Regularized Long Wave Equation Research Articles

Floquet theory and stability analysis for Hamiltonian PDEs

Abstract We analyze Floquet theory as it applies to the stability and instability of periodic traveling waves in Hamiltonian PDEs. Our investigation focuses on several examples of such PDEs, including the generalized KdV and BBM equations (third order), the nonlinear Schrödinger and Boussinesq equations (fourth order), and the Kawahara equation (fifth order). Our analysis reveals that the characteristic polynomial of the monodromy matrix inherits symmetry from the underlying PDE, enabling us to determine the essential spectrum along the imaginary axis and bifurcations of the spectrum away from the axis, employing the Floquet discriminant. We present numerical evidence to support our analytical findings.

Numerical simulation of the generalized modified Benjamin–Bona–Mahony equation using SBP-SAT in time

Numerical simulation of the generalized modified Benjamin–Bona–Mahony equation using SBP-SAT in time

Exploring Soliton Solutions and Chaotic Dynamics in the (3+1)-Dimensional Wazwaz–Benjamin–Bona–Mahony Equation: A Generalized Rational Exponential Function Approach

This paper investigates the explicit, accurate soliton and dynamic strategies in the resolution of the Wazwaz–Benjamin–Bona–Mahony (WBBM) equations. By exploiting the ensuing wave events, these equations find applications in fluid dynamics, ocean engineering, water wave mechanics, and scientific inquiry. The two main goals of the study are as follows: Firstly, using the dynamic perspective, examine the chaos, bifurcation, Lyapunov spectrum, Poincaré section, return map, power spectrum, sensitivity, fractal dimension, and other properties of the governing equation. Secondly, we use a generalized rational exponential function (GREF) technique to provide a large number of analytical solutions to nonlinear partial differential equations (NLPDEs) that have periodic, trigonometric, and hyperbolic properties. We examining the wave phenomena using 2D and 3D diagrams along with a projection of contour plots. Through the use of the computational program Mathematica, the research confirms the computed solutions to the WBBM equations.

Analysis of Lie symmetry, bifurcations with phase portraits, sensitivity and diverse W − M-shape soliton solutions for the (2 + 1)-dimensional evolution equation

This research investigates the Kadomtsev-Petviashvili-Benjamin-Bona-Mahony (KP-BBM) equation, a key model in nonlinear wave dynamics, particularly for shallow water waves. By integrating the features of the KP and BBM equations, it offers a comprehensive framework for studying the propagation, stability, and interactions of long waves. Using Lie group analysis for symmetry reductions and bifurcation with phase portraits to explore dynamic behaviour, the study also applies chaos theory to examine system features. Additionally, the new extended direct algebraic method (NEDAM) is employed to derive novel soliton solutions, including dark, bright-dark, W-shape, singular, periodic, and mixed forms. Sensitivity analysis is performed, and 3D, 2D, contour, and density plots visually demonstrate the results. These findings show that NEDAM effectively extracts exact solitons, providing a basis for further studies in nonlinear wave theory and its applications in engineering and physics.

Global attractor for the damped BBM equation in the sharp low regularity space

Global attractor for the damped BBM equation in the sharp low regularity space

Analysis of Two Hybrid Schemes to Solve the Benjamin-Bona-Mahony (BBM) Equation

In this research, we compare and contrast two numerical methods, the Laplace Adomian Decomposition Method (LADM) and the Elzaki Projected Differential Transform Method (EPDTM), for solving the Benjamin-Bona-Mahony (BBM) equation, and this helps in evaluating their performance, assessing robustness, gaining insights into solution behavior, validating results, generalizing their applicability, and advancing the field. This rigorous comparisons validates the efficacy of the methods as we compare the exact results and the results of LADM and EPDTM, plotting the convergent of the two methods to identify the method that is highly convergent and accurate. Results identify the Elzaki Projected Differential Transform Method (EPDTM) as having a simple and accurate numerical method for solving the BBM equation. EPDTM enhances accessibility, computational efficiency, accuracy, robustness, and validation of the solutions, supporting the understanding and analysis of wave phenomena described by the BBM equation. This study contributes to the advancement of the field of numerical analysis and computational science as practical applications of BBM equation in various fields, including coastal engineering, wave modeling, and wave stability analysis cannot be over emphasize.

Analysis of two hybrid schemes to solve the Benjamin-Bona-Mahony (BBM) equation

In this research, we compare and contrast two numerical methods, the Laplace Adomian Decomposition Method (LADM) and the Elzaki Projected Differential Transform Method (EPDTM), for solving the Benjamin-Bona-Mahony (BBM) equation, and this helps in evaluating their performance, assessing robustness, gaining insights into solution behavior, validating results, generalizing their applicability, and advancing the field. This rigorous comparisons validates the efficacy of the methods as we compare the exact results and the results of LADM and EPDTM, plotting the convergent of the two methods to identify the method that is highly convergent and accurate. Results identify the Elzaki Projected Differential Transform Method (EPDTM) as having a simple and accurate numerical method for solving the BBM equation. EPDTM enhances accessibility, computational efficiency, accuracy, robustness, and validation of the solutions, supporting the understanding and analysis of wave phenomena described by the BBM equation. This study contributes to the advancement of the field of numerical analysis and computational science as practical applications of BBM equation in various fields, including coastal engineering, wave modeling, and wave stability analysis cannot be over emphasize.

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.

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

Uncertainties in regularized long-wave equation and its modified form: A triangular fuzzy-based approach

This article explores the solution of the regularized long-wave equation (RLWE) and modified RLWE (MRLWE) using a semi-analytical approach known as the homotopy perturbation transform method (HPTM), revealing the characteristics of shallow water waves and ion-acoustic plasma waves. The effectiveness and accuracy of the technique are demonstrated by solving scenarios involving a single solitary wave (SSW) and two solitary waves (TSW) presented and compared with the exact solution of the RLWE. Furthermore, we introduced a fuzzy model for both RLWE and MRLWE, considering uncertainties in the coefficients related to the wave amplitude, and to understand the behavior of both fuzzy RLWE (FRLWE) and fuzzy MRLWE (FMRLWE) in the SSW by examining various numerical results using MATLAB.

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 new approach to the Benjamin-Bona-Mahony equation via ultraspherical wavelets collocation method

Abstract In this paper, 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 efficiency and accuracy of the proposed technique. The effectiveness of this practical approach is 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 literature. As we know, many partial differential equations (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 also 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

THE FRACTAL ZAKHAROV–KUZNETSOV–BENJAMIN–BONA–MAHONY EQUATION: GENERALIZED VARIATIONAL PRINCIPLE AND THE SEMI-DOMAIN SOLUTIONS

By means of He’s fractal derivative, a new fractal (2 + 1)-dimensional Zakharov–Kuznetsov–Benjamin–Bona–Mahony equation is extracted in this paper. The semi-inverse method is employed to establish the generalized fractal variational principle. The generalized fractal variational principle can show the conservation laws through the energy form in the fractal space. Moreover, some semi-domain solutions are also explored by applying the variational approach and the one-step method namely Wang’s direct mapping method-II. The dynamics of the extracted solutions on the Cantor set are unveiled graphically. The findings of this study are expected to provide some new insights into the exploration of the fractal PDEs.

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.