General Burgers Research Articles

Group Classification Invariant Solutions of Burgers' Equation

The aims of the present paper is to solve the problem of the group classification of the general Burgers’ equation u_t=f(x,u) u_x^2+g(x,u)u_xx, where f and g are arbitrary smooth functions of the variables x and u, by using Lie method. The paper is one of the few applications of an algebraic approach to the problem of group classification: We followed the analysis mathematical method using the method of preliminary group classification. A number of new interesting nonlinear invariant models which have nontrivial invariance algebras are obtained. The result of the work is a wide class of equations summarized in table form.

Dislocation content of grain boundary phase junctions and its relation to grain boundary excess properties

We analyze the dislocation content of grain boundary (GB) phase junctions, i.e., line defects separating two different GB phases coexisting on the same GB plane. While regular GB disconnections have been characterized for a variety of interfaces, GB phase junctions formed by GBs with different structures and different numbers of excess atoms have not been previously studied. We apply a general Burgers circuit analysis to calculate the Burgers vectors b of junctions in two {\Sigma}5 Cu boundaries previously simulated with molecular dynamics. The Burgers vectors of these junctions cannot be described by the displacement shift complete (DSC) lattice alone. We show that, in general, the normal component of b is not equal to the difference in the GB excess volumes, but contains another contribution from the numbers of GB atoms per unit area {\Delta}N required to transform one GB phase into another. In the boundaries studied, the latter component dominates and even changes the sign of b. We derive expressions for the normal and tangential components of b in terms of the DSC lattice vectors and the non-DSC part due to {\Delta}N and additional GB excess properties, including excess volume and shears. These expressions provide a connection between GB phase transformations driven by the GB free energy difference and the motion of GB junctions under applied normal and shear stresses. The proposed analysis quantifies b and therefore makes it possible to calculate the elastic part of the energy of these defects, evaluate their contribution to the nucleation barrier during GB phase transformations, and treat elastic interactions with other defects.

Soliton molecules and abundant interaction solutions of a general high-order Burgers equation

In this work, many kinds of resonant excitations to the N-solitary waves solution are constructed for a (1+1)-dimensional general high-order Burgers (HB) equation. And the complexiton solution, half periodic kink (HPK) and soliton molecules of the (1+1)-dimensional HB equation are found. In particular, for a fourth-order Burgers equation, the explicit expressions and forming conditions for the soliton molecule and HPK are revealed. The three-soliton molecule, two-soliton molecule and HPK molecule are also proposed and depicted. The soliton molecule-kink type and HPK molecule-kink type interaction solutions can exhibit abundant fusion and fission phenomena.

Group analysis of general Burgers–Korteweg–de Vries equations

The complete group classification problem for the class of (1+1)-dimensional rth order general variable-coefficient Burgers–Korteweg–de Vries equations is solved for arbitrary values of r greater than or equal to two. We find the equivalence groupoids of this class and its various subclasses obtained by gauging equation coefficients with equivalence transformations. Showing that this class and certain gauged subclasses are normalized in the usual sense, we reduce the complete group classification problem for the entire class to that for the selected maximally gauged subclass, and it is the latter problem that is solved efficiently using the algebraic method of group classification. Similar studies are carried out for the two subclasses of equations with coefficients depending at most on the time or space variable, respectively. Applying an original technique, we classify Lie reductions of equations from the class under consideration with respect to its equivalence group. Studying alternative gauges for equation coefficients with equivalence transformations allows us not only to justify the choice of the most appropriate gauge for the group classification but also to construct for the first time classes of differential equations with nontrivial generalized equivalence group such that equivalence-transformation components corresponding to equation variables locally depend on nonconstant arbitrary elements of the class. For the subclass of equations with coefficients depending at most on the time variable, which is normalized in the extended generalized sense, we explicitly construct its extended generalized equivalence group in a rigorous way. The new notion of effective generalized equivalence group is introduced.

Analytical solutions for wall slip effects on magnetohydrodynamic oscillatory rotating plate and channel flows in porous media using a fractional Burgers viscoelastic model

The theoretical analysis of magnetohydrodynamic (MHD) incompressible flows of a Burgers fluid through a porous medium in a rotating frame of reference is presented. The constitutive model of a Burgers fluid is used based on a fractional calculus formulation. Hydrodynamic slip at the wall (plate) is incorporated and the fractional generalized Darcy model deployed to simulate porous medium drag force effects. Three different cases are considered: namely, the flow induced by a general periodic oscillation at a rigid plate, the periodic flow in a parallel plate channel and, finally, the Poiseuille flow. In all cases the plate(s) boundary(ies) are electrically non-conducting and a small magnetic Reynolds number is assumed, negating magnetic induction effects. The well-posed boundary value problems associated with each case are solved via Fourier transforms. Comparisons are made between the results derived with and without slip conditions. Four special cases are retrieved from the general fractional Burgers model, viz. Newtonian fluid, general Maxwell viscoelastic fluid, generalized Oldroyd-B fluid and the conventional Burgers viscoelastic model. Extensive interpretation of graphical plots is included. We study explicitly the influence of the wall slip on primary and secondary velocity evolution. The model is relevant to MHD rotating energy generators employing rheological working fluids.

Finite amplitude standing waves in resonators terminated by a general impedance.

A general inhomogeneous Burgers equation describing finite-amplitude standing waves in resonators terminated by a general impedance is derived. This model equation can be used for modeling of nonlinear acoustic processes connected with some methods for enhancement of Q-factor of acoustic resonators. One of them is the method based on using a second-harmonics absorber. For better understanding of this method, it is convenient to know at least an approximate analytical solution of the model equation. This work presents some approximate solutions, which improve and extend the solutions that have been published previously. The solutions are compared with results obtained by numerical integration of the corresponding equations.

A New Exact Solution of Burgers’ Equation with Linearized Solution

This paper considers a general Burgers’ equation with the nonlinear term coefficient being an arbitrary constant. Two identical solutions of the general Burgers’ equation are separately derived by a direct integration method and the simplest equation method with the Bernoulli equation being the simplest equation. The proposed exact solutions overcome the long existing problem of discontinuity and can be successfully reduced to linearity, while the nonlinear term coefficient approaches zero. In addition, a general Cole-Hopf transform is introduced. Finally, the proposed derived solution is compared with the perturbation solution and other existing exact solutions. A new phenomenon, which we named “kink sliding,” is observed.

Line-integral representations for the elastic displacements, stresses and interaction energy of arbitrary dislocation loops in transversely isotropic bimaterials

The elastic displacements, stresses and interaction energy of arbitrarily shaped dislocation loops with general Burgers vectors in transversely isotropic bimaterials (i.e. joined half-spaces) are expressed in terms of simple line integrals for the first time. These expressions are very similar to their isotropic full-space counterparts in the literature and can be easily incorporated into three-dimensional (3D) dislocation dynamics (DD) simulations for hexagonal crystals with interfaces/surfaces. All possible degenerate cases, e.g. isotropic bimaterials and isotropic half-space, are considered in detail. The singularities intrinsic to the classical continuum theory of dislocations are removed by spreading the Burgers vector anisotropically around every point on the dislocation line according to three particular spreading functions. This non-singular treatment guarantees the equivalence among different versions of the energy formulae and their consistency with the stress formula presented in this paper. Several numerical examples are provided as verification of the derived dislocation solutions, which further show significant influence of material anisotropy and bimaterial interface on the elastic fields and interaction energy of dislocation loops.

Transient dislocation emission from a crack tip in an anisotropic elastic material

The emission of a dislocation with a general Burgers vector from the tip of a stationary semi-infinite crack in an anisotropic elastic material is examined. The dislocation is assumed to leave the crack tip along the crack extension plane at constant speed. Explicit expressions for the transient shielding stress intensity factors at the crack tip and the drag forces on the dislocations are derived. Numerical results for a class of cubic materials and two hexagonal crystals, zinc and cobalt, are given. Dislocation emission under plane stress wave loading is discussed.

The recursion operator method for generalized symmetries and Bäcklund transformations of the Burgers’ equations

In this paper, the generalized symmetries of the second-order Burgers’ equation are obtained through the symmetry transformation method. The Backlund transformations (BTs) of the two equations are constructed by the recursion operator method. Then, the infinite number of exact solutions to these equations are investigated in terms of the generalized symmetries and Backlund transformations. Furthermore, the Backlund transformations and conservation law of the general Burgers’ equations are discussed.

Symmetry group classification for general Burgers’ equation

The present paper solves the problem of the group classification of the general Burgers’ equation ut=f(x,u)ux2+g(x,u)uxx, where f and g are arbitrary smooth functions of the variable x and u, by using Lie method. The paper is one of the few applications of an algebraic approach to the problem of group classification that is called preliminary group classification. Looking the adjoint representation of GE on its Lie algebra g5, we will deal with the construction of the optimal system of its one-dimensional subalgebras. The result of the work is a wide class of equations summarized in table form.

A modified tanh–coth method for solving the general Burgers–Fisher and the Kuramoto–Sivashinsky equations

In this work we use a modified tanh–coth method to solve the general Burgers–Fisher and the Kuramoto–Sivashinsky equations. The main idea is to take full advantage of the Riccati equation that the tanh-function satisfies. New multiple travelling wave solutions are obtained for the general Burgers–Fisher and the Kuramoto–Sivashinsky equations.

New multiple soliton solutions to the general Burgers–Fisher equation and the Kuramoto–Sivashinsky equation

A generalized tanh function method is used for constructing exact travelling wave solutions of nonlinear partial differential equations in a unified way. The main idea of this method is to take full advantage of the Riccati equation which has more new solutions. More new multiple soliton solutions are obtained for the general Burgers–Fisher equation and the Kuramoto–Sivashinsky equation.

Transient motion due to a moving dislocation in a general anisotropic solid

The elastic fields of a dislocation with general Burgers vector starting from rest and moving with constant velocity in a general anisotropic solid are given in closed form. The analysis is made using a sextic formalism that does not require integral transforms. An explicit expression for the energy flux radiated from the dislocation is derived. The drag force on the dislocation is calculated for a class of cubic materials.

The Forces Exerted on Dislocations and the Stress Fields Produced by Them

It is shown that the force $d\mathbf{F}$ exerted on a line element $\ensuremath{\nu}d\ensuremath{\sigma}$ of a dislocation with Burgers vector f by a stress $\ensuremath{\tau}$ is given by $d\mathbf{F}=\ensuremath{-}\ensuremath{\nu}\ifmmode\times\else\texttimes\fi{}(\mathbf{f}\ifmmode\cdot\else\textperiodcentered\fi{}\ensuremath{\tau})d\ensuremath{\sigma}$. An analogy is drawn between the behavior of a closed line dislocation in a stress field and the behavior of a closed current-carrying loop in a field of magnetic induction. Then formulas for the stress components caused at any point of an infinite elastically isotropic crystal by a line element of a general Burgers dislocation are deduced from Burger's expressions for the displacements (see Sec. III(C)). These formulas bear a close analogy to the Biot-Savart formula of electromagnetic theory. Both of these results taken together constitute a complete system for the investigation of the mutual interaction of dislocations in an infinite elastically isotropic crystal.