Nonclassical Symmetries Research Articles

Reductions and exact solutions of some nonlinear partial differential equations under four types of generalized conditional symmetries

AbstractThe generalized conditional symmetry method is applied to study the reduction to finite-dimensional dynamical systems and construction of exact solutions for certain types of nonlinear partial differential equations which have many physically significant applications in physics and related sciences. The exact solutions of the resulting equations are derived via the compatibility of the generalized conditional symmetries and the considered equations, which reduces to solving some systems of ordinary differential equations. For some unsolvable systems of ordinary differential equations, the dynamical behavior and qualitative properties are also considered. To illustrate that the approach has wide application, the exact solutions of a number of nonlinear partial differential equations are also given. The method used in this paper also provides a symmetry group interpretation to some known results in the literature which cannot be obtained by the nonclassical symmetry method due to Bluman and Cole.

Nonclassical symmetry reductions of a porous medium equation with convection

In this paper new symmetry reductions and exact solutions are presented for the porous medium equation with convection Those spatial forms for which the equation can be reduced to an ordinary differential equation are studied. The symmetry reductions and exact solutions presented are derived by using the nonclassical method developed by Bluman and Cole and are unobtainable by Lie classical method. The asymptotic behaviour of some of the new solutions is analysed.

On certain symmetries of the nonlinear Schrödinger equation

In this paper we reduce the problem of symmetries for the Ginzburg-Landau equation to a specific problem of non-classical symmetries for the linear free Schrodinger equation, with the help of an immersion result. This new problem is related to the local U(1)-invariance of the model, and allows us to construct projective representations for arbitrary Lie groups acting on the space-time manifold of the related sigma-model. To achieve this result we propose an enlargement of the usual notion of non-classical symmetry.

MathLie a program of doing symmetry analysis

We discuss the capabilities of MathLie in supporting the symmetry analysis of differential equations. MathLie is a computer algebra program written in Mathematica 3.0 and capable to carry out different types of symmetry calculations for ordinary as well as for partial differential equations. MathLie determines point symmetries, non-classical symmetries, potential symmetries, approximate symmetries, and generalized symmetries for partial differential equations. Three examples, the Harry–Dyme, the turbulent burst equations, and the Zabolotskaya–Khoklov equation demonstrate the application of MathLie in connection with point symmetries. We also present similarity solutions of these equations.

Conditional symmetry of a porous medium equation

We obtain an exhaustive description of conditional symmetries admitted by a nonlinear porous medium equation in one spatial dimension. It is proved that solving the determining equations for the conditional symmetries either yields classical (Lie symmetries) or is equivalent to solving the porous medium equation itself. The second possibility is shown to be typical for an arbitrary evolution-type partial differential equation in one spatial variable. We show that this very property is responsible for the well-known phenomenon when using non-classical symmetries yield exact solutions which prove to be invariant ones.

Nonclassical symmetry reductions of the three-dimensional incompressible Navier-Stokes equations

The nonclassical reduction method as pioneered by Bluman and Cole ( J. Meth. Mech. 18 1025-42) is used to examine symmetries of the full three-dimensional, unsteady, incompressible Navier-Stokes equations of fluid mechanics. The procedure, when applied to a system of partial differential equations, yields reduced sets of equations with one fewer independent variables. We find eight possibilities for reducing the Navier-Stokes equations in the three spatial and one temporal dimensions to sets of partial differential equations in three independent variables. Some of these reductions are derivable using the Lie-group method of classical symmetries but the remainder are genuinely nonclassical. Further investigations of one of our eight forms shows how it is possible to derive novel exact solutions of the Navier-Stokes equations by the nonclassical method.

Non-classical symmetries and the singular manifold method: a further two examples

This paper discusses two equations with the conditional Painleve property. The usefulness of the singular manifold method as a tool for determining the non-classical symmetries that reduce the equations to ordinary differential equations with the Painleve property is confirmed once more

Semiclassical mechanics of periodic motion: I. General scheme

A unified treatment of quantum mechanical oscillatory motion in one dimension is presented in a phase space formalism which is especially adapted to the semiclassical energy domain and closely related to a `naive' quantization of action and angle variables. Characteristics of this scheme are non-classical symmetry properties of the phase space functions representing density operators and observables and the inclusion of half-Bohr orbits besides the familiar Bohr orbits. Over long time intervals the quantum evolution can be well approximated by a Hamiltonian flow along these distinguished classical orbits. The interplay of this reduced classical evolution and the symmetry properties of the phase space functions results in a consistent quantitative description of quantum interference effects which are most clearly seen in the revivals of wavepackets.

Non-classical symmetries and the singular manifold method revisited

The connection between the singular manifold method (Painleve expansions truncated at the constant term) and symmetry reductions of two members of a family of Cahn-Hilliard equations is considered. The conjecture that similarity information for a nonlinear partial differential equation may always be fully recovered from the singular manifold method is violated for these equations, and is thus shown to be invalid in general. Given that several earlier examples demonstrate the connection between the two techniques in some cases, it now becomes necessary to establish when such a relationship exists - a question related to a deeper understanding of Painleve analysis. This issue is also briefly discussed.

Nonclassical symmetries of a porous medium equation with absorption

In this paper new symmetry reductions and exact solutions are presented for the porous medium equation with absorption . Those spatial forms for which the equation can be reduced to an ordinary differential equation are studied. The symmetry reductions and exact solutions presented are derived by using the nonclassical method developed by Bluman and Cole and are unobtainable by Lie classical method.

Involution and symmetry reductions

After reviewing some notions of the formal theory of differential equations, we discuss the completion of a given system to an involutive one. As applications to symmetry theory, we study the effects of local solvability and of gauge symmetries, respectively. We consider nonclassical symmetry reductions and more general reductions using differential constraints.

The role of symmetries in solving differential equations

A review of the role of symmetries in solving differential equations is presented. After showing some recent results on the application of classical Lie point symmetries to problems in fluid draining, meteorology, and epidemiology of AIDS, the nonclassical symmetries method is presented. Finally, it is shown that iterations of the nonclassical symmetries method yield new non-linear equations, which inherit the Lie symmetry algebra of the given equation. Invariant solutions of these equations supply new solutions of the original equation. Furthermore, the equations yield both partial symmetries as given by Vorobev, and differential constraints as given by Vorobev and by Olver. Some examples are given. The importance of using ad hoc interactive REDUCE programs is underlined.

Magnetic properties of Al-based icosahedral alloys

One of the fundamental questions which has not been answered sufficiently well till today is whether a nonclassical symmetry such as an icosahedral one would induce any special and distinctive features in the physical properties of the systems. Most theoretical investigations indicate that there are unique features in the electronic structure on account of icosahedral symmetry which could for example, influence the magnetic properties significantly. Experimental studies, particularly, on Al-based icosahedral alloys, have shown a large variation in magnetic properties ranging from diamagnetic to Curie like and Pauli paramagnetism on one hand and ferromagnetic and spin glass like behaviour on the other. In the present article, we will concentrate on some aspects of moment formation and variation in magnetic properties in Al-Mn based icosahedral systems and their dependence on the method of preparation which presumably controls and introduces the variable disorder. We will also try to answer some pertinent questions related to magnetism in icosahedral systems such as extent and origin of moment formation existence of magnetic and nonmagnetic Mn sites and distribution of interatomic distances reponsible for variation in magnetic properties.

The velocity equations for dilatant granular flow and a new exact solution

For axially symmetric flow of dilatant granular materials, the velocity equations uncouple from the stress equations in certain plastic regimes, and assuming dilatant double shearing a set of three first order partial differential equations are obtained. These equations turn out to be deceptive because although simple in appearance, the determination of simple exact solutions is non-trivial. Here we show that all the known functional forms of existing solutions also arise systematically by consideration of the "optimal systems" of the classical Lie symmetries which indicates that any further solution types most likely arise from non-classical symmetries. For one of the known families we present a special case which admits a particularly simple closed form expression, which has not been previously given in the literature. For this particular special case the integral curves (streamlines) can be readily obtained as well as a simple analytical "approximation" for the particle paths. The streamlines and the validity of the analytical approximation are shown graphically.

Iterations of the non-classical symmetries method and conditional Lie - Bäcklund symmetries

Iterations of the non-classical symmetries method give rise to new nonlinear equations, which inherit the Lie point symmetry algebra of the original equation. Invariant solutions of these heir equations supply new solutions of the original equation. We show that particular cases of such invariant solutions correspond to Zhdanov's conditional Lie - Backlund symmetries.

New exact solutions toN-dimensional radially symmetric nonlinear diffusion equations

New exact solutions toN-dimensional radially symmetric nonlinear diffusion equations with power-law diffusivities are constructed in terms of the generalized conditional symmetry method, which can be thought as a generalization of the nonclassical symmetry method due to Bluman and Cole.

LIE, a PC program for Lie analysis of differential equations

LIE is a self-contained PC program for the Lie analysis of ordinary or partial differential equations, either a single equation or a simultaneous set. It is written in the symbolic mathematics language MUMATH and will run on any PC. It comes as a complete program that incorporates the necessary parts of MUMATH and is ready to run. The previous version was for classical Lie analysis, finding the point symmetries of well-posed differential equations. This is now extended to contact, Lie-Backlund and nonclassical symmetries. Memory utilization has been improved and it can analyse the equations of Magneto-Hydrodynamics, a set of 9 partial differential equations in 12 variables.

Symmetries and Heir Equations for the Laminar Boundary Layer Model

Iterations of the nonclassical symmetries method applied to both the Prandtl equation and Prandtl system give rise to new nonlinear equations, which inherit the Lie symmetry algebra of the original Prandtl equation (system). Invariant solutions of these heir equations supply new solutions of the original equation (system).

Nonclassical Symmetry Reduction and Riemann Wave Solutions

In this paper we employ the “nonclassical symmetry method” in order to obtain Riemann multiple wave solutions of a system of first-order quasilinear differential equations. We show how to construct a Lie module of vector fields which are symmetries of the system supplemented by certain first-order differential constraints. We demonstrate the usefulness of our approach on several examples.

Weak and Partial Symmetries of Nonlinear PDE in Two Independent Variables

Abstract Nonclassical infinitesimal weak symmetries introduced by Olver and Rosenau and partial symmetries introduced by the author are analyzed. For a family of nonlinear heat equations of the form u t = (k(u) u x)x + q(u), pairs of functions (k(u), q(u)) are pointed out such that the corresponding equations admit nontrivial two-dimensional modules of partial symmetries. These modules yield explicit solutions that look like u(t, x) = F (θ(t) x + φ(t)) or u(t, x) = G(f(x) + g(t)).