Boussinesq Equations Research Articles

Stability for the 2D magnetic Bénard system with partial dissipation and thermal damping near an equilibrium

In this paper, we focus on the Boussinesq equations with magnetohydrodynamics convection, in which the temperature obeys degenerate damped wave equations. Under the influence of an equilibrium, we prove that the 2D magnetic Bénard system remains stable in the whole space . In particular, we obtain the decay rate and large‐time behavior. Without the coupling, the 2D MHD system with only vertical dissipation is not known to be stable. It is revealed that both the thermal damping and the horizontal background magnetic field are indispensable.

Solving Time-Space Fractional Boussinesq Equation Using Homotopy Perturbation Method

This paper aims to implement the homotopy perturbation technique to solve the time-space fractional Boussinesq equation, a significant model in the analysis of nonlinear wave propagation. Through the application of the homotopy perturbation technique, we derive analytical expressions for the solutions of the time-space fractional Boussinesq equation and validate these solutions through comparisons with numerical methods. Obtained results demonstrate the efficiency and accuracy of the homotopy perturbation method in solving the time-space fractional Boussinesq equation.

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.

Asymptotic behavior of the Boussinesq equation with nonlocal weak damping and arbitrary growth nonlinear function

In this paper, we consider the asymptotic behavior of the Boussinesq equation with nonlocal weak damping when the nonlinear function is arbitrary polynomial growth. We firstly prove the well‐posedness of solution by means of the monotone operator theory. At the same time, we obtain the dissipative property of the dynamical system associated with the problem in the space and , respectively. After that, the asymptotic smoothness of the dynamical system and the further quasi‐stability are demonstrated by the energy reconstruction method. Finally, we show not only the existence of the finite global attractor but also the existence of the generalized exponential attractor.

Double parametric based solution of fuzzy unconfined aquifer problem using Laplace transforms method

The Boussinesq equation describes the model for horizontal water flow in unconfined aquifers without precipitation, a topic that has been extensively studied in the literature. However, the parameters, as well as the initial and boundary conditions, are often assumed to be exact. In reality, these conditions may be incomplete or uncertain due to limited knowledge, insufficient information, or errors introduced by humans or machines. The fuzzy set theory has recently been successfully employed to model such uncertainties. This article investigates the analytical solution of the one-dimensional Boussinesq equation in a fuzzy environment. The objective of this research is to investigate the recharge and discharge of a semi-infinite unconfined aquifer adjacent to a lake. For the present investigation, uncertainties in terms of fuzzy are considered only for the involved initial and boundary conditions of the problem, whereas other parameters are considered as crisp or exact. The analysis employed the double parametric form of a fuzzy number alongside Laplace transform techniques. The obtained solutions were then compared with existing results in specific cases to validate their accuracy.

The soliton resolution conjecture for the Boussinesq equation

The soliton resolution conjecture for the Boussinesq equation

Numerical study of wave disturbances of the high order nonlinear Boussinesq equations arising in engineering

Numerical study of wave disturbances of the high order nonlinear Boussinesq equations arising in engineering

Filtered Lie-Trotter splitting for the “good” Boussinesq equation: Low regularity error estimates

We investigate a filtered Lie-Trotter splitting scheme for the “good” Boussinesq equation and derive an error estimate for initial data with very low regularity. Through the use of discrete Bourgain spaces, our analysis extends to initial data in H s H^{s} for 0 > s ≤ 2 0>s\leq 2 , overcoming the constraint of s > 1 / 2 s>1/2 imposed by the bilinear estimate in smooth Sobolev spaces. We establish convergence rates of order τ s / 2 \tau ^{s/2} in L 2 L^2 for such levels of regularity. Our analytical findings are supported by numerical experiments.

Bright-dark solitary wave solutions to the nonlinear fractional Boussinesq and breaking soliton equations

Bright-dark solitary wave solutions to the nonlinear fractional Boussinesq and breaking soliton equations

Interaction of lump, periodic, bright and kink soliton solutions of the (1+1)-dimensional Boussinesq equation using Hirota-bilinear approach

In this paper, we explore the characteristics of lump and interaction solutions for a (1+1) dimensional Boussinesq equation. By employing the Hirota bilinear method, we derive and analyze the exact solutions of this equation. Specifically, we achieve the lump with bright-bright soliton solution, 1-lump,2-lumps and 3-lumps with single bright soliton solution, lump with periodic, kink, and anti-kink soliton solutions. Alongside deriving these solutions, we also illustrate their dynamic properties through graphical simulations. The Boussinesq equation holds significant importance due to its applications in various domains, such as water wave modeling, coastal engineering, and the numerical simulation of water wave dynamics in harbors and shallow seas. Our research shows that the employed method is straightforward, easy to understand, and highly efficient, providing valuable insights into the equation’s nature and its practical applications.

An explicit time integration method for Boussinesq approximation

AbstractThis paper presents a novel approach to the discretization of the Boussinesq equation using finite elements, coupled with an explicit pressure correction scheme. The entire numerical scheme is formulated exclusively in terms of matrix‐vector products, enabling a highly efficient numerical solution. The key advantage of this approach lies in its suitability for fast computations and seamless parallelization on graphics processing units (GPUs). We present the numerical scheme and discuss different computational examples.

Dynamics of Double Time-Delayed Newton–Boussinesq Equations on Unbounded Domains

Dynamics of Double Time-Delayed Newton–Boussinesq Equations on Unbounded Domains

Approximate Solutions of the Boussinesq Equation for Horizontal Unconfined Aquifers During Pure Drainage Phase

In this work, conceptual approximations of the Boussinesq equation were introduced and analyzed, resulting into a very accurate and well-applicable model for horizontal unconfined aquifers during the pure drainage phase, without any recharge and zero-inflow conditions. The model was constructed by employing a variety of methods that included wave solution, variable separation, and series expansion, and its analysis and performance against the Boussinesq equation, at early and later times, providing fruitful insights enlightening the main mechanisms and physical characteristics of the drainage phase. The modeled non-linear forms were finally linearized, concluding with explicit analytical expressions that accurately incorporated most of the basic characteristics regarding the evolution of the water table and the outflow from the exact Boussinesq equation under different initial conditions. The endeavors of this work can be utilized for theoretical and modeling purposes related to this problem.

A Smoothed Particle Hydrodynamics Approach for One-Dimensional Dam Break Flow Simulation with Boussinesq Equations

A Smoothed Particle Hydrodynamics Approach for One-Dimensional Dam Break Flow Simulation with Boussinesq Equations

Self-induced transparency of long water waves over bathymetry: the dispersive shock mechanism

The shoreline hazard posed by ocean long waves such as tsunamis and meteotsunamis critically depends on the fraction of energy transmitted across the shallow near-shore shelf. In linear setting, bathymetric inhomogeneities of length comparable to the incident wavelength act as a protective high-pass filter, reflecting long waves and allowing only shorter waves to pass through. Here, we show that, for weakly nonlinear waves, the transmitted energy flux fraction can significantly depend on the amplitude of the incoming wave. The basis of this mechanism is the formation of dispersive shock waves (DSWs), a salient feature of nonlinear evolution of long water waves, often observed in tidal bores and tsunami/meteotsunami evolution. Within the framework of the Boussinesq equations, we show that the DSWs efficiently transfer wave energy into the high wavenumber band, where reflection is negligible. This is phenomenologically similar to self-induced transparency in nonlinear optics: small amplitude long waves are reflected by the bathymetric inhomogeneity, while larger amplitude waves that develop DSWs blueshift into the transparency regime and pass through. We investigate this mechanism in a simplified setting that retains only the key processes of DSW disintegration and reflection, while the effects such as bottom dissipation and breaking are ignored. The results suggests that the phenomenon is a robust, order-one effect. In contrast, the increased transmission due to the growth of bound harmonics associated with the steepening of the wave is weak. The results of the simplified modelling are validated by simulations with the FUNWAVE-TVD Boussinesq model.

Pattern dynamics of higher-order rogue waves in the nonlinear Schrödinger–Boussinesq equation

Pattern dynamics of higher-order rogue waves in the nonlinear Schrödinger–Boussinesq equation

Chaos analysis and traveling wave solutions for fractional (3+1)-dimensional Wazwaz Kaur Boussinesq equation with beta derivative

Wazwaz Kaur Boussinesq (WKB) equation can effectively simulate the behavior of water waves in shallow water, including the nonlinear effect and dispersion phenomenon of waves, which is of great significance for understanding the dynamic process of ocean, river and other water bodies. To enrich the wave equation theory, the (3+1)-dimensional integer order derivative of WKB equation is changed to the fractional one with beta derivative. The current work deals with the fractional (3+1)-dimensional WKB equation for discussing its chaotic behavior and establishing some new analytic solutions. The chaotic properties of the equation are verified by the trend of evolution along with time, Lyapunov exponents and initial sensitivity analysis. And then complete discrimination system for polynomial method is applied to derive some trigonometric, hyperbolic, Jacobi elliptic and other solutions. The graphical demonstrations are provided for part of these solutions. From these visualized graphs, the solitary, periodic and quasi-periodic wave are shown and the effect of fractional derivatives on the equation can be seen intuitively.

A hybrid SPH model of the Navier-Stokes and Boussinesq equations for simulating wave propagation and deformation

A two-way hybrid model based on the smoothed particle hydrodynamics (SPH) method is proposed in this paper to simulate wave propagation and deformation effectively. The hybrid model is composed of two submodels: the first submodel is based on the Boussinesq equations (B-SPH) for simulating wave propagation, and the second submodel is based on the Navier-Stokes (N-S) equations (NS-SPH) for simulating wave deformation caused by interactions with structures. The exchange of information between the two submodels is realized through an open-boundary method: the results derived from the B-SPH submodel are transferred to the open boundaries of the NS-SPH submodel through a relaxation function; the information derived from the NS-SPH submodel is matched to the open boundaries of the B-SPH submodel through the continuity equation. To obtain a stable pressure field in the NS-SPH submodel, the particle shifting technique (PST) is implemented. The subsequent validation of the hybrid model includes four typical numerical examples related to solitary and regular waves. The agreement between the numerical results and the experimental data indicates that the proposed hybrid model is suitable for simulating the propagation and deformation of shallow water waves. Compared with the traditional NS-SPH model, which uses N-S equations to describe the whole fluid domain, the hybrid model has greater computational efficiency.

Exact Solutions of Boussinesq Equations By Hirota Direct Method

Boussinesq Equations (BSQ) are the focus of this article. First, we provide a basic overview of Hirota's D operator, which is used to build multi-soliton solutions for equations involving nonlinear evolution. After that, some details regarding fourth-order BSQ are provided, and we use Hirota's direct method to find a one-solution solution. Hirota's bilinear approach is also used to solve a class of sixth-order Boussinesq-like equations with nonlinear evolution. The outcomes verified that this approach requires complete integrability.

Convergence rates and central limit theorem for 3-D stochastic fractional Boussinesq equations with transport noise

This work investigates the 3-D stochastic modified fractional Boussinesq equations driven by transport noise on the torus. The existence of weak solutions is initially established through Galerkin approximation and compactness techniques. Subsequently, we demonstrate that the weak solutions of stochastic modified fractional Boussinesq equations converge to the unique solution of a deterministic Boussinesq model with fractional dissipation, subject to an appropriate noise scaling, along with providing the quantitative estimates with explicit convergent rates. Finally, the central limit theorem with an explicit convergence rate is established based on the aforementioned scaling limit. It is noteworthy that the scaling limit result of stochastic modified fractional Boussinesq equations to deterministic Boussinesq models with additional fractional dissipation implies that the transport noise regularizes the modified fractional Boussinesq equations, thus leading to “approximate weak uniqueness”.