Search

This paper focuses on the three‐dimensional (3D) incompressible anisotropic Boussinesq system while the velocity of fluid only involves horizontal dissipation and the temperature has a damping term. By utilizing the structure of the system, the energy methods and the means of bootstrapping argument, we prove the global stability property in the Sobolev space Hk(ℝ3)(k≥3)$$ {H}^k\left({\mathrm{\mathbb{R}}}^3\right)\left(k\ge 3\right) $$ of perturbations near the hydrostatic equilibrium. Moreover, we take an effective approach to obtain the optimal decay rates for the global solution itself as well as its derivatives. In this paper, we aim to reveal the mechanism of how the temperature helps stabilize the fluid. Additionally, exploring the stability of perturbations near hydrostatic equilibrium may provide valuable insights into specific severe weather phenomena. [ABSTRACT FROM AUTHOR]

The KP‐I equation has family of solutions which decay to zero at space infinity. One of these solutions is the classical lump solution, which is a traveling wave, and the KP‐I equation in this case reduces to the Boussinesq equation. In this paper we classify all the 'lump‐type' solutions of the Boussinesq equation. Using a robust inverse scattering transform developed by Bilman–Miller for the Schrödinger equation, we show that the lump‐type solutions are rational and their τ$\tau $ functions have to be polynomials of degree k(k+1)$k(k+1)$ for some integer k$k$. In particular, this implies that the lump solution is the unique ground state of the KP‐I equation (as conjectured by Klein–Saut). The problem studied in this paper was mentioned in Airault–McKean–Moser, our result can be regarded as a two‐dimensional analogy of their theorem on the classification of rational solutions for the KdV equation. [ABSTRACT FROM AUTHOR]

To analyze and study the behaviour of the shallow water waves, the perturbed Boussinesq equation has acquired fundamental importance. The principal objective of this paper is to manifest the exact traveling wave solution of the perturbed Boussinesq equation by two well known techniques named as, two variables (G′G,1G) expansion method and generalized projective Riccati equations method. A diverse array of soliton solutions, encompassing periodic, bright solitons, singular solitons and bright singular solitons are obtained by the applications of proposed techniques. The constraint conditions for newly constructed solutions are also specified. To enhance comprehension, the numerical illustrations of constructed solutions have been represented using surface plots, 2D plots and density plots. The results delineated in this paper transcend existing analysis, offering a novel, well-structured, and modern perspective. The solutions obtained not only enrich understanding of shallow water wave models but also exhibit efficacy in providing detailed descriptions of their dynamics. [ABSTRACT FROM AUTHOR]

In this paper, we are inspired by a famous result by Constantin and Fefferman who proved that a simple geometrical assumption on the direction of the vorticity leads to the regularity of weak solutions of the 3D Navier-Stokes equations. We show that the same result can be achieved if the vorticity direction is replaced by the velocity direction. We further strengthen this result and prove that in fact it is not necessary to consider the velocity direction in all close space points but only in the points whose distance equals to a small positive number dependant on the data. In the second part of the paper, we extend a result by Berselli and Córdoba concerning the role of the helicity for the regularity of the weak solutions of the Navier-Stokes equations. [ABSTRACT FROM AUTHOR]

The paper is concerned with the three-dimensional Boussinesq-Coriolis equations with Caputo time-fractional derivatives. Specifically, by striking new balances between the dispersion effects of the Coriolis force and the smoothing effects of the Laplacian dissipation involving with a time-fractional evolution mechanism, we obtain the global existence of mild solutions to Cauchy problem of three-dimensional time-fractional Boussinesq-Coriolis equations in Besov spaces. [ABSTRACT FROM AUTHOR]

We study a system describing the dynamics of a two-phase flow of incompressible viscous fluids influenced by the convective heat transfer of Caginalp-type. The separation of the fluids is expressed by the order parameter which is of diffuse interface and is known as the Cahn–Hilliard model. We shall consider a nonlocal version of the Cahn–Hilliard model which replaces the gradient term in the free energy functional into a spatial convolution operator acting on the order parameter and incorporate with it a potential that is assumed to satisfy an arbitrary polynomial growth. The order parameter is influenced by the fluid velocity by means of convection; the temperature affects the interface via a modification of the Landau–Ginzburg free energy. The fluid is governed by the Navier–Stokes equations which is affected by the order parameter and the temperature by virtue of the capillarity between the two fluids. The temperature on the other hand satisfies a parabolic equation that considers latent heat due to phase transition and is influenced by the fluid via convection. The goal of this paper is to prove the global existence of weak solutions and show that, for an appropriate choice of sequence of convolutional kernels, the solutions of the nonlocal system converge to its local version. [ABSTRACT FROM AUTHOR]

In this paper, exact periodic wave solutions for the perturbed Boussinesq equation with power law nonlinearity are obtained for different nonlinear strengths n. When n = 1 , the periodic traveling wave solutions can be found by the definition of the Jacobian elliptic function. When n ≥ 1 , we construct a transformation to solve for the power law nonlinearity, and the periodic traveling wave solutions can be obtained by applying the extended trial equation method. In addition, we consider the limiting case where the periodicity of the periodic traveling wave solutions vanishes, and we obtain the soliton solution for n = 1 . Numerical simulations show the periodicity of the solution for the perturbed Boussinesq equation. [ABSTRACT FROM AUTHOR]

This paper studies traveling waves with nonzero wave speed (angular traveling waves) of the high‐dimensional Boussinesq equation that have not been studied before. We analyze the properties of these waves and demonstrate that, unlike the unique stationary solution, they lack positivity, radial symmetry, and exponential decay. By employing variational and geometric approaches, along with perturbation theory, we establish the orbital (in)stability and strong instability of these traveling waves. [ABSTRACT FROM AUTHOR]

Variable-coefficient equations can be used to describe certain phenomena when the inhomogeneous media and nonuniform boundaries are taken into consideration. It is meaningful to solve the exact solution of variable-coefficient equations. In this paper, a (2+1)-dimensional variable-coefficient Boussinesq equation is investigated. The integrability is firstly examined by the Painlevé analysis method. Secondly, the Bäcklund transformations, one- and two-soliton solutions of the (2+1)-dimensional variable-coefficient Boussinesq equation are studied by virtue of the Hirota bilinear method. Propagation characteristics and interaction behaviors of the solitons are discussed: (i) soliton shapes and interaction behaviors are affected by the variable coefficients, and (ii) the two-soliton interaction is elastic, and the shape and velocity does not change after the collision, and only the shift changes. [ABSTRACT FROM AUTHOR]

The finding of new integrable coupling systems has become an important area of research in mathematical physics and their study will aid in the classification of multi-component integrable systems. A basic method for generating integrable coupling systems is algebraic expansion, for example, the Frobenius algebra, the Lie algebra, the superalgebra, and so on. In this paper, we introduce a Frobenius Boussinesq equation based on the Frobenius algebra, and then we present a Lax pair of it. It follows that we give a Bäcklund transformation of the Frobenius Boussinesq equation. Furthermore, the lattice equation of the Frobenius Boussinesq equation is presented by using three Bäcklund transformations, and then obtain the exact solutions. Additionally, we obtain the conservation laws of the Frobenius Boussinesq equation via the Bäcklund transformation. Strongly coupled and weakly coupled systems physically represent strong and weak interactions, respectively. In this paper, we introduce a weakly coupled Degasperis–Procesi (DP) equation, and construct a Lax pair of it. In addition, the Bäcklund transformation and superposition principle are applied to investigate the weakly coupled DP equation. We also obtain the conservation laws of the weakly coupled DP equation. Then, we introduce a strongly coupled DP equation, and use the same method to study the strongly coupled DP equation. The exact solutions of these two equations are obtained. Moreover, we introduce a Z n -DP equation. Considering the superposition principle, we obtain the solution of an associated Z n -DP equation by using Bäcklund transformations. These new multi-component integrable systems can enrich the existing integrable models and possibly describe new nonlinear phenomena. [ABSTRACT FROM AUTHOR]

Groundwater evapotranspiration (ETg) is a significant component of water and energy balance analyses in arid and semiarid environments; however, it is also one of the most challenging items to estimate. Estimation of ETg using diurnal groundwater-level fluctuation is considered cost-effective and straightforward, while more accurate estimation technology continues to be developed due to the complex set of factors that influence the variation of groundwater level. This paper proposes a method for calculating ETg that combines the analytical solution of the Boussinesq equation and dynamic harmonic regression analysis, which is a nonstationary approach to groundwater-level signal extraction. The presented approach can calculate ETg values with a high time resolution consistent with the groundwater-level monitoring frequency, and it is robust enough to handle large amounts of measurement data. More importantly, this method eliminates the influence of recharge during the groundwater-level-rise stage. The method is tested in bushland dominated by Sminthopsis psammophila in Mu Us Desert (northern China), which has a shallow water table. The results show that this method is easy to perform and it obtains a reasonable estimate of ETg in the unfrozen seasons, regardless of the disturbance of recharge that results in a sharp rise in groundwater level. However, the method does not work well in the frozen seasons because of the physical mechanisms of change in groundwater movement in the vadose zone and a strong barometric effect. Overall, the method provides a step toward accuracy of ETg estimation using diurnal groundwater-level fluctuation, and this paper provides guidelines for its use. [ABSTRACT FROM AUTHOR]

We prove the local well-posedness of the 3D free-boundary incompressible ideal magnetohydrodynamics (MHD) equations with surface tension, which describe the motion of a perfect conducting fluid in an electromagnetic field. We adapt the ideas developed in the remarkable paper [11] by Coutand and Shkoller to generate an approximate problem with artificial viscosity indexed by κ > 0 whose solution converges to that of the MHD equations as κ → 0. However, the local well-posedness of the MHD equations is no easy consequence of Euler equations thanks to the strong coupling between the velocity and magnetic fields. This paper is the continuation of the second and third authors' previous work [38] in which the a priori energy estimate for incompressible free-boundary MHD with surface tension is established. But the existence is not a trivial consequence of the a priori estimate as it cannot be adapted directly to the approximate problem due to the loss of the symmetric structure. [ABSTRACT FROM AUTHOR]

The paper is devoted to investigating the well-posedness, stability and large-time behavior near the hydrostatic balance for the 2D Boussinesq equations with partial dissipation. More precisely, the global well-posedness is obtained in the case of partial viscosity and without thermal diffusion for the initial data belonging to H δ (R 2) × H s (R 2) for δ ∈ [ s − 1 , s + 1 ] if s ∈ R , s > 2 , for δ ∈ (1 , s + 1 ] if s ∈ (0 , 2 ] and for δ ∈ [ 0 , 1 ] if s = 0. In addition, if one has either horizontal or vertical thermal diffusion then the stability and large-time behavior are provided in H m (R 2) , m ∈ N and in H ˙ m − 1 (R 2) with m ∈ N , m ≥ 2 , respectively. [ABSTRACT FROM AUTHOR]

The current paper concentrates on discovering the exact solutions of the singular time-fractional Boussinesq equation and coupled time-fractional Boussinesq equation by presenting a new technique known as the double Sumudu–generalized Laplace and Adomian decomposition method. Here, two main theorems are addressed that are very useful in this work. Moreover, the mentioned method is effective in solving several problems. Some examples are presented to check the precision and symmetry of the technique. The outcomes show that the proposed technique is precise and gives better solutions as compared to existing methods in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we propose an upwind compact difference method with fourth-order spatial accuracy and second-order temporal accuracy for solving the streamfunction-velocity formulation of the two-dimensional unsteady incompressible Navier–Stokes equations. The streamfunction and its first-order partial derivatives (velocities) are treated as unknown variables. Three types of compact difference schemes are employed to discretize the first-order partial derivatives of the streamfunction. Specifically, these schemes include the fourth-order symmetric scheme, the fifth-order upwind scheme, and the sixth-order symmetric scheme derived by combining the two parts of the fifth-order upwind scheme. As a result, the fourth-order spatial discretization schemes are established for the Laplacian term, the biharmonic term, and the nonlinear convective term, along with the Crank–Nicolson scheme for the temporal discretization. The unconditional stability characteristic of the scheme for the linear model is proved by discrete von Neumann analysis. Moreover, six numerical experiments involving three test problems with the analytic solutions, and three flow problems including doubly periodic double shear layer, lid-driven cavity flow, and dipole-wall interaction are carried out to demonstrate the accuracy, robustness, and efficiency of the present method. The results indicate that the present method not only has good numerical performance but also exhibits quite efficiency. [ABSTRACT FROM AUTHOR]

In this paper, we construct an example of temperature patch solutions for the two-dimensional, incompressible Boussinesq system with kinematic viscosity such that both the curvature and perimeter grow to infinity over time. The presented example consists of two disjoint, simply connected patches. The rates of growth for both curvature and perimeter in this example are at least algebraic. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the global well-posedness for the 3D Boussinesq equations. Firstly, we prove the global existence and uniqueness of the solution in critical space for small initial data. Then, aimed at the large initial data giving rise to global solution, we obtain a certain class of large solutions with initial data slowly varying in one direction. The proof relies on the special structure of the equation and some new refined estimates for the velocity and temperature. [ABSTRACT FROM AUTHOR]

In this paper, we study the Navier–Stokes–Korteweg equations governed by the evolution of compressible fluids with capillarity effects. We first investigate the global well-posedness of solution in the critical Besov space for large initial data. Contrary to pure parabolic methods in Charve et al. (Indiana Univ Math J 70:1903–1944, 2021), we also take the strong dispersion due to large capillarity coefficient κ into considerations. By establishing a dissipative–dispersive estimate, we are able to obtain uniform estimates and incompressible limits in terms of κ simultaneously. Secondly, we establish the large time behaviors of the solution. We would make full use of both parabolic mechanics and dispersive structure which implicates our decay results without limitations for upper bound of derivatives while requiring no smallness for initial assumption. [ABSTRACT FROM AUTHOR]

This present paper is dedicated to the study of the Cauchy problem of the two-dimensional Euler-Boussinesq-Bénard equations which couple the incompressible Euler equations for the velocity and a transport equation with critical dissipation for the temperature. We show that there is a global unique solution to this model with Yudovich's type data. This settles the global regularity problem which was remarked by Wu and Xue (2012) [44]. [ABSTRACT FROM AUTHOR]

We consider the Boussinesq equation on the line for a broad class of Schwartz initial data for which (i) no solitons are present, (ii) the spectral functions have generic behavior near ± 1, and (iii) the solution exists globally. In a recent work, we identified 10 main sectors describing the asymptotic behavior of the solution, and for each of these sectors we gave an exact expression for the leading asymptotic term. In this paper, we give a proof for the formula corresponding to the sector ... . [ABSTRACT FROM AUTHOR]

In this paper, a two-dimensional numerical model for simulating the generation and propagation of tsunami waves caused by upthrust bed movement is developed. To consider the nonlinearity as well as save the computational cost, a Navier–Stokes equation solver is used for the generation zone, and a Serre equation solver is adopted for the downstream evolution of the tsunami waves. The solution of the Navier–Stokes equation solver is extracted and transferred as the initial solution of the Serre solver, which means a one-way coupling is achieved. In this way, a one-way coupled Navier–Stokes-Serre model is obtained. After a detailed validation of the individual solvers, the coupled model is utilized for simulating the generation and propagation of tsunami waves caused by the upthrust bed movement in shallow water of uniform depth. It is found that the coupled model is comparable to the traditional Boussinesq equation model. Finally, the capacity of the coupled model for simulating wave-breaking cases is demonstrated. [ABSTRACT FROM AUTHOR]

Singularity formation of the 3D incompressible Euler equations is known to be extremely challenging (Majda and Bertozzi in Vorticity and incompressible flow, Cambridge University Press, Cambridge, vol 27, 2002; Gibbon in Physica D 237(14):1894–1904, 2008; Kiselev, in: Proceedings of the international congress of mathematicians, vol 3, 2018; Drivas and Elgindi in EMS Surv Math Sci 10(1):1–100, 2023; Constantin in Bull Am Math Soc 44(4):603–621, 2007). In Elgindi (Ann Math 194(3):647–727, 2021) (see also Elgindi et al. in Camb J Math 9(4), 2021), Elgindi proved that the 3D axisymmetric Euler equations with no swirl and C 1 , α initial velocity develops a finite time singularity. Inspired by Elgindi's work, we proved that the 3D axisymmetric Euler and 2D Boussinesq equations with C 1 , α initial velocity and boundary develop a stable asymptotically self-similar (or approximately self-similar) finite time singularity (Chen and Hou in Commun Math Phys 383(3):1559–1667, 2021) in the same setting as the Hou-Luo blowup scenario (Luo and Hou in Proc Natl Acad Sci 111(36):12968–12973, 2014; Luo and Hou in SIAM Multiscale Model Simul 12(4):1722–1776, 2014). On the other hand, the authors of Vasseur and Vishik (Commun Math Phys 378(1):557–568, 2020) and Lafleche et al. (Journal de Mathématiques Pures et Appliquées 155:140–154, 2021) recently showed that blowup solutions to the 3D Euler equations are hydrodynamically unstable. The instability results obtained in Vasseur and Vishik (2020) and Lafleche et al. (2021) require some strong regularity assumption on the initial data, which is not satisfied by the C 1 , α velocity field. In this paper, we generalize the analysis of Elgindi (Ann Math 194(3):647–727, 2021), Chen and Hou (Commun Math Phys 383(3):1559–1667, 2021), Vasseur and Vishik (2020) and Lafleche et al. (2021) to show that the blowup solutions of the 3D Euler and 2D Boussinesq equations with C 1 , α velocity are unstable under the notion of stability introduced in Vasseur and Vishik (2020) and Lafleche et al. (2021). These two seemingly contradictory results reflect the difference of the two approaches in studying the stability of 3D Euler blowup solutions. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the global existence and finite time blow-up of solution for the Cauchy problem of one-dimensional fifth-order Boussinesq equation with logarithmic nonlinearity. Fist we prove the existence and uniqueness of local mild solutions in the energy space by means of the contraction mapping principle. Further under some restriction on the initial data, we establish the results on existence and uniqueness of global solutions and finite time blow-up of solutions by using the potential well method. Moreover, the sufficient and necessary conditions of global existence and finite time blow-up of solutions are given. [ABSTRACT FROM AUTHOR]

The Boussinesq equation has some application in fluid dynamics, water sciences and so forth. In the current paper, we study an improved Boussinesq model. First, a finite difference approximation is employed to discrete the derivative of the temporal variable. Then, we study the existence and uniqueness of solution of the semi‐discrete scheme according to the fixed point theorem. In addition, the unconditional stability and convergence of the semi‐discrete scheme are presented. Then, we construct the fully discrete formulation based upon the radial basis function‐finite difference method. The convergence rate and stability of the fully‐discrete scheme are analyzed. In the end, some examples in 1D and 2D cases are studied to corroborate the capability of the proposed scheme. [ABSTRACT FROM AUTHOR]

Parametric excitation is a unique type of excitation that arises from the time-varying parameters of a dynamical system, typically induced by the periodic movements of the system components. However, the structure-moving subsystems problem creates a new type of parametric excitation where the subsystems do not have exact periodic movements. Under the circumstances, dimension of the mathematical model of such a system exhibits time dependency, which prohibits the employment of conventional analytical methods to predict its stability. In this paper, we propose a novel Generalized Sequential State Equation Method that provides a distinct solution framework for analysis of the subsystem-induced parametric resonance. With the time-domain system partition and state mapping, the proposed method reconfigures the coupled system with a sequence of state space equations and solves them chronologically. The proposed method relaxes the single-subsystem-assumption in its original form and applies to a general scenario where the subsystems travel with arbitrary characteristic period. With the proposed method, occurrences of the subsystem-induced parametric resonance are precisely predicted via a set of analytical stability criteria and the steady state response is determined in an exact analytical form. • Novel generalized sequential state equation method for parametric resonance with arbitrary characteristic period. • The proposed method advances the existing method by lifting the constraints on the inter-distance between subsystems. • Determining the stability of a moving subsystem induced parametric vibration via analytical stability criteria. • Delivering the exact waveform of the steady state of a parametrically excited structure in a general manner. • Investigating the characteristics of the subsystem induced parametric resonance in a multidimensional parameter space. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the global well-posedness of solutions to three-dimensional incompressible Boussinesq equations with temperature-dependent viscosity under the smallness assumption of initial velocity fields u0 in the critical space B ̇ 3 , 1 0 . The key ingredients here lie in the decomposition of the velocity fields and the regularity theory of the Stokes system, which are crucial to get rid of the smallness restricition of the initial temperature θ0. In addition, we mention that the improved decay estimates in time is also necessary. [ABSTRACT FROM AUTHOR]

This paper is devoted to the analytical study of traveling wave solutions to a generalized Boussinesq equation with nonlinear dispersion. Utilizing the bifurcation method of dynamical systems, the existence of compacton and peakon solutions are established. Under certain parameter conditions, the compacton and peakon solutions can be bifurcated by smooth periodic wave solutions, smooth solitary wave solutions, and singular cusp solutions. Numerical simulations are supplied to corroborate the analytical results. Some previous results are extended. [ABSTRACT FROM AUTHOR]

In 1904, Prandtl introduced his famous boundary layer in order to describe the behavior of solutions of Navier Stokes equations near a boundary as the viscosity goes to 0. His Ansatz has later been justified for analytic data by R.E. Caflisch and M. Sammartino. In this paper, we prove that his expansion is false, up to O (ν 1 / 4) order terms in L ∞ norm, in the case of solutions with Sobolev regularity, even in cases where the Prandlt's equation is well posed in Sobolev spaces. In addition, we also prove that monotonic boundary layer profiles, which are stable when ν = 0 , are nonlinearly unstable when ν > 0 , provided ν is small enough, up to O (ν 1 / 4) terms in L ∞ norm. [ABSTRACT FROM AUTHOR]

In this paper, we study the global C ∞ regularity of solutions to the boundary layer equations for two-dimensional steady compressible flow under the favorable pressure gradient. To our knowledge, the difficulty of the proof is the degeneracy near the boundary. By using the regularity theory and maximum principles of parabolic equations together with the von Mises transformation, we give a positive answer to it. When the outer flow and the initial data satisfied appropriate conditions, we prove that Oleinik type solutions smooth up the boundary y = 0 for any x > 0 . In this paper, we study the global regularity of solutions to the boundary layer equations for two-dimensional steady compressible flow under the favorable pressure gradient. To our knowledge, the difficulty of the proof is the degeneracy near the boundary. By using the regularity theory and maximum principles of parabolic equations together with the von Mises transformation, we give a positive answer to it. When the outer flow and the initial data satisfied appropriate conditions, we prove that Oleinik type solutions smooth up the boundary for any . [ABSTRACT FROM AUTHOR]

The atmosphere and ocean are described by highly oscillatory PDEs that challenge both our understanding of their dynamics and their numerical approximation. This paper presents a preliminary numerical study of one type of phase averaging applied to mean flows in the 2D Boussinesq equations that also has application to numerical methods. The phase averaging technique, well-known in dynamical systems theory, relies on a mapping using the exponential operator, and then an averaging over the phase. The exponential operator has connections to the Craya–Herring basis pioneered by Jack Herring to study the fluid dynamics of oscillatory, nonlinear fluid dynamics. In this paper, we perform numerical experiments to study the effect of this averaging technique on the time evolution of the solution. We explore its potential as a definition for mean flows. We also show that, as expected from theory, the phase-averaging method can reduce the magnitude of the time rate of change in the PDEs, making them potentially suitable for time stepping methods. [ABSTRACT FROM AUTHOR]

In this paper, the authors study the global regularity of the 3D magnetohydrodynamics system in terms of one velocity component. In particular, they establish a new Prodi-Serrin type regularity criterion in the framework of weak Lebesgue spaces both in time and space variables. [ABSTRACT FROM AUTHOR]

The Tollmien–Schlichting (T‐S) waves play a key role in the early stages of boundary layer transition. In a breakthrough work, Grenier, Guo, and Nguyen gave the first rigorous construction of the T‐S waves of temporal mode for the incompressible fluid. Yang and Zhang recently made an important contribution by constructing the compressible T‐S waves of temporal mode for certain boundary layer profiles with Mach number m<13$m<\frac{1}{\sqrt 3}$. In this paper, we construct the T‐S waves of both temporal mode and spatial mode to the linearized compressible Navier–Stokes system around the boundary layer flow in the whole subsonic regime m<1$m<1$, including the Blasius profile. Our approach is based on a novel iteration scheme between the quasi‐incompressible and quasi‐compressible systems, with a key ingredient being the solution of an Orr–Sommerfeld‐type equation using a new Airy–Airy–Rayleigh iteration instead of Rayleigh–Airy iteration introduced by Grenier, Guo, and Nguyen. We believe that the method developed in this work can be applied in solving other related problems for subsonic flows. [ABSTRACT FROM AUTHOR]

The Boussinesq and Camassa-Holm equations are, respectively, used to describe the bidirectional and unidirectional motions of small-amplitude waves on shallow water surfaces, but their wave dynamics remain a challenge for almost all conventional numerical methods. In this paper, we use physicsinformed neural networks (PINNs), a mesh-free deep learning method, to accurately predict the soliton (peakon) interaction or rogue wave behaviours of both equations with only a few initial and boundary data, providing a method for solving certain numerically unstable fluid systems or extreme wave solutions of regular fluid systems. It is revealed that by decomposing both of the equations into lower-order coupled systems, one can obtain higher-precision wave behaviours, especially for the Camassa-Holm equation. To retrieve certain unknown hydraulic parameters based on the observed wave data, e.g. the water depth, we use a multiple PINN method and stably identify the dynamic parameters in the Boussinesq equation through the association of localized soliton and rogue wave solutions. Furthermore, we compare the PINNs with conventional high-precision timesplitting Fourier spectral (TSFS) method and find that to achieve the same split feature of the Y-shaped soliton of the Boussinesq equation, PINNs require only one-third of the initial and boundary data of the TSFS method. [ABSTRACT FROM AUTHOR]

Power flow problems can be solved in a variety of ways by using the Newton–Raphson approach. The nonlinear power flow equations depend upon voltages V i and phase angle δ . An electrical power system is obtained by taking the partial derivatives of load flow equations which contain active and reactive powers. In this paper, we present an efficient seventh-order iterative scheme to obtain the solutions of nonlinear system of equations, with only three steps in its formulation. Then, we illustrate the computational cost for different operations such as matrix–matrix multiplication, matrix–vector multiplication, and LU-decomposition, which is then used to calculate the cost of our proposed method and is compared with the cost of already seventh-order methods. Furthermore, we elucidate the applicability of our newly developed scheme in an electrical power system. The two-bus, three-bus, and four-bus power flow problems are then solved by using load flow equations that describe the applicability of the new schemes. [ABSTRACT FROM AUTHOR]

This paper explores the dynamic behavior of the (2 + 1) -dimensional Boussinesq equation, which is a nonlinear water wave equation used to model wave packets in dispersive media with weak nonlinearity. Specifically, we investigate the equation's traveling wave solutions using the Riccati equation mapping method. Our results include solitary and soliton solutions, each with their own set of parameter values. To provide a comprehensive understanding of these solutions, we present them in general form and visualize their significance using various graphs, such as 3D, 2D, and contour plots. The computational effort and resulting outcomes highlight the efficacy of our approach, which has the potential to be applied to other nonlinear physical problems in fields such as mathematical physics, engineering, and nonlinear science. [ABSTRACT FROM AUTHOR]

The primary objective of this paper is to investigate the error analysis for the damped Boussinesq equation with the Ciarlet-Raviart mixed finite element method, focusing on achieving superconvergence. By employing a skillful mean technique and a high precision estimate of the bilinear element, as well as combining with projection and interpolation technique, we first obtain a superclose error estimate between interpolation of the exact solution and numerical solution. Subsequently, by applying an interpolation post-processing approach, we efficiently derive a global superconvergence result. Finally, several numerical results are presented to support our theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, we consider the stochastic Boussinesq equations on T³ with transport noise and rough initial data. We first prove the existence and uniqueness of the local pathwise solution with initial data in Lp(Ω;Lp) for p>5. By assuming additional smallness on the initial data and the noise, we establish the global existence of the pathwise solution. [ABSTRACT FROM AUTHOR]

In this paper, we examine the optical solitons for the nonlinear doubly dispersive equation and the modified Boussinesq equation, which explain the flow of shallow water in a small-amplitude surface system. We realize a variety of solitons using the Sardar sub-equation approach, including bright solitons, dark solitons, singular solitons, mixed bright-singular solitons, periodic, exponential, and rational solutions. The generated optical solutions can be used to simulate water waves and the free movement of a fluid surface, both of which are important in computing models of nonlinear partial differential equations in science, engineering, and mathematical physics. For the physical interpretation of the data, the well-known symbolic software Mathematica 12 was employed. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a newly introduced integrable (2 + 1)-dimensional Boussinesq equation. Solutions of this equation are obtained by Lie symmetry reductions and direct integration. We achieve diverse solitary wave solutions of the equation among which are non-topological soliton as well as Jacobi elliptic function solutions. Moreover, we generate some closed-form solutions of the equation which are in the form of bright, singular and non-singular periodic solitons. Power series solution of the equation is also generated. In a bid to have a sound understanding of the physical phenomena of the underlying model, we exhibited graphically the motion of the secured results. Besides, we discuss the obtained results as well as their respective graphs. Conclusively, we construct conservation laws of the aforementioned equation by employing the general multiplier approach. [ABSTRACT FROM AUTHOR]

In this paper, two semidiscrete low regularity exponential-type integrators are proposed and analyzed for the "good" Boussinesq equation, including a first-order integrator and a second-order one. Compared to the existing numerical methods, the convergence rate can be achieved under weaker regularity assumptions on the exact solution. Specifically, the first-order integrator is convergent linearly in $H^r$ for solutions in $H^{r+1}$ if $r>1/2$ , i.e., the boundedness of one additional derivative of the solution is required to achieve the first-order convergence. When $r>7/6$ , we can even prove linear convergence in $H^r$ for solutions in $H^{r+2/3}$. What's more, half-order convergence is established in $H^{r}(r>3/2)$ for any solutions in $H^r$ , i.e., no additional smoothness assumptions are needed. For the second-order integrator, the quadratic convergence in $H^{r}$ $(r>1/2)$ (or $L^2$) is demonstrated, when the solutions belong to $H^{r+2}$ (or $H^{9/4}$). Numerical examples illustrating the convergence analysis are included. A comparison with other methods demonstrates the superiority of the newly proposed exponential-type integrators for rough data. [ABSTRACT FROM AUTHOR]

This paper presents a new efficient algorithm based on the spectral-Galerkin numerical approximations complemented by a magnetohydrodynamics–Boussinesq model and a new solver for studying the development of a Rayleigh–Taylor (RT) instability. We use the Shenfun computational framework in the Cartesian coordinates, which gives the spectral order and accuracy for the considered model based on the magnetohydrodynamics equations and the Boussinesq conjecture. The numerical simulations were conducted for each two- and three-dimensional case, both with and without an external static magnetic field. The validity of the numerical results was examined by comparing the calculated squared L2-norm of the density parameter with the linear stability analysis. We also examined the effects of a uniform tangential magnetic field on the onset and growth of an RT instability at different magnetic field strengths. The analysis of the effectiveness of the presented method suggests that it can be modified for further research on two-component plasma. [ABSTRACT FROM AUTHOR]

The study of solitons including their stability, presence, validity and strength with constant increasing research is not forgettable in this era. In this research paper, we explore solitary wave solutions and dromions for Boussineq equation with the help of Hirota bilinear method (HBM) and Sine-Gordon expansion (SGE) technique. We will enlist these schemes for the study of different types of solitons by implementing HBM and SGE. We will acquire anti dark, multi-solitons, V-shaped, M-shaped, S-shaped and dark travelling wave solutions with their effects during transformations and collision. [ABSTRACT FROM AUTHOR]

This paper uses the theory of planar dynamic systems and the knowledge of reaction-diffusion equations, and then studies the bounded traveling wave solution of the generalized Boussinesq equation affected by dissipation and the influence of dissipation on solitary waves. The dynamic system corresponding to the traveling wave solution of the equation is qualitatively analyzed in detail. The influence of the dissipation coefficient on the solution behavior of the bounded traveling wave is studied, and the critical values that can describe the magnitude of the dissipation effect are, respectively, found for the two cases of b3 < 0 and b3 > 0 in the equation. The results show that, when the dissipation effect is significant (i.e., r is greater than the critical value in a certain situation), the traveling wave solution to the generalized Boussinesq equation appears as a kink-shaped solitary wave solution; when the dissipation effect is small (i.e., r is smaller than the critical value in a certain situation), the traveling wave solution to the equation appears as the oscillation attenuation solution. By using the hypothesis undetermined method, all possible solitary wave solutions to the equation when there is no dissipation effect (i.e., r = 0) and the partial kink-shaped solitary wave solution when the dissipation effect is significant are obtained; in particular, when the dissipation effect is small, an approximate solution of the oscillation attenuation solution can be achieved. This paper is further based on the idea of the homogenization principles. By establishing an integral equation reflecting the relationship between the approximate solution of the oscillation attenuation solution and the exact solution obtained in the paper, and by investigating the asymptotic behavior of the solution at infinity, the error estimate between the approximate solution of the oscillation attenuation solution and the exact solution is obtained, which is an infinitesimal amount that decays exponentially. The influence of the dissipation coefficient on the amplitude, frequency, period, and energy of the bounded traveling wave solution of the equation is also discussed. [ABSTRACT FROM AUTHOR]

The paper concentrates on the solitary waves that are retrievable from the generalized Boussinesq equation. The numerical simulations are displayed in the paper that gives a visual perspective to the model studied in neurosciences. The Laplace–Adomian decomposition scheme makes this visualization of the solitons possible. The numerical simulations are being reported for the first time using an elegant approach. The results would be helpful for neuroscientists and clinical studies in Medicine. The novelty lies in the modeling that is successfully conducted with an impressively small error measure. In the past, the model was integrated analytically only to recover soliton solutions and its conserved quantities. [ABSTRACT FROM AUTHOR]

The coupled Boussinesq equation (CBE) acts a key role in modeling the shallow water waves of two-layer fluid flow. However, it becomes powerless for the nonsmooth boundary. So, a fractal modification of the CBE is suggested in this paper. Aided by the semi-inverse method, we successfully construct its fractal generalized variational principle. In addition, three recent techniques, namely, the Sardar-subequation method, He's frequency formulation method and subequation method, combined with the two-scale fractal dimension transform, are utilized to find abundant wave solutions. By these methods, various solutions expressed by the generalized hyperbolic functions, generalized trigonometric functions, hyperbolic functions and cosine function are obtained. With the aid of the mathematical software, the three-dimensional contours and two-dimensional curves are plotted to present the dynamic behaviors of the solutions. The results in this paper demonstrate that the proposed methods are powerful and effective to construct the traveling wave solutions of the fractal nonlinear evolution equations. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to extend the versatile mixed methods originally developed by Chen and Williams for isothermal flows in "Versatile Mixed Methods for the Incompressible Navier-Stokes Equations," Computers & Mathematics with Applications, Volume 80, Number 6, 2020, and use them to simulate non-isothermal incompressible flows. These new mixed methods are particularly interesting, as with only minor modifications they can be applied to a much broader range of flows, including non-isothermal weakly-compressible flows, and fully-compressible flows. In the main body of this paper, we carefully develop these mixed methods for solving the Boussinesq model equations. Thereafter, we establish the L2-stability of the discrete temperature and velocity fields. In addition, we obtain rigorous error estimates for the temperature and velocity fields. Finally, we assess the practical behavior of the methods by applying them to a set of well-known convection problems. [ABSTRACT FROM AUTHOR]

This paper presents decoupled second‐order accurate algorithm based on Crank–Nicolson LeapFrog (CNLF) scheme for the evolution Boussinesq equations. The proposed algorithm deals with the spatial discretization by finite element method and treat the temporal discretization by CNLF method. For the nonlinear term in the Boussinesq equations, the semi‐implicit method is used. Unconditional stability and error estimate of the numerical algorithm are proven. Some numerical tests are presented to justify the theoretical analysis. [ABSTRACT FROM AUTHOR]