Regularized Long Wave Equation Research Articles

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.

Extended lifespan of the fractional BBM equation

For 0 < α < 1, N ⩾ 2 and with initial data ‖ u 0 ‖ H N + α 2 = ε, sufficiently small, we show that the existence time for solutions of the fractional BBM equation ∂ t u + ∂ x u + u ∂ x u + | D | α ∂ t u = 0, can be extended from the hyperbolic existence time 1 ε to 1 ε 2 . For the proof we use a quasilinear modified energy method, based on a normal form transformation as in Hunter, Ifrim, Tataru, Wong (Proc. Am. Math. Soc., 143(8) (2015) 3407–3412).

Exact Solutions of the Nonlinear Modified Benjamin-Bona-Mahony Equation by an Analytical Method

The current manuscript investigates the exact solutions of the modified Benjamin-Bona-Mahony (BBM) equation. Due to its efficiency and simplicity, the modified auxiliary equation method is adopted to solve the problem under consideration. As a result, a variety of the exact wave solutions of the modified BBM equation are obtained. Furthermore, the findings of the current study remain strong since Jacobi function solutions generate hyperbolic function solutions and trigonometric function solutions, as liming cases of interest. Some of the obtained solutions are illustrated graphically using appropriate values for the parameters.

Analysis of Fractional-Order Regularized Long-Wave Models via a Novel Transform

A new integral transform method for regularized long-wave (RLW) models having fractional-order is presented in this study. Although analytical approaches are challenging to apply to such models, semianalytical or numerical techniques have received much attention in the literature. We propose a new technique combining integral transformation, the Elzaki transform (ET), and apply it to regularized long-wave equations in this study. The RLW equations describe ion-acoustic waves in plasma and shallow water waves in seas. The results obtained are extremely important and necessary for describing various physical phenomena. This work considers an up-to-date approach and fractional operators in this context to obtain satisfactory approximate solutions to the proposed problems. We first define the Elzaki transforms of the Caputo fractional derivative (CFD) and Atangana-Baleanu fractional derivative (ABFD) and implement them for solving RLW equations. We can readily obtain numerical results that provide us with improved approximations after only a few iterations. The derived solutions were found to be in close contact with the exact solutions. Furthermore, the suggested procedure has attained the best level of accuracy. In fact, when compared to other analytical techniques for solving nonlinear fractional partial differential equations, the present method might be considered one of the finest.

Investigation of optical solitons to the nonlinear complex Kundu-Eckhaus and Zakharov-Kuznetsov-Benjamin-Bona-Mahony equations in conformable.

This research manuscript focuses on the extraction of dark-bright solitons and periodic wave distributions of two models, namely, the Zakharov–Kuznetsov–Benjamin–Bona–Mahony equation and complex Kundu–Eckhaus equation with conformable derivative. To reach these important properties, the generalized exponential rational function method is considered. To observe wave distributions in periodic and singular sense, dynamical behaviour modulus of solutions are also archived. Strain conditions for validity of results obtained in this paper are also reported.

A Legendre spectral element method for the family of regularized long wave equations

Regularized Long Wave (RLW) equation is one of the most important nonlinear PDEs. It is related to the famous KdV equation and acts as an alternative for scrutinizing soliton phenomena. In this paper, we use the Legendre Spectral Element Method (LSEM) for the numerical study of the family of the RLW equations containing RLW, modified RLW, and generalized RLW. We perform the space discretization by the LSEM, and the Crank–Nicolson method is applied to discretize the time. To confirm the accuracy and efficiency of the proposed method, the conservation properties of the equations are checked, and various types of errors are reported.

Uniformly accurate splitting schemes for the Benjamin-Bona-Mahony equation with dispersive parameter

We propose a new class of uniformly accurate splitting methods for the Benjamin–Bona-Mahony equation which converge uniformly in the dispersive parameter $$\varepsilon $$ . The proposed splitting schemes are furthermore asymptotic convergent and preserve the KdV limit. We carry out a rigorous convergence analysis of the splitting schemes exploiting the smoothing properties in the system. This will allow us to establish improved error bounds with gain either in regularity (for non smooth solutions) or in the dispersive parameter $$\varepsilon $$ . The latter will be interesting in regimes of a small dispersive parameter. We will in particular show that in the classical BBM case $$P(\partial _x) = \partial _x$$ our Lie splitting does not require any spatial regularity, i.e, first order time convergence holds in $$H^{r}$$ for solutions in $$H^{r}$$ without any loss of derivative. This estimate holds uniformly in $$\varepsilon $$ . In regularizing regimes $$\varepsilon =\mathscr {O}(1) $$ we even gain a derivative with our time discretisation at the cost of loosing in terms of $$\frac{1}{\varepsilon }$$ . Numerical experiments underline our theoretical findings.

Lie group analysis, optimal system and analytic solutions of a (3+1)-dimensional generalized Kadomtsev–Petviashvili–Benjamin–Bona–Mahony equation for the fluid flow around an offshore structure

Lie group analysis, optimal system and analytic solutions of a (3+1)-dimensional generalized Kadomtsev–Petviashvili–Benjamin–Bona–Mahony equation for the fluid flow around an offshore structure

Cauchy problem for the BBM equation in [formula omitted

It is shown that the BBM equation is globally well-posed in lqL2(R) for all 2≤q<∞, and locally well-posed in l∞L2(R). These imply that the BBM equation is well-posed in larger spaces, and thus improve the results in references.

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.

Generalized Exp-Function Method to Find Closed Form Solutions of Nonlinear Dispersive Modified Benjamin–Bona–Mahony Equation Defined by Seismic Sea Waves

Using the new generalized exp-function method, we were able to derive significant novel closed form solutions to the nonlinear dispersive modified Benjamin–Bona–Mahony (DMBBM) equation. The general framework of the new generalized exp-function method has been given. Many novel closed form solutions have been obtained in the form of hyperbolic, trigonometric, and rational function solutions. Using the computer application Wolfram Mathematica 10, we plotted 2D, 3D, and contour surfaces of closed form solutions found in this work. In the form of a table, the acquired results are compared to the known solutions in the existing literature.

Quasi-invariance of Gaussian measures for the periodic Benjamin-Ono-BBM equation

The BBM equation is a Hamiltonian PDE which revealed to be a very interesting test-model to study the transformation property of Gaussian measures along the flow, after Tzvetkov (Sigma 3:e28-35). In this paper we study the BBM equation with critical dispersion (which is a Benjamin-Ono type model). We prove that the image of the Gaussian measures supported on fractional Sobolev spaces of increasing regularity are absolutely continuous, but we cannot identify the density, for which new ideas are needed.

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.

Higher order Haar wavelet method integrated with strang splitting for solving regularized long wave equation

In this paper, we are going to utilize newly developed Higher Order Haar wavelet method (HOHWM) and classical Haar wavelet method (HWM) to numerically solve the Regularized Long Wave (RLW) equation. Spatial variable of the RLW equation is treated with HOHWM and HWM separately. On the other hand temporal variable is discretized by finite differences combined with Strang splitting approach. The presented methods applied to three different test problems and the obtained results are given in tables as well as depicted in figures. The obtained results are compared with analytical results wherever they exist. The error norms L2 and L∞ and invariants I1, I2 and I3 are used to show the accuracy of the methods when comparing the present results with those in the literature.

KdV and BBM models in open cylindrical channel under precession

The Korteweg–de Vries partial differential equation that has nonlinear-dispersion balance was derived under potential conditions to match the case of a single Kelvin mode that was noticed revolving on the outer periphery of an open cylindrical channel under precession conditions, which is assumed the solitary wave case in the channel. This led to a new version of the equation with a forcing term that includes the tilt effect with coefficients include the rotation effect. It was solved numerically using Fourier transformation methods for space discretization and the fourth order Runge–Kutta method for time discretization; the results were in a good match with the experiment. The rotational case led to a new Benjamin–Bona–Mahony equation that has variable coefficients with time and space mainly coming from the Coriolis effect in the axial direction of motion, with a forcing term comes from the gravity force. It was also solved numerically using a simple implicit finite difference scheme. This equation has two versions, one in terms of the velocity and one in terms of the amplitude. The first was compared with the bore velocity signal, which reflected the cnoidal type of waves, and the results were in a satisfactory match with the extracted signals; the second one was tracked with time to see the role Coriolis and gravity forces play on the single Kelvin wave form.

On the stability of solitary wave solutions for a generalized fractional Benjamin–Bona–Mahony equation

In this paper we establish a rigorous spectral stability analysis for solitary waves associated to a generalized fractional Benjamin–Bona–Mahony type equation. Besides the well known smooth and positive solitary wave with large wave speed, we present the existence of smooth negative solitary waves having small wave speed. The spectral stability is then determined by analysing the behaviour of the associated linearized operator around the wave restricted to the orthogonal of the tangent space related to the momentum at the solitary wave. Since the analytical solution is not known, we generate the negative solitary waves numerically by using Petviashvili method. We also present some numerical experiments to observe the stability properties of solitary waves for various values of the order of nonlinearity and fractional derivative. Some remarks concerning the orbital stability are also celebrated.

Instability, index theorem, and exponential trichotomy for Linear Hamiltonian PDEs

Consider a general linear Hamiltonian system∂tu=JLu\partial _{t}u=JLuin a Hilbert spaceXX. We assume that L:X→X∗\ L:X\rightarrow X^{\ast }induces a bounded and symmetric bi-linear form⟨L⋅,⋅⟩\left \langle L\cdot ,\cdot \right \rangleonXX, which has only finitely many negative dimensionsn−(L)n^{-}(L). There is no restriction on the anti-self-dual operatorJ:X∗⊃D(J)→XJ:X^{\ast }\supset D(J)\rightarrow X. We first obtain a structural decomposition ofXXinto the direct sum of several closed subspaces so thatLLis blockwise diagonalized andJLJLis of upper triangular form, where the blocks are easier to handle. Based on this structure, we first prove the linear exponential trichotomy ofetJLe^{tJL}. In particular,etJLe^{tJL}has at most algebraic growth in the finite co-dimensional center subspace. Next we prove an instability index theorem to relaten−(L)n^{-}\left ( L\right )and the dimensions of generalized eigenspaces of eigenvalues of JL\ JL, some of which may be embedded in the continuous spectrum. This generalizes and refines previous results, where mostlyJJwas assumed to have a bounded inverse. More explicit information for the indexes with pure imaginary eigenvalues are obtained as well. Moreover, when Hamiltonian perturbations are considered, we give a sharp condition for the structural instability regarding the generation of unstable spectrum from the imaginary axis. Finally, we discuss Hamiltonian PDEs including dispersive long wave models (BBM, KDV and good Boussinesq equations), 2D Euler equation for ideal fluids, and 2D nonlinear Schrödinger equations with nonzero conditions at infinity, where our general theory applies to yield stability or instability of some coherent states.

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.

Lower bound on the radius of analyticity of solution for fifth order KdV–BBM equation

We show that the uniform radius of spatial analyticity \(\sigma (t)\) of solution at time t for the fifth order KdV–BBM equation cannot decay faster than 1/t for large \(t>0\), given initial data that is analytic with fixed radius \(\sigma _0\). This significantly improves a recent result by Carvajal and Panthee (On the radius of analyticity for the solution of the fifth order KdV–BBM model, 2020. arXiv:2009.09328) , where they established an exponential decay of \(\sigma (t)\) for large t.

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.