F. M. Mahomed

Relationship between Symmetries andConservation Laws

The fundamental relation between Lie-Bäcklund symmetry generators andconservation laws of an arbitrary differential equation is derived without regardto a Lagrangian formulation of the differential equation. This relation is used inthe construction of conservation laws for partial differential equations irrespectiveof the knowledge or existence of a Lagrangian. The relation enables one toassociate symmetries to a given conservation law of a differential equation.Applications of these results are illustrated for a range of examples.

Symmetry Lie algebras of nth order ordinary differential equations

Abstract We show that an nth (n ⩾ 3) order linear ordinary differential equation has exactly one of n + 1, n + 2, or n + 4 (the maximum) point symmetries. The Lie algebras corresponding to the respective numbers of point symmetries are obtained. Then it is shown that a necessary and sufficient conditon for an nth (n ⩾ 3) order equation to be linearizable via a point transformation is that it must admit the n dimensional Abelian algebra nA1 = A1 ⊕ A1 ⊕ … ⊕ A1. We discuss in detail the symmetry realizations of (n − 1)A1 ⊕s A1. Finally, we prove that an nth (n ⩾ 3) order equation q(n) = H(t, q, …, qn − 1) cannot admit exactly an n + 3 dimensional algebra of point symmetries which is a subalgebra of nA1 ⊕, gl(2, R ).

Comparison of different approaches to conservation laws for some partial differential equations in fluid mechanics

Abstract The conservation laws for second order scalar partial differential equations and systems of partial differential equations which occur in fluid mechanics are constructed using different approaches. The direct method, Noether’s theorem, the characteristic method, the variational approach (multiplier approach) for arbitrary functions as well as on the solution space, symmetry conditions on the conserved quantities, the direct construction formula approach, the partial Noether approach and the Noether approach for the equation and its adjoint are discussed and explained with the help of an illustrative example on a non-linear field equation describing the relaxation to a Maxwellian distribution. The conservation laws for the non-linear diffusion equation for the spreading of an axisymmetric thin liquid drop, the system of two partial differential equations governing flow in a laminar two-dimensional jet and the system of two partial differential equations governing flow in a laminar radial jet are discussed via these approaches.

A Basis of Conservation Laws for Partial Differential Equations

Abstract The classical generation theorem of conservation laws from known ones for a system of differential equations which uses the action of a canonical Lie–Bäcklund generator is extended to include any Lie–Bäcklund generator. Also, it is shown that the Lie algebra of Lie–Bäcklund symmetries of a conserved vector of a system is a subalgebra of the Lie–Bäcklund symmetries of the system. Moreover, we investigate a basis of conservation laws for a system and show that a generated conservation law via the action of a symmetry operator which satisfies a commutation rule is nontrivial if the system is derivable from a variational principle. We obtain the conservation laws of a class of nonlinear diffusion-convection and wave equations in (1 + 1)-dimensions. In fact we find a basis of conservation laws for the diffusion equations in the special case when it admits proper Lie–Bäcklund symmetries. Other examples are presented to illustrate the theory.

Lie–Bäcklund and Noether Symmetries with Applications

New identities relating the Euler–Lagrange, Lie–Bäcklund and Noether operators are obtained. Some important results are shown to be consequences of these fundamental identities. Furthermore, we generalise an interesting example presented by Noether in her celebrated paper and prove that any Noether symmetry is equivalent to a strict Noether symmetry, i.e. a Noether symmetry with zero divergence. We then use the symmetry based results deduced from the new identities to construct Lagrangians for partial differential equations. In particular, we show how the knowledge of a symmetry and its corresponding conservation law of a given partial differential equation can be utilised to construct a Lagrangian for the equation. Several examples are given.

Noether symmetry approach in f(R)–tachyon model

Abstract In this Letter by utilizing the Noether symmetry approach in cosmology, we attempt to find the tachyon potential via the application of this kind of symmetry to a flat Friedmann–Robertson–Walker (FRW) metric. We reduce the system of equations to simpler ones and obtain the general class of the tachyonʼs potential function and f ( R ) functions. We have found that the Noether symmetric model results in a power law f ( R ) and an inverse fourth power potential for the tachyonic field. Further we investigate numerically the cosmological evolution of our model and show explicitly the behavior of the equation of state crossing the cosmological constant boundary.

Lie algebras associated with scalar second‐order ordinary differential equations

Second‐order ordinary differential equations are classified according to their Lie algebra of point symmetries. The existence of these symmetries provides a way to solve the equations or to transform them to simpler forms. Canonical forms of generators for equations with three‐point symmetries are established. It is further shown that an equation cannot have exactly r ∈{4,5,6,7} point symmetries. Representative(s) of equivalence class(es) of equations possessing s ∈{1,2,3,8} point symmetry generator(s) are then obtained.

Linearization criteria for a system of second-order ordinary differential equations

Abstract Firstly, we prove two linearization criteria for a system of two second-order ordinary differential equations (odes). The first states that a system of two non-linear second-order odes is reducible via a point transformation to a linear system if and only if it admits the four-dimensional abelian Lie algebra L 4,1 : [X i ,X j ]=0, i, j=1,…, 4 . The second states that in order for a system of two non-linear second-order odes to be linearizable, it is necessary and sufficient that it admits the four-dimensional Lie algebra L 4,2 : [X i ,X j ]=0, [X i ,X 4 ]=X i , i, j=1, 2, 3 . The approach used is constructive and enables one to explicitly work out the transformation that leads to linearization. Secondly, we give conditions under which a system of two second-order non-linear odes is reducible to the free particle equations x″=0, y″=0 . These linearization criteria are then generalized to a system of n(n>2) second-order odes. Finally, we give examples of how one can effect linearization for a system.

Noether gauge symmetry approach in f(R) gravity

We discuss the f(R) gravity model in which the origin of dark energy is identified as a modification of gravity. The Noether symmetry with gauge term is investigated for the f(R) cosmological model. By utilization of the Noether Gauge Symmetry (NGS) approach, we obtain two exact forms f(R) for which such symmetries exist. Further it is shown that these forms of f(R) are stable.

Symmetry Breaking for a System of Two Linear Second-Order Ordinary Differential Equations

A new canonical form for a system of two linear second-orderordinary differential equations (odes) is obtained. The latter isdecisive in unravelling symmetry structure of a system of two linearsecond-order odes. Namely we establish that the point symmetry Liealgebra of a system of two linear second-order odes can be5-, 6-, 7-, 8- or 15-dimensional. This result enhances both the richness andthe complexity of the symmetry structure of linear systems.