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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 (피,S(t)) associated with the problem in the space H02(Ω)×L2(Ω)$$ {H}_0^2\left(\Omega \right)\times {L}^2\left(\Omega \right) $$ and DA34×H01(Ω)$$ D\left({A}^{\frac{3}{4}}\right)\times {H}_0^1\left(\Omega \right) $$, respectively. After that, the asymptotic smoothness of the dynamical system (피,S(t)) 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. [ABSTRACT FROM AUTHOR]

The Korteweg-de Vries (KdV) equation is known as a universal equation describing various long waves in dispersive systems. In this article, we prove that in a certain scaling regime, a large class of rough solutions to the Boussinesq equation are approximated by the sums of two counter-propagating waves solving the KdV equations. It extends the earlier result [25] to slightly more regular than L 2 -solutions. Our proof is based on robust Fourier analysis methods developed for the low regularity theory of nonlinear dispersive equations. [ABSTRACT FROM AUTHOR]

Use of the finite difference method has produced successful solutions to the general partial differential equations due to its efficiency and effectiveness with wide applications. For example, the 2D Boussinesq paradigm equation can be numerically studied using a linear‐implicit finite difference scheme based on the Crank‐Nicolson/Adams‐Bashforth technique. First, conservative quantities are derived and preserved through numerical scheme. Then, the convergence and stability analysis is then provided to simulate a numerical solution whose existence and uniqueness are proved based on the boundedness of the numerical solution. Analysis of spatial accuracy is found to be second order on a uniform grid. Numerical results from simulations indicate that these proposed scheme provide satisfactory second‐order accuracy both in time and space with an L∞$$ {L}^{\infty } $$‐norm, and also preserve discrete invariants. Additionally, previous scientific literature review has provided little evidence of studied terms with dispersive effect in 2D Boussinesq paradigm equation. The current study explores solution behavior by applying the proposed scheme to numerically analyze initial Gaussian condition. [ABSTRACT FROM AUTHOR]

The emergence of higher-dimensional evolution equations in dissimilar scientific arenas has been on the rise recently with a vast concentration in optical fiber communications, shallow water waves, plasma physics, and fluid dynamics. Therefore, the present study deploys certain improved analytical methods to perform a solitonic analysis of the newly introduced three-dimensional nonlinear dynamical equations (all within the current year, 2024), which comprise the new (3 + 1) Kairat-II nonlinear equation, the latest (3 + 1) Kairat-X nonlinear equation, the new (3 + 1) Boussinesq type nonlinear equation, and the new (3 + 1) generalized nonlinear Korteweg–de Vries equation. Certainly, a solitonic analysis, or rather, the admittance of diverse solitonic solutions by these new models of interest, will greatly augment the findings at hand, which mainly deliberate on the satisfaction of the Painleve integrability property and the existence of solitonic structures using the classical Hirota method. Lastly, this study is relevant to contemporary research in many nonlinear scientific fields, like hyper-elasticity, material science, optical fibers, optics, and propagation of waves in nonlinear media, thereby unearthing several concealed features. [ABSTRACT FROM AUTHOR]

The presented work concerns with some novel solutions of the (2+1)-dimensional Boussinesq equation (BE), which acts as an important model for shallow water wave. Some resonant soliton solutions such as the X -shape soliton (XSS) and Y -shape soliton (YSS) solutions are developed via imposing the resonant conditions on the N -soliton solutions (N -SSs) that developed by the Hirota bilinear approach(HBA). Based on the XSS and YSS solutions, the novel hybrid interactions including the interaction between the 1-soliton and the YSS, the interaction between two YSS solutions are extracted. In addition, the complex N -SSs are also explored and discussed. Finally, the travelling wave solutions including the bright solitary and kinky solitary wave solutions are studied by employing the Bernoulli sub-equation function method (BSEFM). The graphs of the attained solutions are drawn to show the physical properties. The findings of this study are expected to help us apprehend the dynamics of the (2+1)-dimensional BE better. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The two‐way propagation of a certain class of long‐crested water waves is governed approximately by systems of equations of the Boussinesq type. These equations have been put forward in various forms by many authors and their higher‐order generalizations arise when modeling the propagation of waves on large lakes, ocean, and in other contexts. Considered here is a class of such system which couple two higher‐order Korteweg–de‐Vries type equations. Our aim is to investigate the controllability properties of the linearized model posed on a periodic interval. By using the classical duality approach and some theorems on nonharmonic Fourier series, we prove that the system is exactly controllable in certain well‐chosen Sobolev spaces by means of suitable boundary controls. [ABSTRACT FROM AUTHOR]

A reduced-order model (ROM) of the global oceans is developed by projecting the hydrostatic Boussinesq equations of motion onto a proper orthogonal decomposition (POD) basis. Three-dimensional POD modes are calculated from the ocean fields of an ensemble climate reanalysis dataset. The coefficients in the POD ROM are calculated using a regression approach. The performance of various POD ROM configurations are assessed. Each configuration is derived from an alternate sea-water equation of state, linking the density and temperature fields. POD ROM variants incorporating an equation of state in which density is a quadratic function of temperature, are able to reproduce the statistics of the large-scale structures at a fraction of the computational cost required to numerically simulate this flow. Due to the speed and efficiency of calculation, such reduced-order models of the global geophysical system will enable researchers and policy makers to assess the physical risk for a broader range of potential future climate scenarios. [ABSTRACT FROM AUTHOR]

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The Boussinesq equation and its related types are able to provide a significant explanation for a variety of different physical processes that are relevant to plasma physics, ocean engineering, and fluid flow. Within the framework of shallow water waves, the aim of this research is to find solutions for solitary waves using newly developed nonlocal models of Boussinesq’s equations. The extraction of bright and dark solitary wave solutions along with bright–dark hybrid solitary wave solutions is accomplished through the implementation of two integration algorithms. The general projective Riccati equations method and the enhanced Kudryashov technique are the ones that have been implemented as techniques. The enhanced Kudryashov method combines the benefits of both the original Kudryashov method and the newly developed Kudryashov method, which may generate bright, dark, and singular solitons. The Projective Riccati structure is determined by two functions that provide distinct types of hybrid solitons. The solutions get increasingly diverse as these functions are combined. The techniques that were applied are straightforward and efficient enough to provide an approximation of the solutions discovered in the research. Furthermore, these techniques can be utilized to solve various kinds of nonlinear partial differential equations in mathematical physics and engineering. In addition, plots of the selected solutions in three dimensions, two dimensions, and contour form are provided. [ABSTRACT FROM AUTHOR]

In this paper, we apply the powerful sine-Gordon expansion method (SGEM), along with a computational program, to construct some new traveling wave soliton solutions for two models, including the higher-order nonlinear Boussinesq dynamical wave equation, which is a well-known nonlinear evolution model in mathematical physics, and the (1+1)-dimensional framework of the Van der Waals gas system. This study presents some new complex traveling wave solutions, as well as logarithmic and complex function properties. The 3D and 2D graphical representations of all obtained solutions, unveiling new properties of the considered model are simulated. Additionally, several simulations, including contour surfaces of the results, are performed, and we discuss their physical implications. A comprehensive conclusion is provided at the end of this paper. [ABSTRACT FROM AUTHOR]

This paper investigated the application of analytical methods, specifically the Mohand transform iterative method (MTIM) and the Mohand residual power series method (MRPSM), to solve the fractional Boussinesq equation. Utilizing the Caputo operator to manage fractional derivatives, these semi-analytical approaches provide accurate solutions to complex fractional differential equations. Through convergence analysis and error estimation, the study validated the efficacy of these methods by comparing numerical solutions to known exact solutions. Graphical and tabular representations illustrated the accuracy of the proposed methods, highlighting their performance for varying fractional orders. The findings demonstrated that both MTIM and MRPSM offer reliable, efficient solutions, making them valuable tools for addressing fractional differential systems in fields such as applied mathematics, engineering, and physics. [ABSTRACT FROM AUTHOR]

This study investigated the effects of fluctuating boundary water levels and surface recharge on groundwater flow within unconfined aquifers. We aimed to understand how changes in recharge patterns and variable boundary water levels, such as those from rivers or canals, affect groundwater levels over time and space. To achieve this, we solved the linearized Boussinesq equation using the time-marching method alongside the generalized integral transformation method. Our analysis focused on how different types of recharge affect groundwater level variations and flow dynamics. We found that boundary effects on groundwater level change propagate from the edges toward the aquifer's center, becoming more pronounced with increased boundary water levels. Over time, the system stabilizes, leading to a steady water table height and flow rate, which depend on the disparity between the boundary water levels. Our analytical model demonstrated flexibility and practical applicability by allowing for the consideration or omission of various influencing factors, thus facilitating complete knowledge about groundwater variations and offering future strategic insights for sustainable groundwater resource management. [ABSTRACT FROM AUTHOR]

We consider the global well-posedness and decay rates for solutions of 3D incompressible micropolar equation in the critical Besov space. Spectrum analysis allows us to find not only parabolic behaviors of solutions, but also damping effect of angular velocity in the low frequencies. Based on this observation, we establish the global well-posedness with more general regularity on the initial data. The approach concerns large time behaviors is so-called the Gevrey method which bases on the parabolic mechanics of the micropolar system. This method enables us even to derive decay of arbitrary higher order derivatives by transforming it into the control of the radius of analyticity under a particular regularity. To bound the growth of the radius of analyticity in general L p Besov spaces, we shall develop some new techniques concern Gevrey estimates, especially an extended Coifman-Meyer theory, to cover those endpoint Lebesgue framework. [ABSTRACT FROM AUTHOR]

This work introduces two (3+1)-dimensional expansions of the Korteweg–de Vries (KdV) and modified KdV (mKdV) equations. These extensions incorporate a second-order time-derivative term, similar to the Boussinesq equation. The Painlevé test is utilized to verify the integrability of each extended model. The bilinear form is employed to investigate the existence of multiple-soliton (MS) solutions for each system under consideration. Furthermore, we provide solutions in the form of lumps for the extended KdV equation. The multidimensional KdV-type equations surpass the standard KdV equation. However, they offer enhanced accuracy by representing a broader spectrum of nonlinear phenomena in plasma physics, fluid mechanics, tsunami phenomena, and other science disciplines. Furthermore, the aforementioned equations can be employed to analyze the characteristics of various acoustic waves (AW) in different plasma models, such as their amplitude, width, frequency, and dispersion, as well as the phase shifts after collisions. [ABSTRACT FROM AUTHOR]

We introduce a low-order dynamical system to describe thermal convection in an annular domain. The model derives systematically from a Fourier–Laurent truncation of the governing Navier–Stokes Boussinesq equations and accounts for spatial dependence of the flow and temperature fields. Comparison with fully resolved direct numerical simulations (DNS) shows that the model captures parameter bifurcations and reversals of the large-scale circulation (LSC), including states of (i) steady circulating flow, (ii) chaotic LSC reversals and (iii) periodic LSC reversals. Casting the system in terms of the fluid's angular momentum and centre of mass (CoM) reveals equivalence to a damped pendulum with forcing that raises the CoM above the fulcrum. This formulation offers a transparent mechanism for LSC reversals, namely the inertial overshoot of a forced pendulum, and it yields an explicit formula for the frequency $f^*$ of regular LSC reversals in the high-Rayleigh-number (Ra) limit. This formula is shown to be in excellent agreement with DNS and produces the scaling law $f^* \sim {Ra}^{0.5}$. [ABSTRACT FROM AUTHOR]

We prove the strong ill-posedness of the two-dimensional Boussinesq system in vorticity form in L∞(R²) without boundary, building upon the method that Elgindi and Shikh Khalil [Strong Ill-Posedness in L∞ for the Riesz Transform Problem, arXiv:2207.04556v1, 2022] developed for scalar equations. We provide examples of initial data with vorticity and density gradient of small L∞(R²) size, for which the horizontal density gradient L∞(R²) has a strong L∞(R²)-norm inflation in infinitesimal time, while the vorticity and the vertical density gradient remain bounded. Furthermore, exploiting the three-dimensional version of Elgindi's decomposition of the Biot-Savart law, we apply our method to the three-dimensional axisymmetric Euler equations with swirl and away from the vertical axis, showing that a large class of initial data with vorticity uniformly bounded and small in L∞(R²) provides a solution whose gradient of the swirl has a strong L∞(R²)-norm inflation in infinitesimal time. The norm inflation is quantified from below by an explicit lower bound which depends on time and the size of the data and is valid for small times. [ABSTRACT FROM AUTHOR]

Boussinesq equations are used as mathematical models for investigating the behaviour of small-amplitude long waves on water surfaces. In this paper, we focus on the logarithmic Boussinesq equation. We first generalize the model to consider different model coefficients. Secondly, we provide analytical and numerical solutions to the direct problem. Thirdly, we applied the modulating function method to solve coefficients inverse problems in the generalized logarithmic Boussinesq equation. Finally, numerical simulations are given to show the stability, accuracy, and effectiveness of the proposed methods for solving the posed direct and inverse problems. [ABSTRACT FROM AUTHOR]

We consider a third-order non-self-adjoint operator which is an -operator in the Lax pair for the Boussinesq equation on the circle. We construct a mapping from the set of operator coefficients to the set of spectral data, similar to the corresponding mapping for the Hill operator constructed by E. Korotyaev. We prove that, in a neighborhood of zero, our mapping is analytic and one-to-one. [ABSTRACT FROM AUTHOR]

Good Boussinesq equations are considered in this work. First, we apply three combined compact schemes to approximate spatial derivatives of good Boussinesq equations. Then, three fully discrete schemes are developed based on a symplectic scheme in the time direction, which preserves the symplectic structure. Meanwhile, the convergence and conservation of the fully discrete schemes are analyzed. Finally, we present numerical experiments to confirm our theoretical analysis. Both our analysis and numerical tests indicate that the fully discrete schemes are efficient in solving the spatial derivative mixed equation. [ABSTRACT FROM AUTHOR]

In this study, we propose a modified version of section-averaged Boussinesq equations of Winckler-Liu. The model is reformulated in conservative variables, allowing a decoupling of the Shallow-Water equations and the dispersive problem. An appropriate hybrid finite volume and finite element discretisations are performed and verified with a solitary wave solution derived for the typical case of prismatic channels with a trapezoidal cross-section. For the finite volume step we compare upwind and energy conservative numerical fluxes. The impact of this choice on the long time dynamics for Favre waves is thoroughly investigated. The proposed model and numerical approximations can accurately reproduce the main features of the wave train's free surface. The impact of the numerical dissipation introduced by upwind fluxes is discussed, emphasising the need for precaution in their applications to evaluate quantities of engineering interest such as maximum wave amplitudes. • A reformulation of Winckler and Liu's model is proposed to study wave propagation in channels with arbitrary cross-sections. • The model simplifies dispersive terms integration into existing shallow water solver, enhancing computational flexibility. • Hybrid discretization method, combining finite volume and finite element is proposed. • Two numerical flux formulations and their dissipative behaviours are investigated over long propagation times of Favre waves. • Comparing with Favre waves laboratory experiments, the numerical model captures key propagation phenomena. [ABSTRACT FROM AUTHOR]

We consider the pointwise in space $ L^p $-type regularity for elliptic and parabolic equations of-order $ m $ in $ \mathbb{R}^{n} $. We provide pointwise Schauder estimates for the general range of $ L^{p} $ exponents, extending previous results from $ p>n/m $ to $ 1

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 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 the stability problem of steady solutions to the semi-stationary Boussinesq equations in the strip domain R 2 × (0 , 1). For an equilibrium state with any general steady solution θe which satisfies ϑe > m > 0, we show the global existence and asymptotic behavior of solutions to the system with the no-slip boundary condition when the initial temperature is close enough to it. Thus such a steady solution is asymptotically stable, which reflects the well-known phenomenon of Rayleigh-Taylor stability. [ABSTRACT FROM AUTHOR]

Motivated by the question how to describe long‐wave dynamics on periodic networks, we consider a Boussinesq equation posed on the infinite periodic necklace graph. For the description of long‐wave traveling waves, we derive the KdV equation and establish the validity of this formal approximation by providing estimates for the error. The proof is based on suitable energy estimates. As a consequence of the approximation result, the soliton dynamics present in the KdV equation can approximately be seen for the original system, too. There are no serious obstacles to transfer the analysis to other dispersive systems or to other periodic quantum graphs for which the KdV equation can be derived. However, a general KdV validity theory on quantum graphs would be extremely abstract and of minor use. [ABSTRACT FROM AUTHOR]

This document is a correction notice for an article titled "On the two-dimensional Boussinesq equations with temperature-dependent thermal and viscosity diffusions in general Sobolev spaces." The authors make several corrections to the original article, including removing certain results and estimates, correcting the proof of the uniqueness result, and providing corrected proofs for the three endpoint regularity cases. The corrected proofs are provided using the same notations and equation numbering as in the original article. The document concludes by stating that the original article can be found online. The article discusses the uniqueness result of the two-dimensional Boussinesq equations with temperature-dependent thermal and viscosity diffusions in general Sobolev spaces. The authors mention that the uniqueness result is not expected below a certain regularity assumption. The article is available under the Creative Commons Attribution 4.0 International License. The publisher remains neutral regarding jurisdictional claims and institutional affiliations. The article is authored by Zihui He and Xian Liao from the Institute of Analysis at the Karlsruhe Institute of Technology in Germany. [Extracted from the article]

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 research communication, the (3 + 1)-dimensional Wazwaz Kaur Boussinesq equation is studied and solved by the direct algebraic method, namely the F expansion method. The F expansion method converts the nonlinear partial differential equation into an ordinary differential equation and finally into an algebraic equation. The Jacobi elliptic solutions of the Wazwaz Kaur Boussinesq equation are reported under six sets of classifications, followed by the existence condition and elliptic modulus limiting cases. The three-dimensional graphs for the few Jacobi solutions are drawn. [ABSTRACT FROM AUTHOR]

We investigate the existence and stability of traveling waves for the sixth‐order Boussinesq equation, considering a broad class of nonlinearities adhering to power‐like scaling relations. Employing the Nehari manifold method, we establish the existence of traveling waves and derive variational criteria for assessing their stability or instability. Subsequently, we develop a numerical approach based on these variational principles to delineate regions of stability and instability. Finally, for homogeneous nonlinearities, we establish a sufficient condition for strong instability. [ABSTRACT FROM AUTHOR]

The space-time fractional Burger-like equation and the space-time coupled Boussinesq equation with conformable derivative, both of which are significant in fluid dynamics, are investigated in this work using the improved G′/G method. The process works effectively and produces soliton solutions. The method was successfully and consistently implemented with Maple, a symbolic computing tool. The solutions also contain a few of graphics. Numerous novel exact solutions to these equations, distinct from those found earlier with the proposed approach, have been provided. The study’s findings add to the body of knowledge by offering insightful justifications for various types of nonlinear systems. The results demonstrated the value of the proposed method as a mathematical tool and the ease, dependability, and speed increases that result from carrying out these tasks using a symbolic computing program. Notably, it applies to many nonlinear evolution problems in mathematical physics. [ABSTRACT FROM AUTHOR]

The dispersion relation of the magnetized Kelvin–Helmholtz (KH) instability driven by shear flow with zero thickness of the shear layer is derived theoretically based on a set of magnetohydrodynamic equations in the presence of a transverse magnetic field and a radiation field. The influence of the magnetic field strength, the radiation field strength, and the density ratio of the two sides of the shear layer on KH instability is analyzed by solving the dispersion equation. The results indicate that the presence of radiation and transverse magnetic fields can destabilize the KH instability due to the resulting increase in Mach number, which in turn reduces the compressibility of the system. Also, the extent of the destabilizing effect of the magnetic field can be affected by the magnetoacoustic Mach number M 1 f and the Mach number M 2. The growth rates vary more significantly for relatively small values of both parameters. Finally, the stabilizing effect of a large density ratio is considered, and it is found that as the density ratio increases, the effect of the radiation field is more significant at larger Mach number M 2. These results can be applied to astrophysical phenomena with velocity shear, such as flows across the transition layer between an H ii region and a molecular cloud, accretion flows, and shear flows of cosmic plasmas. [ABSTRACT FROM AUTHOR]

This study employs precise, accurate, and efficient analytical and numerical techniques to obtain novel and accurate solitary wave solutions for the (2+1)-dimensional Ablowitz–Kaup–Newell–Segur (AKNS) equation. The Khater II (Khat II) method and extended cubic-B-spline (ECBS) scheme are utilized as recent computational and accurate numerical techniques, respectively, for investigating the AKNS equation and constructing highly precise solitary wave solutions. The stability characteristics of the obtained solutions are also analyzed to ensure their applicability. These solutions are characterized by their peak amplitudes, wave speeds, and widths. Through comparisons with previous numerical techniques, our proposed approach demonstrates superior effectiveness and reliability. The research is significant due to the application of the (2+1)-dimensional AKNS equation in modeling various physical phenomena, such as nonlinear optics, water waves, and plasma physics. The precise and efficient determination of solitary wave solutions not only enhances the understanding of these phenomena but also provides a valuable framework for practical applications. The results have substantial implications for comprehending complex wave dynamics in physical systems and serve as a valuable tool for further research and analysis in the field of wave dynamics. [ABSTRACT FROM AUTHOR]

We revisit the problem of nonlinear water wave propagation in the presence of an abrupt depth transition. To this end, we use an asymptotic approach conducted to order 3 with respect to the shallowness parameter, in order to capture the first nonlinear and dispersive contributions. However, the discontinuity of bathymetry, as opposed to slowly varying bathymetry, requires the use of a consistent three-scale analysis framework and the consideration of different regions, far from the step and free surface, near the free surface, and near the step. This framework enables consistent navigation, ultimately providing Boussinesq equations supplemented by jump conditions at the depth discontinuity that encompass the effect of step on wave propagation. [ABSTRACT FROM AUTHOR]

The purpose of this work is to propose and analyze a hybridized discontinuous Galerkin (HDG) method for the generalized Boussinesq equations with singular heat source. We use polynomials of order k, k − 1 and k to approximate the velocity, the pressure and the temperature. By introducing Lagrange multipliers for the pressure, the approximate velocity field obtained by the HDG method is shown to be exactly divergence-free and H(div)-conforming. Under a smallness assumption on the problem data and solutions, we prove by Brouwer's fixed point theorem that the discrete system has a solution in two dimensions. If the viscosity and thermal conductivity are further assumed to be positive constants, a priori error estimates with convergence rate O(h) and efficient and reliable a posteriori error estimates are derived. Finally numerical examples illustrate the theoretical analysis and show the performance of the obtained a posteriori error estimator. [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]

This investigation employs advanced computational techniques to ascertain novel and precise solitary wave solutions of the Camassa–Holm ( ℋ) equation, a partial differential equation governing wave phenomena in one-dimensional media. Originally designed for the representation of shallow water waves, the ℋ equation has exhibited versatility across various disciplines, including nonlinear optics and elasticity theory. It intricately delineates the interplay between nonlinear and dispersive effects in wave systems, with nonlinearity arising from component interactions and dispersion rooted in the temporal spreading of waves. Furthermore, the ℋ equation governs the spatiotemporal evolution of wave profiles, encompassing both nonlinear and dispersive influences. Notably, the equation allows for soliton solutions — localized wave packets sustaining their form over extended distances. The identification of precise solitary wave solutions holds paramount significance for comprehending the ℋ equation's behavior in diverse physical contexts, such as fluid dynamics and nonlinear optics. Moreover, this study establishes a correlation between the investigated model and plasma physics, demonstrating the efficacy and efficiency of the employed computational techniques through benchmarking against alternative computational methods. This augmentation underscores the broader relevance of the ℋ equation, extending its applicability to provide insights into wave phenomena analogous to those encountered in plasma physics. [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]

The dynamics of a bilayer system filling a rectangular cuvette subjected to external heating is studied. The influence of two types of thermal exposure on the flow pattern and on the dynamic contact angle is analyzed. In particular, the cases of local heating from below and distributed thermal load from the lateral walls are considered. The simulation is carried out within the frame of a two-sided evaporative convection model based on the Boussinesq approximation. A benzine–air system is considered as reference system. The variation in time of the contact angle is described for both heating modes. Under lateral heating, near-wall boundary layers emerge together with strong convection, whereas the local thermal load from the lower wall results in the formation of multicellular motion in the entire volume of the fluids and the appearance of transition regimes followed by a steady-state mode. The results of the present study can aid the design of equipment for thermal coating or drying and the development of methods for the formation of patterns with required structure and morphology. [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]

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 present study deals with shear wave propagation in a fully coupled Magneto-Electro-Elastic (MEE) multi-laminated periodic structure having non-uniform and imperfect interfaces. As a solution methodology, we applied a more general low-frequency dynamic asymptotic homogenization technique where the solution will be single-frequency dependent and the obtained results generalize those published in Chaki and Bravo-Castillero (Compos Struct 322:117410, 2023b) where the perfect contact case was studied. Effective homogenized dispersive equations of motion in second- and fourth-order approximations, also known as "Good" Boussinesq equations in elastic case, are derived. Local problems, closed-form expression of dispersion equations in second and fourth-order approximations and closed-form solutions of first and second local problems in second-order approximation for tri-laminated MEE periodic structure have been obtained and also validated for elastic laminates with imperfect contact case and MEE laminates with perfect contact case. The effect of non-uniform and imperfect contact, angle of incidence, unit cell size, volume fraction and ME-coupling on the wave propagation is illustrated through dispersion graphs. The effect of non-uniform and imperfect contact on dispersion curve serves as the highlight of the present work. [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]

On 18 March 2021 an earthquake of magnitude M w = 6.0 occurred offshore the Algerian coasts and generated a tsunami with offshore amplitudes smaller than a few millimetres crossing the western Mediterranean Sea. The objective of this study is threefold: first, to determine whether seismic sources calculated in the context of tsunami early warning are relevant; secondly, to determine whether tsunami simulations are able to reproduce tide-gauge observations and thirdly, to define the sensitivity of simulations to the grid resolutions and tsunami parameters. In the Mediterranean Sea, a very small number of available coastal tide gauges recorded the tsunami. Among them, a few French tide gauge stations recorded water waves with amplitudes smaller than a few centimetres and with periods ranging from 5 to 20 min associated to harbour or bay resonances. Numerical simulations of the tsunami are performed by the operational code Taitoko for seven different source fault models. Three of them allow for a rapid source detection and characterization in the framework of tsunami warning at CENALT (Centre National d'Alerte aux Tsunamis, France). The integrated code Taitoko uses a system of multiple nested grids. Standard Boussinesq equations are solved in the Mediterranean grid, whereas non-linear shallow water equations are solved in coastal and harbour grids with 25 and 5 m resolutions, respectively. Whatever the fault model, the observed time-series of water heights are reproduced satisfactorily both in phase and amplitude by the model at Nice and Monaco but poorly at Port Mahon (Minorca) and Toulon. [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 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 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]

This study explores optical soliton solutions within the framework of the Caudrey–Dodd–Gibbon equation (CDGE) and the ( 1 + 1 )-dimensional Boussinesq equation, utilizing the modified Sardar sub-equation method (MSSEM). The derived solutions are meticulously verified using symbolic software Mathematica and encompass a rich array of mathematical functions, including trigonometric, hyperbolic, and exponential functions. Visualization techniques, such as 3D plots, 2D plots, density graphs, and contour graphs, effectively illustrate the diverse behaviors exhibited by these soliton solutions. These equations have been studied previously, their comprehensive traveling wave solutions are relatively unexplored. By employing the modified Sardar sub-equation method, which has demonstrated effectiveness in solving nonlinear evolution equations, this study aims to bridge this gap. The soliton solutions obtained encompass a wide spectrum of behaviors, including bright, dark, periodic, singular, dark-periodic, bright-periodic, dark-singular, compactons, and bright-singular soliton solutions, as well as other complex phenomena. The comprehensive exploration of soliton solutions not only enhances our understanding of fundamental wave phenomena but also provides valuable insights for addressing nonlinear equations across various scientific disciplines, including optical fiber communications, plasma physics, and nonlinear dynamics. [ABSTRACT FROM AUTHOR]