Keywords
- Canonical Form
- Supersonic Flow
- Invariant Transformation
- Large Amplitude Wave
- Backlund Transformation
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, access via your institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
A.V. BÄCKLUND, Ueber Flächentransformationen, Math. Ann. 9 (1876), 297–320.
-, Zur theorie der partiellen Differentialgleichung erster ordnung, Math. Ann. 17 (1880), 285–328.
-, Zur theorie der Flächentransformationen, Math. Ann 19 (1882), 387–422.
C. LOEWNER, A transformation theory of partial differential equations of gasdynamics, NACA Tech. Note 2065, 1950.
-, Generation of solutions of systems of partial differential equations by composition of infinitesimal Baecklund transformations, J. Analyse Math. 2 (1952), 219–242.
A. SEEGER, Theorie der Gitterfehlstellen, Handbuch der Physik 7, no. 1, 383–665, Springer, Berlin, 1955.
A.C. SCOTT, Propagation of flux on a long Josephson tunnel junction, Nuovo Cimento B 69 (1970), 241–261.
G.L. LAMB, JR., Analytical descriptions of ultrashort optical pulse propagation in a resonant medium, Rev. Modern Phys. 43 (1971), 99–124.
T.W. BARNARD, 2Nπ Ultrashort light pulses, Phys. Rev. A 7 (1973), 373–376.
R.M. MIURA, Korteweg-deVries equation and generalizations. I. A remarkable explicit nonlinear transformation, J. Math. Phys. 9 (1968), 1202–1204.
G.B. WHITHAM, Linear and Nonlinear Waves, John Wiley and Sons, New York, N.Y., 1974.
A.C. SCOTT, F.Y.F. CHU,-AND D.W. MCLAUGHLIN, The soliton: a new concept in applied science, Proc. IEEE 61 (1973), 1443–1483.
E. HOPF, The partial differential equation ut+uux=μuxx, Comm. Pure Appl. Math. 3 (1950), 201–230.
J.D. COLE, On a quasi-linear parabolic equation occurring in aerodynamics, Quart. Appl. Math. 9 (1951), 225–236.
A.A. NIKOL’SKII, Invariant transformation of the equations of motion of an ideal monatomic gas and new classes of their exact solutions, Prikl. Mat. Meh. 27 (1963), 740–756.
E.D. TOMILOV, On the method of invariant transformations of the gasdynamics equations, Prikl. Mat. Meh. 29 (1965), 959–960.
L.A. MOVSESIAN, On an invariant transformation of equations of one-dimensional unsteady motion of an ideal compressible fluid, Prikl. Mat. Meh. 31 (1967), 137–141.
M.D. USTINOV, Transformation and some solutions of the equation of motion of an ideal gas, Izv. Akad. Nauk SSSR Ser. Meh. Zidk. Gaza 3 (1966), 68–74.
-, Ideal gas flow behind an infinite amplitude shock wave, Izv. Akad. Nauk SSSR Ser. Meh. Zidk. Gaza 4 (1967), 88–90.
V.A. RYKOV, On an exact solution of the equations of magnetogasdynamics of finite conductivity, Prikl. Mat. Meh. 29 (1965), 178–181.
A. HAAR, Über adjungierte Variationsprobleme und adjungierte Extremalflächen, Math. Ann. 100 (1928), 481–502.
H. BATEMAN, The lift and drag functions for an elastic fluid in two-dimensional irrotational flow, Proc. Nat. Acad. Sci. U.S.A. 24 (1938), 246–251.
H.S. TSIEN, Two-dimensional subsonic flow of compressible fluids, J. Aero. Sci. 6 (1939), 399–407.
H. BATEMAN, The transformation of partial differential equations, Quart. Appl. Math. 1 (1943–44), 281–295.
G. POWER-AND P. SMITH, Application of a reciprocal property to subsonic flow, Appl. Sci. Res. A8 (1959), 386–392.
-, Reciprocal properties of plane gas flows, J. Math. Mech. 10 (1961), 349–361.
R.C. PRIM, Steady rotational flow of ideal gases, J. Rat. Mech. Anal. 1 (1952), 425–497.
C. ROGERS, The construction of invariant transformations in plane rotational gasdynamics, Arch. Rational Mech. Anal. 47 (1972), 36–46.
C. ROGERS, S.P. CASTELL-AND J.G. KINGSTON, On invariance properties of conservation laws in non-dissipative planar magneto-gasdynamics, J. Mécanique 13 (1974), 343–354.
C. ROGERS, Reciprocal relations in non-steady one-dimensional gasdynamics, Z. Angew. Math. Phys. 19 (1968), 58–63.
-, Invariant transformations in non-steady gasdynamics and magnetogasdynamics, Z. Angew. Math. Phys. 20 (1969), 370–382.
S.P. CASTELL-AND C. ROGERS, Application of invariant transformations in one-dimensional non-steady gasdynamics, Quart. Appl. Math. 32 (1974), 241–251.
P. SMITH, An extension of the substitution principle to certain unsteady gas flows, Arch. Rational Mech. Anal. 15 (1964), 147–153.
G. POWER-AND C. ROGERS, Substitution principles in non-steady magnetogasdynamics, Appl. Sci. Res. 21 (1969), 176–184.
G. POWER, C. ROGERS,-AND R.A. OSBORN, Baecklund and generalised Legendre transformations in gasdynamics, Z. Angew. Math. Mech. 49 (1969), 333–340.
T. VON KÁRMÁN, Compressibility effects in aerodynamics, J. Aero. Sci. 8 (1941), 337–356.
J. PÉRÈS, Quelques transformations des équations du mouvement d’un fluide compressible, Comptes Rendus Acad. Sciences Paris 219 (1944), 501–504.
-Sur l’integration des équations qui regissent le mouvement d’un fluide compressible, Proc. 7th Int. Cong. Appl. Mech. 2 (1948), 382–387.
N. COBURN, The Kármán-Tsien pressure-volume relation in the two-dimensional supersonic flow of compressible fluids, Quart. Appl. Math. 3 (1945), 106–116.
F.I. FRANKL’, On Chaplygin’s problem for mixed sub-and supersonic flows, Izv. Akad. Nauk. SSSR 9 (1945), 121–143.
S.A. KHRISTIANOVICH, On supersonic gas flows, Trudy TSAGI No. 543, 1941.
R. SAUER, Unterschallströmungen um Profile bei quadratisch approximierter Adiabate, Bayer. Akad. Wiss. Math.-Natur. Kl. S.-B. 9 (1951), 65–71.
W. MÜLLER, Gasströmungen bei quadratisch angenäherter Adiabate. Diss. Th. München, 1953.
G.A. DOMBROVSKII, Approximation methods in the theory of plane adiabatic gas flow, Moscow, 1964.
C. ROGERS, Application of Bäcklund transformations in aligned magnetogasdynamics, Acta Physica Austriaca 31 (1970), 80–88.
C. ROGERS, J.G. KINGSTON,-AND S.P. CASTELL, The reduction to canonical form of hodograph equations in elliptic diabatic flow, Acta Physica Austriaca 34 (1971), 242–250.
S.P. CASTELL, Approximate solutions and applications of hodograph equations in elliptic diabatic flow, Acta Physica Austriaca 37 (1973), 193–204.
A. WEINSTEIN, Generalised axially symmetric potential theory, Bull. Amer. Math. Soc. 59 (1953), 20–38.
K.B. RANGER, Some integral transformation formulae for the Stokes-Beltrami equations, J. Math. Mech. 12 (1963), 663–673.
C. ROGERS-AND J.G. KINGSTON, Application of Baecklund transformations to the Stokes-Beltrami equations, J. Austral. Math. Soc. 15 (1973), 179–189.
D.L. CLEMENTS-AND C. ROGERS, On the application of a Baecklund transformation to linear isotropic elasticity, J. Inst Math. Appl. 14 (1974), 23–30.
A.P.S. SELVADURAI-AND A.J.M. SPENCER, Second order elasticity with axial symmetry—I General Theory, Internat. J. Engrg. Sci. 10 (1972), 97–114.
C. ROGERS-AND D.L. CLEMENTS, On the reduction of the hodograph equations for one-dimensional elastic-plastic wave propagation, Quart. Appl. Math. 32 (1975), 469–474.
R. COURANT-AND K.O. FRIEDRICHS, Supersonic Flow and Shock Waves, Interscience, New York, N.Y., 1948.
H.M. CEKIRGE-AND E. VARLEY, Large amplitude waves in bounded media. I. Reflexion and transmission of large amplitude shockless pulses at an interface, Philos. Trans. Roy. Soc. London Ser. A 273 (1973), 261–313.
J.Y. KAZAKIA-AND E. VARLEY, Large amplitude waves in bounded media. II. The deformation of an impulsively loaded slab: the first reflexion; III. The deformation of an impulsively loaded slab: the second reflexion, Philos. Trans. Roy. Soc. London Ser. A 277 (1974), 191–250.
C. ROGERS, Iterated Baecklund-type transformations and the propagation of disturbances in non-linear elastic materials, J. Math. Anal. Appl. 49 (1975), 638–648.
G. EASON, Wave propagation in inhomogeneous elastic media, Bull. Seism. Soc. Amer. 57 (1967), 1267–1277.
-, Wave propagation in inhomogeneous elastic media; normal leading of spherical and cylindrical surfaces, Appl. Sci. Res. 21 (1970), 467–477.
D.L. CLEMENTS-AND C. ROGERS, On wave propagation in inhomogeneous elastic media, Internat. J. Solids and Structures 10 (1974), 661–669.
-, Wave propagation in (N+1)-dimensional spherically symmetric inhomogeneous elastic media, Canad. J. Phys. 52 (1974), 1246–1252.
T. BRYANT MOODIE, C. ROGERS, AND D.L. CLEMENTS, Large wave-length pulse propagation in curved elastic rods, to appear J. Acoustical Soc. Amer.
C. ROGERS, T. BRYANT MOODIE, AND D.L. CLEMENTS, Les ondes de cisaillement à symétrie sphérique en (I+1)-dimensions pour un matériel viscoélastique inhomogène et isotropique, submitted for publication.
C. ROGERS-AND D.L. CLEMENTS, Wave propagation in fluid filled elastic tubes, Acta. Mech. 22 (1975), 1–9.
D.L. CLEMENTS AND C. ROGERS, Analytic solution of the linearized shallow-water wave equations for certain continuous depth variations, to appear J. Austral. Math. Soc.
M. KURASHIGE, Radial propagation of rotary shear waves in a finitely deformed elastic solid, Internat. J. Engrg. Sci. 12 (1974), 585–596.
D.L. CLEMENTS-AND C. ROGERS, On the theory of stress concentration for shear strained prismatical bodies with a non-linear stress-strain law, Mathematika 22 (1975), 34–42.
C. ROGERS AND J. SWETITS, On a class of non-linear filtration laws, to appear Rheologica Acta.
D.L. CLEMENTS, C. ATKINSON, AND C. ROGERS, Anti-plane crack problems for an inhomogeneous elastic material, submitted for publication.
L.P. EISENHART, A Treatise on the Differential Geometry of Curves and Surfaces, Dover Publications, New York, N.Y., 1960.
K.W. BAUER-AND C. ROGERS, Zur infinitesimalen Deformation von Flachen, Mathematische-Statistiche Sektion Forschungszentrum Graz 31 (1975), 1–15.
D. RESCH, Some Baeckland Transformations of Partial Differential Equations of Second Order, Syracuse University Thesis, Syracuse, 1950.
L. BERS, Mathematical Aspects of Subsonic and Transonic Gasdynamics, Wiley, New York, N.Y., 1958.
C. ROGERS-AND J.G. KINGSTON, Non-dissipative magnetohydrodynamic flows with magnetic and velocity field lines orthogonal geodesics on a normal congruence, SIAM J. Appl. Math. 26 (1974), 183–195.
H. GRAD, Reducible problems in magneto-fluid dynamic steady flows Rev. Mod. Phys 32 (1950), 830–846.
K. STEWARTSON, The dispersion of the current on the surface of a highly conducting fluid, Proc. Cambridge Philos. Soc. 53 (1957), 774–775.
I.M. IUR’EV, On a solution to the equations of magneto-gasdynamics, J. Appl. Math. Mech. 24 (1960), 233–237.
C. ROGERS, H.M. CEKIRGE, AND A. ASKAR, Electromagnetic wave propagation in non-linear dielectric media, submitted for publication.
D.L. CLEMENTS AND C. ROGERS, On anti-plane contact problems involving an inhomogeneous half-space, submitted for publication.
S. BERGMAN, Integral Operators in the Theory of Linear Partial Differential Equations, Springer-Verlag, New York, N.Y., 1968.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1976 Springer-Verlag
About this paper
Cite this paper
Rogers, C. (1976). On applications of generalized Bäcklund transformations to continuum mechanics. In: Miura, R.M. (eds) Bäcklund Transformations, the Inverse Scattering Method, Solitons, and Their Applications. Lecture Notes in Mathematics, vol 515. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0081165
Download citation
DOI: https://doi.org/10.1007/BFb0081165
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-07687-2
Online ISBN: 978-3-540-38220-1
eBook Packages: Springer Book Archive
