Non-local symmetries and conservation laws for one-dimensional gas dynamics equations
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 bywhere u(1),u(2),…,u(r) represent the collections of all first-, second- to rth-order partial derivatives of u,with i,i1,…,ir=1,…,m, α=1,…,n. We use another compact notation, uαi,…,uαi1…ik, for partial derivatives ∂uα/∂xi,…,∂kuα/∂xi1∂xi2⋯∂xik.
The system (1) admits a non-local symmetry1 generated byif
Here are the non-local variables defined as integrals of functions of the local variables (x,u,u(1),…,u(k)). The notation means=0 on the solution space of (1).
The prolongation of X is
The coefficient functions of this operator arewhereare the Lie characteristic functions which are in terms of ũ, Di is the total derivative operator with respect to the independent variable xi:andis the total derivative operator with respect to xi of the covering system
The added equations in (4) define the non-local variables ũ in terms of conservation laws of (1)where T=(T1,…,Tm) is called a conserved vector of (1) and is the vector space of all differential functions of finite orders, as defined in [7], [8].
For the case m=2 we can usesince it preserves the contact condition .
The covering system becomes
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 be the extension of a non-local operator to act on all relevant non-local variables and be the total derivative operator with respect to xi also extended to act on non-local variables. These operators satisfy the relation Proof Thus
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 . 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), be conserved vectors of the system with associated non-local variables , i.e.and
Application: non-local symmetries for equations from gas dynamics
The relationship between the symmetries of the equations of one-dimensional gas flow in Lagrange variablesand the system in Euler variableswhich are connected by the Bäcklund transformation s=t, , 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.,with 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 systemwith 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
References (16)
- et al.
A basis of conservation laws for partial differential equations
J. Nonlinear Math. Phys.
(2002) - et al.
Symmetries and differential equations
(1989) Differential Equations: Their Solutions Using Symmetries
(1989)Applications of Lie groups to differential equations
(1986)Continuous Symmetries, Lie Algebras, Differential Equations and Computer Algebra
(1996)- N.H. Ibragimov (Ed.), CRC Handbook of Lie Group Analysis of Differential Equations, vol. 1–3, Chemical Rubber Company,...
- N. Ibragimov, Sur i’équivalence des équations d’évolution qui admettent une algèbre de Lie-Bäcklund infinie, C.R....
- et al.
Lie–Bäcklund transformation groups
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