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Reciprocal Transformations of Two-Component Hyperbolic Systems and Their Invariants
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.
228, 365]376 Ž1998.
AY986131
Reciprocal Transformations of Two-Component Hyperbolic Systems and Their Invariants E. V. Ferapontov* Centre for Nonlinear Studies, Landau Institute for Theoretical Physics Academy of Science of Russia, 117940 Moscow, GSP-1, Russia
and C. Rogers and W. K. Schief School of Mathematics, The Uni¨ ersity of New South Wales, Sydney 2052, New South Wales, Australia Submitted by William F. Ames Received August 24, 1998
Criteria for the equivalence under reciprocal transformations of two-component hyperbolic systems are established in terms of reciprocal invariants. These results are applied to a 1 q 1-dimensional isentropic gasdynamics system to determine reciprocal equivalence to the canonical system with a polytropic gas law. The four-parameter model pressure-density relations for which this reciprocal equivalence holds are parameterized in terms of the Lagrangian signal speed. Q 1998 Academic Press
1. INTRODUCTION Reciprocal-type transformations have a long history. In 1928, Haar w1x in a paper devoted to adjoint variational problems, established a reciprocaltype invariance of a plane potential gasdynamics system. A decade later, Bateman w2x introduced a less restricted class of invariant transformations that subsequently were termed reciprocal relations w3x. Application of invariance properties to approximation theory in subsonic gasdynamics had *Present address: Fachbereich Mathematik, SFB 288, Technische Universitat, ¨ Berlin, 10623 Berlin, Germany. E-mail: [email protected]. 365 0022-247Xr98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.
366
FERAPONTOV, ROGERS, AND SCHIEF
been noted as early as 1939 by Tsien w4x. That the reciprocal relations of Bateman constitute a Backlund transformation was established in w5x. ¨ Invariance of nonlinear gasdynamic, magnetogasdynamic, and general hydrodynamic systems under reciprocal-type transformations has been extensively studied w6]18x. The application of reciprocal transformations to both stationary and moving boundary problems in soil mechanics and nonlinear heat conduction has likewise been the subject of much research w19]25x. Accounts of reciprocal-type transformations and some of their many applications in continuum mechanics are to be found in the monographs by Rogers and Shadwick w26x and Meirmanov et al. w27x. Reciprocal transformations also have an important role in soliton theory. Thus, conjugated with appropriate gauge transformations, they provide a link between the AKNS and WKI inverse scattering schemes w28x. Moreover, invariance of certain integrable hierarchies under reciprocal transformations induces auto-Backlund transformations w29]31x. ¨ Here, our concern is with two-component hyperbolic systems related by reciprocal transformations. Key invariants are introduced, in terms of which conditions for equivalence under reciprocal transformations may be set down. The procedure is applied to link a 1 q 1-dimensional gasdynamic system for a four-parameter class of model constitutive laws to the canonical system when the polytropic gas law prevails.
2. RECIPROCAL TRANSFORMATIONS OF TWO-COMPONENT HYPERBOLIC SYSTEMS: RECIPROCAL INVARIANTS Two-component hyperbolic systems, u it s ¨ ji Ž u . u xj ,
i , j s 1, 2
Ž 2.1.
arise naturally in diverse areas of continuum mechanics such as isentropic gasdynamics, chromatography, and plasticity. They are distinguished by the existence of so-called Riemann invariants R s Ž R1, R 2 ., in terms of which Ž2.1. adopts the diagonal form R1t s ¨ 1 Ž R . R1x ,
R 2t s ¨ 2 Ž R . R 2x ,
Ž 2.2.
thereby considerably simplifying their investigation. Here, we shall assume the system Ž2.2. to be linearly nondegenerate, i.e., 1¨ 1 / 0, 2¨ 2 / 0. Any system Ž2.2. possesses infinitely many conservation laws, h Ž R . dx q g Ž R . dt,
Ž 2.3.
TWO-COMPONENT HYPERBOLIC SYSTEMS
367
with density hŽ R . and flux g Ž R . governed by the equations
i g s ¨ i i h,
i s 1, 2
Ž 2.4.
Ž i s r R i ., equivalent to the condition h t s g x associated with the closedness of the 1-form Ž2.3.. The compatibility condition 1 2 g s 2 1 g applied to Ž2.4. results in the second-order equation
1 2 h s
2¨ 1 ¨2 y¨
hq 1 1
1¨ 2 ¨1 y ¨ 2
2 h
Ž 2.5.
for the conserved densities of system Ž2.2.. Let us select two particular conservation laws, B Ž R . dx q AŽ R . dt and Ž N R . dx q M Ž R . dt, and introduce new independent variables X, T via the relations dX s B Ž R . dx q A Ž R . dt,
dT s N Ž R . dx q M Ž R . dt.
Ž 2.6.
Under this change of variable x, t ª X, T, the diagonal system Ž2.2. is mapped to the new diagonal system R1T s V 1 Ž R . R1X ,
R T2 s V 2 Ž R . R 2X ,
Ž 2.7.
where the new characteristic velocities V i are given by Vis
¨ i B Ž R . y AŽ R .
M Ž R . y ¨ iN Ž R .
.
Ž 2.8.
Mappings of the type in Ž2.6., predicated on the existence of conservation laws, are termed reciprocal transformations and have extensive applications w1]31x. Following Ferapontov w13x, we introduce two fundamental reciprocal invariants, namely, the symmetric 2-form Žmetric.,
1¨ 1 2¨ 2
Ž¨1 y ¨ 2.
2
dR1dR 2 ,
Ž 2.9.
and the differential, dV ,
Ž 2.10.
of the 1-form, Vs
ž
1 2¨ 2 2¨ 2
q
1¨ 2 ¨1 y ¨ 2
/
dR1 q
ž
1 2¨ 1 1¨ 1
q
2¨ 2 ¨ 2 y ¨1
/
dR 2 . Ž 2.11.
368
FERAPONTOV, ROGERS, AND SCHIEF
Neither of the invariants Ž2.9., Ž2.10. changes under a reparameterization R1 ª f 1 Ž R1 ., R 2 ª f 2 Ž R 2 .. It was recently demonstrated in w32x that these invariants are closely related to the invariants of surfaces in Lie sphere geometry. To establish the invariance of Ž2.9. and Ž2.10. under arbitrary reciprocal transformations, it suffices to check their invariance under the elementary reciprocal transformations dX s Bdx q Adt,
dT s dt
Ž 2.12.
and dX s dt,
dT s dx.
Ž 2.13.
The invariance of Ž2.9. and Ž2.10. under these elementary transformations is readily established by direct calculation. Since any reciprocal transformation is a composition of elementary ones, the required invariance follows. It is remarkable that the invariants Ž2.9. and Ž2.10. form a complete set in the sense that if the invariants of one two-component system can be mapped to those of another by an appropriate reparameterization of the Riemann invariants Ri ª f i Ž Ri . ,
i s 1, 2,
Ž 2.14.
then the two-component systems can be related to each other by a reciprocal transformation. In what follows, this result is applied to determine reciprocally related two-component systems in 1 q 1-dimensional isentropic gasdynamics.
3. RECIPROCAL TWO-COMPONENT SYSTEMS IN ISENTROPIC GASDYNAMICS Let us consider the Eulerian equations of 1 q 1-dimensional isentropic gasdynamics, namely,
r t q Ž r u . x s 0,
Ž 3.1.
r Ž u t q uu x . q p x s 0,
Ž 3.2.
with an arbitrary barotropic law p s PŽ r . ,
c 2 s P9 Ž r . ) 0.
Ž 3.3.
In the above, p, r designate the gas pressure and density, while c is the local speed of sound and u is the gas speed. The usual Riemann invariants
TWO-COMPONENT HYPERBOLIC SYSTEMS
369
R1 , R 2 of the system Ž3.1. ] Ž3.3. are given by the relations dR1 s
cŽ r .
r
d r q du,
dR 2 s
cŽ r .
r
d r y du,
Ž 3.4.
so that, up to a constant, us
R1 y R 2 2
,
Ž 3.5.
while r can be expressed in terms of R1 q R 2 via the implicit relation r
cŽ j .
0
j
Hr
dj s
R1 q R 2 2
.
Ž 3.6.
In the Riemann invariants R1 , R 2 , the gasdynamics system Ž3.1. ] Ž3.3. assumes the diagonal form, R1t q ¨ 1 R1x s 0,
R 2t q ¨ 2 R 2x s 0,
Ž 3.7.
where the characteristic velocities ¨ 1 s u q c, ¨ 2 s u y c can be expressed in terms of R1 , R 2 via Ž3.5., Ž3.6.. A direct calculation shows that
1¨ 1 s y 2¨ 2 s
1 2
ž
1q
c9 r c
/
Ž c9 s dcrd r . ,
,
Ž 3.8.
whence the metric Ž2.9. adopts the form
1¨ 1 2¨ 2 2 2
Ž¨ y¨ . 1
dR1dR 2 s y
1 16
ž
1 c
q
c9 r c2
/
dR1 dR 2 ,
Ž 3.9.
while the second invariant dV is identically zero for arbitrary barotropic law Ž3.3.. In the particular case of the polytropic gas law, p s rg ,
g / 1,
Ž 3.10.
Ž R1 q R 2 . ,
Ž 3.11.
Ž3.6. yields cs
gy1 4
370
FERAPONTOV, ROGERS, AND SCHIEF
so that ¨1 s ¨2 s
R1 y R 2 2 R1 y R 2 2
q y
gy1 4
gy1 4
Ž R1 q R 2 . , Ž 3.12. Ž R1 q R 2 . ,
and the metric Ž3.9. adopts the explicit form
1¨ 1 2¨ 2
dR dR s y 1
Ž¨1 y ¨ 2.
2
2
1 4
ž
gq1 gy1
2
/
dR1 dR 2
Ž R1 q R 2 .
2
.
Ž 3.13.
In the case g s 1, the metric Ž3.9. becomes
1¨ 1 2¨ 2 2 2
Ž¨ y¨ . 1
dR1dR 2 s y
1 16
dR1 dR 2 .
Ž 3.14.
The Gaussian curvature of the metric 2 F Ž R1 , R 2 . dR1 dR 2
Ž 3.15.
K s y Ž 1 2 ln F . rF ,
Ž 3.16.
is given by
whence we conclude that, in the case g / 1, the metric Ž2.9. has constant Gaussian curvature, K s 16
ž
gy1 gq1
2
/
,
Ž 3.17.
while, in the case g s 1, the metric is flat, that is, K s 0. Accordingly, the following results obtain: THEOREM 1. The diagonal two-component system Ž2.2. is reciprocally related to the 1 q 1-diagonal isentropic gasdynamics system with pŽ r . s r g , g / 1, iff the in¨ ariant Ž2.10. ¨ anishes and the metric Ž2.9. has constant positi¨ e Gaussian cur¨ ature K. Proof. Let the metric of the two-component system with dV s 0 and Gaussian curvature K represented by Ž3.17. be of the form Ž3.15.. Then, F satisfies the Liouville equation KF , 1 2 ln F s yK
TWO-COMPONENT HYPERBOLIC SYSTEMS
371
with general solution Fs
y2 Ž f 1 Ž R1 . . 9 Ž f 2 Ž R 2 . . 9 K Ž f 1 Ž R1 . q f 2 Ž R 2 . .
2
.
Accordingly, the metric adopts the form y
1 4
ž
gq1 gy1
f 19f 2 9
2
/Ž
f qf 1
2 2
dR1dR 2 ,
.
which reduces to y
1 4
ž
gq1 gy1
2
/
dR1 dR 2
Ž R1 q R 2 .
2
,
following the reparameterization Ž2.14.. Since the two-component system under study has the same reciprocal invariants Ž2.9., Ž2.10. as the 1 q 1-dimensional isentropic gasdynamics system with p s r g , g / 1, they are reciprocally related. THEOREM 2. The diagonal two-component system Ž2. is reciprocally related to the 1 q 1-dimensional gasdynamic system with p s r iff the in¨ ariant Ž2.10. ¨ anishes and the metric Ž2.9. has zero Gaussian cur¨ ature. Proof. The metric Ž3.15. has zero curvature iff 1 2 Žln F . s 0, so that F s Ž f 1 Ž R1 . . 9 Ž f 2 Ž R 2 . . 9, whence, the corresponding metric reduces to y
1 16
dR1dR 2 ,
after reparameterization of the type Ž2.14.. Hence, the two-component system has the same reciprocal invariants as the 1 q 1-dimensional gasdynamic system with g s 1, so that they are reciprocally related. THEOREM 3. The diagonal two-component system Ž2. is reciprocally related to the 1 q 1-dimensional isentropic gasdynamic system with an arbitrary pressure-density relation p s pŽ r . iff the in¨ ariant Ž2.10. ¨ anishes and the metric Ž2.9. possesses a nontri¨ ial isometry. Proof. The necessity of the isometry requirement results from the relation Ž3.9., valid for arbitrary equation of state p s pŽ r .. Thus, since r is a function of R1 q R 2 , the metric Ž3.9. possesses the obvious isometry
372
FERAPONTOV, ROGERS, AND SCHIEF
r R1 y r R 2 . To establish the sufficiency, let us assume that the metric Ž3.15. possesses an isometry m Ž R . r R1 q n Ž R . r R 2 . Since isometry must preserve the null directions of the metric, it must be of the form m s m Ž R1 ., n s n Ž R 2 ., so that, after appropriate reparameterization of the Riemann invariants, it becomes r R1 y r R 2 . Hence, F s F Ž R1 q R 2 ., and the metric 2 F Ž R1 q R 2 . dR1 dR 2 coincides with Ž3.9. for an appropriate class of equations of state p s pŽ r ..
4. RECIPROCAL CONSTITUTIVE LAWS Here, we determine the class of pressure-density relations for which the corresponding 1 q 1-dimensional gasdynamics system is reciprocally related to that with p s r g. The case g s 1 and g / 1 are considered separately. The case g s 1. The zero Gaussian curvature condition on the metric Ž3.9. shows that the local speed of sound cŽ r . is constrained by the equation c 1 c9 r 9 ln q 2 sa , Ž 4.1. c r c
ž
/
where a is an arbitrary constant of integration. It now proves convenient to introduce the Lagrangian signal speed given by A Ž e . s Ž yry1 0 dprde .
1r2
,
Ž 4.2.
where e s r 0rr y 1 is the stretch and r 0 is the density of the gas in a reference state with zero stretch. Here, we set r 0 s 1 so that A s r c,
Ž 4.3.
and Ž4.1. yields pyds
H0
A
ds
a ln s q b
,
Ž 4.4.
where b and d are additional constants of integration. On the other hand, use of Ž4.2. gives eyes
H0
A
d Ž 1rs . ya ln Ž 1rs . q b
.
Ž 4.5.
TWO-COMPONENT HYPERBOLIC SYSTEMS
373
Accordingly, the four-parameter class of Ž p, r .-relations for which the 1 q 1-dimensional gasdynamic system is reciprocally related to that for p s r is given parametrically in terms of the Lagrangian signal speed A by the logarithmic integrals pyds
z a
Li Ž Arz . ,
eyesy
1
za
Li Ž zrA . ,
Ž 4.6.
where z s eyb r a and a / 0. If a s 0, then Ž4.1. integrates to give the class of explicit pressure-density relations pŽ r . s
lr q m nr q j
j / 0,
,
Ž 4.7.
reciprocally related to the gasdynamic system with g s 1. It is noted that gas law and the case j s 0 produces the well-known Karman]Tsien ´ ´ corresponds to the excluded linearly degenerate case when both of the reciprocal invariants vanish. The case g / "1. The constant curvature K condition for the metric Ž3.9. yields
r r c
c
žž ln
1 c
q
c9 r c
2
//
9 9
s
K 1 8
ž
c
q
c9 r c2
/
,
Ž 4.8.
whence, on integration thrice, it is seen that pyds
H0
ds
A
b q a ln s q
K 16
,
Ž ln s .
Ž 4.9.
2
while, on use of Ž4.2., it is seen that eyes
H0
A
d Ž 1rs .
b y a ln Ž 1rs . q
K 16
.
Ž ln s .
Ž 4.10.
2
Thus, if we set p* s p y d , e* s e y e , then the relation p* Ž 1rA; a , b . s e* Ž A; ya , b .
Ž 4.11.
results. It remains to determine constraints on the parameters a and b such that the above constitutive laws are appropriate for gasdynamics. Now, if T s yp denotes the strain, then dTrde s A2 and d 2 Trde 2 - 0 for a material that exhibits soft elastic behavior and d 2 Trde 2 ) 0 for a
374
FERAPONTOV, ROGERS, AND SCHIEF
hard elastic material. For the class of materials under discussion with model constitutive laws given parametrically by Ž4.9. ] Ž4.10., it is seen that d 2 Trde 2 s y2 A3 b q a ln A q
K 16
2 Ž ln A . ,
Ž 4.12.
so that soft elastic material behavior typical of gasdynamics is ensured if a , b ) 0. In the case a s 0, corresponding to the class of pressure-density relations Ž4.7., the choice of parameters such that psn q
l2r
2
r-
,
1ymr 2
1
m2
- rmax
Ž 4.13.
guarantees the gasdynamic conditions dprd r ) 0,
d 2 prd r 2 ) 0.
Ž 4.14.
It is interesting to note that the three-parameter class of model laws Ž4.13. is evidentially superior to the celebrated Karman]Tsien approximation ´ ´ psAy
B
r
B ) 0,
,
Ž 4.15.
in that the latter has the disadvantage that it exhibits the nonphysical behavior d 2 prd r 2 - 0. It is readily seen that a reciprocal transformation that maps the primed 1 q 1-dimensional gasdynamics system Ž3.1. with gas law p9 s n 2 q
l2r 9
Ž 4.16.
1 y m2r 9
to the corresponding unprimed gasdynamic system with gas law p s l2r is given by
r 9 s r Ž 1 q m2r .
y1
,
u9 s u,
dx9 s Ž 1 q m2r . dx y m2r udt,
p9 s p q n 2 , dt9 s dt.
Ž 4.17.
Thus, the change in independent variable becomes, up to translation constants, x9 s x y m2 X ,
t9 s t ,
Ž 4.18.
where X is the usual Lagrangian material coordinate. The class of reciprocal transformations Ž4.17. is of a type originally introduced by Movsesian w33x in an analysis of the flow of a compressible gas behind a piston.
TWO-COMPONENT HYPERBOLIC SYSTEMS
375
ACKNOWLEDGMENTS The support of the Australian Research Council is gratefully acknowledged ŽC. Rogers and W. K. Schief..
REFERENCES ¨ 1. A. Haar, Uber Adjungierte Variationsprobleme und Adjungierte Extremalflachen, Math. Ann. 100 Ž1928., 481]502. 2. H. Bateman, The lift and drag functions for an elastic fluid in two-dimensional irrotational flow, Proc. Nat. Acad. Sci. U.S. A. 24 Ž1938., 246]251. 3. G. Power and P. Smith, Reciprocal properties of plane gas flows, J. Math. Mech. 10 Ž1961., 349]361. 4. H. S. Tsien, Two-dimensional subsonic flow of compressible fluids, J. Aeronaut. Sci. 6 Ž1939., 399]407. 5. J. A. Baker and C. Rogers, Invariance properties under a reciprocal Backlund transfor¨ mation in gasdynamics, J. Mecanique Theor. ´ ´ Appl. 1 Ž1982., 563]578. 6. C. Rogers, Reciprocal relations in non-steady gasdynamics and magneto-gasdynamics, Z. Angew. Math. Phys. 19 Ž1968., 58]63. 7. C. Rogers, Invariant transformations in non-steady gasdynamics and magnetogasdynamics, Z. Angew. Math. Phys. 20 Ž1969., 370]382. 8. C. Rogers, The construction of invariant transformations in plane rotational gasdynamics, Arch. Rat. Mech. Anal. 47 Ž1972., 36]46. 9. S. P. Castell and C. Rogers, Application of invariant transformations in one-dimensional non-steady gasdynamics, Quart. Appl. Math. 32 Ž1974., 241]251. 10. C. Rogers, S. P. Castell, and J. G. Kingston, On invariance properties of conservation laws in non-dissipative planar magneto-gasdynamics, J. Mecanique 13 Ž1974., 243]354. ´ 11. C. Rogers and J. G. Kingston, Reciprocal properties in quasi-one-dimensional non-steady oblique field magneto-gasdynamics, J. Mecanique 15 Ž1976., 185]192. ´ 12. C. Rogers, J. G. Kingston, and W. F. Shadwick, On reciprocal-type invariant transformations in magneto-gasdynamics, J. Math. Phys. 21 Ž1980., 395]397. 13. E. V. Ferapontov, Reciprocal transformations and their invariants, Differential Equations 25 Ž1989., 898]905. 14. E. V. Ferapontov, Reciprocal auto-transformations and hydrodynamic symmetries, Differential Equations 27 Ž1991., 1250]1263. 15. E. V. Ferapontov, On integrability of 3 = 3 semi-Hamiltonian hydrodynamic type systems u it s ¨ ji Ž u. u xj which do not possess Riemann invariants, Physica D63 Ž1993., 50]70. 16. E. V. Ferapontov, On the matrix Hopf equation and integrable Hamiltonian systems of hydrodynamic type which do not possess Riemann invariants, Phys. Lett. A179 Ž1993., 391]397. 17. E. V. Ferapontov, Several conjectures and results in the theory of integrable Hamiltonian systems of hydrodynamic type which do not possess Riemann invariants, Teor. Mat. Phys. 99 Ž1994., 257]262. 18. E. V. Ferapontov, Dupin hypersurfaces and integrable Hamiltonian systems of hydrodynamic type which do not possess Riemann invariants, Differential Geom. Appl. 5 Ž1995., 121]152. 19. C. Rogers, M. P. Stallybrass, and D. L. Clements, On two-phase filtration under gravity and with boundary infiltration: Application of a Backlund transformation, J. Nonlinear ¨ Anal. 7 Ž1983., 785]799.
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FERAPONTOV, ROGERS, AND SCHIEF
20. C. Rogers, Application of a reciprocal transformation to a two-phase Stefan problem, J. Phys. A 18 Ž1985., L105]L109. 21. C. Rogers and T. Ruggeri, A reciprocal Backlund transformation and application to a ¨ nonlinear hyperbolic model in heat conduction, Lett. Il Nuo¨ o Cimento 44 Ž1985., 289]296. 22. C. Rogers, On a class of moving boundary problems in nonlinear heat conduction: Application of a Backlund transformation, Int. J. Nonlinear Mech. 21 Ž1986., 249]256. ¨ 23. C. Rogers and P. Broadbridge, On a nonlinear moving boundary problem with heterogeneity: Application of a Backlund transformation, Z. Angew. Math. Phys. 39 Ž1988., ¨ 122]128. 24. C. Rogers and B. Yu Guo, A note on the onset of melting in a class of simple metals. Condition on the applied boundary flux, Acta Math. Sci. 8 Ž1988., 425]430. 25. B. Yu Guo, C. Rogers, and D. Siegel, Application of maximum principles to boundary value problems in non-linear heat conduction, J. Nonlinear Anal. 14 Ž1990., 293]304. 26. C. Rogers and W. F. Shadwick, ‘‘Backlund Transformations and Their Applications,’’ ¨ Academic Press, New York, 1982. 27. A. M. Meirmanov, V. V. Puchnachov, and S. I. Shmarev, ‘‘Evolution Equations and Lagrangian Coordinates,’’ de Gruyter, BerlinrNew York, 1997. 28. C. Rogers and P. Wong, On reciprocal Backlund transformations of inverse scattering ¨ schemes, Phys. Scripta 30 Ž1989., 10]14. 29. C. Rogers and M. C. Nucci, On reciprocal Backlund transformations and the Korteweg]de ¨ Vries hierarchy, Phys. Scripta 33 Ž1986., 289]292. 30. C. Rogers and S. Carillo, On reciprocal properties of the Caudrey]Dodd]Gibbon and Kaup]Kuperschmidt hierarchies, Phys. Scripta 36 Ž1987., 865]869. 31. W. Oevel and C. Rogers, Gauge transformations and reciprocal links in 2 q 1-dimensions, Re¨ . Math. Phys. 5 Ž1993., 299]330. 32. E. V. Ferapontov, Surfaces in Lie sphere geometry and the stationary Davey]Stewartson hierarchy, preprint, SFB N287 Berlin, 1997. 33. L. A. Movsesian, On an invariant transformation of equations of one-dimensional unsteady motion of an ideal compressible fluid, Prikl. Mat. Mekh. 31 Ž1967., 130]333.
228, 365]376 Ž1998.
AY986131
Reciprocal Transformations of Two-Component Hyperbolic Systems and Their Invariants E. V. Ferapontov* Centre for Nonlinear Studies, Landau Institute for Theoretical Physics Academy of Science of Russia, 117940 Moscow, GSP-1, Russia
and C. Rogers and W. K. Schief School of Mathematics, The Uni¨ ersity of New South Wales, Sydney 2052, New South Wales, Australia Submitted by William F. Ames Received August 24, 1998
Criteria for the equivalence under reciprocal transformations of two-component hyperbolic systems are established in terms of reciprocal invariants. These results are applied to a 1 q 1-dimensional isentropic gasdynamics system to determine reciprocal equivalence to the canonical system with a polytropic gas law. The four-parameter model pressure-density relations for which this reciprocal equivalence holds are parameterized in terms of the Lagrangian signal speed. Q 1998 Academic Press
1. INTRODUCTION Reciprocal-type transformations have a long history. In 1928, Haar w1x in a paper devoted to adjoint variational problems, established a reciprocaltype invariance of a plane potential gasdynamics system. A decade later, Bateman w2x introduced a less restricted class of invariant transformations that subsequently were termed reciprocal relations w3x. Application of invariance properties to approximation theory in subsonic gasdynamics had *Present address: Fachbereich Mathematik, SFB 288, Technische Universitat, ¨ Berlin, 10623 Berlin, Germany. E-mail: [email protected]. 365 0022-247Xr98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.
366
FERAPONTOV, ROGERS, AND SCHIEF
been noted as early as 1939 by Tsien w4x. That the reciprocal relations of Bateman constitute a Backlund transformation was established in w5x. ¨ Invariance of nonlinear gasdynamic, magnetogasdynamic, and general hydrodynamic systems under reciprocal-type transformations has been extensively studied w6]18x. The application of reciprocal transformations to both stationary and moving boundary problems in soil mechanics and nonlinear heat conduction has likewise been the subject of much research w19]25x. Accounts of reciprocal-type transformations and some of their many applications in continuum mechanics are to be found in the monographs by Rogers and Shadwick w26x and Meirmanov et al. w27x. Reciprocal transformations also have an important role in soliton theory. Thus, conjugated with appropriate gauge transformations, they provide a link between the AKNS and WKI inverse scattering schemes w28x. Moreover, invariance of certain integrable hierarchies under reciprocal transformations induces auto-Backlund transformations w29]31x. ¨ Here, our concern is with two-component hyperbolic systems related by reciprocal transformations. Key invariants are introduced, in terms of which conditions for equivalence under reciprocal transformations may be set down. The procedure is applied to link a 1 q 1-dimensional gasdynamic system for a four-parameter class of model constitutive laws to the canonical system when the polytropic gas law prevails.
2. RECIPROCAL TRANSFORMATIONS OF TWO-COMPONENT HYPERBOLIC SYSTEMS: RECIPROCAL INVARIANTS Two-component hyperbolic systems, u it s ¨ ji Ž u . u xj ,
i , j s 1, 2
Ž 2.1.
arise naturally in diverse areas of continuum mechanics such as isentropic gasdynamics, chromatography, and plasticity. They are distinguished by the existence of so-called Riemann invariants R s Ž R1, R 2 ., in terms of which Ž2.1. adopts the diagonal form R1t s ¨ 1 Ž R . R1x ,
R 2t s ¨ 2 Ž R . R 2x ,
Ž 2.2.
thereby considerably simplifying their investigation. Here, we shall assume the system Ž2.2. to be linearly nondegenerate, i.e., 1¨ 1 / 0, 2¨ 2 / 0. Any system Ž2.2. possesses infinitely many conservation laws, h Ž R . dx q g Ž R . dt,
Ž 2.3.
TWO-COMPONENT HYPERBOLIC SYSTEMS
367
with density hŽ R . and flux g Ž R . governed by the equations
i g s ¨ i i h,
i s 1, 2
Ž 2.4.
Ž i s r R i ., equivalent to the condition h t s g x associated with the closedness of the 1-form Ž2.3.. The compatibility condition 1 2 g s 2 1 g applied to Ž2.4. results in the second-order equation
1 2 h s
2¨ 1 ¨2 y¨
hq 1 1
1¨ 2 ¨1 y ¨ 2
2 h
Ž 2.5.
for the conserved densities of system Ž2.2.. Let us select two particular conservation laws, B Ž R . dx q AŽ R . dt and Ž N R . dx q M Ž R . dt, and introduce new independent variables X, T via the relations dX s B Ž R . dx q A Ž R . dt,
dT s N Ž R . dx q M Ž R . dt.
Ž 2.6.
Under this change of variable x, t ª X, T, the diagonal system Ž2.2. is mapped to the new diagonal system R1T s V 1 Ž R . R1X ,
R T2 s V 2 Ž R . R 2X ,
Ž 2.7.
where the new characteristic velocities V i are given by Vis
¨ i B Ž R . y AŽ R .
M Ž R . y ¨ iN Ž R .
.
Ž 2.8.
Mappings of the type in Ž2.6., predicated on the existence of conservation laws, are termed reciprocal transformations and have extensive applications w1]31x. Following Ferapontov w13x, we introduce two fundamental reciprocal invariants, namely, the symmetric 2-form Žmetric.,
1¨ 1 2¨ 2
Ž¨1 y ¨ 2.
2
dR1dR 2 ,
Ž 2.9.
and the differential, dV ,
Ž 2.10.
of the 1-form, Vs
ž
1 2¨ 2 2¨ 2
q
1¨ 2 ¨1 y ¨ 2
/
dR1 q
ž
1 2¨ 1 1¨ 1
q
2¨ 2 ¨ 2 y ¨1
/
dR 2 . Ž 2.11.
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FERAPONTOV, ROGERS, AND SCHIEF
Neither of the invariants Ž2.9., Ž2.10. changes under a reparameterization R1 ª f 1 Ž R1 ., R 2 ª f 2 Ž R 2 .. It was recently demonstrated in w32x that these invariants are closely related to the invariants of surfaces in Lie sphere geometry. To establish the invariance of Ž2.9. and Ž2.10. under arbitrary reciprocal transformations, it suffices to check their invariance under the elementary reciprocal transformations dX s Bdx q Adt,
dT s dt
Ž 2.12.
and dX s dt,
dT s dx.
Ž 2.13.
The invariance of Ž2.9. and Ž2.10. under these elementary transformations is readily established by direct calculation. Since any reciprocal transformation is a composition of elementary ones, the required invariance follows. It is remarkable that the invariants Ž2.9. and Ž2.10. form a complete set in the sense that if the invariants of one two-component system can be mapped to those of another by an appropriate reparameterization of the Riemann invariants Ri ª f i Ž Ri . ,
i s 1, 2,
Ž 2.14.
then the two-component systems can be related to each other by a reciprocal transformation. In what follows, this result is applied to determine reciprocally related two-component systems in 1 q 1-dimensional isentropic gasdynamics.
3. RECIPROCAL TWO-COMPONENT SYSTEMS IN ISENTROPIC GASDYNAMICS Let us consider the Eulerian equations of 1 q 1-dimensional isentropic gasdynamics, namely,
r t q Ž r u . x s 0,
Ž 3.1.
r Ž u t q uu x . q p x s 0,
Ž 3.2.
with an arbitrary barotropic law p s PŽ r . ,
c 2 s P9 Ž r . ) 0.
Ž 3.3.
In the above, p, r designate the gas pressure and density, while c is the local speed of sound and u is the gas speed. The usual Riemann invariants
TWO-COMPONENT HYPERBOLIC SYSTEMS
369
R1 , R 2 of the system Ž3.1. ] Ž3.3. are given by the relations dR1 s
cŽ r .
r
d r q du,
dR 2 s
cŽ r .
r
d r y du,
Ž 3.4.
so that, up to a constant, us
R1 y R 2 2
,
Ž 3.5.
while r can be expressed in terms of R1 q R 2 via the implicit relation r
cŽ j .
0
j
Hr
dj s
R1 q R 2 2
.
Ž 3.6.
In the Riemann invariants R1 , R 2 , the gasdynamics system Ž3.1. ] Ž3.3. assumes the diagonal form, R1t q ¨ 1 R1x s 0,
R 2t q ¨ 2 R 2x s 0,
Ž 3.7.
where the characteristic velocities ¨ 1 s u q c, ¨ 2 s u y c can be expressed in terms of R1 , R 2 via Ž3.5., Ž3.6.. A direct calculation shows that
1¨ 1 s y 2¨ 2 s
1 2
ž
1q
c9 r c
/
Ž c9 s dcrd r . ,
,
Ž 3.8.
whence the metric Ž2.9. adopts the form
1¨ 1 2¨ 2 2 2
Ž¨ y¨ . 1
dR1dR 2 s y
1 16
ž
1 c
q
c9 r c2
/
dR1 dR 2 ,
Ž 3.9.
while the second invariant dV is identically zero for arbitrary barotropic law Ž3.3.. In the particular case of the polytropic gas law, p s rg ,
g / 1,
Ž 3.10.
Ž R1 q R 2 . ,
Ž 3.11.
Ž3.6. yields cs
gy1 4
370
FERAPONTOV, ROGERS, AND SCHIEF
so that ¨1 s ¨2 s
R1 y R 2 2 R1 y R 2 2
q y
gy1 4
gy1 4
Ž R1 q R 2 . , Ž 3.12. Ž R1 q R 2 . ,
and the metric Ž3.9. adopts the explicit form
1¨ 1 2¨ 2
dR dR s y 1
Ž¨1 y ¨ 2.
2
2
1 4
ž
gq1 gy1
2
/
dR1 dR 2
Ž R1 q R 2 .
2
.
Ž 3.13.
In the case g s 1, the metric Ž3.9. becomes
1¨ 1 2¨ 2 2 2
Ž¨ y¨ . 1
dR1dR 2 s y
1 16
dR1 dR 2 .
Ž 3.14.
The Gaussian curvature of the metric 2 F Ž R1 , R 2 . dR1 dR 2
Ž 3.15.
K s y Ž 1 2 ln F . rF ,
Ž 3.16.
is given by
whence we conclude that, in the case g / 1, the metric Ž2.9. has constant Gaussian curvature, K s 16
ž
gy1 gq1
2
/
,
Ž 3.17.
while, in the case g s 1, the metric is flat, that is, K s 0. Accordingly, the following results obtain: THEOREM 1. The diagonal two-component system Ž2.2. is reciprocally related to the 1 q 1-diagonal isentropic gasdynamics system with pŽ r . s r g , g / 1, iff the in¨ ariant Ž2.10. ¨ anishes and the metric Ž2.9. has constant positi¨ e Gaussian cur¨ ature K. Proof. Let the metric of the two-component system with dV s 0 and Gaussian curvature K represented by Ž3.17. be of the form Ž3.15.. Then, F satisfies the Liouville equation KF , 1 2 ln F s yK
TWO-COMPONENT HYPERBOLIC SYSTEMS
371
with general solution Fs
y2 Ž f 1 Ž R1 . . 9 Ž f 2 Ž R 2 . . 9 K Ž f 1 Ž R1 . q f 2 Ž R 2 . .
2
.
Accordingly, the metric adopts the form y
1 4
ž
gq1 gy1
f 19f 2 9
2
/Ž
f qf 1
2 2
dR1dR 2 ,
.
which reduces to y
1 4
ž
gq1 gy1
2
/
dR1 dR 2
Ž R1 q R 2 .
2
,
following the reparameterization Ž2.14.. Since the two-component system under study has the same reciprocal invariants Ž2.9., Ž2.10. as the 1 q 1-dimensional isentropic gasdynamics system with p s r g , g / 1, they are reciprocally related. THEOREM 2. The diagonal two-component system Ž2. is reciprocally related to the 1 q 1-dimensional gasdynamic system with p s r iff the in¨ ariant Ž2.10. ¨ anishes and the metric Ž2.9. has zero Gaussian cur¨ ature. Proof. The metric Ž3.15. has zero curvature iff 1 2 Žln F . s 0, so that F s Ž f 1 Ž R1 . . 9 Ž f 2 Ž R 2 . . 9, whence, the corresponding metric reduces to y
1 16
dR1dR 2 ,
after reparameterization of the type Ž2.14.. Hence, the two-component system has the same reciprocal invariants as the 1 q 1-dimensional gasdynamic system with g s 1, so that they are reciprocally related. THEOREM 3. The diagonal two-component system Ž2. is reciprocally related to the 1 q 1-dimensional isentropic gasdynamic system with an arbitrary pressure-density relation p s pŽ r . iff the in¨ ariant Ž2.10. ¨ anishes and the metric Ž2.9. possesses a nontri¨ ial isometry. Proof. The necessity of the isometry requirement results from the relation Ž3.9., valid for arbitrary equation of state p s pŽ r .. Thus, since r is a function of R1 q R 2 , the metric Ž3.9. possesses the obvious isometry
372
FERAPONTOV, ROGERS, AND SCHIEF
r R1 y r R 2 . To establish the sufficiency, let us assume that the metric Ž3.15. possesses an isometry m Ž R . r R1 q n Ž R . r R 2 . Since isometry must preserve the null directions of the metric, it must be of the form m s m Ž R1 ., n s n Ž R 2 ., so that, after appropriate reparameterization of the Riemann invariants, it becomes r R1 y r R 2 . Hence, F s F Ž R1 q R 2 ., and the metric 2 F Ž R1 q R 2 . dR1 dR 2 coincides with Ž3.9. for an appropriate class of equations of state p s pŽ r ..
4. RECIPROCAL CONSTITUTIVE LAWS Here, we determine the class of pressure-density relations for which the corresponding 1 q 1-dimensional gasdynamics system is reciprocally related to that with p s r g. The case g s 1 and g / 1 are considered separately. The case g s 1. The zero Gaussian curvature condition on the metric Ž3.9. shows that the local speed of sound cŽ r . is constrained by the equation c 1 c9 r 9 ln q 2 sa , Ž 4.1. c r c
ž
/
where a is an arbitrary constant of integration. It now proves convenient to introduce the Lagrangian signal speed given by A Ž e . s Ž yry1 0 dprde .
1r2
,
Ž 4.2.
where e s r 0rr y 1 is the stretch and r 0 is the density of the gas in a reference state with zero stretch. Here, we set r 0 s 1 so that A s r c,
Ž 4.3.
and Ž4.1. yields pyds
H0
A
ds
a ln s q b
,
Ž 4.4.
where b and d are additional constants of integration. On the other hand, use of Ž4.2. gives eyes
H0
A
d Ž 1rs . ya ln Ž 1rs . q b
.
Ž 4.5.
TWO-COMPONENT HYPERBOLIC SYSTEMS
373
Accordingly, the four-parameter class of Ž p, r .-relations for which the 1 q 1-dimensional gasdynamic system is reciprocally related to that for p s r is given parametrically in terms of the Lagrangian signal speed A by the logarithmic integrals pyds
z a
Li Ž Arz . ,
eyesy
1
za
Li Ž zrA . ,
Ž 4.6.
where z s eyb r a and a / 0. If a s 0, then Ž4.1. integrates to give the class of explicit pressure-density relations pŽ r . s
lr q m nr q j
j / 0,
,
Ž 4.7.
reciprocally related to the gasdynamic system with g s 1. It is noted that gas law and the case j s 0 produces the well-known Karman]Tsien ´ ´ corresponds to the excluded linearly degenerate case when both of the reciprocal invariants vanish. The case g / "1. The constant curvature K condition for the metric Ž3.9. yields
r r c
c
žž ln
1 c
q
c9 r c
2
//
9 9
s
K 1 8
ž
c
q
c9 r c2
/
,
Ž 4.8.
whence, on integration thrice, it is seen that pyds
H0
ds
A
b q a ln s q
K 16
,
Ž ln s .
Ž 4.9.
2
while, on use of Ž4.2., it is seen that eyes
H0
A
d Ž 1rs .
b y a ln Ž 1rs . q
K 16
.
Ž ln s .
Ž 4.10.
2
Thus, if we set p* s p y d , e* s e y e , then the relation p* Ž 1rA; a , b . s e* Ž A; ya , b .
Ž 4.11.
results. It remains to determine constraints on the parameters a and b such that the above constitutive laws are appropriate for gasdynamics. Now, if T s yp denotes the strain, then dTrde s A2 and d 2 Trde 2 - 0 for a material that exhibits soft elastic behavior and d 2 Trde 2 ) 0 for a
374
FERAPONTOV, ROGERS, AND SCHIEF
hard elastic material. For the class of materials under discussion with model constitutive laws given parametrically by Ž4.9. ] Ž4.10., it is seen that d 2 Trde 2 s y2 A3 b q a ln A q
K 16
2 Ž ln A . ,
Ž 4.12.
so that soft elastic material behavior typical of gasdynamics is ensured if a , b ) 0. In the case a s 0, corresponding to the class of pressure-density relations Ž4.7., the choice of parameters such that psn q
l2r
2
r-
,
1ymr 2
1
m2
- rmax
Ž 4.13.
guarantees the gasdynamic conditions dprd r ) 0,
d 2 prd r 2 ) 0.
Ž 4.14.
It is interesting to note that the three-parameter class of model laws Ž4.13. is evidentially superior to the celebrated Karman]Tsien approximation ´ ´ psAy
B
r
B ) 0,
,
Ž 4.15.
in that the latter has the disadvantage that it exhibits the nonphysical behavior d 2 prd r 2 - 0. It is readily seen that a reciprocal transformation that maps the primed 1 q 1-dimensional gasdynamics system Ž3.1. with gas law p9 s n 2 q
l2r 9
Ž 4.16.
1 y m2r 9
to the corresponding unprimed gasdynamic system with gas law p s l2r is given by
r 9 s r Ž 1 q m2r .
y1
,
u9 s u,
dx9 s Ž 1 q m2r . dx y m2r udt,
p9 s p q n 2 , dt9 s dt.
Ž 4.17.
Thus, the change in independent variable becomes, up to translation constants, x9 s x y m2 X ,
t9 s t ,
Ž 4.18.
where X is the usual Lagrangian material coordinate. The class of reciprocal transformations Ž4.17. is of a type originally introduced by Movsesian w33x in an analysis of the flow of a compressible gas behind a piston.
TWO-COMPONENT HYPERBOLIC SYSTEMS
375
ACKNOWLEDGMENTS The support of the Australian Research Council is gratefully acknowledged ŽC. Rogers and W. K. Schief..
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20. C. Rogers, Application of a reciprocal transformation to a two-phase Stefan problem, J. Phys. A 18 Ž1985., L105]L109. 21. C. Rogers and T. Ruggeri, A reciprocal Backlund transformation and application to a ¨ nonlinear hyperbolic model in heat conduction, Lett. Il Nuo¨ o Cimento 44 Ž1985., 289]296. 22. C. Rogers, On a class of moving boundary problems in nonlinear heat conduction: Application of a Backlund transformation, Int. J. Nonlinear Mech. 21 Ž1986., 249]256. ¨ 23. C. Rogers and P. Broadbridge, On a nonlinear moving boundary problem with heterogeneity: Application of a Backlund transformation, Z. Angew. Math. Phys. 39 Ž1988., ¨ 122]128. 24. C. Rogers and B. Yu Guo, A note on the onset of melting in a class of simple metals. Condition on the applied boundary flux, Acta Math. Sci. 8 Ž1988., 425]430. 25. B. Yu Guo, C. Rogers, and D. Siegel, Application of maximum principles to boundary value problems in non-linear heat conduction, J. Nonlinear Anal. 14 Ž1990., 293]304. 26. C. Rogers and W. F. Shadwick, ‘‘Backlund Transformations and Their Applications,’’ ¨ Academic Press, New York, 1982. 27. A. M. Meirmanov, V. V. Puchnachov, and S. I. Shmarev, ‘‘Evolution Equations and Lagrangian Coordinates,’’ de Gruyter, BerlinrNew York, 1997. 28. C. Rogers and P. Wong, On reciprocal Backlund transformations of inverse scattering ¨ schemes, Phys. Scripta 30 Ž1989., 10]14. 29. C. Rogers and M. C. Nucci, On reciprocal Backlund transformations and the Korteweg]de ¨ Vries hierarchy, Phys. Scripta 33 Ž1986., 289]292. 30. C. Rogers and S. Carillo, On reciprocal properties of the Caudrey]Dodd]Gibbon and Kaup]Kuperschmidt hierarchies, Phys. Scripta 36 Ž1987., 865]869. 31. W. Oevel and C. Rogers, Gauge transformations and reciprocal links in 2 q 1-dimensions, Re¨ . Math. Phys. 5 Ž1993., 299]330. 32. E. V. Ferapontov, Surfaces in Lie sphere geometry and the stationary Davey]Stewartson hierarchy, preprint, SFB N287 Berlin, 1997. 33. L. A. Movsesian, On an invariant transformation of equations of one-dimensional unsteady motion of an ideal compressible fluid, Prikl. Mat. Mekh. 31 Ž1967., 130]333.