KdV Burgers Research Articles

Effects of viscosity and induced magnetic fields on weakly nonlinear wave transmission in a viscoelastic tube using physics-informed neural networks

This study presents an advanced deep learning methodology that utilizes physics-informed neural networks (PINNs), to analyze the transmission of weakly nonlinear waves in a prestressed viscoelastic arterial tube. Using the long wave approximation, a mathematical model is constructed to replicate the propagation of weakly nonlinear waves in a viscoelastic arterial tube filled with viscous nanofluid, taking into account the influence of an induced magnetic field. The perturbed Burger, perturbed Korteweg–de Vries, and perturbed Korteweg–de Vries-Burgers equations are formulated based on the combined effects of nanofluid viscosity and the applied magnetic field using the reductive perturbation technique. Semi-supervised physics-informed neural network models are utilized to solve these perturbed evolutionary equations, trained on a limited dataset within their rectangular domain of definition. Gaussian process-based Bayesian optimization is used to determine the hyperparameters of the neural network, ensuring optimal model is performance. The effectiveness of the optimal models is evaluated by calculating the residual losses associated with the perturbed partial differential equations (PDEs). Visual representations of weakly non-linear wave propagation, considering nanofluid viscosity and induced magnetic fields, enhance the comprehension of dissipative effects in the cardiovascular system. These insights aid in obtaining precise measurements of pulse wave velocity for cardiovascular health monitoring. Consequently, the application of PINN proves to be a valuable tool for solving real-world PDEs and highlights its importance in advancing medical machine learning fields.

Radical Petrov–Galerkin Approach for the Time-Fractional KdV–Burgers’ Equation

Radical Petrov–Galerkin Approach for the Time-Fractional KdV–Burgers’ Equation

Investigating the dynamics, synchronization of a novel chaotic system reduced from a modified KdV–Burgers–Kuramoto equation

Investigating the dynamics, synchronization of a novel chaotic system reduced from a modified KdV–Burgers–Kuramoto equation

On asymptotics of the solution to the Cauchy problem for a singularly perturbed operator-differential transport equation with weak diffusion in the case of several space variables

A formal asymptotic expansion is constructed for the solution of the Cauchy problem for a singularly perturbed operator differential transport equation with small nonlinearities and weak diffusion in the case of several space variables. Under the conditions imposed on the data of the problem, the leading asymptotic term is described by the multidimensional generalized Burgers–Korteweg–de Vries equation. Under certain conditions, the remainder is estimated with respect to the residual.

Uniform solitary wave theory for viscous flow over topography

The flow of a density stratified fluid over obstacles has been intensively explored from a natural and scientific point of view. This flow has been successfully governed by using the forced Korteweg–de Vries-Burgers equation that generated solitons in a viscous flow. This is done by adding the viscous term beyond the Korteweg–de Vries approximation. It is based on the conservation laws of the Korteweg–de Vries-Burgers equation for mass and energy, and assumes that the upstream wavetrains are composed of solitary waves. Our results show that the influence of viscosity plays a key role in determining the upstream solitary wave amplitude of the bore. A good comparison is obtained between the numerical and analytical solutions.

Electron-acoustic solitary waves via suprathermal populations in Saturn's magnetosphere: Bifurcation, sensitivity, and stability analysis

Cassini and Voyager space missions observed non-thermal electron populations (with varying characteristics) in Saturn's magnetosphere, which can be correctly described using kappa distributions. Based on these observations, our objective is to inspect the evolution of electron-acoustic solitary waves (EASWs) within Saturn's magnetosphere. The propagation of weakly nonlinear (EASWs) in a collisional plasma system comprising a cold electron fluid, hot electrons following a kappa distribution, and stationary ions is investigated. By employing the reductive perturbation technique, the Korteweg–de Vries Burgers (KdV–B) equation is derived. An exact solution of the KdV–B equation, with a conformable time-derivative, is found using the unified method. It is observed that plasma current-induced collision between electrons and ions leads to remarkable dissipation, generating EASWs. Furthermore, when studying the sensitivity of the system, the appearance of a positive potential is depicted as external forces vanish, which may be due to stationary ions. Additionally, bifurcation, stability, and significant influence of plasma characteristics are considered.

Evolution of water wave packets by wind in shallow water

We use the Korteweg–de Vries (KdV) equation, supplemented with several forcing/friction terms, to describe the evolution of wind-driven water wave packets in shallow water. The forcing/friction terms describe wind-wave growth due to critical level instability in the air, wave decay due to laminar friction in the water at the air–water interface, wave stress in the air near the interface induced by a turbulent wind and wave decay due to a turbulent bottom boundary layer. The outcome is a modified KdV–Burgers equation that can be a stable or unstable model depending on the forcing/friction parameters. To analyse the evolution of water wave packets, we adapt the Whitham modulation theory for a slowly varying periodic wave train with an emphasis on the solitary wave train limit. The main outcome is the predicted growth and decay rates due to the forcing/friction terms. Numerical simulations using a Fourier spectral method are performed to validate the theory for various cases of initial wave amplitudes and growth and/or decay parameter ranges. The results from the modulation theory agree well with these simulations. In most cases we examined, many solitary waves are generated, suggesting the formation of a soliton gas.

Boundary controllability for the Korteweg–de Vries–Burgers equation on a finite domain

Boundary controllability for the Korteweg–de Vries–Burgers equation on a finite domain

Impact of Ion Pressure Anisotropy in Collisional Quantum Magneto-Plasma with Heavy and Light Ions

We have examined collisional degenerate plasma composed of charged state of heavy positive ion and light positive as well as negative ion. Employing the reductive perturbation method, we derived the damped Korteweg-de Vries-Burgers (dKdV-B) equation and by using its standard solution we analyze the characteristics of the solitary-shock profile under varying parameters. Furthermore, with the application of planar dynamical systems bifurcation theory, the phase portraits have been analyzed. This dynamical system analysis allowed us to extract important information on the stability of these structures as represented by the dKdV-B equation.

Boundary stabilisation of stochastic delay impulsive Korteweg–de Vries–Burgers equations

Boundary stabilisation is addressed for stochastic delay impulsive Korteweg–de Vries–Burgers (SDIKdVB) equations. Initially, a novel boundary controller is devised. Utilizing the Lyapunov–Krasovskii functional approach and the impulsive theory, a sufficient condition is derived for achieving mean-square exponential stabilisation (MSES) of SDIKdVB equations. Secondly, the scenario is investigated in which uncertainties exist in the system parameters, and a sufficient condition is presented to attain the robust MSES. Additionally, the investigation is conducted on the impact of the diffusion term, the dispersion term, impulsive strengths and impulsive intervals on MSES. Lastly, the effectiveness of the theoretic findings is demonstrated via practical applications in both biomechanics and physics.

Diverse variety of exact solutions for some nonlinear models via the [formula omitted]-expansion method

In this article, we explore several significant nonlinear physical models, including the Benjamin–Bona–Mahony–Peregrine–Burgers (BBMPB) equation, the Burgers–Korteweg–De Vries (BK) equation, the one-dimensional Oskolkov (OSK) equation, the Klein–Gordon (KG) equation with quadratic non-linearity, and the improved Boussinesq (IB) equation. Utilizing the (G′G)-expansion method ansatz, we derive new exact traveling wave solutions for these models. These solutions, expressed in the forms of rational, hyperbolic, and trigonometric functions, present a novel contribution distinct from existing literature. The physical dynamics of these solutions are elucidated through Mathematica simulations.

Boundary Stabilization of the Korteweg-de Vries-Burgers Equation with an Infinite Memory-Type Control and Applications: A Qualitative and Numerical Analysis

Boundary Stabilization of the Korteweg-de Vries-Burgers Equation with an Infinite Memory-Type Control and Applications: A Qualitative and Numerical Analysis

Magnetosonic shock waves in degenerate electron–positron–ion plasma with separated spin densities

This study investigates magnetosonic shock waves in a spin-polarized three-component quantum plasma using the quantum magnetic hydrodynamic model. We explore the influence of spin effects, specifically spin magnetization current and spin pressure, on shock wave behavior. Numerical analysis of the linear dispersion relation under varying parameters such as positron imbalance, spin polarization ratio, plasma beta, quantum diffraction, and magnetic diffusivity reveals differential impacts, with diffusion exerting significant influence on the plasma frequency. Our findings highlight the sensitivity discrepancy between the real and imaginary parts of the dispersion relation. Furthermore, nonlinear behavior of magnetosonic shock waves is examined via the Korteweg–de Vries–Burgers equation, showcasing transitions between oscillatory and monotonic wave patterns based on changes in dimensionless parameters. Notably, we observe the combined effects of spin-up and spin-down positrons with spin-up and spin-down electrons on shock wave dynamics, contributing to a deeper understanding of spin-plasma interactions with implications across various fields.

Exploring dust-ion acoustic shocks in a plasma in the light of phase plane analysis

This paper presents a comprehensive exploration of shock waves in a dusty plasma, incorporating the dynamics of inertial dust and ions with Maxwellian distributed electrons. By deriving the Korteweg-de Vries-Burgers (KdV-B) equation and by using its standard solution we analyze the characteristics of the shock front under varying parameters. Furthermore, we conduct a phase plane analysis to elucidate the system's behavior and dynamics by tuning in the parameters that provide various types of perturbation in the system. The investigation sheds light on the intricate behavior of shock waves in dusty plasmas and contributes valuable insights to the understanding of plasma dynamics.

Convergence rate toward shock wave under periodic perturbation for generalized Korteweg–de Vries–Burgers equation

In this paper, a viscous shock wave under space-periodic perturbation of generalized Korteweg–de Vries–Burgers equation is investigated. It is shown that if the initial periodic perturbation around the viscous shock wave is small, then the solution time asymptotically tends to a viscous shock wave with a shift partially determined by the periodic oscillations. Moreover the exponential time decay rate toward the viscous shock wave is also obtained for some certain perturbations.

Innovative coupling of s-stage one-step and spectral methods for non-smooth solutions of nonlinear problems

The behavior of nonlinear dynamical systems arising in mathematical physics through numerical tools is a challenging task for researchers. In this context, an efficient semi-spectral method is proposed and applied to observe the robust solutions for the mathematical physics problems. Firstly, the space variable is approximated by the Vieta-Lucas polynomials and then the s-stage one-step method is applied to discretize the temporal variable which transfers the problem in the form Cn+1=Cn+Δtϕ(x,t,Cn,F(un)). Novel operational matrices of integer order are developed to replace the spatial derivative terms presented in the discussed problem. Related theorems are included in the study to validate the approach mathematically. The proposed semi-spectral schemes convert the considered nonlinear problem to a system of linear algebraic equations which is easier to tackle. We also accomplish an investigation on the error bound and convergence to confirm the mathematical formulation of the computational algorithm. To show the accuracy and effectiveness of the suggested computational method numerous test problems, such as the advection-diffusion problem, generalized Burger-Huxley, sine-Gordon, and modified KdV–Burgers’ equations are considered. An inclusive comparative examination demonstrates the currently suggested computational method in terms of credibility, accuracy, and reliability. Moreover, the coupling of the spectral method with the fourth-order Runge-Kutta method seems outstanding to handle the nonlinear problem to examine the precise smooth and non-smooth solutions of physical problems. The computational order of convergence (COC) is computed numerically through numerous simulations of the proposed schemes. It is found that the proposed schemes are in exponential order of convergence in the spatial direction and the COC in the temporal direction validates the studies in the literature.

Exact solutions of stochastic Burgers–Korteweg de Vries type equation with variable coefficients

We will present exact solutions for three variations of the stochastic Korteweg de Vries–Burgers (KdV–Burgers) equation featuring variable coefficients. In each variant, white noise exhibits spatial uniformity, and the three categories include additive, multiplicative, and advection noise. Across all cases, the coefficients are time-dependent functions. Our discovery indicates that solving certain deterministic counterparts of KdV–Burgers equations and composing the solution with a solution of stochastic differential equations leads to the exact solution of the stochastic Korteweg de Vries–Burgers (KdV–Burgers) equations.

Instability of periodic waves for the Korteweg–de Vries–Burgers equation with monostable source

In this paper, it is proved that the KdV–Burgers equation with a monostable source term of Fisher–KPP type has small-amplitude periodic traveling wave solutions with finite fundamental period. These solutions emerge from a subcritical local Hopf bifurcation around a critical value of the wave speed. Moreover, it is shown that these periodic waves are spectrally unstable as solutions to the PDE, that is, the Floquet (continuous) spectrum of the linearization around each periodic wave intersects the unstable half plane of complex values with positive real part. To that end, classical perturbation theory for linear operators is applied in order to prove that the spectrum of the linearized operator around the wave can be approximated by that of a constant coefficient operator around the zero solution, which intersects the unstable complex half plane.

Bifurcation analysis of electron-acoustic waves in electron beam-plasma with suprathermal electrons

It has investigated the properties of nonlinear electron-acoustic waves propagating in a four-component collisionless, unmagnetized plasma. The system consists of cool inertial background electrons of temperature T c , the cool inertial electron beam of temperature T b , hot inertialess suprathermal electrons modeled by a κ-distribution of temperature T h , and ions. The nonlinear modified Korteweg-de Vries-Burgers (mKdV-Burgers) equation that characterizes the electron acoustic waves in such a plasma model is derived. The wave solutions of the resultant mKdV-Burgers equation are investigated using planar dynamical system bifurcation theory. The unperturbed hot, cool, and beam electron densities and the superthermal electron parameter are revealed to have a considerable influence on the nonlinear electron-acoustic traveling waves and the associated electric fields. The system evolves also monotonic shock waves when the dissipation term dominates over the dispersion term. The current model might be useful for understanding the generation of nonlinear electron-acoustic waves in a variety of astrophysical plasma settings, including the Earth's magnetosphere.

Magnetoacoustic waves in spin-1/2 dense quantum degenerate plasma: nonlinear dynamics and dissipative effects

Abstract Within the confines of a two-fluid quantum magnetohydrodynamic model, the investigation of magnetoacoustic shock and solitary waves is conducted in an electron-ion magnetoplasma that considers electrons of spin 1/2. When the plasma system is nonlinearly investigated using the reductive perturbation approach, the Korteweg de Vries-Burgers (KdVB) equation is produced. Sagdeev’s potential is created, revealing the presence of solitary solutions. However, when dissipative terms are included, intriguing physical solutions can be obtained. The KdVB equation is further investigated using the phase plane theory of a planar dynamical system to demonstrate the existence of periodic and solitary wave solutions. Predicting several classes of traveling wave solutions is advantageous due to various phase orbits, which manifest as soliton-shock waves, and oscillatory shock waves. The presence of a magnetic field, the density of electrons and ions, and the kinematic viscosity significantly alter the properties of magnetoacoustic solitary and shock waves. Additionally, electric fields have been identified. The outcomes obtained here can be applied to studying the nature of magnetoacoustic waves that are observed in compact astrophysical environments, where the influence of quantum spin phenomena remains significant, and also in controlled laboratory plasma experiments.