Regularized Long Wave Equation Research Articles

Soliton solutions to the conformable time-fractional generalized Benjamin–Bona–Mahony equation using the functional variable method

Soliton solutions to the conformable time-fractional generalized Benjamin–Bona–Mahony equation using the functional variable method

Soliton, quasi-soliton, and their interaction solutions of a nonlinear (2 + 1)-dimensional ZK–mZK–BBM equation for gravity waves

Abstract The ZK–mZK–BBM equation plays a crucial role in actually depicting the gravity water waves with the long wave region. In this article, the bilinear forms of the (2 + 1)-dimensional ZK–mZK–BBM equation were derived using variable transformation. Then, the multiple soliton solutions of the ZK–mZK–BBM equation are obtained by bilinear forms and symbolic computation. Under complex conjugate transformations, quasi-soliton solutions and mixed solutions composed of one-soliton and one-quasi-soliton are derived from soliton solutions. These solutions are further studied graphically to observe the propagation characteristics of gravity water waves. The results enrich the research of gravity water wave in fluid mechanics.

The traveling wave solutions for generalized Benjamin–Bona–Mahony equation

The traveling wave solutions for generalized Benjamin–Bona–Mahony equation

Convergence of solutions of the BBM and BBM-KP model equations

Convergence of solutions of the BBM and BBM-KP model equations

Numerical solution of nonlinear Hunter-Saxton equation, Benjamin-Bona Mahony equation, and Klein-Gordon equation using Hosoya polynomial method

Models of nonlinear partial differential equations are essential in the theoretical sciences, especially Physics and Mathematics. It is exceedingly challenging to arrive at an analytical solution. For this reason, scientists are constantly looking for new computational techniques. For specific nonlinear mathematical models, such as the extensively researched Hunter-Saxton equation (HSE), Benjamin-Bona-Mahony equation (BBME), and Klein-Gordon equation (KGE), we are putting forth a novel and effective graph theoretic polynomial method named the Hosoya polynomial collocation method (HPCM). The orthonormalized Hosoya polynomials of the path graph and their operational matrices serve as the functional foundation for the new HPCM. The considered nonlinear models were transformed into a system of nonlinear algebraic equations using the operational matrix approximation and an appropriate collocation approach to reach the approximate solution. Six numerical examples verified the HPCM's productivity. Compared to the current numerical approaches in the literature, the results are significantly more efficient and almost match the analytical solution.

Consistent travelling wave characteristic of space–time fractional modified Benjamin–Bona–Mahony and the space–time fractional Duffing models

Study on solitary wave phenomenon are closely related on the dynamics of the plasma and optical fiber system, which carry on broad range of wave propagation. The space–time fractional modified Benjamin–Bona–Mahony equation and Duffing model are important modeling equations in acoustic gravity waves, cold plasma waves, quantum plasma in mechanics, elastic media in nonlinear optics, and the damping of material waves. This study has effectively developed analytical wave solutions to the aforementioned models, which may have significant consequences for characterizing the nonlinear dynamical behavior related to the phenomenon. Conformable derivatives are used to narrate the fractional derivatives. The expanded tanh-function method is used to look into such kinds of resolutions. An ansatz for analytical traveling wave solutions of certain nonlinear evolution equations was originally a power sequence in tanh. The discovered explanations are useful, reliable, and applicable to chaotic vibrations, problems of optimal control, bifurcations to global and local, also resonances, as well as fusion and fission phenomena in solitons, scalar electrodynamics, the relation of relativistic energy–momentum, electromagnetic interactions, theory of one-particle quantum relativistic, and cold plasm. The solutions are drafted in 3D, contour, listpoint, and 2D patterns, and include multiple solitons, bell shape, kink type, single soliton, compaction solitary wave, and additional sorts of solutions. With the aid of Maple and MATHEMATICA, these solutions were verified and discovered that they were correct. The mentioned method applied for solving NLFPDEs has been designed to be practical, straightforward, rapid, and easy to use.

A novel approach for Benjamin-Bona-Mahony equation via ultraspherical wavelets collocation method

Abstract We develop a precise and efficient ultraspherical wavelet method for a famous Benjamin-Bona-Mahony (BBM) mathematical model. The suggested technique uses the collocation method and ultraspherical wavelets. The proposed scheme is applied to linear and nonlinear BBM equations to inspect the proposed technique’s efficiency and accuracy. This practical approach’s effectiveness has been verified. Moreover, the method based on the ultraspherical wavelets is simple, accurate, fast, flexible, and convenient. The results are analyzed using tables and graphs and compared with other methods in the literature. As we know, Many PDEs don’t have exact solutions, and some semi-analytical methods work based on controlling parameters, but this is a controlling parameter-free technique. Also, it is pretty simple to implement and consumes less time to execute the programs. The recommended wavelet-based numerical approach is interesting, productive, and efficient. The proposed technique’s convergence analysis is presented through the theorem.

Asymptotically Autonomous Robustness of Random Attractors for 3D BBM Equations Driven by Nonlinear Colored Noise

Asymptotically Autonomous Robustness of Random Attractors for 3D BBM Equations Driven by Nonlinear Colored Noise

New analytic wave solutions to (2 + 1)-dimensional Kadomtsev–Petviashvili–Benjamin–Bona–Mahony equation using the modified extended mapping method

New analytic wave solutions to (2 + 1)-dimensional Kadomtsev–Petviashvili–Benjamin–Bona–Mahony equation using the modified extended mapping method

Machine Learning Study through Physics-Informed Neural Networks: Analysis of the Stable Vortices in Quasi-Integrable Systems

Vortices in the nonlinear equations, including Zakharov-Kuznetsov (ZK) equation and the regularized long-wave (RLW) equation are studied. The Physics-Informed Neural Networks solve these equations in the forward process and obtain the solutions. In the inverse process, the proper equations can successfully be derived from a given training data. However, between the ZK equation and the RLW equation, sometimes serious misidentification occurs. In order to improve the resolution of the identification, we introduce two methods: a friction method and deformations of the initial profile which offers a nice discrimination of the equations.

Unconditional superconvergence analysis of an energy-preserving finite element scheme for nonlinear BBM equation

An energy-preserving Crank-Nicolson (CN) fully-discrete scheme is developed with the nonconforming Quasi-Wilson element for the nonlinear Benjamin-Bona-Mahony (BBM) equation. The existence and uniqueness of the numerical solution are demonstrated by the Brouwer fixed point theorem. Then with the help of the special character of this element, that is, the consistency error can reach order O(h2), one order higher than its interpolation error, and the interpolation post-processing technique, the unconditional supercloseness and superconvergence behavior on quadrilateral meshes are derived rigorously without the restriction between mesh size h and time step τ. Finally, some numerical experiments are carried out to confirm the theoretical analysis.

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.

An efficient wavelet‐based approximation method for solving nonlinear fractional‐time long wave equations: An operational matrix approach

This research paper develops an efficient Gegenbauer wavelet‐based approximation method to solve nonlinear fractional‐time regularized long wave (RLW) equations arising in ocean engineering. The operational matrices of derivatives, as well as the fractional order integration, are engaged in the proposed method. Using the operational matrix method and collocation point, the nonlinear RLW equations are converted into a system of algebraic equations, and these equations are further solved. Some results regarding the error‐bound estimation of this method have been developed. The method's accuracy and efficiency are confirmed by error estimation. The obtained results are compared with other available results and exact solutions. Moreover, the use of Gegenbauer wavelets is found to be a simple, straightforward forward and efficient tool for solving nonlinear PDEs arising in ocean engineering.

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.

Bernoulli Wavelets Numerical Approach for the Nonlinear Klein–Gordon and Benjamin–Bona–Mahony Equation

Bernoulli Wavelets Numerical Approach for the Nonlinear Klein–Gordon and Benjamin–Bona–Mahony Equation

Existence of periodic wave of a BBM equation with delayed convection and weak diffusion

Existence of periodic wave of a BBM equation with delayed convection and weak diffusion

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.

On the applications of collocation method for numerically analyzing the nonlinear Degasperis–Procesi and Benjamin–Bona–Mahony equations

This research uses the Sinc Collocation method to numerically examine the Degasperis–Procesi and Benjamin–Bona–Mahony equations, achieving a high level of precision and accuracy on computational grounds with a variety of mesh points. The proposed technique involves global collocation using Sinc bases function (SBF) as an activation function. Initially, the time derivative discretization has been performed through finite difference scheme and the remaining spatial derivatives are approximated by [Formula: see text]-weighted scheme to minimize the complexity in numerical computations. The main contribution of research is in modifying the proposed scheme to overcome the complexity involved in the whole process of finding solutions with a cardinal expansion of Sinc functions in nonlinear and singular systems. The solutions of Degasperis–Procesi and Benjamin–Bona–Mahony stiff nonlinear systems are illustrated numerically and graphically and the validation and verification of the obtained results are examined through stability analysis by calculating the spectral radius by bounding the value of [Formula: see text]. A comparative study is presented with some of the well-known numerical techniques, which assured the accuracy and reliability of our technique.

Optical Solitary Waves and Soliton Solutions of the (3+1)-Dimensional Generalized Kadomtsev–Petviashvili–Benjamin–Bona–Mahony Equation

Optical Solitary Waves and Soliton Solutions of the (3+1)-Dimensional Generalized Kadomtsev–Petviashvili–Benjamin–Bona–Mahony Equation

Optical Solitary Waves and Soliton Solutions of the (3 + 1)-Dimensional Generalized Kadomtsev–Petviashvili–Benjamin–Bona–Mahony Equation

Optical Solitary Waves and Soliton Solutions of the (3 + 1)-Dimensional Generalized Kadomtsev–Petviashvili–Benjamin–Bona–Mahony Equation