Boussinesq-type Equations Research Articles

Novel (3 + 1)-Dimensional Variable-Coefficients Boussinesq-Type Equation: Exploring Integrability, Wronskian, and Grammian Solutions

Abstract In this paper, we introduce a novel (3+1)-dimensional variable-coefficients Boussinesq-type equation. We
analyze its integrability using the Painlev´e test and the N -soliton solution, demonstrating that both tests yield identical
conditions. Using the Hirota bilinear form of the equation, we derive Wronskian and Grammian determinant solutions
utilizing Pl¨ucker relations and the Jacobi identity for determinants. In particular, we use elementary transformation
and long wave limit to get the determinant expression of mth-order lump solutions from the 2mth-order Wronskian
determinant solutions. Furthermore, we reveal a variety of novel semi-rational solutions using the Hirota method and
Grammian determinant techniques.

On a non-local problem for a fractional differential equation of the Boussinesq type

In recent years, the fractional partial differential equation of the Boussinesq type has attracted much attention from researchers due to its practical importance. In this paper, we study a non-local problem for the Boussinesq type equation Dtαu(t)+A Dtαu(t)+ν2Au(t) =0, 0<t<T, 1<α<3∕2, where Dtα is the Caputo fractional derivative, and A is an abstract operator. In the classical case, i.e., when α = 2, this problem has been studied previously, and an interesting effect has been discovered: the existence and uniqueness of a solution depend significantly on the length of the time interval and the parameter ν. In this note, we show that in the case of a fractional equation, there is no such effect: a solution of the problem exists and is unique for any T and ν.

NUMERICAL SOLUTIONS OF BOUSSINESQ TYPE EQUATIONS BY MESHLESS METHODS

In this paper, two different meshfree method with radial basis functions (RBFs) is proposed to solve Boussinesq-type (Bq) equations. The basic conservative properties of the equation are investigated by computing the numerical values of the motion’s invariants. The accuracy of the method is tested using computational tests to simulate solitary waves in terms of L_∞ error norm. The outcomes are contrasted with analytical solution and a few other earlier studies in the literature. The results show that meshless methods are very effective and accurate.

Analyzing wave structure and bifurcation in geophysical Boussinesq-type equations

Analyzing wave structure and bifurcation in geophysical Boussinesq-type equations

Revisiting solutions of the Adler–Bobenko–Suris lattice equations and lattice Boussinesq-type equations

Revisiting solutions of the Adler–Bobenko–Suris lattice equations and lattice Boussinesq-type equations

Revisiting solutions of the Adler-Bobenko-Suris lattice equations and lattice Boussinesq-type equations

C использованием подхода матриц Коши заново рассматриваются решения всех решеточных уравнений из списка Адлера-Бобенко-Суриса, за исключением уравнения Q4, а также решения некоторых решеточных уравнений типа Буссинеска. Путем введения "ложного" неавтономного плосковолнового множителя получены солитонные, осциллирующие и частично осциллирующие решения рассматриваемых решеточных уравнений. В отличие от обычных солитонных решений осциллирующие решения принимают постоянные значения на всех элементарных четырехугольниках в плоскости $\mathbb{Z}^2$, что показывает их периодическую структуру.

Some novel different solutions for Boussinesq-type models including bright, singular, and dark soliton ones

Some new different kinds of one-soliton solutions for various forms of Boussinesq-type equations are presented in thispaper to describe the nonlinear wave phenomena in coastal and ocean areas such as tsunami waves. These one-solitonsolutions include bright, dark, and singular ones. The property of each solution in coastal and ocean engineering isexplained.

Boussinesq-type equations of hydroelastic waves in shallow water

Accurate computation of hydroelastic waves in shallow water is critical because many hydroelastic wave applications are nearshores, such as sea-ice and floating infrastructures. In this paper, Boussinesq assumptions for shallow water are employed to derive nonlinear Boussinesq-type equations of hydroelastic waves, in which non-uniform distribution of structural stiffness and varying water depth are considered rigorously. Application of Boussinesq assumptions enables complicated three-dimensional problems to be reduced and formulated on the two-dimensional horizontal plane, therefore the proposed Boussinesq-type models are straightforward and versatile for a wide range of hydroelastic wave applications. Two configurations, a floating plate and a submerged plate, are studied. The first-order linear governing equations are solved analytically with periodic conditions assuming constant depth and uniform stiffness, and the linear dispersion relations are subsequently derived for both configurations. For flexural-gravity waves of a floating plate, unique behaviours of flexural-gravity waves different from shallow-water waves are discussed, and a generalized solitary wave solution is investigated. A nonlinear numerical solver is developed, and nonlinear flexural-gravity waves are found to have smaller wavelength and celerity than their linear counterparts. For hydroelastic waves of a submerged plate, dual-mode analytical solutions are discovered for the first time. Numerical computation has demonstrated that a plate with decreasing submerged depth is able to transfer wave energy from the deeper water to the surface layer.

Landslide-generated wave simulation using coupled multi-phase flow and Boussinesq-type models

The effect of landslide-generated tsunami waves on coastal areas can be initially understood through a numerical simulation. Moreover, an efficient numerical simulation would be beneficial for predicting tsunami waves. However, a single numerical model that can capture the landslide and wave propagation is still impractical, especially in a large-scale domain. A domain-partitioning technique or coupling model would be more practical. This paper reports a numerical study of a landslide-generated wave, which uses a multi-phase CFD model for the landslide phenomena and a surface wave model based on Boussinesq-type equations for wave propagation. The proposed coupling model that sits between the multi-phase CFD and wave models validated by laboratory experiments is shown to be effective. The computation domain of the CFD model can be defined as small as it covers the impacted area of the landslide. The rest of the domain can be handled by the wave model that propagates the wave accurately with a low computation cost.

On the Non-local Problem for a Boussinesq Type Equations

On the Non-local Problem for a Boussinesq Type Equations

On the Painlevé integrability and nonlinear structures to a (3 + 1)-dimensional Boussinesq-type equation in fluid mediums: Lumps and multiple soliton/shock solutions

This work examines the Painlevé integrability of a (3 + 1)-dimensional Boussinesq-type equation. Using the Mathematica program, we rigorously establish Painlevé's integrability for the suggested problem. By utilizing Hirota's bilinear technique, we obtain the dispersion relations and phase shifts, which enable us to derive multiple soliton solutions. In addition, we systematically derive a wide range of lump solutions using the Maple symbolic computation. The investigation extends to encompass a variety of exact solutions with distinct structural features, including kink, periodic, singular, and rational solutions. This comprehensive analysis illustrates the profound richness of the model's dynamics and its potential to elucidate diverse nonlinear wave phenomena across various physical contexts. Therefore, the results that we will obtain play a vital role in understanding the mechanism of generation and propagation of many mysterious phenomena that arise in various scientific fields, including plasma physics, fluid mechanics, and the propagation of waves on the surfaces of seas and oceans to optical fibers.

Strong earthquakes and tsunami potential in the Hellenic Subduction Zone

The tsunamigenic potential of an earthquake depends on its size, source depth and focal mechanism. The Hellenic Subduction Zone (HSZ) has been selected in the paper to study this important issue. The HSZ was ruptured by 11 strong (Mw6.0) earthquakes in the time period 2009–2023. One earthquake ruptured onshore but only three out of ten offshore earthquakes produced tsunamis: 1 July 2009 (Mw6.4), 25 October 2018 (Mw6.8), 5 May 2020 (Mw6.6). For each one of the two more recent earthquakes of 5 May 2020 (tsunamigenic, thrust faulting) and 12 October 2021 (non-tsunamigenic, strike-slip faulting) we developed heterogeneous fault models from the inversion of teleseismic P-waveforms, and homogeneous fault models from published moment-tensor solutions. For each fault model tsunami generation and propagation was numerically simulated based on an advanced phase-resolving wave model with the use of higher-order Boussinesq-type equations. The modelled tsunami mareograms are consistent with tide records of the small tsunami (height ∼30 cm) produced by the 2020 earthquake. For the 2021 earthquake the modelled mareograms showed tsunami-like disturbance with amplitude not exceeding the noise level. The tsunamigenic earthquakes of 2009, 2018 and 2020 shared magnitude Mw≥ 6.4, shallow depth (h<20 km), moderate-to-high dip-angle and thrust faulting or oblique slip with significant thrust component. In the remaining seven non-tsunamigenic earthquakes, including the 2021 one, at least one of these features is missing. The results obtained help to better understand the seismic mechanisms of tsunami generation in the HSZ. Further investigation is needed for the central HSZ segment to the south of Crete Island, which historically has not been ruptured by large (Mw>7.0) tsunamigenic earthquakes. In contrast, the western and eastern HSZ segments ruptured by the large 365 AD and 1303 AD tsunamigenic earthquakes.

Families of exact solutions of a Generalized (2+1)-dimensional Boussinesq type equation

In this paper, we study a Generalized (2+1)-dimensional Boussinesq-type equation. Using the Hirota bilinear method, we present the N-order bright soliton solutions and dark soliton solutions. For the one-soliton solution, the bright soliton solution and the dark soliton solution share the same limit line but have different extreme values. Building on the soliton solutions, we derive higher-order bright and dark breather solutions as well as mixed solutions. The dynamic behavior is characterized using visual representations. Furthermore, through the long-wave limit method, we obtain the bright and dark lump solutions. Notably, they share the same extreme points but have different extreme values. Additionally, we derive two semi-rational solutions as lump-soliton and lump-breather. It is found that the lump moves along the peak amplitude of the soliton wave.

Numerical Modeling of Sediment Transport and Bed Evolution in Nonuniform Open-Channel Flows

Abstract The shallow-water and multilayer hydrostatic models have been commonly used to analyze the problems of a sediment-laden, plane open-channel flow. The models are adequate to solve a quasi-hydrostatic flow problem, but their accuracy deteriorates as the e ects of the vertical acceleration gain in significance. Herein, a higher-order numerical model for treating the problems of unsteady, plane open-channel flow over a movable bed is proposed. In this model, the flow hydrodynamics is governed by the depth-averaged Boussinesq-type equations, and the bed morphodynamics is determined by an Exner-type equation and additional equations describing the non-equilibrium transports of suspended load and bed load. A hybrid finite-volume and finite-di erence scheme was used to discretize and solve the governing equations, yielding solutions that are in satisfactory agreement with the experimental data. Overall, the results of the proposed model for the temporal free-surface profile and bed evolution were fairly adequate. For the two particular cases considered, however, the quality of its results was moderately a ected by the e ects of the three-dimensional characteristics of the dam-break flow and the sliding of the dike body due to sub-surface flow. The results of this investigation highlight the importance of including a higher-order Boussinesq-type correction for refining sediment transport computations.

High-Order Boussinesq Equations for Water Wave Propagation in Porous Media

To accurately capture wave dynamics in porous media, the higher-order Boussinesq-type equations for wave propagation in deep water are derived in this paper. Starting with the Laplace equations combined with the linear and nonlinear resistance force of the dynamic conditions on the free surface, the governing equations were formulated using various independent velocity variables, such as the depth-averaged velocity and the velocity at the still water level and at an arbitrary vertical position in the water column. The derived equations were then improved, and theoretical analyses were carried out to investigate the linear performances with respect to phase celerity and damping rate. It is shown that Boussinesq-type models with Padé [4, 4] dispersion can be applied in deep water. A numerical implementation for one-dimensional equations expressed with free surface elevation and depth-averaged velocity is presented. Solitary wave propagation in porous media was simulated, and the computed results were found to be generally in good agreement with the measurements.

Detecting delamination via nonlinear wave scattering in a bonded elastic bar

In this paper we examine the effect of delamination on wave scattering, with the aim of creating a control measure for layered waveguides of various bonding types. Previous works have considered specific widths of solitary waves for the simulations, without analysing the effect of changing the soliton parameters. We consider two multi-layered structures: one containing delamination ‘sandwiched’ by perfect bonding and one containing delamination but ‘sandwiched’ by soft bonding. These structures are modelled by coupled Boussinesq-type equations. Matched asymptotic multiple-scale expansions lead to coupled Ostrovsky equations in soft bonded regions and Korteweg-de Vries equations in the perfectly bonded and delaminated region. We use the Inverse Scattering Transform to predict the behaviour in the delaminated regions. In both cases, numerical analysis shows that we can predict the delamination length by changes in the wave structure, and that these changes depend upon the Full Width at Half Magnitude (FWHM) of the incident soliton. In the case of perfect bonding, we derive a theoretical prediction for the change and confirm this numerically. For the soft bonding case, we numerically identify a similar relationship using the change in amplitude. Therefore we only need to compute one curve to determine the behaviour for any incident solitary wave, creating a framework for designing measurement campaigns for rigorously testing the integrity of layered structures.

On the solvability of a nonlocal problem for a Boussinesq-type differential equation

We study a nonlocal problem for a differential Boussinesq-type equations in a multidimensional domain. Conditions for the existence and uniqueness of the solution are established, and a spectral decomposition of the solution is obtained.

Dispersive transverse waves for a strain-limiting continuum model

It is well known that propagation of waves in homogeneous linearized elastic materials of infinite extent is not dispersive. Motivated by the work of Rubin, Rosenau, and Gottlieb, we develop a generalized continuum model for the response of strain-limiting materials that are dispersive. Our approach is based on both a direct inclusion of Rivlin–Ericksen tensors in the constitutive relations and writing the linearized strain in terms of the stress. As a result, we derive two coupled generalized improved Boussinesq-type equations in the stress components for the propagation of pure transverse waves. We investigate the traveling wave solutions of the generalized Boussinesq-type equations and show that the resulting ordinary differential equations form a Hamiltonian system. Linearly and circularly polarized cases are also investigated. In the case of unidirectional propagation, we show that the propagation of small-but-finite amplitude long waves is governed by the complex Korteweg–de Vries (KdV) equation.

An Unorthodox Arrangement of Boussinesq-Type Wave Equations for Accurate and Robust Numerical Treatment

A set of Boussinesq-type wave equations with enhanced dispersion characteristics is presented for accurate, efficient, and robust numerical treatment. New arrangement uses three different velocity variables simultaneously in order to keep continuity and momentum equations in simplest conservation forms while improving the dispersion characteristics. This approach allows us to retain all the nonlinear contributions with minimum number of terms. Spatial and time-dependent variations of the seabed are fully accounted for and the effect of external free surface pressure is included. A numerical scheme based on finite differences is developed, and various well-known experimental cases are simulated for testing the performance of the proposed set of equations. Comparisons of simulations with measurements reveal quite satisfactory agreements and, hence, bolster confidence in the wave model.

On the Solvability of a Nonlocal Problem for a Boussinesq-Type Differential Equation

On the Solvability of a Nonlocal Problem for a Boussinesq-Type Differential Equation