Bäcklund transformations for systems of conservation laws
Authors:
J. G. Kingston and C. Rogers
Journal:
Quart. Appl. Math. 41 (1984), 423-431
MSC:
Primary 58G37; Secondary 35Q20, 58F05, 76S05
DOI:
https://doi.org/10.1090/qam/724053
MathSciNet review:
724053
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: A class of Bäcklund transformations is introduced for systems of conservation laws. Applications are given in non-steady gasdynamics and two-phase filtration through a porous medium.
- Alfred Haar, Über adjungierte Variationsprobleme und adjungierte Extremalflächen, Math. Ann. 100 (1928), no. 1, 481–502 (German). MR 1512497, DOI https://doi.org/10.1007/BF01448858
H. Bateman, The lift and drag functions for an elastic fluid in two-dimensional irrotational flow, Proc. Nat. Acad. Sci. U.S.A. 24, 246–251 (1938)
- G. Power and Peter Smith, Application of a reciprocal property to subsonic flow, Appl. Sci. Res. A 8 (1959), 386–392. MR 105969, DOI https://doi.org/10.1007/bf00411764
- G. Power and P. Smith, Reciprocal properties of plane gas flows, J. Math. Mech. 10 (1961), 349–360. MR 0128246
- Hsue-Shen Tsien, Two-dimensional subsonic flow of compressible fluids, J. Aeronaut. Sci. 6 (1939), 399–407. MR 3135
- M. H. Martin, The propagation of a plane shock into a quiet atmosphere, Canad. J. Math. 5 (1953), 37–39. MR 53714, DOI https://doi.org/10.4153/cjm-1953-004-2
- Monroe H. Martin, A new approach to problems in two dimensional flow, Quart. Appl. Math. 8 (1950), 137–150. MR 49731, DOI https://doi.org/10.1090/S0033-569X-1950-49731-4
- Charles Loewner, A transformation theory of the partial differential equations of gas dynamics, Tech. Notes Nat. Adv. Comm. Aeronaut. 1950 (1950), no. 2065, 56. MR 0044712
- Charles Loewner, Generation of solutions of systems of partial differential equations by composition of infinitesimal Baecklund transformations, J. Analyse Math. 2 (1953), 219–242. MR 57426, DOI https://doi.org/10.1007/BF02825638
C. Rogers, Reciprocal relations in non-steady one-dimensional gasdynamics and magneto-gasdynamics, Z. angew Math. Phys. 19, 59–63 (1968)
C. Rogers, Invariant transformations in non-steady gasdynamics and magneto-gasdynamics, Z. angew Math. Phys. 20, 370–382 (1969)
- C. Rogers, The construction of invariant transformations in plane rotational gasdynamics, Arch. Rational Mech. Anal. 47 (1972), 36–46. MR 343760, DOI https://doi.org/10.1007/BF00252187
S. P. Castell and C. Rogers, Application of invariant transformations in one-dimensional non-steady gasdynamics, Quart. Appl. Math. 32, 241–251 (1974)
- 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 (English, with French summary). MR 373452
- C. Rogers and J. G. Kingston, Reciprocal properties in quasi one-dimensional non-steady oblique field magneto-gasdynamics, J. Mécanique 15 (1976), no. 1, 185–192 (English, with French summary). MR 411348
C. Rogers. J. G. Kingston and W. F. Shadwick, On reciprocal-type invariant transformations in magneto-gasdynamics, J. Math. Phys. 21, 395–397 (1980)
- J. A. Baker and C. Rogers, Invariance properties under a reciprocal Bäcklund transformation in gasdynamics, J. Méc. Théor. Appl. 1 (1982), no. 4, 563–578 (English, with French summary). MR 682179
- C. Rogers and W. F. Shadwick, Bäcklund transformations and their applications, Mathematics in Science and Engineering, vol. 161, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1982. MR 658491
L. A. Movsesian, On an invariant transformation of equations of one-dimensional unsteady motion of an ideal compressible fluid, P.M.M. 31, 131–141 (1967)
M. D. Ustinov, Transformation and some solutions of the equation of motion of an ideal gas, Izv. Akad. Nauk SSSR, Mekanika Zhidkosti i Gaza 3, 68–74 (1966)
- A. S. Fokas and Y. C. Yortsos, On the exactly solvable equation$\ S_{t}=[(\beta S+\gamma )^{-2}S_{x}]_{x}+\alpha (\beta S+\gamma )^{-2}S_{x}$ occurring in two-phase flow in porous media, SIAM J. Appl. Math. 42 (1982), no. 2, 318–332. MR 650227, DOI https://doi.org/10.1137/0142025
- Julian D. Cole, On a quasi-linear parabolic equation occurring in aerodynamics, Quart. Appl. Math. 9 (1951), 225–236. MR 42889, DOI https://doi.org/10.1090/S0033-569X-1951-42889-X
- Eberhard Hopf, The partial differential equation $u_t+uu_x=\mu u_{xx}$, Comm. Pure Appl. Math. 3 (1950), 201–230. MR 47234, DOI https://doi.org/10.1002/cpa.3160030302
A. Haar, Über adjungierte Variationsprobleme und adjungierte Extremalflachen, Math. Ann. 100, 481–502 (1928)
H. Bateman, The lift and drag functions for an elastic fluid in two-dimensional irrotational flow, Proc. Nat. Acad. Sci. U.S.A. 24, 246–251 (1938)
G. Power and P. Smith, Application of a reciprocal property to subsonic flow, Appl. Sci. Res. A4, 386–392 (1959)
G. Power and P. Smith, Reciprocal properties of plane gas flows, J. Math. Mech. 10, 349–361 (1961)
H. S. Tsien, Two-dimensional subsonic flow of compressible fluids, J. Aeronaut. Sci. 6, 399–407 (1939)
M. H. Martin, The propagation of a plane shock into a quiet atmosphere, Canad J. Math. 5, 37–39 (1953)
M. H. Martin, A new approach to problems in two dimensional flow, Quart. Appl. Math. 8, 137–150 (1951)
C. Loewner, A transformation theory of partial differential equations of gasdynamics, Nat. Advis. Comm. Aeronaut. Tech. Notes 2065, 1–56 (1950)
C. Loewner, Generation of solutions of systems of partial differential equations by composition of infinitesimal Bäcklund transformations, J. Anal. Math. 2, 219–242 (1952)
C. Rogers, Reciprocal relations in non-steady one-dimensional gasdynamics and magneto-gasdynamics, Z. angew Math. Phys. 19, 59–63 (1968)
C. Rogers, Invariant transformations in non-steady gasdynamics and magneto-gasdynamics, Z. angew Math. Phys. 20, 370–382 (1969)
C. Rogers, The construction of invariant transformations in plane rotational gasdynamics, Arch. Rat. Mech. Anal. 47, 36–46 (1972)
S. P. Castell and C. Rogers, Application of invariant transformations in one-dimensional non-steady gasdynamics, Quart. Appl. Math. 32, 241–251 (1974)
C. Rogers, S. P. Castell and J. G. Kingston, On invariance properties of conservation laws in non-dissipative planar magneto-gasdynamics, J. Mécanique 13, 343–354 (1974)
C. Rogers and J. G. Kingston, Reciprocal properties in one-dimensional oblique field magneto-gasdynamics, J. Mecanique, 15, 185–192 (1976)
C. Rogers. J. G. Kingston and W. F. Shadwick, On reciprocal-type invariant transformations in magneto-gasdynamics, J. Math. Phys. 21, 395–397 (1980)
J. A. Baker and C. Rogers, Invariance properties under a reciprocal Bäcklund transformation in gasdynamics, J. Mécanique (to appear). Theorque et Appliquée 1, 563–578 (1982)
C. Rogers and W. F. Shadwick, Bäcklund transformations and their applications, Academic Press, New York, 1982
L. A. Movsesian, On an invariant transformation of equations of one-dimensional unsteady motion of an ideal compressible fluid, P.M.M. 31, 131–141 (1967)
M. D. Ustinov, Transformation and some solutions of the equation of motion of an ideal gas, Izv. Akad. Nauk SSSR, Mekanika Zhidkosti i Gaza 3, 68–74 (1966)
A. S. Fokas and Y. C. Yortsos, On the exactly soluble equation ${S_t} = {\left [ {{{(\beta S + \gamma )}^{ - 2}}{S_x}} \right ]_x} + {(\beta S + \gamma )^{ - 2}}{S_x}$ occurring in two phase flow in porous media, Soc. Ind. Appl. Math. J. Math. 42, 318–332 (1982)
J. D. Cole, On a quasilinear parabolic equation occurring in aerodynamics, Quart. Appl. Math. 9, 225–236 (1951)
E. Hopf, The partial differential equation ${u_t} + u{u_x} = u{u_{xx}}$, Comm. Pure Appl. Math. 3, 201–230 (1950)
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC:
58G37,
35Q20,
58F05,
76S05
Retrieve articles in all journals
with MSC:
58G37,
35Q20,
58F05,
76S05
Additional Information
Article copyright:
© Copyright 1984
American Mathematical Society
