Benjamin–Bona–Mahony Research Articles

On integrability of a new dynamical system associated with the BBM‐type hydrodynamic flow

This article explores the exceptional integrability property of a family of higher‐order Benjamin–Bona–Mahony (BBM)‐type nonlinear dispersive equations. Here, we highlight its deep relationship with a generalized infinite hierarchy of the integrable Riemann‐type hydrodynamic equations. A previous Lie symmetry analysis revealed a particular case which was conjectured to be integrable. Namely, a Lie–Bäcklund symmetry exists, thus highlighting another associated dynamical system. Here, we investigate these two equations using the gradient‐holonomic integrability scheme. Moreover, we construct their infinite hierarchy of conservation laws analytically, using three compatible Poisson structures to prove the complete integrability of both dynamical systems. We investigate these two equations using the so‐called gradient‐holonomic integrability scheme. Based on this scheme, applied to the equation under consideration, we have analytically constructed its infinite hierarchy of conservation laws, derived two compatible Poisson structures, and proved its complete integrability.

Computational approach and convergence analysis for interval-based solution of the Benjamin–Bona–Mahony equation with imprecise parameters

PurposeTo provide a new semi-analytical solution for the nonlinear Benjamin–Bona–Mahony (BBM) equation in the form of a convergent series. The results obtained through HPTM for BBM are compared with those obtained using the Sine-Gordon Expansion Method (SGEM) and the exact solution. We consider the initial condition as uncertain, represented in terms of an interval then investigate the solution of the interval Benjamin–Bona–Mahony (iBBM).Design/methodology/approachWe employ the Homotopy Perturbation Transform Method (HPTM) to derive the series solution for the BBM equation. Furthermore, the iBBM equation is solved using HPTM to the initial condition has been considered as an interval number as the coefficient of it depends on several parameters and provides lower and upper interval solutions for iBBM.FindingsThe obtained numerical results provide accurate solutions, as demonstrated in the figures. The numerical results are evaluated to the precise solutions and found to be in good agreement. Further, the initial condition has been considered as an interval number as the coefficient of it depends on several parameters. To enhance the clarity, we depict our solutions using 3D graphics and interval solution plots generated using MATLAB.Originality/valueA new semi-analytical convergent series-type solution has been found for nonlinear BBM and interval BBM equations with the help of the semi-analytical technique HPTM.

Long‐time asymptotics and the radiation condition with time‐periodic boundary conditions for linear evolution equations on the half‐line and experiment

AbstractThe asymptotic Dirichlet‐to‐Neumann (D‐N) map is constructed for a class of scalar, constant coefficient, linear, third‐order, dispersive equations with asymptotically time/periodic Dirichlet boundary data and zero initial data on the half‐line, modeling a wavemaker acting upon an initially quiescent medium. The large time asymptotics for the special cases of the linear Korteweg‐de Vries and linear Benjamin–Bona–Mahony (BBM) equations are obtained. The D‐N map is proven to be unique if and only if the radiation condition that selects the unique wave number branch of the dispersion relation for a sinusoidal, time‐dependent boundary condition holds: (i) for frequencies in a finite interval, the wave number is real and corresponds to positive group velocity, and (ii) for frequencies outside the interval, the wave number is complex with positive imaginary part. For fixed spatial location , the corresponding asymptotic solution is (i) a traveling wave or (ii) a spatially decaying, time‐periodic wave. The linearized BBM asymptotics are found to quantitatively agree with viscous core‐annular fluid experiments.

Traveling waves in a quintic BBM equation under both distributed delay and weak backward diffusion

In this paper, we focus on the existence of periodic and solitary waves for a quintic Benjamin–Bona–Mahony (BBM) equation with distributed delay and diffused perturbation. The corresponding traveling wave equation is transformed into a three-dimensional dynamical system, which is regarded as a singularly perturbed system for small perturbation parameter. By geometric singular perturbation theory, the three-dimensional dynamical system can be reduced to a near-Hamiltonian planar system and a locally invariant manifold diffeomorphic to the critical manifold with normally hyperbolicity can be constructed. Further, the existence of periodic wave and solitary wave are established by proving the persistence of periodic and homoclinic orbits of the near-Hamiltonian planar system for sufficiently small perturbation, and their existence conditions for the delayed quintic BBM equation have also been obtained. The uniqueness of periodic wave is established by analyzing the monotonicity of ratios of Abelian integrals form a Chebyshev set. Moreover, the monotonicity of wave speed is proved, and the supremum and infimum of wave speed are gotten. Numerical simulations are in complete agreement with the theoretical predictions.

Unconditional superconvergence analysis of an energy conservation scheme with Galerkin FEM for nonlinear Benjamin–Bona–Mahony equation

In this paper, a general energy conservation Crank–Nicolson (CN) fully-discrete finite element method (FEM) scheme is developed for solving the nonlinear Benjamin–Bona–Mahony (BBM) equation, and the unconditional supercloseness and superconvergence behavior are discussed with the nonconforming EQ1rot element. Different from the time–space splitting technique, the boundedness of numerical solution in the broken H1-norm is obtained directly based on the above energy conservation property, and the well-posedness of numerical solution is proved rigorously through the Brouwer fixed point theorem. Then, with the help of the special characters of this element and the interpolation post-processing technique, the unconditional superclose and superconvergence results are deduced strictly without any restriction between mesh size h and time step τ. Further, our analysis is extended to some other popular conforming and nonconforming finite elements. Finally, two numerical experiments are implemented to confirm the theoretical analysis. It is worth mentioning that this paper not only improve the previous results, but also can be regard as a framework for solving the BBM equation and some other PDEs with Galerkin FEMs.

Transport of Gaussian measures with exponential cut-off for Hamiltonian PDEs

We show that introducing an exponential cut-off on a suitable Sobolev norm facilitates the proof of quasi-invariance of Gaussian measures with respect to Hamiltonian PDE flows and allows us to establish the exact Jacobi formula for the density. We exploit this idea in two different contexts, namely the periodic fractional Benjamin–Bona–Mahony (BBM) equation with dispersion β > 1 and the periodic one-dimensional quintic defocussing nonlinear Schrödinger equation (NLS). For the BBM equation we study the transport of the cut-off Gaussian measures on fractional Sobolev spaces, while for the NLS equation we study the measures based on the modified energies introduced by Planchon–Visciglia and the third author. Moreover, for the BBM equation we also show almost sure global well-posedness for data in {C^alpha }(mathbb{T}) for arbitrarily small α > 0 and invariance of the Gaussian measure associated with the {H^{beta /2}}(mathbb{T}) norm.

OPTICAL SOLITARY WAVES AND SOLITON SOLUTIONS OF THE (3 + 1)-DIMENSIONAL GENERALIZED KADOMTSEV–PETVIASHVILI–BENJAMIN–BONA–MAHONY EQUATION

In this investigation, an appropriate traveling wave transformation has been employed to analyze the fourth-order nonlinear (3+1)-dimensional generalized Kadomtsev–Petviashvili–Benjamin–Bona–Mahony (KP-BBM) equation for an offshore structure. In addition to being integrable and having constant coefficients, the examined model represents fluid flow in the context of an offshore structure. The aforesaid nonlinear system is subjected to the initial application of two trustworthy and dependable approaches, namely the improved Bernoulli sub-equation function method and the modified extended-function method. Investigating and obtaining certain explicitly exact traveling waves, periodic waves, and soliton solutions is the major objective. The generated solutions take the form of trigonometric hyperbolic functions, exponential functions, rational functions, and multiple forms of trigonometric functions. The proposed solutions are both novel and important in that they provide light on the relevant aspects of the physical phenomena. The characteristics of the solutions have been displayed in a variety of figures, including two- and three-dimensional ones, for the best visual optical discernment.

Periodic Behaviors of a Linear Fourth-Order Difference Solution to the Benjamin–Bona–Mahony-Type Equation with Time-Periodic Boundaries

The periodic behaviors of a linear fourth-order difference solution to the Benjamin–Bona–Mahony (BBM)-type equation with time-periodic boundaries are analyzed in this paper. Firstly, we employ a variable transformation to change the original BBM-type equation with time-periodic boundaries into a new BBM-type equation with zero boundaries. We then construct a fourth-order linear finite difference method to discrete the new BBM-type equation. The solvability, convergence, stability and accuracy of the approximating solution are discussed. The computation procedure of the present method is given in detail. Numerical results show that the proposed difference method is reliable and efficient for time-periodic simulation.

Modulation instability analysis and soliton solutions of the modified BBM model arising in dispersive medium

In this article, a nonlinear dispersive structure namely the modified Benjamin–Bona–Mahony (BBM) model is studied by employing three different approaches to develop some new analytical and semi-analytical solutions. The modified extended tan hyperbolic function method, the improved F-expansion method and the Adomian decomposition method are implemented to obtain various interesting wave structures. The comparison among various types of solutions is analyzed to study the corresponding absolute error. Furthermore, the stability of the results along with the modulation instability are successfully deliberated. The 3D, contour and 2D graphs are visualized for several fascinating exact solutions to comprehend their behavior.

Convergence analysis and an efficient numerical technique for the solution of Benjamin Bona Mahony partial differential equation

This study proposed a collocation based modified scheme of trigonometric cubic b-spline approach (MCTB-spline) to approximate the solution of the Benjamin Bona Mahony (BBM) partial differential equation. The MCTB-spline collocation method is used to discretise the space derivatives terms of the partial differential equation, while the finite difference method (FDM) is used to make discretised the time-variant derivatives of the BBM partial differential equation. This study also describes the convergency of the present method, which helps to illustrate the accuracy of the present scheme for the partial differential equation. A Rubin Graves linearisation process linearises BBM partially differential equations into nonlinear terms. An example has been selected for demonstration of numerical approximations and provided a comparative study with the previous scheme of different types of error norms. To validate the stability, Von-Neumann stability condition was applied on the proposed scheme. Results suggested that the present scheme achieved better results and outperformed the previous techniques used. We will apply the generalisation of the present scheme to higher order and coupled partial differential equations in the future.

Convergence Analysis and an Efficient Numerical Technique for the Solution of Benjamin Bona Mahony (BBM) Partial Differential Equation

Convergence Analysis and an Efficient Numerical Technique for the Solution of Benjamin Bona Mahony (BBM) Partial Differential Equation

Unique Continuation and Time Decay for a Higher-Order Water Wave Model

This work is devoted to prove the exponential decay for the energy of solutions of a higher order Korteweg–de Vries (KdV)–Benjamin–Bona–Mahony (BBM) equation on a periodic domain with a localized damping mechanism. Following the method in [L. Rosier and B.-Y. Zhang, J. Diff. Equ. 254 (2013) 141–178], which combines energy estimates, multipliers and compactness arguments, the problem is reduced to prove the Unique Continuation Property (UCP) for weak solutions of the model. Then, this is done by deriving Carleman estimates for a system of coupled elliptic-hyperbolic equations.

A NEW FRACTAL MODIFIED BENJAMIN–BONA–MAHONY EQUATION: ITS GENERALIZED VARIATIONAL PRINCIPLE AND ABUNDANT EXACT SOLUTIONS

In this paper, we derive a new fractal modified Benjamin–Bona–Mahony equation (MBBME) that can model the long wave in the fractal dispersive media of the optical illusion field based on He’s fractal derivative. First, we apply the semi-inverse method (SIM) to develop its fractal generalized variational principle with the aid of the fractal two-scale transforms. The obtained fractal generalized variational principle reveals the conservation laws via the energy form in the fractal space. Second, Wang’s Bäcklund transformation-based method, which combines the Bäcklund transformation and the symbolic computation with the ansatz function schemes, is used to study the abundant exact solutions. Some new solutions in the form of the rational function-type, double-exp function-type, Sin-Cos function-type and the Sinh-Cosh function-type are successfully constructed. The impact of the fractal orders on the behaviors of the different solutions is elaborated in detail via the 3D plots, 2D contours and 2D curves, where we can find that: (1) When the fractal order [Formula: see text], the direction of wave propagation tends to be more vertical to the [Formula: see text]-axis, on the other hand, it tends to be more parallel to the [Formula: see text]-axis when [Formula: see text]; (2) The fractal order cannot impact the peak amplitude of the waveform; (3) For the periodic waveform, the fractal orders can affect its period, that is, the period becomes smaller when the fractal order [Formula: see text]. The obtained results show that the proposed methods are effective and powerful, and can construct the abundant exact solutions, which are expected to give some new enlightenment to study the variational theory and traveling wave solutions of the fractal partial differential equations.

Numerical and analytical results of the 1D BBM equation and 2D coupled BBM-system by finite element method

In this paper, the numerical solutions are proposed for the 1D Benjamin–Bona–Mahony (BBM) equation and 2D coupled BBM system by using Galerkin finite element technique. In this regard, the cubic B-splines and linear triangular elements are used, respectively. In 1D space, a proposed numerical scheme is implemented to a test problem including the motion of a single solitary solution. To verify practicality and robustness of our new procedure, the error norms [Formula: see text], [Formula: see text] and three constants [Formula: see text] and [Formula: see text] are evaluated. Stability analysis of the linearized technique indicates that it is unconditionally stable. Moreover, a tsunami wave in 2D space is used to investigate the efficiency of the considered method. Also, the improved [Formula: see text]-[Formula: see text] function technique (IThChT) and the combined [Formula: see text]-[Formula: see text] function technique (ITCT) are obtained in the mentioned BBM equation. The presented methods are seen to be robust, impressive and economical to employment as compared to the existing finite difference techniques and other earlier papers for discovering the numerical solutions for numerous types of linear and nonlinear PDEs.

DYNAMICAL BEHAVIOR OF THE LONG WAVES IN THE NONLINEAR DISPERSIVE MEDIA THROUGH ANALYTICAL AND NUMERICAL INVESTIGATION

This paper studies the well-known mathematical model’s analytical wave solutions (modified Benjamin–Bona–Mahony (BBM) equation), which demonstrates the propagation of long waves in the nonlinear dispersive media in a visual illusion. Six recent analytical and semi-analytical schemes (extended simplest equation (ESE) method, modified Kudryashov (MKud) method, sech–tanh expansion method, Adomian decomposition (ADD) method, El Kalla (EK) expansion method, variational iteration (VI) method) are applied to the considered model for constructing abundant analytical and semi-analytical novel solutions. This variety of solutions aims to investigate the analytical techniques’ accuracy by calculating the absolute error between analytical and semi-analytical solutions that shows the matching between them. The analytical results are sketched through two-dimensional (2D), three-dimensional (3D), contour plot, spherical plot and polar plot. The stability characterization of the analytical solutions is investigated through the Hamiltonian system’s features. The originality and novelty of this paper are discussed, along with previously published papers.

In (1 + 1)–dimension; inelastic interaction of long-surface gravity waves of small-amplitude unidirectional propagation

This article investigate novel waves’ structures for the modified Benjamin–Bona–Mahony (BBM) equation and the coupled Klein–Gordon (CKG) equations through two recent analytical schemes. The BBM and CKG equations describe respectively the unidirectional promulgation of long waves in certain nonlinear dispersive media and limited purview of the relativistic energy-momentum relation. These models have many applications in various fields such as fluid mechanics, optical fiber, and many other distinct areas. The investigated models get many solitary wave solutions by employing the extended simple equation (ESE) and generalized Kudryashov (GKud) techniques. The assembled results are articulated through nonidentical graphs in two-, three-dimensional, and contour formulations. The novelty of our article is explained by comparing our solutions with those obtained in previously published articles. The employed techniques’ performance is investigated to show their effectiveness and power.

Hyperbolic approximation of the BBM equation

It is well known that the Benjamin–Bona–Mahony (BBM) equation can be seen as the Euler–Lagrange equation for a Lagrangian expressed in terms of the solution potential. We approximate the Lagrangian by a two-parameter family of Lagrangians depending on three potentials. The corresponding Euler–Lagrange equations can be then written as a hyperbolic system of conservations laws. The hyperbolic BBM system has two genuinely nonlinear eigenfields and one linear degenerate eigenfield. Moreover, it can be written in terms of Riemann invariants. Such an approach conserves the variational structure of the BBM equation and does not introduce the dissipation into the governing equations as it usually happens for the classical relaxation methods. The state-of-the-art numerical methods for hyperbolic conservation laws such as the Godunov-type methods are used for solving the ‘hyperbolized’ dispersive equations. We find good agreement between the corresponding solutions for the BBM equation and for its hyperbolic counterpart.

Analysis of BBM solitary wave interactions using the conserved quantities

In this paper, a simple, robust, fast and effective method based on the conserved quantities is developed to approximate and analyse the shape, structure and interaction characters of the solitary waves described by the Benjamin–Bona–Mahony (BBM) equation. Due to the invariant character of the conserved quantities, there is no need to solve the related complex nonlinear partial differential BBM equation to simulate the interactions between the solitary waves at the most merging instance. Good accuracy of the proposed method has been found when compared with the numerical method for the solitary wave interactions with different initial incoming wave shapes. The conserved quantity method developed in this work can serve as an ideal tool to benchmark numerical solvers, to perform the stability analysis, and to analyse the interacting phenomena between solitary waves.

Norm Inflation for Benjamin–Bona–Mahony Equation in Fourier Amalgam and Wiener Amalgam Spaces with Negative Regularity

We consider the Benjamin–Bona–Mahony (BBM) equation of the form ut+ux+uux−uxxt=0,(x,t)∈M×R where M=T or R. We establish norm inflation (NI) with infinite loss of regularity at general initial data in Fourier amalgam and Wiener amalgam spaces with negative regularity. This strengthens several known NI results at zero initial data in Hs(T) established by Bona–Dai (2017) and the ill-posedness result established by Bona–Tzvetkov (2008) and Panthee (2011) in Hs(R). Our result is sharp with respect to the local well-posedness result of Banquet–Villamizar–Roa (2021) in modulation spaces Ms2,1(R) for s≥0.

Singular solutions of the BBM equation: analytical and numerical study

We show that the Benjamin–Bona–Mahony (BBM) equation admits stable travelling wave solutions representing a sharp transition from a constant state to a periodic wave train. The constant state is determined by the parameters of the periodic wave train: the wave length, amplitude and phase velocity, and satisfies both the generalized Rankine–Hugoniot conditions for the exact BBM equation and for its wave averaged counterpart. Such stable shock-like travelling structures exist if the phase velocity of the periodic wave train is not less than the solution wave averaged. To validate the accuracy of the numerical method, we derive the (singular) solitary limit of the Whitham system for the BBM equation and compare the corresponding numerical and analytical solutions. We find good agreement between analytical results and numerical solutions.