Boussinesq Type Research Articles

Solitonic Analysis of the Newly Introduced Three-Dimensional Nonlinear Dynamical Equations in Fluid Mediums

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.

Ground state solutions for elliptic Kirchhoff–Boussinesq type problems with supercritical exponential growth

In this work, we are interested in studying the existence of a ground state solution for the problem Δ(wβ(x)Δu)±div(wβ(x)|∇u|p−2∇u)=f(x,u) in B,andu=∂u∂ν=0on∂B,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\Delta (w_{\\beta}(x)\\Delta u ) \\pm \ ext{div}(w_{\\beta} (x) | \ abla u |^{p-2} \ abla u)= f(x,u) \ ext{ in }B,\\quad \ ext{and} \\ u= \\dfrac{\\partial u}{\\partial \ u}=0 \\ \ ext{on}\\ \\partial B, $$\\end{document} where B is the unit ball in R4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathbb{R}^{4}$\\end{document}, wβ(x)=(loge|x|)β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$w_{\\beta}(x)=\\big(\\log \\frac{e}{|x|}\\big)^{\\beta}$\\end{document} or wβ(x)=(log(1|x|)β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$w_{\\beta}(x)=\\big(\\log (\\frac{1}{|x|}\\big)^{\\beta}$\\end{document} for β∈(0,1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\beta \\in (0,1)$\\end{document}, 2<p<4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$2< p < 4$\\end{document} and f:R→R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$f:\\mathbb{R}\\rightarrow \\mathbb{R}$\\end{document} is a superlinear continuous class function with supercritical exponential growth. We utilize the Nehari manifold method to establish the existence result.

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&lt;t&lt;T, 1&lt;α&lt;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.

The impact of modulational instability on coastal wave forecasting using quadratic models

Coastal wave forecasting over large spatial scales is essential for many applications (e.g., coastal safety assessments, coastal management and developments, etc.). This demand explains the necessity for accurate yet effective models. A well-known efficient modelling approach is the quadratic approach (often referred to as frequency-domain models, nonlinear mild-slope models, amplitude models, etc.). The efficiency of this approach stems from a significant modelling reduction of the original governing equations (e.g., Euler equations). Most significantly, the description of wave nonlinearity essentially collapses into a single mode coupling term determined by the quadratic interaction coefficients. As a result, it is expected that the efficiency achieved by the quadratic approach is accompanied by a decrease in prediction accuracy. In order to gain further insight into the predictive capabilities of this modelling approach, this study examines six different quadratic formulations, three of which are of the Boussinesq type and the other three are referred to as fully dispersive. It is found that while the Boussinesq formulations reliably predict the evolution of coastal waves, the predictions by the fully dispersive formulations tend to be affected by false developments of modulational instability. Consequently, the predicted wave fields by the fully dispersive formulations are characterized by unexpectedly strong modulations of the sea-swell part and associated unexpected infragravity response. The impact of the modulational instability on wave prediction based on the quadratic approach is further demonstrated using existing laboratory results of bichromatic and irregular wave conditions.

On the Non-local Problem for a Boussinesq Type Equations

On the Non-local Problem for a Boussinesq Type Equations

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.

An operator-splitting approach with a hybrid finite volume/finite difference scheme for extended Boussinesq models

In this paper, we present a hybrid numerical scheme combining finite volume and finite difference methods (FV/FD), for solving two extended Boussinesq-type (BT) models. We propose two new reformulations specially designed to allow a separation between the hyperbolic part and dispersive part of the equations. Taking advantage of the proposed reformulations, we employ a splitting scheme specifically tailored to comprehensively capture the broad spectrum of natural events that are observed in the study of coastal oceanography. We employ a high-order FV well-balanced scheme to effectively handle the hyperbolic portion of the equations, guaranteeing non-negativity of water depth and accommodating dry regions. We solve the remaining dispersive component by employing a traditional FD approach. Several test cases in one horizontal dimension are validated showing that the proposed approach is able to accurately replicate coastal wave dynamics. In comparison to the non-linear shallow water equations (NSWE), the BT models exhibit significantly enhanced accuracy in simulating highly dispersive waves across varying water depths.

Oscillatory and regularized shock waves for a dissipative Peregrine–Boussinesq system

Abstract We consider a dissipative, dispersive system of the Boussinesq type, which describes wave phenomena in scenarios where dissipation plays a significant role. Examples include undular bores in rivers or oceans, where turbulence-induced dissipation significantly influences their behavior. In this study, we demonstrate that the proposed system admits traveling wave solutions known as diffusive-dispersive shock waves. These solutions can be categorized as oscillatory and regularized shock waves, depending on the interplay between dispersion and dissipation effects. By comparing numerically computed solutions with laboratory data, we observe that the proposed model accurately captures the behavior of undular bores over a broad range of phase speeds. Traditionally, undular bores have been approximated using the original Peregrine system, which, even though it doesn’t possess these as traveling wave solutions, tends to offer accurate approximations within suitable time scales. To shed light on this phenomenon, we demonstrate that the discrepancy between the solutions of the dissipative Peregrine system and the non-dissipative counterpart is proportional to the product of the dissipation coefficient and the observation time.

On an elliptic Kirchhoff–Boussinesq type problems with exponential growth

In this paper, we prove an existence result of nontrivial solutions for the problem where is a smooth bounded domain, and is a superlinear continuous function with exponential subcritical or critical growth. We apply the Nehari manifold method in order to prove the main result.

FREQUENCY DISPERSION IN DEPTH-INTEGRATED MODELS THROUGH MACHINE LEARNING SURROGATES

Boussinesq- type wave models have the accuracy to resolve wave propagation in coastal zones, having the ability to capture nearshore dynamics that include both nonlinear and dispersive effects for relatively short waves. The accuracy of Boussinesq type models over their counterparts which utilize the non- linear shallow water (NLSW) equations provides a clear advantage in studying nearshore processes. However, the computational expense of finding the Boussinesq solution over the NLSW solution hinders fast and/ or real time simulation using Boussinesq type models.

Numerical approaches for Boussinesq type equations with its application in Kampar River, Indonesia

In this study, we examine numerous numerical approaches for solving the 1-Dimensional Boussinesq problem. The proposed methods to be discussed include Mohapatra and Chaudhry’s two-four finite difference scheme, the modified two-four finite difference scheme, and the staggered finite volume scheme. The modified two-four finite difference scheme and staggered finite volume scheme has a shorter computational time than the Mohapatra–Chaudhry scheme, which reduces the computational cost. We compare the performance of each numerical scheme against the analytical solutions and approximate solutions using different type of numerical method that is called the MUSCL4 scheme. Furthermore, the calculated results are compared and utilized to assess the contribution of each individual Boussinesq term. Each Boussinesq term was evaluated to examine its affect on the Boussinesq equation’s calculated result. Further, we also implement the numerical schemes that we formulate to investigate the undular bore phenomena in Kampar River. This finding may be useful to those who use the Boussinesq equation to study fluid or wave phenomena.

On the Fourth-Order Lattice Gel'fand-Dikii Equations

The fourth-order lattice Gel'fand-Dikii equations in quadrilateral form are investigated. Utilizing the direct linearization approach, we present some equations of the extended lattice Gel'fand-Dikii type. These equations are related to a quartic discrete dispersion relation and can be viewed as higher-order members of the extended lattice Boussinesq type equations. The resulting lattice equations given here are in five-component form, and some of them are multi-dimensionally consistent by introducing extra equations. Lax integrability is discussed both by direct linearization scheme and also through multi-dimensional consistent property. Some reductions of the five-component lattice equations to the four-component forms are considered.

Comparative Study on Numerical Simulation of Wave-Current Nonlinear Interaction Based on Improved Mass Source Function

In coastal waters, wave propagation is often affected by rivers and tides. The wave current interaction increases the complexity of the wave propagation. In this study, we consider the Boussinesq type equation with an improved dispersion term as the governing equation and establish a numerical model of wave propagation in the coexistence of wave current environment. Firstly, we use the MIKE 21 BW model to simulate the propagation of dual-frequency waves. The Navier–Stokes equation wave model is used to verify the results and the Fourier transform is used to analyze and discuss the dual-frequency waves. Our findings show that the numerical model established by the Boussinesq equation can better describe the nonlinear interaction between waves more accurately at a much higher computational efficiency compared with the Navier–Stokes equation wave model. In addition, we set the constant current source point in the wave numerical model and conduct the numerical simulation of waves in the current environment, by improving the mass source wave generation method. The numerical simulation of wave-current interactions between uniform and variable water depths is performed, thus demonstrating its capability to describe accurately the influence of water flow on wave propagation.

Optical-terahertz solitons with tilted wave fronts of system of the Zakharov–Boussinesq type equations

The generation of terahertz radiation in a quadratically nonlinear medium by optical pulses with tilted wavefronts is investigated. A system of equations describing this process is derived. Solutions of this system in the form of optical-terahertz solitons are obtained. The possibility of the existence of a new type of solitons—dispersionless solitons is shown.

Optical-Terahertz Solitons with Tilted Wave Fronts of a System of Equations of the Zakharov–Boussinesq Type

Optical-Terahertz Solitons with Tilted Wave Fronts of a System of Equations of the Zakharov–Boussinesq Type

Elliptic Solutions of Boussinesq Type Lattice Equations and the Elliptic N{textrm{th}} Root of Unity

We establish an infinite family of solutions in terms of elliptic functions of the lattice Boussinesq systems by setting up a direct linearisation scheme, which provides the solution structure for those equations in the elliptic case. The latter, which contains as main structural element a Cauchy kernel on the torus, is obtained from a dimensional reduction of the elliptic direct linearisation scheme of the lattice Kadomtsev–Petviashvili equation, which requires the introduction of a novel technical concept, namely the ‘elliptic cube root of unity’. Thus, in order to implement the reduction we define, more generally, the notion of ellipticN{textrm{th}}root of unity, and discuss some of its properties in connection with a special class of elliptic addition formulae. As a particular concrete application we present the class of elliptic multi-soliton solutions of the lattice Boussinesq systems.

Integrability and solutions for a fourth-order lattice Gel’fand–Dikii equation

In this letter, we investigate a lattice fourth-order Gel’fand–Dikii quadrilateral system. It is an extension of Hietarinta’s (B-2) equation of the discrete Boussinesq type. We show that the system is multidimensionally consistent, which determines its discrete integrability and gives rise to a Bäcklund transformation. A seed solution and a one-soliton solution are derived using the Bäcklund transformation. The plane wave factor in the one-soliton solution indicates this system is related to a quartic polynomial, which characterizes discrete dispersion relations. Dynamics and asymptotic behaviors are illustrated.

Multiple-order rogue wave solutions to a (2+1)-dimensional Boussinesq type equation

In this paper, based on the Hirota bilinear method and symbolic computation approach, multiple-order rogue waves of (2+1)-dimensional Boussinesq type equation are constructed. The reduced bilinear form of the equation is deduced by the transformation of variables. Three kinds of rogue wave solutions are derived by means of bilinear equation. The maximum and minimum values of the first-order rogue wave solution are given at a specific moment. Furthermore, the second-order and third-order rogue waves are explicitly derived. The dynamic characteristics of three kinds of rogue wave solutions are shown by three-dimensional plot.

Complex solutions to the higher-order nonlinear boussinesq type wave equation transform

Complex solutions to the higher-order nonlinear boussinesq type wave equation transform