Non-local symmetries and conservation laws for one-dimensional gas dynamics equations

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Abstract

The theory on the generation of conservation laws [Symmetry conditions and construction of conservation laws for differential equations, Preprint] for systems of partial differential equations by the use of Lie–Bäcklund symmetries is generalised to include the use of non-local symmetries. It is shown how the action of a symmetry on the appropriate conservation laws can be utilised to extend the symmetry generator with respect to a covering system. Systems of differential equations that describe one-dimensional gas flow are used to demonstrate the direct calculation of new non-local symmetries as well as the generation of new conservation laws. Also a solution that is invariant under a non-local symmetry is found.

Introduction

Consider an rth order system of p partial differential equations of m independent variables x=(x1,x2,…,xm) with components xi and n dependent variables u=(u1,u2,…,un) with components uα byFβ(x,u,u(1),…,u(r))=0,β=1,…,p,where u(1),u(2),…,u(r) represent the collections of all first-, second- to rth-order partial derivatives of u,uαxi,2uαxi1xi2,…,ruαxi1xi2xirwith i,i1,…,ir=1,…,m, α=1,…,n. We use another compact notation, uαi,…,uαi1ik, for partial derivatives ∂uα/∂xi,…,∂kuα/∂xi1xi2⋯∂xik.

The system (1) admits a non-local symmetry1 generated byX=ξi(x,u,ũ)xiα(x,u,ũ)uαifX(r)Fβ≙0.

Here ũ=(ũ1,…,ũñ) are the non-local variables defined as integrals of functions of the local variables (x,u,u(1),…,u(k)). The notation =̂0 means=0 on the solution space of (1).

The prolongation of X isX=ξixiαuαiαuiαi1i2αui1i2α+⋯

The coefficient functions of this operator areζiα=Di(Wα)+ξjuijα,ζi1i2α=Di1Di2(Wα)+ξiui1i2iα,…,whereWαα−ξjujαare the Lie characteristic functions which are in terms of ũ, Di is the total derivative operator with respect to the independent variable xi:Di=xi+uiαuα+uii1αui1α+⋯+uii1i2…ikαui1i2…ikα+⋯andDi=Di+ũiγũγ,γ=1,…ñis the total derivative operator with respect to xi of the covering systemFβ(x,u,u(1),…,u(r))=0,β=1,…,p,Gδ(x,u,u(1),…,u(k),ũ,ũ(1))=0,δ=1,…,p̃.

The added equations in (4) define the non-local variables ũ in terms of conservation laws of (1)DiTi≙0,TiA,where T=(T1,…,Tm) is called a conserved vector of (1) and A is the vector space of all differential functions of finite orders, as defined in [7], [8].

For the case m=2 we can useTγ1=ũ2γ,Tγ2=−ũ1γ,since it preserves the contact condition D1Tγ1+D2Tγ2=D1ũ2γ−D2ũ1γ=0.

The covering system becomesFβ(x,u,u(1),…,u(r))=0,β=1,…,p,ũ1γ+Tγ2=0,ũ2γ−Tγ1=0,γ=1,…,ñ.

In this case non-local symmetries can be calculated directly without the use of the covering system.

Section snippets

Generation theorem for non-local symmetries

We present a non-local form of a classical result (see, e.g. [4], [3], [9]).

Theorem 2.1

Let X be the extension of a non-local operator to act on all relevant non-local variables ũγ and Di be the total derivative operator with respect to xi also extended to act on non-local variables. These operators satisfy the relation[Di,X]=Dik)Dk.

Proof

DiX=Dik)xk+Diα)uα+Diũγ)ũγ+⋯,XDi=(Diα)−Dik)uαk)uα+(Diũγ)−Dik)ũγk)ũγ+⋯

ThusDiXXDi=Dik)xk+Dik)uαkuα+Dik)ũγkũγ+⋯=Dik)Dk.

The

Extension of symmetry generators

Suppose that we have the non-local symmetry X of a system but for calculation purposes we need the extension of this symmetry X. Then there is a way to extend X by the use of Theorem 2.2 presented above for the case m=2.

Theorem 3.1

Suppose that X is a symmetry of a system of PDEs (1) with two independent variables x=(x1,x2) and n dependent variables u=(u1,u2,…,un). Let Tγ=(Tγ1,Tγ2), γ=1,2,…,ñ be conserved vectors of the system with associated non-local variables ũγ, i.e.Tγ2=−D1ũγ=−ũ1γ,Tγ1=D2ũγ=ũ2γand

Application: non-local symmetries for equations from gas dynamics

The relationship between the symmetries of the equations of one-dimensional gas flow in Lagrange variablesqs−vy=0,vs+py=0,ps+B(p,q)vy=0and the system in Euler variablesρt+vρx+ρvx=0,ρ(vt+vvx)+px=0,ρ(pt+vpx)+Bp,1ρvx=0which are connected by the Bäcklund transformation s=t, y=∫ρdx, q=1/ρ, were studied in [10] and quasi-local symmetries were found.

For this example we consider the case where B(p,q)=λp/q, i.e.,qs−vy=0,vs+py=0,ps+λpqvy=0with λ a constant.

These are the equations that govern the motion

Invariance under a non-local symmetry

It is possible to use non-local symmetries that are local in some covering system to transform the system of differential equations that admit these symmetries. Although we do not necessarily find a reduced system in this way, it leads to invariant solutions.

We use X11 to transform the system in Lagrange coordinates since it is a non-local symmetry which is local for the covering systemqs−vy=0,vs+py=0,ps+λpqvy=0,ũ3s=−p,ũ3y=vwith λ a constant.

We need to find new variables r0, r1, u0, u1, u2, u3

Some remarks

The method of direct calculation gives all non-local symmetries dependent on the non-local variables included in the calculation, irrespective of their status (local or non-local) in possible covering systems. In contrast, the calculation of quasi-local symmetries as well as that of potential symmetries yield only those that are local in the chosen covering system. Symmetries that are local in some covering system are useful in reduction methods of the differential equations. The symmetries

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