Group analysis for a linear hyperbolic equation arising from a quasilinear reducible system

The aim of this paper is to carry out infinitesimal group analysis for a second order linear hyperbolic equation in canonical form. To such a kind of equation one could be led in considering the hodograph transformation for a first order reducible system. In the first part of the paper we characterize the most general expression for the generator of the Lie group, which has an infinite dimensional algebra. Then we calculate the invariant surfaces in some special cases of interest, and point out the corresponding ordinary differential equations whose integration allows us to determine possible classes of solutions for the equation we are considering.     

Invariance groups and reduction of the unsteady transonic small disturbance equation

A geometric approach is undertaken towards the solution of the unsteady transonic small disturbance equation describing the low frequency flow field about a thin airfoil. From group properties given in Anderson and Ibragimov, Lie-Bäcklund Transformations in Applications (1979) we derive three invariance groups for the equation. Based on these groups a reduction of the equation is performed. The reduction leads to a steady equation and to an ordinary differential equation from which group invariant solutions of the unsteady transonic small disturbance equation can be obtained.     

Lie symmetry analysis and invariant solutions of $$\varvec{}$$-dimensional Calogero–Bogoyavlenskii–Schiff equation

Lie group analysis is applied to carry out the similarity reductions of the \((3+1)\)-dimensional Calogero–Bogoyavlenskii–Schiff (CBS) equation. We obtain generators of infinitesimal transformations of the CBS equation and each of these generators depend on various parameters which give us a set of Lie algebras. For each of these Lie algebras, Lie symmetry method reduces the \((3+1)\)-dimensional CBS equation into a new \((2+1)\)-dimensional partial differential equation and to an ordinary differential equation. In addition, we obtain commutator table of Lie brackets and symmetry groups for the CBS equation. Finally, we obtain closed-form solutions of the CBS equation by using the invariance property of Lie group transformations.     

Methods of Quantum Field Theory in the Physics of Subsurface Solute Transport

The stochastic theory of subsurface solute transport has received stimulus recently from modeling techniques originating in quantum field theory (QFT), resulting in new calculations of the solute macrodispersion tensor that derive from the solving Dyson equation with a subsequent renormalization group analysis. In this paper, we offer a critical evaluation of these techniques as they relate specifically to the derivation of a field-scale advection–dispersion equation. An approximate Dyson equation satisfied by the ensemble-average solute concentration for tracer movement in a heterogeneous porous medium is derived and shown to be equivalent to a truncated cumulant expansion of the standard stochastic partial differential equation which describes the same phenomenon. The full Dyson equation formalism, although exact, is of no importance to the derivation of an improved field-scale advection–dispersion equation. Similarly, renormalization group analysis of the macrodispersion tensor has not yet provided results that go beyond what is available currently from the cumulant expansion approach.     

Equivalence Classes and Symmetries of the Variable Coefficient Kadomtsev—Petviashvili Equation

We classify the variable coefficient Kadomtsev—Petviashvili (VCKP) equation into equivalence classes under the group of local point transformations, leaving the equation form invariant but changing the coefficient functions. We list the representatives of all equivalence classes with the corresponding transformations. Then, we obtain the symmetry group of the VCKP equation and in particular discuss how to use these transformations to classify low-dimensional symmetry algebras in the generic case. We conclude with a discussion of the implications of the present article.     

Optimal system,group invariant solutions and conservation laws of the CGKP equation

In this paper, the (2 + 1)-dimensional cubic generalized Kadomtsev–Petviashvili (CGKP) equation that is derived from the Maxwell–Bloch equations is investigated. By means of Lie symmetry analysis method, we obtain the Lie point symmetries for the equation and the optimal system of the symmetry algebra. Based on the optimal system, a lot of group invariant solutions are obtained. In addition, explicit conservation laws of the equation are studied.     

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.     

The dispersion of shear wave in multilayered magnetoelastic self-reinforced media

In this paper, we study the propagation of shear waves in a magnetoelastic self-reinforced medium using finite difference technique. Dispersion equation has been deduced for the case when (n ? 1) layers lie over a half space. It is observed that the obtained dispersion equation is in assertion with the classical Love wave equation for both the cases when a single and double layer lies over a half space. The stability condition for the used finite difference scheme and the expression for the phase and group velocity have been derived. The dispersion curve for different values of magnetoelastic coupling parameter, phase and group velocity variation for different values of stability ratio has been depicted by means of graphs.     

Replacing ordinary derivatives by gauge derivatives in the continuum theory of dislocations to compensate the action of the symmetry group

When the symmetry group of a body is continuous it plays a fundamental role on the nonlinear continuum theory of dislocations: it induces a non-uniqueness to the field that describes the defects – the uniform reference – and affects also other fundamental ingredients of the theory. The purpose of the present paper is to examine how certain important quantities of the dislocation theory are affected from symmetry's group action. Apart from the uniform reference we study how the deformation gradient, the first and second Piola–Kirchhoff stress tensors, the elasticities of the material and the momentum equation are affected from the action of the symmetry group. This action is inhomogeneous, namely, differs from point to point. A similar inhomogeneous action of a group may be found in gauge theories. Prompt by the gauge approach, we propose the use of the gauge covariant exterior derivative to compensate for the action of the symmetry group on the uniform reference. The main advantage of using this derivative is that the momentum equation for the static case retains its divergence form. It remains an open question how the Yang–Mills potentials may be determined for the present theory.     

On symmetries, conservation laws and invariant solutions of the foam-drainage equation

This study deals with symmetry group properties and conservation laws of the foam-drainage equation. Firstly, we study the classical Lie symmetries, optimal systems, similarity reductions and similarity solutions of the foam-drainage equation which are obtained through the Lie group method of infinitesimal transformations. Secondly, using the new general theorem on non-local conservation laws and partial Lagrangian approach, local and non-local conservation laws are also studied and, finally, non-classical symmetries are derived.     

Symmetry-based optimal portfolio for a DC pension plan under a CEV model with power utility

In this article, explicit representation of solution for the Hamilton–Jacobi–Bellman (HJB) equation associated with the portfolio optimization problem for an investor who seeks to maximize the expected power (CRRA) utility of the terminal wealth in a defined-contribution pension plan under a constant elasticity of variance model is derived based on the application of the Lie symmetry method to the partial differential equation and its associated terminal condition. Compared with the ingenious ansatz techniques used before, here we present a group theoretical analysis of the terminal value problem for the solution following the algorithmic procedure of the Lie symmetry analysis. It shows that the interesting properties of the group structures of the original HJB equation and its successive similarity reduced equations lead to an elegant resolution of the problem. Moreover, we identify the meaningful range of risk aversion coefficient which is ignored in the previous work. At last, the properties and sensitivity analysis of the derived optimal strategy are demonstrated by numerical simulations and several figures. The method used here is quite general and can be applied to other equations obtained in financial mathematics.

     

Evolution Equations on Non-Flat Waveguides

We investigate the dispersive properties of evolution equations on waveguides with a non-flat shape. More precisely, we consider an operator $$H=-\Delta_{x}-\Delta_{y}+V(x,y)$$ with Dirichlet boundary conditions on an unbounded domain ??, and we introduce the notion of a repulsive waveguide along the direction of the first group of variables, x. If ?? is a repulsive waveguide, we prove a sharp estimate for the Helmholtz equation Hu???u?=?f. As consequences, we prove smoothing estimates for the Schr?dinger and wave equations associated to H, and Strichartz estimates for the Schr?dinger equation. Additionally, we deduce that the operator H does not admit eigenvalues.     

Group classification and exact solutions of a radially symmetric porous-medium equation

In this paper we perform a group classification for the generalized radial porous-medium equation. We also classify symmetry reductions of the equation to first- or second-order ordinary differential equations (ODEs) and hence construct invariant solutions in a systematic manner. We show that the reduced second-order equations are invariant under either a two-parameter or one-parameter Lie groups. In the first case, they are completely integrated by a pair of quadratures. In the latter, they are often reduced to first-order ODEs of Abel type.     

Renormalization group methods for a Mathieu equation with delayed feedback

This paper presents the application of the renormalization group (RG) methods to the delayed differential equation. By analyzing the Mathieu equation with time delay feedback, we get the amplitude and phase equations, and then obtain the approximate solutions by solving the corresponding RG equations. It shows that the approximate solutions obtained from the RG method are superior to those from the conventionally perturbation methods.     

Fer’s expansion for generalized Hamiltonian system based on Lie transformation technique

In the present paper, we present a new method for integrating the ordinary differential equation, especially for the ordinary differential equation derived from explicitly time-dependent generalized Hamiltonian dynamic system, which is based on taking a factorization of the evolution operator as an infinite product of the exponentials of Lie operators. The above process is a Lie group (algebraic) method that retains the structural intrinsic properties of the exact solution when truncated and is used to analyze the main features of the so-called Fer’s expansion. The numerical examples are presented at the end of this paper.     

New solutions for surface tension driven spreading of a thin film

The standard fourth-order non-linear PDE modelling the flow of thin fluid film subject to surface tension is studied. The Lie group method is used to reduce the model equation from a fourth-order PDE to a fourth-order ODE. Analytical solutions are obtained for certain cases. Where analytical progress cannot be made, we determine numerical solutions.     

On group classification of the spatially homogeneous and isotropic Boltzmann equation with sources

In Nonenmacher (1984) [1] an admitted Lie group of transformations for the spatially homogeneous and isotropic Boltzmann equation with sources was studied. In fact, the author is Nonenmacher (1984) [1] considered the equation for a generating function of the power moments of the Boltzmann equation solution. However, this equation is still a non-local partial differential equation, and this property was not taken into account there. In the present paper the admitted Lie group of this equation is studied, using our original method developed for group analysis of equations with non-local operators (Grigoriev and Meleshko, 1986; Meleshko, 2005; Grigoriev et al., 2010 [2], [3], [4]). The Lie groups obtained are compared with Nonenmacher (1984) [1]. The deficiency of Nonenmacher (1984) [1] is corrected.     

A theoretical Taylor–Galerkin model for first‐order hyperbolic equation

This paper presents a class of Taylor–Galerkin (TG) finite‐element models for solving the first‐order hyperbolic equation which admits discontinuities. Five parameters are introduced for purposes of controlling stability, monotonicity and accuracy. In this paper, the total variation diminishing concept and the theory of M‐matrix are applied to construct a monotonic TG model for capturing discontinuities. To avoid making the scheme overly diffusive, we apply a flux‐corrected transport (FCT) technique of Boris and Book to overcome the difficulty with anti‐diffusive flux. In smooth flow regions, our strategyof developing the temporal and spatial high‐order TG finite‐element model is based on modified equation analysis. In regions where discontinuity is encountered, we resort to two dispersively more accurate models to make the prediction accuracy as high as that obtained in smooth cases. These models are developed using the entropy‐increasing principle and the theory of group velocity. Guided by this theory, a slower group velocity should be used ahead of the shock. To avoid a train of post‐shocks, free parameters should be chosen properly to obtain a group velocity which takes on a larger value than the exact phase velocity. In this paper, we also apply the entropy‐increasing principle to determine free parameters introduced in the finite‐element model. Under the entropy‐increasing requirement, it is mandatory that coefficients of the even and odd derivative terms shown in the modified equation should change signs alternatively in order to avoid non‐physical wiggles. Several benchmark problems have been investigated to confirm the integrity of these proposed characteristic models. Copyright © 2003 John Wiley & Sons, Ltd.     

Geometrical theory of fluid flows and dynamical systems

Various dynamical systems have often common geometrical structures and can be formulated on the basis of Riemannian geometry and Lie group theory, provided that a dynamical system has a group symmetry, namely it is invariant under group transformations, and further that the group manifold is endowed with a Riemannian metric. The basic ideas and tools are described, and their applications are presented for the following five problems: (a) free rotation of a rigid body, which is a well-known system in mechanics and presented as an illustrative example of the geometrical theory; (b) geodesic equation and KdV equation on the group of diffeomorphisms of a circle and its extended group; (c) a self-gravitating system of a finite number of points masses and a geometrical interpretation of chaos of Hénon–Heiles system; (d) geometrical formulation of hydrodynamics of an incompressible ideal fluid on the group of volume preserving diffeomorphisms, where an interpretation of the origin of Riemannian curvatures of the fluid flow is given; (e) geodesic equation on a loop group and the local induction equation for the motion of a vortex filament, where the geodesic equation on its extended group is found to be equivalent to the equation for a vortex filament with an axial flow along it.

It is remarkable that the present geometrical formulations are successful for all the problems considered here and give an insight into the deep background common to the diverse physical systems. Furthermore, the geometrical formulation opens a new approach to various dynamical systems, which is rewarded with new results.