel PINN based on velocity potential field division and single time slice focus

Modified Benney-Luke equation (mBL equation) is a three-dimensional temporal-spatial equation with complex structures, that is a high-dimensional partial differential equation (PDE), it is also a new equation of the physical ocean field, and its solution is important for studying the internal wave-wave interaction of inclined seafloor. For conventional PDE solvers such as the pseudo-spectral method, it is difficult to solve mBL equation with both accuracy and speed. Physics-informed neural network (PINN) incorporates physical prior knowledge in deep neural networks, which can solve PDE with relative accuracy and speed. However, PINN is only suitable for solving low-dimensional PDE with simple structures, and not suitable for solving high-dimensional PDE with complex structures. This is mainly because high-dimensional PDEs usually have complex structures and high-order derivatives and are likely to be high-dimensional non-convex functions, and the high-dimensional non-convex optimization problem is an NP-hard problem, resulting in the PINN easily falling into inaccurate local optimal solutions when solving high-dimensional PDEs. Therefore, we improve the PINN for the characteristics of mBL equation and propose “DF-ParPINN: parallel PINN based on velocity potential field division and single time slice focus” to solve mBL equation with large amounts of data. DF-ParPINN consists of three modules: temporal-spatial division module of overall velocity potential field, data rational selection module of multiple time slices, and parallel computation module of high-velocity fields and low-velocity fields. The experimental results show that the solution time of DF-ParPINN is no more than 0.5s, and its accuracy is much higher than that of PINN, PIRNN, cPINN, and DeepONet. Moreover, the relative error of DF-ParPINN after deep training 1000000 epochs can be reduced to less than 0.1. The validity of DF-ParPINN proves that the improved PINN also can solve high dimensional PDE with complex structures and large amounts of data quickly and accurately, which is of great significance to the deep learning of the physical ocean field.

A Study of Extreme Water Waves Using a Hierarchy of Models Based on Potential-Flow Theory

The formation of extreme waves arising from the interaction of three line-solitons with equal far-field amplitudes is examined through a hierarchy of water-wave models. The Kadomtsev–Petviashvili equation (KPE) is first used to prove analytically that its exact three-soliton solution has a ninefold maximum amplification that is achieved in the absence of spatial divergence. Reproducing this ninefold maximum paves the way for simulations based on both the Benney–Luke equations (BLE) and more advanced potential-flow equations (PFE). To preserve (for the sake of computations) the region of interaction, exact KPE solutions on an infinite domain are used to yield initial conditions that seed the BLE and PFE models within a periodic domain. The above strategies are realised by directly implementing the corresponding time-discretised variational principles within the finite-element environment Firedrake, one aim being automation of the generation of the algebraically cumbersome weak formulations. In the case of three-soliton interactions, it is found numerically that an amplification factor in the interval circa (7.6, 9) can be achieved within the BLE framework, whereas in the PFE framework, this falls to circa 7.8.

Symmetry analysis, closed-form invariant solutions and dynamical wave structures of the Benney-Luke equation using optimal system of Lie subalgebras

Nonlinear evolution equations (NLE-Es) arise in a variety of domains, including fluid dynamics, biological sciences, the study of solid states, optical fibres, coastal engineering, ocean physics, and nonlinear complex physical systems. This work examines the Lie symmetry study of the Benney-Luke (B-L) equation relying on two nonzero real parameters. The Lie infinitesimal generators, the one-dimensional optimal system, and the geometric vector fields are all obtained using the Lie symmetry technique. To start, we find close-form invariant solutions by employing symmetry reduction of Lie subalgebras. In some reduction cases, we transform the B-L equation into a variety of non-linear ordinary differential equations (NL-ODEs), which have the benefit of providing a significant number of closed-form solitary wave solutions. The traveling wave solutions contain special functional parameter solutions, trigonometric function solutions, rational function solutions, and hyperbolic trigonometric function solutions. The dynamical profile of closed-form wave solutions exhibits periodic solitons, dark peakon solitons, bright solitons, and bright solitons (bell shape) which we reveal in our study for the first time. By using Noether’s method, we also determine the conservation laws. Finally, three-dimensional diagrams are used to reveal the dynamical investigation of some known solutions.

On Features of the Solution of a Boundary-Value Problem for the Multidimensional Integro-Differential Benney–Luke Equation with Spectral Parameters

In this paper, we consider the problems on the solvability and constructing solutions of one nonlocal boundary-value problem for the multidimensional, fourth-order, integro-differential Benney–Luke equation with degenerate kernel and spectral parameters. For various values of spectral parameters, necessary and sufficient conditions of the existence of a solution are obtained. The Fourier series for solutions of the problem corresponding to various sets of spectral parameters are obtained. For regular values of spectral parameters, the absolute and uniform convergence of the series and the possibility of their termwise differentiation with respect to all variables are proved. The problem is also examined studied for cases of irregular values of spectral parameters.

On the exact controllability for the Benney-Luke equation in a bounded domain

We study the exact boundary controllability of the nonlinear - Benney-Luke equation at the right endpoint of a bounded domain, applying the control first to the first spatial derivative and to the second spatial derivative. We use the Hilbert Uniqueness Method to get the exact controllability of the linearized problem. From this fact and the Banach contraction fixed point principle, we establish the exact boundary controllability for the nonlinear Benney-Luke equation in bounded domains, for sufficiently small initial and final states.

Numerical Experiments on Extreme Waves Through Oblique–Soliton Interactions

Extreme water-wave motion is investigated analytically and numerically by considering two-soliton and three-soliton interactions on a horizontal plane. We successfully determine numerically that soliton solutions of the unidirectional Kadomtsev–Petviashvili equation (KPE), with equal far-field individual amplitudes, survive reasonably well in the bidirectional and higher-order Benney–Luke equations (BLE). A well-known exact two-soliton solution of the KPE on the infinite horizontal plane is used to seed the BLE at an initial time, and we confirm that the KPE-fourfold amplification approximately persists. More interestingly, a known three-soliton solution of the KPE is analysed further to assess its eight- or ninefold amplification, the latter of which exists in only a special and difficult-to-attain limit. This solution leads to an extreme splash at one point in space and time. Subsequently, we seed the BLE with this three-soliton solution at a suitable initial time to establish the maximum amplification: it is approximately 7.8 for a KPE amplification of 8.4. Herein, the computational domain and solutions are truncated approximately to a fully periodic or half-periodic channel geometry of sufficient size, essentially leading to cnoidal-wave solutions. Moreover, special geometric (finite-element) variational integrators in space and time have been used in order to eradicate artificial numerical damping of, in particular, wave amplitude.

On diffraction and oblique interactions of horizontally two-dimensional internal solitary waves

In the context of three-dimensional oceanic internal waves, taking topographic effects into account, a modified Benney–Luke equation is proposed for describing internal wave–wave interactions on a sloping bottom. The derived equation is characterised by isotropy and bi-directional propagation, which are absent in the widely used Kadomtsev–Petviashvili equation. Indeed, these disparities are confirmed by numerical results of the diffraction of a truncated internal solitary wave and the evolution of a partially bent solitary wave. However, a good agreement between the numerical results of the modified Benney–Luke equation and those of the primitive equations confirms the validity of our simplified model. Because the stratification in a realistic ocean environment is usually continuous, in contrast to the assumption of a sharp density discontinuity used here, to maintain the kinematical equivalence, a layering scheme for determining the density and thickness of each layer from a continuous stratification is proposed. In addition, the occasionally observed but rarely examined X-shaped internal wave–wave interactions are shown to feature novel wave patterns, where topographic effects modulate the propagation speed, amplitude and waveform.

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.

Study on the explicit solutions of the Benney–Luke equation via the variational direct method

In this current work, we main to find the abundant explicit solutions of the Benney–Luke equation by a novel approach, which is called the variational direct method (VDM). The VDM based on the variational theory and Ritz‐like method can reduce the order of the differential equation to make the equation simpler and obtain the optimal solution by the stationary conditions. The explicit solutions in terms of dark solitary, dark‐like solitary, kinky‐dark solitary, and periodic wave solutions are presented through the numerical results in the form of 3‐D plot, 2‐D curve, and 2‐D contour. The results show that the VDM is simple, straightforward, and powerful, which is expected to open new perspectives for the traveling wave theory.

Exact solutions of the Benney–Luke equation via (1/G')-expansion method

In this study, the (1/G') -expansion method was implemented to solve the Benney–Luke (BL) equation. Exact solutions of the BL equation were obtained via this method. The solutions obtained from the BL equation were in hyperbolic form. 3D, 2D and contour graphs of obtained solutions are presented. Results show that the (1/G') -expansion method provides an efficient and straightforward mathematical instrument for finding solutions of nonlinear evolution equations (NLEEs).

Lagrangian Formulation, Conservation Laws, Travelling Wave Solutions: A Generalized Benney-Luke Equation

The aim of this paper is to find the Noether symmetries of a generalized Benney-Luke equation. Thereafter, we construct the associated conserved vectors. In addition, we search for exact solutions for the generalized Benney-Luke equation through the extended tanh method. A brief observation on equations arising from a Lagrangian density function with high order derivatives of the field variables, is also discussed.

On the orbital stability of traveling waves that behave such as particles

Properties that solitary waves share with particles have contributed significantly to the development of new theories and technological advances in different areas of knowledge. In this sense, the study of orbital stability of solitary waves is key in solitary wave dynamics. Although the definition of orbital stability is relatively simple, the mathematical analysis required to verify it is quite complex. However, the theory of Grillakis, Shatah and Strauss provides us with a very useful criterion to verify orbital stability. In this work, we present their theory and apply it to analyse the orbital stability of Generalized Korteweg-de Vries equation, Compressible fluid equation, and one-dimensional Benney-Luke equation. For the first two equations, the criterion guaranteed the orbital stability of the solitary waves. For the third one, it was guaranteed only for certain ranges of its parameters.

On the new type of solutions to Benney-Luke equation

In this study, the Benney-Luke equation is considered. In order to derive new type of solutions, the sn-ns method is applied to this equation. Then, we introduce trigonometric and elliptic functions solutions in addition to the hyperbolic ones which are gained by tanh-coth. Three types of solutions are derived at the same time with the help of this method. Therefore, it can be said that this method is convenient to obtain more solutions to many kinds of nonlinear partial differential equations.

Cnoidal wave solutions for the optical Benney-Luke equation

At proper scale of the asymptotic analysis, the optical Benney-Luke equation (OBLE) explains the propagation of a continuous wave (cw) laser beam in a nonlocal non-linear defocusing media. By means of an OBLE and appropriate boundary conditions, we have obtained non-linear periodic wave solutions. The corresponding analytical expressions and numerical plots of the cnoidal waves in the phase plane are presented using the Sagdeev pseudo-potential approach. Our results are written in terms of the media nonlocality through the optical surface tension parameter.

The derivation of an isotropic model for internal waves and its application to wave generation

In contrast to two-dimensional oceanic internal waves which have been extensively investigated, there are relatively few theoretical studies on three-dimensional internal waves. The most remarkable theory describing three-dimensional internal waves is the Kadomtsev–Petviashvili (KP) equation. Nevertheless, two shortcomings – unidirectional propagation and anisotropy – limit its application in some general cases. In the current paper, via an asymptotic analysis, we derive an isotropic and bidirectional model, the modified Benney–Luke equation, for a two-layer fluid with bottom topography which can vary both in time and space, therefore tides and currents can be incorporated into the internal wave problem due to the relative motion. In the derivation, the assumption of incompressibility and long waves are invoked and the effects of the Earth’s rotation and dissipation are ignored. Based on this model, the generation of nonlinear internal waves by background currents and barotropic tides flowing over topography are investigated. The resulting equation aims to facilitate dynamical analyses, as well as the interpretation of in-situ observational data and laboratory experimental results.