SIGMA 8 (2012), 071, 16 pages arXiv:1210.3673
Contribution to the Special Issue “Geometrical Methods in Mathematical Physics”
Conservation Laws, Hodograph Transformation and Boundary Value Problems of Plane Plasticity
Sergey I. Senashov a and Alexander Yakhno b
a) Siberian State Aerospace University, Krasnoyarsk, Russia
b) Departamento de Matemáticas, CUCEI, Universidad de Guadalajara, 44430, Mexico
Received April 18, 2012, in final form September 29, 2012; Published online October 13, 2012
For the hyperbolic system of quasilinear first-order partial differential equations, linearizable by hodograph transformation, the conservation laws are used to solve the Cauchy problem. The equivalence of the initial problem for quasilinear system and the problem for conservation laws system permits to construct the characteristic lines in domains, where Jacobian of hodograph transformations is equal to zero. Moreover, the conservation laws give all solutions of the linearized system. Some examples from the gas dynamics and theory of plasticity are considered.
conservation laws; hodograph transformation; Riemann method; plane plasticity; boundary value problem.
pdf (534 kb)
tex (162 kb)
- Annin B.D., Bytev V.O., Senashov S.I., Group properties of equations of
elasticity and plasticity, Nauka, Novosibirsk, 1985.
- Barbashov B.M., Chernikov N.A., Solution of the two plane wave scattering
problem in a nonlinear scalar field theory of the Born-Infeld type,
Sov. Phys. JETP 24 (1967), 437-442.
- Bocharov A.V., Chetverikov V.N., Duzhin S.V., Khor'kova N.G., Krasil'shchik
I.S., Samokhin A.V., Torkhov Yu.N., Verbovetsky A.M., Vinogradov A.M.,
Symmetries and conservation laws for differential equations of mathematical
physics, Translations of Mathematical Monographs, Vol. 182, American
Mathematical Society, Providence, RI, 1999.
- Born M., Infeld L., Foundation of a new field theory, Proc. R. Soc.
Lond. Ser. A 144 (1934), 425-451.
- Chakrabarty J., Theory of plasticity, 3rd ed., Elsevier, Heinemann, 2006.
- Chirkunov Yu.A., Group properties and conservation laws for second-order
quasilinear differential equations, J. Appl. Mech. Tech. Phys.
50 (2009), 413-418.
- Clarkson P.A., Fokas A.S., Ablowitz M.J., Hodograph transformations of
linearizable partial differential equations, SIAM J. Appl. Math.
49 (1989), 1188-1209.
- Currò C., Oliveri F., Reduction of nonhomogeneous quasilinear 2×2
systems to homogeneous and autonomous form, J. Math. Phys.
49 (2008), 103504, 11 pages.
- Fushchych W.I., Tychynin V.A., Hodograph transformations and generating of
solutions for nonlinear differential equations, Rep. NAS of Ukraine
(1993), no. 10, 52-58.
- Grundland A.M., Riemann invariants, in Wave Phenomena: Modern Theory and
Applications (Toronto, 1983), North-Holland Math. Stud., Vol. 97,
North-Holland, Amsterdam, 1984, 123-152.
- Ibragimov N.K., The experience of group analysis of ordinary differential
equations, Current Life, Science and Technology: Series "Mathematics
and Cybernetics", Vol. 91, Znanie, Moscow, 1991 (in Russian).
- Jeffrey A., Quasilinear hyperbolic systems and waves, Research Notes in
Mathematics, No. 5, Pitman Publishing, London - San Francisco, Calif. -
- Kachanov L.M., Fundamentals of the theory of plasticity, Dover Publications,
- Khristianovich S.A., The plane problem of mathematical plasticity theory for the closed contour, loaded by external forces, Sb. Math. 1 (1936),
511-534 (in Russian).
- Martin M.H., Riemann's method and the problem of Cauchy, Bull. Amer.
Math. Soc. 57 (1951), 238-249.
- Meleshko S.V., Methods for constructing exact solutions of partial differential
equations, Mathematical and Analytical Techniques with Applications to
Engineering, Springer, New York, 2005.
- Men'shikh O.F., Interaction of finite solitary waves for equations of
Born-Infeld type, Theoret. and Math. Phys. 79 (1989),
- Nowacki W.K., Stress waves in non-elastic solids, Pergamon Press, Oxford, 1978.
- Ovsiannikov L.V., Group analysis of differential equations, Academic Press
Inc., New York, 1982.
- Peradzynski Z., Geometry of interactions of Riemann waves, in Advances in
Nonlinear Waves, Vol. II, Res. Notes in Math., Vol. 111, Pitman,
Boston, MA, 1985, 244-285.
- Rozdestvenski B.L., Janenko N.N., Systems of quasilinear equations
and their applications to gas dynamics, Translations of Mathematical
Monographs, Vol. 55, American Mathematical Society, Providence, RI, 1983.
- Senashov S.I., Vinogradov A.M., Symmetries and conservation laws of
2-dimensional ideal plasticity, Proc. Edinburgh Math. Soc. (2)
31 (1988), 415-439.
- Senashov S.I., Yakhno A., 2-dimensional plasticity: boundary problems and
conservation laws, reproduction of solutions, in Proceedinds of Fifth
International Conference "Symmetry in Nonlinear Mathematical Physics" (June
23-29, 2003, Kyiv), Proceedings of Institute of Mathematics,
Kyiv, Vol. 50, Part 1, Editors A.G. Nikitin, V.M. Boyko, R.O. Popovych, I.A. Yehorchenko, Institute of Mathematics, Kyiv, 2004, 231-237.
- Senashov S.I., Yakhno A., Reproduction of solutions of bidimensional ideal
plasticity, Internat. J. Non-Linear Mech. 42 (2007),
- Senashov S.I., Yakhno A., Yakhno L., Deformation of characteristic curves of
the plane ideal plasticity equations by point symmetries, Nonlinear
Anal. 71 (2009), e1274-e1284.
- Sokolovski V.V., Jones D.H., Schofield A.N., Statics of soil media,
Butterworths Scientific Publications, London, 1960.
- Tsarëv S.P., The geometry of Hamiltonian systems of hydrodynamic type.
The generalized hodograph method, Math. USSR-Izv. 37
- von Mises R., Mathematical theory of compressible fluid flow, Applied
Mathematics and Mechanics, Vol. 3, Academic Press Inc., New York, 1958.
- Yakhno A., Yakhno L., 'Homotopy' of Prandtl and Nadai solution,
Internat. J. Non-Linear Mech. 45 (2010), 793-799.
- Zabusky N.J., Exact solution for the vibrations of a nonlinear continuous model
string, J. Math. Phys. 3 (1962), 1028-1039.