Burgers System Research Articles

Lie symmetries, exact solutions and conservation laws of time fractional Boussinesq–Burgers system in ocean waves

In this paper, the Lie symmetry analysis method is applied to the time-fractional Boussinesq–Burgers system which is used to describe shallow water waves near an ocean coast or in a lake. We obtain all the Lie symmetries admitted by the system and use them to reduce the fractional partial differential equations with a Riemann–Liouville fractional derivative to some fractional ordinary differential equations with an Erdélyi–Kober fractional derivative, thereby getting some exact solutions of the reduced equations. For power series solutions, we prove their convergence and show the dynamic analysis of their truncated graphs. In addition, the new conservation theorem and the generalization of Noether operators are developed to construct the conservation laws for the equations studied.

Solving the System of Two-Dimensional Coupled Burgers' Equations byModified Variational Iteration Method with Genetic Algorithm

Solving the System of Two-Dimensional Coupled Burgers' Equations byModified Variational Iteration Method with Genetic Algorithm

Higher-dimensional integrable deformations of the classical Boussinesq–Burgers system* *This work is supported by the National Natural Science Foundation of China (Grant Nos. 11871396, 12271433).

In this paper, the (1+1)-dimensional classical Boussinesq–Burgers (CBB) system is extended to a (4+1)-dimensional CBB system by using its conservation laws and the deformation algorithm. The Lax integrability, symmetry integrability and a large number of reduced systems of the new higher-dimensional system are given. Meanwhile, for illustration, an exact solution of a (1+1)-dimensional reduced system is constructed from the viewpoint of Lie symmetry analysis and the power series method.

A numerical method for two-dimensional transient nonlinear convection-diffusion equations

In this research, we have developed the half-boundary method (HBM) for nonlinear convection-diffusion equations (CDEs), which hold significant importance in nuclear power engineering. The HBM employs a variable relationship between the nodes within the computational domain and the nodes located on half of the boundaries. This approach offers notable benefits, includingthe reduction ofthe maximum matrix order and the optimization the maximum memory storage for calculations. Moreover, the HBM is an efficient and streamlined approach to directly handle Neumann boundary conditions, thanks to the utilization of mixed variables. We primarily investigate the memory usage and accuracy of the proposed algorithm in the unsteady-state CDEs, in context of material nonlinear CDEs, the Burgers’ equation and the system of Burgers' equations. The numerical results obtained demonstrate the method’s potential in simulating flow and heat transfer phenomena, particularly in situations where convection is dominant.

Invariant analysis, approximate solutions, and conservation laws for the time fractional higher order Boussinesq–Burgers system

In this paper, we employ the time fractional higher order nonlinear Boussinesq–Burgers system to investigate shallow water waves based on the Riemann–Liouville and Caputo fractional partial derivatives. The Lie algebra of infinitesimal generators associated with the symmetry groups that leave the studied system invariant is systematically constructed, and the similarity reductions are established. Furthermore, conserved quantities of the system under consideration are formulated. Next, the power series solution, including the convergence analysis, is obtained. Based on the fractional Sumudu–Adomian decomposition method, some approximate solutions are derived. Finally, in order to show the efficiency of the considered methods, the effect of the fractional order, as well as the dynamical behavior of solutions, numerical validation and graphical representations are designed.

Universality in coupled stochastic Burgers systems with degenerate flux Jacobian

In our contribution we study stochastic models in one space dimension with two conservation laws. One model is the coupled continuum stochastic Burgers equation, for which each current is a sum of quadratic nonlinearities, linear diffusion, and spacetime white noise. The second model is a two-lane stochastic lattice gas. As distinct from previous studies, the two conserved densities are tuned such that the flux Jacobian, a 2×2 matrix, has coinciding eigenvalues. In the steady state, spacetime correlations of the conserved fields and the time-integrated currents at the origin are investigated. For a particular choice of couplings, the dynamical exponent 3/2 is confirmed. Furthermore, at these couplings, the continuum stochastic Burgers equation and lattice gas are demonstrated to be in the same universality class.

A Numerical Solution of a Coupling System of Conformable Time-Derivative Two-Dimensional Burgers’ Equations

In this paper, we deal with a numerical solution of a coupling system of fractional conformable time-derivative two-dimensional (2D) Burgers’ equations. The presence of both the fractional time derivative and the nonlinear terms in this system of equations makes solving it more difficult. Firstly, we use the Cole-Hopf transformation in order to reduce the coupling system of equations to a conformable time-derivative 2D heat equation for which the numerical solution is calculated by the explicit and implicit schemes. Secondly, we calculate the numerical solution of the proposed system by using both the obtained solution of the conformable time-derivative heat equation and the inverse Cole-Hopf transformation. This approach shows its efficiency to deal with this class of fractional nonlinear problems. Some numerical experiments are displayed to consolidate our approach

Considering the wave processes in oceanography, acoustics and hydrodynamics by means of an extended coupled (2+1)-dimensional Burgers system

Recent activities in oceanography, acoustics and hydrodynamics are impressive. In respect to oceanography, acoustics and hydrodynamics, the Burgers-type systems/equations attract people’s attention, including an extended coupled (2+1)-dimensional Burgers system describing the wave processes in oceanography, acoustics and hydrodynamics, which we investigate hereby. We construct two families of the hetero-Bäcklund transformations, each from that system to a (2+1)-dimensional Broer-Kaup-Kupershmidt system, as well as a family of the similarity reductions, from that system to a known ordinary differential equation. We take notice of all our results dependent on the coefficients in that system.

Pushed fronts in a Fisher–KPP–Burgers system using geometric desingularization

Pushed fronts in a Fisher–KPP–Burgers system using geometric desingularization

From decoupled integrable models to coupled ones via a deformation algorithm* *Supported by The National Natural Science Foundation (Nos. 12235007, 12090020, 11975131, 12090025).

By using a reconstruction procedure of conservation laws of different models, the deformation algorithm proposed by Lou, Hao and Jia has been used to a new application such that a decoupled system becomes a coupled one. Using the new application to some decoupled systems such as the decoupled dispersionless Korteweg–de Vries (KdV) systems related to dispersionless waves, the decoupled KdV systems related to dispersion waves, the decoupled KdV and Burgers systems related to the linear dispersion and diffusion effects, and the decoupled KdV and Harry–Dym (HD) systems related to the linear and nonlinear dispersion effects, we have obtained various new types of higher dimensional integrable coupled systems. The new models can be used to describe the interactions among different nonlinear waves and/or different effects including the dispersionless waves (dispersionless KdV waves), the linear dispersion waves (KdV waves), the nonlinear dispersion waves (HD waves) and the diffusion effect. The method can be applied to couple all different separated integrable models.

Investigation of exact solutions and conservation laws for nonlinear fractional (2+1)-dimensional Burgers system of equations

Investigation of exact solutions and conservation laws for nonlinear fractional (2+1)-dimensional Burgers system of equations

Analysis of degenerate Burgers' equations involving small perturbation and large wave amplitude

We only focus on the shock waves to Burgers system by considering the porous medium diffusion and singularity in energy estimates. In this paper, the conditions and are considered. Moreover, we establish the existence of shock waves through the phase plane. The weighted function is employed to handle the singularity in the weighted energy estimates under small perturbations and arbitrary wave amplitudes. Finally, the weighted energy estimates are then used to prove the stability of shock waves.

Wave propagation and stabilization in the Boussinesq–Burgers system

This paper considers the existence and stability of traveling wave solutions of the Boussinesq–Burgers system describing the propagation of bores. Assuming the fluid is weakly dispersive, we establish the existence of three different wave profiles by the geometric singular perturbation theory alongside phase plane analysis. We further employ the method of weighted energy estimates to prove the nonlinear asymptotic stability of the traveling wave solutions against small perturbations. The technique of taking antiderivative is utilized to integrate perturbation functions because of the conservative structure of the Boussinesq–Burgers system. Using a change of variable to deal with the dispersion term, we perform numerical simulations for the Boussinesq–Burgers system to showcase the generation and propagation of various wave profiles in both weak and strong dispersions. The numerical simulations not only confirm our analytical results, but also illustrate that the Boussinesq–Burgers system can generate numerous propagating wave profiles depending on the profiles of initial data and the intensity of fluid dispersion, where in particular the propagation of bores can be generated from the system in the case of strong dispersion.

On assorted soliton wave solutions with the higher-order fractional Boussinesq–Burgers system

The purpose of this study is to highlight the shallow water wave patterns along the ocean shore or in lakes with the higher-order Boussinesq–Burgers system possessing a fractional derivative operator. A generic fractional transformation is used, which turns the proposed model into an nonlinear ordinary differential equations (NLODEs) system. For the construction of new solitons of the mentioned coupled system, the auxiliary equation technique is employed. This approach produced numerous soliton solutions such as bright, singular and w-shaped solitons of the aforesaid model successfully. These results are expressed graphically to exemplify their physical appearance with the help of soft computations in Mathematica. All the solutions yielded by this method are novel and have not been derived yet.

Lax integrable higher dimensional Burgers systems via a deformation algorithm and conservation laws

How to obtain higher-dimensional integrable systems from lower ones is a hot topic deserving deeply consideration. Recently, a deformation algorithm with the help of different conservation laws proposed by Lou, Hao and Jia has been applied to establish some higher-dimensional nonlinear systems. In this paper, the method is mainly used to establish the (2+1)-dimensional Burgers equation and higher-dimensional Burgers systems. The Lax pairs, integrable Burgers hierarchy, Lie point symmetries and kink soliton solutions are also investigated.

Half boundary method for two-dimensional steady-state nonlinear convection-diffusion equations

In this research, half-boundary method (HBM) is developed for nonlinear convection-diffusion equations (CDEs) which play an important role in applied mathematics and physics. The HBM is based on the variable relationship between the nodes inside the domain and the nodes on half of the boundaries, making it ideal for reducing the maximum order of matrix and calculation memory storage. Besides, the HBM can solve discontinuous problems directly without adding continuity conditions due to the use of the mixed variables. The effectiveness and accuracy of the proposed algorithm are investigated mainly for the two-dimensional(2D) steady-state Burgers’ equation, the material nonlinear 2D CDEs and the system of 2D Burgers’ equations. The results show the excellent performance of the HBM in simulating flow and heat transfer, especially for convection domination.

An Extended (2 + 1)-dimensional Coupled Burgers System in Fluid Mechanics: Symmetry Reductions; Kudryashov Method; Conservation Laws

The Burgers-type equations are noticed in plasma astrophysics, ocean dynamics, atmospheric science, computational fluid mechanics, cosmology, condensed matter physics, statistical physics, nonlinear acoustics, vehicular traffic, electronic transport, etc. This prompts us to examine an extended (2 + 1)-dimensional coupled Burgers system in fluid mechanics. We determine novel exact solutions by the Lie symmetry method in conjunction with Kurdyshov method. Finally, conservation laws of the abovementioned system are generated. The findings can well mimic complex waves and their dealing dynamics in fluids.

Differentiable physics-enabled closure modeling for Burgers’ turbulence

Data-driven turbulence modeling is experiencing a surge in interest following algorithmic and hardware developments in the data sciences. We discuss an approach using the differentiable physics paradigm that combines known physics with machine learning to develop closure models for Burgers’ turbulence. We consider the one-dimensional Burgers system as a prototypical test problem for modeling the unresolved terms in advection-dominated turbulence problems. We train a series of models that incorporate varying degrees of physical assumptions on an a posteriori loss function to test the efficacy of models across a range of system parameters, including viscosity, time, and grid resolution. We find that constraining models with inductive biases in the form of partial differential equations that contain known physics or existing closure approaches produces highly data-efficient, accurate, and generalizable models, outperforming state-of-the-art baselines. Addition of structure in the form of physics information also brings a level of interpretability to the models, potentially offering a stepping stone to the future of closure modeling.

Hamiltonian structure, optimal classification, optimal solutions and conservation laws of the classical Boussinesq–Burgers system

Shallow water waves phenomena have been vastly studied in the literature for many decades. Understanding their intricacies is not only important in predicting the timing and magnitude of their destructive nature (such as tsunami) but also in knowing how to prevent their occurrences. Among the equations used for gaining accurate and deep understanding of shallow water waves for two-layered fluid flow is the classical Boussinesq–Burgers system. Its solutions have been studied in the literature using various methods; however, no study has considered its connection with (some of) the Painlevé transcendents which lead to infinite rational and special function solutions. The present study explore this avenue through the Lie group method — a powerful and versatile method that has been effectively and extensively used for various analyses of nonlinear differential equations. Invariant, rational and special function solutions of the classical Boussinesq–Burgers equations are derived and non-trivial conserved quantities are also constructed.

Taking into consideration an extended coupled (2+1)-dimensional Burgers system in oceanography, acoustics and hydrodynamics

As for certain wave processes in oceanography, acoustics or hydrodynamics, we investigate an extended coupled (2+1)-dimensional Burgers system by way of finding out that the system passes the Painlevé test and working out two sets of the similarity reductions through symbolic computation. Each set relies on the coefficients in the system, and can develop into a known ordinary differential equation.