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ISSN 2074-1863 Уфимский математический журнал. Том 4. № 4 (2012). С. 196-207.
УДК 517.958: 537.84
ON INFINITESIMAL RECIPROCAL-TYPE TRANSFORMATIONS IN GASDYNAMICS. LIE GROUP CONNECTIONS AND NONLINEAR SELF-ADJOINTNESS
N.H. IBRAGIMOV, C. ROGERS
Abstract. Bateman-type reciprocal transformations are represented as non-local infinitesimal symmetries of the governing equations of steady, two-dimensional, inviscid gasdynamics. In particular, this representation allows the construction of a novel non-local conservation law using the recently introduced concept of nonlinear self-adjointness.
Keywords: Bateman-type reciprocal transformations, gasdynamics, non-local symmetries and conservation laws.
Introduction
This paper is dedicated to Lev Ovsyannikov for his immense contributions to applications of Lie group theory in gasdynamics.
Reciprocal relations in steady two-dimensional gasdynamics have their origin in work of Bateman [1, 2]. These multi-parameter transformations leave invariant the governing equations, up to the equation of state. Loewner [3], in the context of irrotational planar gasdynamics undertook the systematic reduction of hodograph systems via finite matrix Bäcklund transformations to Cauchy-Riemann, Tricomi or classical wave canonical forms in subsonic, transonic and supersonic regimes respectively. Such reduction may be achieved for certain multi-parameter equations of state which may be used to approximate real gas behaviour. A detailed account of that work and reciprocal relations, along with other physical applications is given in [4]. Subsequently Loewner [5] extended his analysis to consider infinitesimal Baäcklund transformations which asymptotically reduce the hodograph equations to appropriate tractable canonical forms. In later work, a remarkable connection has been made in [6, 7] between the class of infinitesimal Backlund transformations introduced in a gasdynamics context in [4] and modern soliton theory of 2+1-dimensions [8, 9]. Thus, if the continuous parameter of the infinitesimal Baäcklund transformations of [4] is regarded as a third independent variable, then a triad of linear matrix equations is evident in the Loewner scheme and its generalisations. The compatibility conditions for the latter generate a broad class of novel 2+1-dimensional integrable non-linear evolution equations wherein the spatial variables have equal standing. The (9-dressing method was described for these systems in [7]. A characteristic feature of use 2+1-dimensional master systems is that they admit a compact representation in terms of a triad of eigenfunction matrices. The class includes, inter alia higher-dimensional
N.H. Ibragimov and C. Rogers, On infinitesimal reciprocal-type transformations in
GASDYNAMICS. Lie GROUP CONNECTIONS AND NONLINEAR SELF-ADJOINTNESS.
© Nail H. Ibragimov and Colin Rogers 2012.
N.H. Ibragimov acknowledges the financial support of the Government of Russian Federation through Resolution No. 220, Agreement No. 11.G34.31.0042.
Поступила 9 ноября 2012 г.
integrable generalizations of the chiral-fields model, non-Abelian sine-Gordon and Toda lattice equations. A 2+1-dimensional integrable generalization of the classical sine-Gordon emerges naturally in this context (see e.g. [10, 11, 12, 13, 14, 15, 16]). With regard to the class of finite Backlund transformations introduced by Loewner in [4], it was established in [17] that these may represented as a composition of gauge and classical Darboux-type transformations. This result as used to construct iterated versions of Loewner-type transformations based on established procedures in soliton theory. It is noted that Bateman-type reciprocal transformations of steady rotational gasdynamics may be derived as a specialisation of the class of finite Loewner transformations [27].
Reciprocal transformations for non-steady physical systems were originally introduced in the setting of gasdynamics and magnetogasdynamics in [18, 19]. These constitute multi-parameter transformations which leave invariant the governing continuity and momentum equations but not, in general, the equation of state. The action of the reciprocal transformations on the gasdynamic or magneto-gasdynamic system is to provide a link to associated gasdynamic (magnetogasdynamic) systems with a new multi-parameter class of equations of state. Such reciprocal and related invariant transformations may be applied to classes of shock propagation problems in the manner described by Ustinov [20]. In recent work [21], the reciprocal transformations of 1+1-dimensional non-steady gasdynamics as derived in [18] were applied to derive a new integrable-differential version of the affinspharen equation. The latter describes the analogues of spheres in affine geometry (Jonas [22], Nomizu and Sasaki [23]). It represents an alternative avatar of the classical Tzitzeica equation [24]. It has been established that the affinspharen equation arises naturally out of a Lagrangian description of 1+1-dimensional anisentropic gasdynamics for a privileged class of constitutive laws [25, 26]. Integrability is preserved under the action of a reciprocal transformation as is supported by the construction of a Lax pair for the resultant deformed affinspharen equation.
Here, infinitesimal versions of reciprocal-type invariance transformations in gasdynamics are presented along with novel Lie group and nonlinear self-adjointness connections.
1. The Bateman-type reciprocal transformations
Here, we consider invariance under reciprocal-type transformations of the governing equations of steady, two-dimensional, inviscid gasdynamics, namely
(pu)x + (pv)y = 0 ,
p(uux + vuy)+ px = 0 , p(uvx + vvy)+ py = 0 , (1.1)
usx + vsy = 0
where
q = uex + vey
is the gas velocity, while p, p and s denote, in turn, the gas pressure, gas density and specific
entropy. To the system (1.1) must be adjoined an appropriate equation of state
> 0 (1.2)
P = P(P,S) ,
The system (1.1) implies the pair of conservation laws
(puv)x + ('P + pv2)y = 0
(1.3)
(p + pu2)x + (puv)y = 0
S
whence, new independent variables may be introduced according to
dx' = ^f1 [ (p + fi2 + pv2)dx — puvdy ] ,
2 (1.4)
dy' = Pf1[—puvdx + (p + p2 + pu2)dy ]
subject to the requirement 0 < |J(x',y',x,y)l < so that
0 < l(p + p2)(p + P2 + pq2)l < ^ . (1.5)
It may be established that, the gasdynamic system (1.1) is invariant under the 4-parameter class of reciprocal transformations [28]
fclU fclV
u =-------------— , V =
, a , P3P(P + fc2) , ,Jl( )
P = -----—^ , P = . fí ,---2 , S =^(S)
p + P2 p + P2 + pq2
:i.6)
This result has its roots in work of Bateman on lift and drag functions in planar irrotational gasdynamics [1]. The subject was elaborated upon by Tsien [29]. Additional invariance properties of reciprocal transformations concerning the equation of state, were investigated in [30].
In more recent work, reciprocal transformations have been applied in [31] to uncover hidden soliton-type integrability in subsonic gasdynamics and to isolate periodic vortex motions which are reciprocally linked to hydrodynamic Stuart vortices [32]. In [33], reciprocal transformations were combined with the double action of an auto-Backlund transformation on an integrable elliptic Tzitzeica model in subsonic gasdynamics to isolate breather-type compressible vortex motions of a generalized Karman-Tsien gas. Doubly periodic motions valid for such a gas were obtained via multi-parameter reciprocal transformations and the Hirota bilinear operator formalism in [34]. In [35], reciprocal transformations were adduced in conjunction with the action of an auto-Backlund transformation to isolate vortex train motions in super-Alfvenic magnetogasdynamics.
2. Infinitesimal Bateman-type reciprocal transformations
It is observed that the one-parameter class of reciprocal relations (1.6) with
=fc2 = fc4 = fc, = 1 (2.1)
reduces to the identity transformation in the limit ¡3 ^ <x>. The reciprocally associated gasdynamic flows may be regarded as a deformation of the original motions. Action of this class on a seed hydrodynamic vortex street was used in [31] to construct a compressible, subsonic version of the Stuart vortex street, valid for a generalised Kármán-Tsien model.
If one sets e = ¡3-l then the one-parameter class of reciprocal transformations becomes
. u . v
u =------- , v =
together with
1 + 1 +
/ P , P(1 + íP) , lTi^ (2.2)
P =------------ , P =-----------;--------^ , s = W(s)
1 + tp 1 + e(p + pq2)
dx' = e[(p + pv2)dx — puvdy] + dx ,
(2.3)
dy/ = e[ —puvdx + (p + pu2)dy] + dy .
2The reciprocal variables x' and y' are up to scaling, the drag and lift functions of Bateman [1]
On expansion and retention of terms, to 0(e2) the relations (2.2) produce the class of infinitesimal transformations
u' = u(1 — ep) , v' = v(1 — ep) , p' = p(1 — ep) , p' = p(1 — tpq2) , s' = Ф(з)
(2.4)
where t is now deemed to be a small deformation parameter. It is readily shown, ab initio that the gasdynamic system is invariant under this class of infinitesimal transformations augmented by (2.3), up to the equation of state.
Thus, with regard to the continuity equations
—p'v'dx' + p'u'dy' =
= —pv[ 1 — t(p + pq2) + 0(e2) ] ■ [ t((p + pv2)dx — puvdy) + dx ] +
+ pu[ 1 — t(p + pq2) + 0(e2) ] [ t(—puvdx + (p + pu2)dy) + dy ] =
= —pvdx + pudy+
+ e[ —pv( (p + pv2)dx — puvdy) + (p + pq2)pvdx] +
+ e[ pu(—puvdx + (p + pu2)dy) —
— e(p + pq2)pudy ] + 0(e2) =
= —pvdx + pudy + 0(e2) .
— (p' + p'v'2)dx' + p'u'v'dy' =
= — [p(1 — ep) + pv2(1 — tpq2 — 2ep) + 0(e2) }dx'+
+ puv(1 — tpq2 — 2tp + 0(e2) )dy' =
= —(p + pv2)dx+
+ e[p2 + p2v2q2 + 2pv2p — (p + pv2)2 ]dx+
+ tpuv(p + pv2)dy + puvdy — tp2u2v2dx+
+ tpuvdy[ — (pq2 + 2p) + p + pu2 ] + 0(e2) =
= —(p + pv2)dx + puvdy + 0(e2) .
Moreover,
Similarly,
(2.5)
—p'u'v'dx' + (p' + p'u'2)dy' =
= —puv(1 — t(pq2 + 2p) + 0(e2) )dx'+
+ [ p(1 — tp) + pu2(1 — t(pq2 + 2p)]+ 0(t2)dy' =
= —puvdx + tpuv(pq2 + 2p)dx—
— puvt(p + pv2)dx — tp2u2v2dy+
+ (p + pu2)dy — e[p2 + pu2(pq2 + 2p) ]dy—
— tpuv(p + pu2)dx + t(p + pu2)2dy + 0(e2) =
= —puvdx + (p + pu2)dy + 0(e2) .
Hence, to 0(e2), the gasdynamic system determined by the conservation laws consisting of the continuity equation augmented by the pair (1.3) is invariant under the class of infinitesimal transformations (2.3) —(2.4).
Further,
d^' = —p'v'dx' + p'u'dy' =
= —pv[ 1 — t(p + pq2) + 0(e2) }dx'+
+ pu[1 — t(jp + pq2) + 0(e2) ]dy' =
= —pvdx + e[ pv(p + pq2) — (p + pv2)pv ]dx + tpuv2dy+
+ pudy + e[ —pu(p + pq2) ]dy — ep2u2vdx+
+ tpu(p + pu2)dy + 0(e2) =
= —pvdx + pudy + 0(e2) =
= d^ + 0(e2)
so that the isentropic condition (1.1)3 is also preserved under the class of infinitesimal transformations.
3. Lie group connection. Reconstruction of the Bateman relations via
initial value problems
The class of reciprocal transformations (2.2) is readily reconstructed from (2.4) via the usual Lie group procedure involving the solution of a system of initial value problems (see e.g. [36]) which, in this case yields
dp /2 /1
— = —p'2 , p' I €=q = p ,
dt
du dv'
— = —u'p', u |e=o , — = —v'p', v' |e=o = v, (3.1)
dt dt
^ = r! 2 a '2 n'\ = n
~dt = —P^ , P ^=0 = P ,
together with
d(dx') = [(p + pv2)dx — puvdy]dt , dx'|e=0 = dx ,
d(dy') = [ —puvdx + (p + pv2)dy ]dt , dy'|e=0 = dy . Thus, solution of the initial value problem (3.1)1 yields the reciprocal pressure
' P
P
(3.2)
1 + tp
while insertion, in turn of this relation into (3.1)2 produces
du' u'p
dt 1 + tp
whence, on integration and use of the initial condition u'|e=0 = u it is seen that
u
u
1 + tp
and, in a similar manner
1 + tp
Accordingly, (3.1)3 yields
dp1 p' 2q2
dt (1 + tp)2
v
whence, on integration and use of the initial condition p'|e=0 we retrieve the reciprocal density relation
, = p(1 + tp)
P 1 + t(p + pq2) .
The reciprocal variable relations (2.3) are likewise retrieved via (3.2).
Thus, the one-parameter class (2.2) of Bateman-type reciprocal transformations is seen to have Lie group origin. It is natural to investigate to what extent this class may be additional boosted by more general Lie group action. In particular, Prim-type substitution invariance principles of gasdynamics and magneto-gasdynamics [37, 38, 39] may be generated by Lie group methods (see [40] —[42] and work cited therein). The imbedding of the one-parameter class of one-parameter reciprocal transformations in a more general Lie-group investigation of invariance of the gasdynamic system (1.1) will be investigated subsequently in Section 5.
It is remarked that, Lie group methods as described in the authoritative monograph by Ovsiannikov [47] have extensive applications in hydrodynamics (Andreev et al [48]). Importantly, such applications are not restricted to the construction of exact solutions. Thus, in particular, key properties such as the time evolution of moments of inertia as described by Ball [49, 50] may, in fact, be generated via Lie-group methods [51, 52]. Variants of such theorems may be shown to be key to isolating hidden integrable Hamiltonian structure of Ermakov-Ray-Reid type in rotating shallow water theory [53], gas cloud evolution [54] and magnetogasdynamics [55].
4. Infinitesimal reciprocal transformations in non-steady gasdynamics
The governing equations of one-dimensional, anisentropic non-steady gasdynamics, neglecting heat conduction and radiation are
Pt + (pu)x = 0 , p(ut + uux) + px = 0 ,
(4.1)
Sf + usx — 0
together with a prevailing gas law of the type (1.2).
In [18] it was established that the gasdynamic system is invariant, up to the equation of state, under the 4-parameter class of reciprocal transformations
@\U
u =
P + 2
/ Q Pl@3 / [hp(p + fc2)
P = P4----------, P ~
P + 32 P + 32 + pu2
s' = ^(s) , (42)
dt' = [3fl[ —pudx + (p + pu2 + /32)dt] , dx' = dx ,
0 < \p + pu2 + [2\ < <x .
The specialisation of the parameters [ as in (2.1) with e = [-1 leads to the one-parameter
subclass of invariant transformations
u
u
1 + ep
, = P / = P(1 + eP) (4.3)
^ 1 + ep , ^ 1 + t(p + pu2)
together with
dt1 = e[ —pudx + (p + pu2)dt ] + dt , dx' = dx . (4.4)
Expansion and retention of terms to 0(e2) in (4.3) leads to the class of infinitesimal transformations
u = u(1 — ) , = (1 — u2) , = (1 — ) ,
2 (4-5)
dt = e[ — pudx + (p + pu2)dt] + dt , dx = dx
augmented by s' = ^(s). It is readily shown that the gasdynamic system (4.1) is invariant to 0(e2) under this class of infinitesimal transformations and that the original reciprocal transformations (4.3) may be reconstructed via the solution of a system of initial value problems analogous to that in (3.1).
5. Representation of reciprocal transformations as non-local symmetries
It has been shown in [43], using the transition from Lagrange to Euler coordinates in gasdynamics, that the equations
1
Vt + VVX + - px = 0,
P
Pt + vpx + pvx = 0, (5.1)
Pt + vpx — pvx = 0
describing the Chaplygin gas have two nonlocal symmetries:
d d p d
1 dx dp p dp ,
, 2 s (5.2)
t2 \ d d d pr d
2 ) dx dv dp p dp
where T,s,a are non-local variables defined by the compatible over-determined systems
Tx = p, T Tt = —vp VT
p , St = — ,
p
1 V
p, Ot = - — ■
p
(5.3)
Equations (5.3) can be equivalently written as the following conservation laws:
Dt(p) + Dx(vp) = 0,
O +«■ (?) = 0' >'-•>
D’(1) + D- (Í) =0
The second conservation equation (5.4) contains the non-local variable r and therefore it is termed a non-local conservation law. The first, second and third conservation equations (5.4) provide the compatibility conditions for the over-determined equations (5.3) for the non-local variables r, s and a, respectively.
If we apply the same approach to the conservation laws (1.3),
(puv)x + (p + pv2)y = 0,
(p + pu2 )x + (puv)y = 0,
we arrive at the non-local variables a and a determined by the equations
ax = p + pv2, ay = —puv (5.5)
and
&x = —puv, Gy = p + pu2, (5.6)
respectively. Using these variables, we can replace the infinitesimal reciprocal transformations (3.2)-(3.3) with the following non-local symmetry generator
V 9 + 9 2 9 9 9 ( 2 + 2) 2 9 (57)
x = a— + a-----------p ----pu-------pv-------------------------------(u+v)p — ■ (5.7)
ox ay op ou ov op
Let us verify that the operator (5.7) is admitted by the steady two-dimensional equations of gasdynamics (1.1),
(pu)x + (pv)y = 0, px + p(uux + vuy) = 0, py + p(uvx + vvy) = 0, usx + vsy = 0,
with an arbitrary equation of state. The usual prolongation formula
<■ = Dt(r,k) — u) D„(?)
gives the following prolongation of the operator (5.7) to the derivatives involved in the equations
(1.1):
X = X+(p— +(p— +(p— +(p— +(s— +(s(5 8)
+^" dpx +^y dpy +^x dux +^y duy +^x dvx +^y dvy +^x dpx +^y dpy +^x dsx +^y dsy , K J
where
= puvpy — (3p + pv 2 )px, c = puvpx — (3p + pu2 )py,
(5.9)
(x = puvuy — (2p + pv )ux — upx, Q = puvux — (2p + pu )uy — upy,
Q = puvvy — (2p + pv2 )vx — vpx, Q = puvvx — (2p + pu2 )vy — vpy,
(x = puvpy — (p + 2pu2 + 3pv2)px — 2p2 (uux + vvx),
= puvpx — (p + 2pv2 + 3pu2)py — 2p2(uuy + vvy),
Q = puvsy — (p + pv2)sx, (y = puvsx — (p + pu2)sy.
The calculation shows that
X [(pu)x + (pv)y ] = — 2[p + p(u2 + v2)][(pu)x + (pv)y ] —
— pu[px + p(uux + vuy)] — pv[py + p(uvx + VVy)],
X [px + p(uux + vuy)] = — [3p + p(u2 + v2)][px + p(uux + vuy)],
X [py + p(uvx + VVy)] = — [3p + p(u2 + v2)][py + p(uvx + VVy))].
Hence, the infinitesimal test for the invariance of Equations (1.1) is satisfied.
We will use the non-local symmetry (5.7) for constructing a conservation law by the method of nonlinear self-adjointness [44] (see also [45]). According to this method, we take the formal Lagrangian of Equations (1.1) in the form
£ = R(pux + pvy + upx + vpy) + S (usx + vsy)+
+ U [px + p(uux + vuy)] + V [py + p(uvx + vvy)]
(5.10)
and write the adjoint system
„ SC n 8C n 8C n 8C n
~5u = , ¿C = , Tp = ’ ¿C = ’ = , )
to Equations (1.1), where 8C/8u is the variational derivative of the formal Lagrangian with respect to the dependent variable u, etc.
The nonlinear self-adjointness implies that the adjoint equations (5.12) are satisfied for all solutions of the system (1.1). In our case this property follows, e.g. from the conservation form of the first equation of the system (1.1). Namely, it is well known that if a function of independent and dependent variables together with partial derivatives of any order has a divergence form, then the variational derivatives of this function vanish (see, e.g. [46]). Using this statement and noting that the expression
pux + p vy + u px + v Py
has a divergence form,
pux + pvy + upx + vpy = Dx(pu) + Dy (pv ), we see that the formal Lagrangian (5.11) with
R =1, U = V = S = 0 (5.13)
solves the equations (5.12).
Now we apply to the non-local symmetry (5.7) the general formula from [45]) for constructing conserved vectors associated with symmetries and obtain
c1 = -w1 — - w2 — - w3 — - w4 — - w5 — ,
- - - - -
oux ovx opx opx O Sx
c2 = -w1 — - w2 — - w3 — - w4 — - w5 — ,
- - - - -
Ouy OVy opy opy O Sy
(5.14)
where
w1 = -pu - aux - auy, w2 = -pv - avx - avy,
= —p — apx — apy, = —(u + v )p — apx — apy, (5.15)
VV — —Q, Sx — (7 Sy.
Inserting in (5.14) the expressions (5.15) for W1 ,...W5 and invoking Equations (5.13) we obtain the following non-local conserved vector:
C1 = ppu + u(u2 + v2) p2 + q(p u)x + 7 (pu)y,
(5.16)
C2 = ppv + v(u2 + v2) p2 + Q(pv)x + 7 (pv)y.
The vector (5.16) satisfies the conservation law in the following form:
Dx (C ^ + Dy (C2) = 2[p + p(u2 + v2)][(pu)x + (pv )y ] + pu[px + p(uux + vuy)] +
+ pv[py + p(uvx + VVy)] + Q[(pu)x + (pv)y]x + 7 [(pu)x + (pv)y]y. (5.17)
A BIOGRAPHICAL NOTE
H. Bateman FRS was educated at Trinity College, Cambridge University and was Senior Wrangler in the Mathematical Tripos of 1903. He was elected to the Royal Society of London in 1928 and to the National Academy of Science (USA) in 1930.
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33. C. Rogers, W.K. Schief, K.W. Chow and C.C. Mak On Tzitzeica vortex streets and their reciprocals in subsonic gas dynamics // Stud. Appl. Math. 114, 2005. P. 271-283.
34. K.W. Chow, C.C. Mak, C. Rogers and W.K. Schief Doubly periodic and multiple pole solutions of the sinh-Poisson equation: application of reciprocal transformations in subsonic gas dynamics // J. Comp & Appl. Math. 190, 2006. P. 114-126.
35. C. Rogers and W.K. Schief Vortex trains in super-Alfvenic magnetogasdynamics. Application of reciprocal-Bäcklund transformations // J. Nonlinear Math. Phys. 12, Suppl 1, In honour of Francesco Calogero, 2005. P. 548-564.
36. C. Rogers and W.F. Ames Nonlinear Boundary Value Problems in Science and Engineering. Academic Press, Mathematics in Science and Engineering Series, New York, 1989.
37. R.C. Prim A note on the substitution principle for steady gas flows // J. Appl. Phys. 20, 1949. P. 448-450.
38. R.C. Prim Steady rotational flow of ideal gases // Arch. Rat. Mech. Anal 1, 1952. P. 425-497.
39. G. Power and C. Rogers Substitution principles in non-steady magneto-gasdynamics // Appl. Sci. Res. 21, 1969. P. 176-184.
40. F. Oliveri On the equations of ideal gasdynamics with a separable equation of state: Lie group analysis and substitution principles // Int. J. Nonlinear Mech. 27, 1992. P. 773-784.
41. F. Oliveri and M.P. Speciale Exact solutions to the equations of ideal gasdynamics by means of the substitution principle // Int. J. Nonlinear Mech. 33, 1998. P. 585-592.
42. F. Oliveri On substitution principles in ideal magneto-gasdynamics by means of Lie group analysis // Nonlinear Dynamics 42, 2005. P. 217-231.
43. I.S. Akhatov, R.K. Gazizov, and N.H. Ibragimov Non-local symmetries: Heuristic approach // Itogi Nauki i Tekhniki. Sovremennie problemy matematiki. Noveishye dostizhenia, vol. 34, 1989. P. 3-84. English transl., Journal of Soviet Mathematics, 55(1), 1991. P. 1401-1450.
44. N.H. Ibragimov Nonlinear self-adjointness in constructing conservation laws // Archives of ALGA, vol. 7/8, 2010-2011. P. 1-99. See also arXiv:1109.1728v1[math-ph], 2011. P. 1-104.
45. N. H. Ibragimov Nonlinear self-adjointness and conservation laws // J. Phys. A: Math. Theor., 44, 2011. P. 432002.
46. R. Courant and D. Hilbert Methods of Mathematical Physics, Vol. I, Interscience Publishers, John Wiley, New York, 1989.
47. L.V. Ovsiannikov Group Analysis of Differential Equations, Academic Press, New York, 1982.
48. V.K. Andreev, O.V. Kaptsov, V.V. Pukhnachev and A.A. Rodionov Application of Group-Theoretical Methods in Hydrodynamics. Mathematics and its Applications. Vol. 450, Springer, Berlin, Heidelberg, 1998.
49. F.K. Ball Some general theorems concerning the finite motion of a shallow liquid lying on a paraboloid // J. Fluid Mech. 17, 1963. P. 240-256.
50. F.K. Ball The effect of rotation on the simpler modes of motion in an elliptic paraboloid // J. Fluid Mech. 22, 1965. P. 529-545.
51. C. Rogers Generation of invariance theorems for non linear boundary value problems in shallow water theory: application of MACSYMA Numerical and Applied Mathematics, IMACS Meeting Proceedings, Paris 69-74, 1989.
52. D. Levi, M.C. Nucci, C. Rogers and P. Winternitz Group theoretical analysis of a rotating shallow liquid in a rigid container // J. Phys. A: Mathematical & General 22, 1989. P. 4743-4767.
53. C. Rogers and H. An Ermakov-Ray-Reid systems in 2+1-dimensional rotating shallow water theory // Stud. Appl. Math. 125, 2010. P. 275-299.
54. C. Rogers and W.K. Schief On the integrability of a Hamiltonian reduction of a 2+1-dimensional non-isothermal rotating gas cloud system // Nonlinearity 24, 2011. P. 3165-3178.
55. C. Rogers and W.K. Schief The pulsrodon in 2+1-dimensional magnetogasdynamics. Hamiltonian-Ermakov reduction // J. Math. Phys. 52, 083701, 2011. 20 pp.
Nail H. Ibragimov,
Laboratory Group analysis of mathematical models in natural and engineering sciences,
Ufa State Aviation Technical University,
K. Marx Str. 12,
450 000 Ufa, Russia
and Research Centre ALGA: Advances in Lie Group Analysis,
Blekinge Institute of Technology, SE-371 79 Karlskrona, Sweden E-mail: nailhib@mail.ru
Colin Rogers,
Australian Research Council Centre of Excellence for Mathematics and Statistics of Complex Systems, School of Mathematics and Statistics,
University of New South Wales, Sydney, Australia.
Currently, Visiting Fellow, Clare Hall,University of Cambridge, UK E-mail: c.rogers@unsw.edu.au
