Group classification and exact solutions of a higher-order Boussinesq equation

Nonlinear Dynamics - We consider a family of sixth-order Boussinesq equations in one space dimension with an arbitrary nonlinearity. The equation was originally derived for a one-dimensional...     

Complete group classification and exact solutions to the extended short pulse equation

In this paper, the complete group classification is performed on the extended short pulse equation (ESPE), which including many important non-linear wave equations as its special cases. In the sense of geometric symmetry, all of the vector fields of the equation are obtained in terms of the arbitrary parameters of the equation. Furthermore, the symmetry reductions and exact solutions to the short pulse types of equations are investigated, and the physical significance of the solutions are considered from the transformation group point of view.     

New exact solutions for a generalized KdV equation

In this paper, we establish a triple-order complete discrimination system to derive the traveling wave solutions of the generalized KdV equation with high power nonlinearities, which consist of solitary patterns solutions, compactons solutions, periodic solutions and Jacobi elliptic functions solutions.     

Bifurcations of traveling wave solutions and exact solutions to generalized Zakharov equation and Ginzburg-Landau equation

This paper studies the dynamic behaviors of some exact traveling wave solutions to the generalized Zakharov equation and the Ginzburg-Landau equation. The effects of the behaviors on the parameters of the systems are also studied by using a dynamical system method. Six exact explicit parametric representations of the traveling wave solutions to the two equations are given.     

Construction of exact solutions of the Boussinesq equation

The known model of nonlinear dispersive waves, which was proposed by Boussinesq in the second half of the nineteenth century, is considered. Solutions of the Boussinesq equation, which are expressed via elementary functions and describe wave packets, their interaction between each other and with solitons, and some other structures are obtained. To construct these solutions, Hirota's bilinear representation and differential relations specified by ordinary differential equations with constant coefficients are used. Computing Center, Siberian Division, Russian Academy of Sciences, Krasnoyarsk 660036. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 3, pp. 74–78, May–June, 1998.     

A class of exact solutions of the Navier-Stokes equation

The unsteady problem of a viscous incompressible flow in free space is investigated, and a class of exact solutions of the Navier-Stokes equation is given for a general initial condition. This flow field represents several shear layers superimposed on an irrotational, three-dimensional straining flow. This solution incorporates the three main features of vortex motion, i.e., stretching, convection and viscous diffusion of vorticity. The solution is exemplified for several kinds of initial condition. One of them represents a flow approaching a steady state in which the above three effects are brought to an equilibrium. Another solution shows collision of two shear layers in various arrangements. Two parallel shear layers merge into a single layer, while two antiparallel shear layers in which the vortices in each layer are in the opposite directions disappear as pair cancellation, and two layers merge like vectors by oblique collision. It is also shown that N shear layers, in general, merge like vectors.     

Symmetry reductions and exact solutions to the ill-posed Boussinesq equation

In this paper, using the Lie symmetry analysis method, we study the ill-posed Boussinesq equation which arises in shallow water waves and non-linear lattices. The similarity reductions and exact solutions for the equation are obtained. Then the exact analytic solutions are considered by the power series method, and the physical significance of the solutions is considered from the transformation group point of view.     

Similarity analysis and exact solutions for a general discrete two-velocity model of Boltzmann equation

Summary This paper concerns with the similarity analysis for a general discrete two-velocity model of the Boltzmann equation introduced by Illner [12]. We find the general groups of invariance and we get some exact solutions, recovering general results which contain either solutions extensively described in the literature or undiscovered ones.

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Sommario In questa nota si applica l'analisi dei gruppi infinitesimi di trasformazione ad un modello generale discreto a due velocità dell'equazione di Boltzmann introdotto da Illner [12]. Si trovano i più generali gruppi di invarianza e si ottengono alcune soluzioni esatte, ritrovando risultati generali che contengono sia soluzioni ampiamente descritte in letteratura che nuove soluzioni.

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Work supported by the C.N.R. through the G.N.F.M.     

Group properties and exact solutions of equations for vibrational convection of a binary mixture

A model of vibrational convection of a binary mixture with the thermodiffusion effect is considered. The symmetries of model equations are found, depending on the values of physical parameters. A partially invariant solution, which describes separation of the binary mixture in a thermodiffusion column, is constructed and studied. The influence of streamwise vibrations on the flow regime and separation of the mixture is investigated.     

Similarity reductions and exact solutions of generalized Bretherton equation with time-dependent coefficients

In this paper, we apply Lie-group formalism to the generalized Bretherton equation with variable coefficients u _tt +α(t)u _xx +β(t)u _xxxx +δ(t)u ^m +θ(t)u ⁿ =0, to investigate the symmetries. We derive the infinitesimals and the admissible forms of the coefficients that admit the classical symmetry group. The ordinary differential equations deduced from the optimal system of subalgebras are further studied and some exact solutions are obtained.     

Hypoplasticity theory for granular materials—II: Three-dimensional axially symmetric exact solutions

In part I of this paper, we consider the governing equations of hypoplasticity theory for two-dimensional steady quasi-static plane strain compressible gravity flow and determine some exact analytical solutions applying for certain special cases. Similarly, for the three-dimensional situation considered here in part II, we undertake a similar mathematical investigation to determine some simple solutions of the governing equations for three-dimensional steady quasi-static axially symmetric compressible gravity flow for hypoplastic granular materials. We again find that for certain special cases, we are able to determine some exact solutions for the stress and velocity profiles. We comment that hypoplasticity theory generally gives rise to complicated systems of coupled non-linear differential equations, for which the determination of any analytical solutions is not a trivial matter, and that the solutions determined here might be exploited as benchmarks for full numerical schemes.     

Group properties and invariant solutions for infinitesimal transformations of a non-linear wave equation

This paper concerns the infinitesimal group analysis for a second order non-linear wave equation involving non-homogeneous processes. In the first part we characterize the most general expression for the generator of the Lie group. Then we calculate the invariant surfaces in some special cases, obtaining the corresponding ordinary differential equations whose integration allow us to get classes of solutions for the original equation.     

The analytic continuation of solutions of the generalized axially symmetric Helmholtz equation

Archive for Rational Mechanics and Analysis - The analytic continuation of a solution of the generalized axially symmetric Helmholtz equationu xx + u yy + (2α/x)u x + k 2 u = 0is examined. A...     

Two exact solutions of the DPL non-Fourier heat conduction equation with special conditions

This paper presents two exact explicit solutions for the three dimensional dual-phase lag （DLP） heat conduction equation, during the derivation of which the method of trial and error and the authors＇ previous experiences are utilized. To the authors＇ knowledge, most solutions of 2D or 3D DPL models available in the literature are obtained by numerical methods, and there are few exact solutions up to now. The exact solutions in this paper can be used as benchmarks to validate numerical solutions and to develop numerical schemes, grid generation methods and so forth. In addition, they are of theoretical significance since they correspond to physically possible situations. The main goal of this paper is to obtain some possible exact explicit solutions of the dual-phase lag heat conduction equation as the benchmark solutions for computational heat transfer, rather than specific solutions for some given initial and boundary conditions. Therefore, the initial and boundary conditions are indeterminate before derivation and can be deduced from the solutions afterwards. Actually, all solutions given in this paper can be easily proven by substituting them into the governing equation.     

A new equation and exact solutions describing focal fields in media with modular nonlinearity

Nonlinear Dynamics - Brand-new equations which can be regarded as modifications of Khokhlov–Zabolotskaya–Kuznetsov or Ostrovsky–Vakhnenko equations are suggested. These equations...     

Group classification for path equation describing minimum drag work and symmetry reductions

The path equation describing the minimum drag work first proposed by Pakdemirli is reconsidered （Pakdemirli, M. The drag work minimization path for a fly- ing object with altitude-dependent drag parameters. Proceedings of the Institution of Mechanical Engineers, Part C, Journal of Mechanical Engineering Science 223（5）, 1113- 1116 （2009））. The Lie group theory is applied to the general equation. The group classi- fication with respect to an altitude-dependent arbitrary function is presented. Using the symmetries, the group-invariant solutions are determined, and the reduction of order is performed by the canonical coordinates.     

Group classification and invariant solutions for the equations of flow and heat transfer of a viscoplastic medium

Exact solutions without major restrictions on the properties of the material are needed in research on the flow (especially viscosity) of metals at high temperatures under nonisothermal conditions. Often the shear resistance is governed mainly by the temperature and the deformation rate. Here are examined the group properties of the equations of flow and heat transfer of a medium whose shear resistance is a function of temperature and rate of shear deformation. The properties specific to metals are not used, so the results are applicable to a variety of media.