Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
E. Infeld and G. Rowlands, Nonlinear Waves, Solitons and Chaos, Cambridge University Press, Cambridge, New York (1st ed. 1990; 2nd revised ed. 1992).
R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. II, Interscience Publ., New York, London, Sydney (1962).
A. Jeffrey and T. Taniuti, Non-linear Wave Propagation, with Applications to Physics and Magnetohydrodynamics, Academic Press, New York, London (1964).
A. Jeffrey, Quasilinear hyperbolic systems and waves, Pitman Publishing, London, San Francisco, Melbourne (1976).
A. Jeffrey, Lectures on nonlinear wave propagation, in Wave Propagation, Corso CIME, Bressanone, 1980, Ed. by G. Ferrarese, Liguori, Napoli (1982).
C. Dafermos, Hyperbolic systems of conservation laws, in Systems of nonlinear partial differential equations. Ed. by J. Ball. NATO ASI series, C, No 111, Reidel Publ. Co. (1983) pp. 25–70.
J. Smoller, Shock waves and reaction-diffusion equations, Springer-Verlag, New York, Heidelberg, Berlin (1983).
D. Serre, Systèmes de lois de conservation, I & II, Diderot, Paris (1996).
B. Sévennec, Géométrie des systèmes hyperboliques de lois de conservation, Mémoire 56, Soc. Math. Fr. (1994).
P. D. Lax, The multiplicities of eigenvalues, Bull. Amer. Math. Soc., 6 (1982) p.213.
S. Friedland, J. W. Robbin and J. H. Sylvester, On the crossing rule, Comm. Pure Appl. Math., 37 (1984) p.19.
J. Hadamard, Leçons sur la propagation des ondes, Hermann, Paris (1903).
C. Reid, Courant in Göttingen and New York. The Story of an Improbable Mathematician, Springer-Verlag, New York (1976) p.279.
R. Courant and P. D. Lax, The propagation of discontinuities in wave motion, Proc. Natl. Acad. Sci. U.S., 42 (1956) pp.872–76.
C. Cattaneo, Elementi di teoria della propagazione ondosa. Lezioni raccolte da S. Pluchino. Quaderni dell’Unione Matematica Italiana 20, Pitagora Editrice, Bologna (1981).
A. Jeffrey, The development of jump discontinuities in nonlinear hyperbolic systems of equations in two independent variables, Arch. Ratl. Mech. Analys., 14 (1963) pp.27–37.
G. Boillat and T. Ruggeri, On the evolution law of weak discontinuities for hyperbolic quasi-linear systems, Wave Motion, 1 (1979) pp.149–152.
P. D. Lax, Asymptotic solutions of oscillatory initial value problem, Duke. Math. Journ., 24 (1957) pp.627–646.
D. Ludwig, Exact and asymptotic solutions of the Cauchy problem, Comm. Pure Appl. Math., 13 (1960) pp.473–508.
Y. Choquet-Bruhat, Ondes asymptotiques pour un système d’équations aux dérivées partielles non linéaires, J. Maths. pure appl., 48 (1969) pp.117–158; Ondes Asymptotiques, in Wave Propagation, Corso CIME. Bressanone, 1980 (op. cit.) pp.99–165.
A. M. Anile and A. Greco, Asymptotic waves and critical time in general relativistic magnetohydrodynamics, Ann. Inst. Henri Poincaré, XXIX (1978) p.257.
J. K. Hunter and J. B. Keller, Weakly nonlinear high frequency waves, Comm. Pure Appl. Math., XXXVI (1983) pp.547–569.
G. Boillat, Ondes asymptotiques non linéaires, Ann. Mat. pura ed appl., CXI (1976) pp.31–44.
D. Serre, Oscillations non-linéaires hyperboliques de grande amplitude: DIM ≥2 in Nonlinear variational problems and partial differential equations. Proc. of the 3rd conf., Isola d’Elba, 1990. Ed. by A. Marino et al., Pitman Res. Notes Math., Ser.320 (1995) pp.245–294.
R. Courant and K. O. Friedrichs, Supersonic flow and shock waves, Interscience Publ., New York (1948).
P. D. Lax, The initial value problem for nonlinear hyperbolic equations in two independent variables, Ann. Math. Studies, Princeton, 33 (1954) pp.211–29.
P. D. Lax, Hyperbolic systems of conservation laws, II, Comm. Pure Appl. Math., 10 (1957) pp.537–566.
G. Boillat, Sur la croissance des ondes simples et l’instabilité de chocs caractéristiques des systèmes hyperboliques avec application à la discontinuité de contact d’un fluide, C. R. Acad. Sci. Paris, 284 A (1977) pp.1481–1484.
G. Boillat, On nonlinear plane waves, in 8th Internat. Conf. on Waves and Stability in Continuous Media, Plaermo, 1995. Ed. by A.M. Greco and S. Rionero (to appear in a special issue of Rend. Circ. Mat. Palermo).
G. Velo and D. Zwanziger, Propagation and quantization of Rarita-Schwinger waves in an external electromagnetic potential, Phys. Rev., 186 (1969) pp.1337–1341.
G. Boillat, Exact plane wave solution of Born-Infeld electrodynamics, Lett. N. Cim., serie 2, 4 (1972) pp.274–276.
G. Boillat, Covariant disturbances and exceptional waves, J. Math. Phys., 14 (1973) pp.973–976.
H. C. Tze, Born duality and strings in hadrodynamics and electrodynamics, N. Cim., 22 A (1974) pp.507–526.
H. Freistühler, Linear degeneracy and shock waves, Math. Z., 207 (1991) pp.583–596.
G. Boillat, Chocs caractéristiques, C.R. Acad. Sci. Paris, 274 A (1972) pp.1018–1021.
G. Boillat and A. Muracchini, Chocs caractéristiques de croisement, C.R. Acad. Sci. Paris, 310 I (1990) pp.229–232.
G. Boillat, Le cône critique et le champ scalaire, C.R. Acad. Sci. Paris, 260 (1965) pp.2427–2429.
W. Heisenberg and H. Euler, Zeitschrift für Physik, 98 (1936) p. 714.
G. B. Witham, Linear and Nonlinear Waves, John Wiley & Sons, New York, London, Sydney, Toronto (1974).
Th. von Kármán, Compressibility effects in aerodynamics, J. Aeron. Sci., 8 (1941) pp. 337–356.
E. Carafoli, High Speed Aerodynamics, Pergamon Press (1956).
Y. Choquet-Bruhat, Théorèmes d’existence globaux pour des fluides ultrarelativistes, C.R. Acad. Sci. Paris, 319 I (1994) pp. 1337–1342.
T. Ruggeri, A. Muracchini and L. Seccia, Shock waves and second sound in a rigid heat conductor: a critical temperature for NaF and Bi, Phys. Rev. Lett., 64 (1990) p. 2640; Continuum approach to phonon gas and shape of second sound via shock waves theory, N. Cim., 16 D (1994) pp. 15–44.
A. Donato and T. Ruggeri, Onde di discontinuità e condizioni di eccezionalità per materiali ferromagnetici, Rend. Accad. Naz. Lincei, Serie VIII, LIII (1972) pp. 289–294.
G. Boillat, La propagation des ondes, Gauthier-Villars, Paris.
T. Ruggeri, Sulla propagazione di onde elettromagnetiche di discontinuità in mezzi non lineari, Ist. Lombardo (Rend. Sc.) A 107 (1973) pp. 283–297.
T. Taniuti, On wave propagation in non-linear fields, Suppl. Progr. Theor. Phys., No9 (1959) pp. 69–128.
M. Born, Proc. Roy. Soc. London, Ser. A, 143 (1933) p. 410.
W. Heisenberg, Zs. f. Phys., 133 (1952) p. 79; 126 (1949) p. 519; 113 (1939) p. 61.
B. M. Barbishov and N. A. Chernikov, Solution of the two plane wave scattering problem in a nonlinear scalar field theory of the Born-Infeld type, Sov. Phys. J. E. T. P., 24 (1966) pp. 437–442.
I. Imai, Progr. Theor. Phys., 2 (1947) p. 97.
T. Taniuti, On the Heisenberg’s non-linear meson equation, Progr. Theor. Phys., 14 (1955) pp. 408–409.
A. Greco, On the exceptional waves in relativistic magnetohydrodynamics, Rend. Accad. Naz. Lincei, serie VIII, LII (1972) pp. 507–512.
M. Born and L. Infeld, Foundations of the new field theory, Proc. Roy. Soc. London, A 144 (1934) pp. 425–451; M. Born, Structure atomique de la matière, Coll. U. Armand Colin, Paris (1971); Atomic Physics, Blackie and Son, Ltd., London.
J. Plebański, Lectures on non-linear electrodynamics, given at the Niels Bohr Institute and NORDITA in 1968, NORDITA, Copenhagen (1970). I. Bialynicki-Birula, Nonlinear Electrodynamics: variations on a theme by Born and Infeld in Quantum Theory and Particles and Fields. B. Jancewicz and J. Lukierski eds. World Scientific, Singapore (1983).
J. Naas and H.L. Schmid, Mathematisches Wörterbuch mit Einbeziehung der theoretischen Physik, Akademie Verlag, Berlin und B.G. Teubner, Stuttgart (1961).
A. Einstein, H. Born and M. Born, Briefwechsel, 1916–1955, kommentiert von Max Born. Geleitwort von Bertrand Russell, Vorwort von Werner Heisenberg. Nymphenburger Verlagshandlung, München, 1969; H. and M. Born, Der Luxus des Gewissens, Ibid., 1969.
G. Boillat, Nonlinear electrodynamics. Lagrangians and equations of motion, J. Math. Phys., 11 (1970) pp. 941–951; Born-Infeld particle; small scale phenomena, Lett. N. Cim., 4 (1970) pp. 773–778; Exact plane-wave solution of Born-Infeld electrodynamics, Ibid., 4 (1972) pp. 274–276; Shock relations in nonlinear electrodynamics, Phys. Lett., 40 A (1972) pp. 9–10.
A. Strumia, Einstein equations, relativistic strings and Born-Infeld electrodynamics, N. Cim., 110 B (1995) pp. 1497–1504.
J. H. Olsen and A. S. Shapiro, Large-amplitude unsteady flow in liquid-filled elastic tubes, J. Fluid Mech., 29 (1967) pp. 513–538.
W. A. Green, The growth of plane discontinuities propagating into a homogeneously deformed elastic material, Arch. Ratl. Mech. Analys., 16 (1964) pp. 79–88.
N. H. Scott, Acceleration waves in incompressible elastic solids, Quart. J. Mec. Appl. Math., 29 (1976) pp. 259–310.
A. Jeffrey and M. Teymur, Formation of shock waves in hyperelastic solids, Acta Mech., 20 (1974) pp. 133–149.
L. Lustman, A note on nonbreaking waves in hyperelastic materials, Stud. Appl. Math., 63 (1980) pp. 147–154.
A. Donato, Legge di evoluzione delle discontinuità e determinazione di una classe di potenziali elastici compatibile con la propagazione di one eccezionali in un mezzo continuo sottoposto a particolari deformazioni finite, ZAMP, 28 (1977) pp. 1059–1066.
G. Boillat and T. Ruggeri, Su alcune classi di potenziali termodinamici come conseguenza dell’esistenza di particolari onde di discontinuità nella meccanica dei continui con deformazioni finite, Rend. Sem. Mat. Univ. Padova, 51 (1974) pp. 293–304.
G. Boillat and S. Pluchino, Onde eccezionali in mezzi iperelastici con deformazioni finite piane, ZAMP, 35 (1984) pp. 363–372.
G. Grioli, On the thermodynamic potential for continuums with reversible transformations—some possible types, Meccanica, 1 (1966) pp. 15–20.
C. Tolotti, Deformazioni elastiche finite: onde ordinarie di discontinuità e casi tipici di solidi elastici isotropi, Rend. Mat. Appl., serie V, 4 (1943) pp. 34–59.
G. Boillat, Le champ scalaire de Monge-Ampère, Det Kgl. Norske Vid. Selsk. Forh., 41 (1968) pp. 78–81.
G. Valiron, Cours d’analyse mathématique. Equations fonctionnelles. Applications, Masson, Paris (1950).
G. Boillat, Sur l’équation générale de Monge-Ampère à plusieurs variables, C. R. Acad. Sci. Paris, 313 I (1991) pp. 805–808.
T. Ruggeri, Su una naturale estensione a tre variabili dell’equazione di Monge-Ampère, Accad. Naz. dei Lincei, LV (1973) pp. 445–449.
A. Donato, U. Ramgulam and C. Rogers, The 3+1 dimensional Monge-Ampère equation in discontinuity wave theory: application of a reciprocal transformation, Meccanica, 27 (1992) pp. 257–262.
G. Boillat, Sur l’équation générale de Monge-Ampère d’ordre supérieur, C. R. Acad. Sci. Paris, 315 I (1992) pp. 1211–1214.
A. Donato and F. Oliveri, Linearization of completely exceptional second order hyperbolic conservative equations, Applicable Analys., 57 (1995) pp. 35–45.
G. Ruppeiner, Riemannian geometry of thermodynamics and critical phenomena, in Advances in Thermodynamics. Ed. by S. Sieniutycz & P. Salomon, Vol. 3, Taylor & Francis, New York (1990) pp. 129–159.
A. Donato and G. Valenti, Exceptionality criterion and linearization procedure for a third order nonlinear partial differential equation, J. Math. Analys. Appls., 186 (1994) pp. 375–382.
G. Boillat, Sur la forme générale du système de Monge-Ampère, Conf. at the CIRAM, Università di Bologna, October 1995 (to appear).
K. O. Friedrichs, Symmetric hyperbolic linear differential equations, Comm. Pure Appl. Math., 7 (1954) pp. 345–392.
A. Fisher and D. P. Marsden, The Einstein evolution equations as a first order quasi-linear symmetric hyperbolic system, Comm. Math. Phys., 28 (1972) pp. 1–38.
T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Ratl. Mech. Analys., 58 (1975) pp. 181–205; Quasi-linear equations of evolution of hyperbolic type with applications to partial differential equations, Lect. Notes in Math., No448, Springer Verlag, New York, Berlin (1975).
S. K. Godunov, An interesting class of quasilinear systems, Sov. Math. Doklady, 2 (1961) pp. 947–949.
K. O. Friedrichs and P. D. Lax, Systems of conservation equations with a convex extension, Proc. Nat. Acad. Sci. USA, 68 (1971) pp. 1686–1688.
B. L. van der Waerden, Euler als Schöpfer der Kontinuums-Mechanik, Naturwissenschaften, 71 (1984) pp. 414–417.
Schweizer Lexikon 91 in sechs Bänden, Verlag Schweizer Lexikon, Luzern (1992–93) (Euler).
S. K. Godunov, The problem of a generalized solution in the theory of quasilinear equations and in gas dynamics, Russian Math. Surveys, 17 (1962) pp. 145–156.
P. Carbonaro, Exceptional relativistic gas dynamics, Phys. Lett. A, 129 (1988) pp. 372–376.
A. Greco, Discontinuités des rayons et stricte exceptionnalité en magnéto-hydrodynamique relativiste, Ann. Inst. Henri Poincaré, XII (1975) pp. 217–227.
A. H. Taub, Relativistic Rankine-Hugoniot equations, Phys. Rev., 74 (1948) pp. 328–334.
A. H. Taub, General relativistic shock waves in fluids for which pressure equals energy density, Comm. math. Phys., 29 (1973) pp. 79–88.
G. Boillat, A moi compte deux mots. A spelling remark, Math. Intelligencer, 4 (1982) p. 2.
T. Ruggeri and A. Strumia, Main field and convex covariant density for quasi-linear hyperbolic systems, Ann. Inst. Henri Poincaré, A XXXIV (1981) pp. 65–84.
G. Boillat, Sur l’existence et la recherche d’équations de conservation supplémentaires pour les systèmes hyperboliques, C. R. Acad. Sci. Paris, 278 A (1974) pp. 909–912.
I-Shih Liu, Method of Lagrange multipliers for exploitation of the entropy principle, Arch. Ratl. Mech. Analys., 46 (1972) pp. 131–148.
A. Strumia, Main field and symmetric hyperbolic form of the Dirac equation, Lett. N. Cim., 36 (1983) pp. 609–613.
G. Boillat and T. Ruggeri, Symmetric form of nonlinear mechanics equations and entropy growth across a shock, Acta Mech., 35 (1980) pp. 271–274.
V. I. Kondaurov, Conservation laws and symmetrization of the equations of the nonlinear theory of elasticity, Sov. Phys. Dokl., 26 (1981) pp. 234–236.
C. Truesdell, Six lectures on modern natural philosophy, Springer-Verlag, New York (1966) p. 72.
I. Müller, The coldness, a universal function in thermoelastic bodies, Arch. Ratl. Mech. Analys., 41 (1971) pp. 319–332.
I. Müller and T. Ruggeri, Extended Thermodynamics, Springer Tracts in Natural Philosophy, Vol. 37, Springer-Verlag, New York, Berlin (1993).
K. O. Friedrichs, On the laws of relativistic electro-magneto-fluid dynamics, Comm. Pure Appl. Math., 27 (1974) pp. 749–808; Conservation equations and the law of motion in classical physics, Ibid., 31 (1978) pp. 123–131.
G. Boillat, Chocs dans les champs qui dérivent d’un principe variationnel: équation de Hamilton-Jacobi pour la fonction génératrice, C. R. Acad. Sci. Paris 283 A (1976) pp. 539–542.
Encyclopaedia of Mathematics, Kluwer Academic Publ., Dordrecht, Boston, London (1988–94).
P. Roman, Theory of elementary particles, North-Holland (1960).
L. L. Foldy, Relativistic wave equations, in D. R. Bates, Quantum Theory, Vol.III, Radiation and High Energy Physics, Academic Press, New York and London (1962) p. 42.
N. N. Bogoliubov and D. V. Shirkov, Introduction to the theory of quantized fields, 3rd. Ed. John Wiley & Sons, New York (1980) p. 41 sqq.
S. De Leo, Duffin-Kemmer-Petiau equation on the quaternion field, Progr. Theor. Phys., 94 (1995) pp. 1109–1120.
G. Boillat, Involutions des systèmes conservatifs, C. R. Acad. Sci. Paris, 307 I (1988) pp. 891–894.
A. Lichnerowicz, Théories relativistes de la gravitation et de l’électromagnétisme, Masson, Paris (1955).
Y. Choquet-Bruhat, Problème de Cauchy pour les modèles gravitationnels avec termes de Gauss-Bonnet, C. R. Acad. Sci. Paris, 306 I (1988) pp. 445–450.
C. M. Dafermos, Quasilinear hyperbolic systems with involutions, Arch. Ratl. Mech. Analys., 94 (1986) pp. 373–389.
G. Boillat, Symétrisation des systèmes d’équations aux dérivées partielles avec densité d’énergie convexe et contraintes, C. R. Acad. Sci. Paris, 295 I (1982) pp. 551–554.
S. K. Godunov, Symmetric form of the equations of magnetohydrodynamics (in Russian), Prepr. Akad. Nauk S.S.S.R. Sib. Otd., Vychisl. Tsentr, 3 No1 (1972) p. 26.
G. Boillat and S. Pluchino, Sopra l’iperbolicità dei sistemi con vincoli e considerazioni sul superfluido e la magnetoidrodinamica, ZAMP 36 (1985) pp. 893–900.
S. Lundquist, Studies in magneto-hydrodynamics, Arkiv för Fysik, 5 (1952) p. 297.
L. Landau and E. Lifchitz, Mécanique des fluides, MIR, Moscou (1971).
G. Grioli, Ed., Macroscopic theories of superfluids, Internat. Meeting, Accad. Naz. dei Lincei, Rome, May 1988. Cambridge University Press (1991).
G. Boillat and A. Muracchini, On the symmetric conservative form of Landau’s superfluid equations, ZAMP, 35 (1984) pp. 282–288; On a special form of the hydrodynamic superfluid equations, Ibid., 36 (1985) pp.901–904; Thermodynamic conditions for a symmetric form of superfluid equations, N. Cim. 9 D (1987) pp.253–259.
G. Boillat and T. Ruggeri, Hyperbolic principal subsystems: entropy, convexity and subcharacteristic conditions, Arch. Ratl. Mech. Analys. (to appear).
T.-P. Liu, Hyperbolic conservation laws with relaxation, Comm. Math. Phys., 108 (1987) pp. 153–175.
G.-Q. Chen, C. D. Levermore and T.-P. Liu, Hyperbolic conservative laws with stiff relaxation terms and entropy, Comm. Pure Appl. Math., 67 (1994) pp.787–830.
W. Weiss, Hierarchie der Erweiterten Thermodynamik, Dissertation TU Berlin (1990).
C. Cercignani and A. Majorana, Analysis of thermal and shear waves according to the B. G. K. kinetic model, ZAMP, 36 (1985) pp. 699–711.
H. Ott, Lorentz-Transformation der Wärme und der Temperatur, Zs. f. Phys., 175 (1963) pp. 70–104.
H. Arzeliès, Relativistic transformation of temperature and some other thermodynamical quantities (in French), N. Cim., 35 (1965) pp. 792–804.
C. Møller, The Theory of Relativity, 2nd Ed., Clarendon Press, Oxford (1972).
J. Putterman, Superfluid hydrodynamics, North Holland, Amsterdam (1974).
J. R. Dorroh, Nonlinear symmetric first-order systems, J. Math. Analys. Appls., 91 (1983) pp. 523–526.
G. Boillat, On symmetrization of partial differential systems, Applicable Analys., 57 (1995) pp. 17–21.
G. Boillat, Limitation des vitesses de choc quand la densité d’énergie est convexe et les contraintes involutives, C. R. Acad. Sci. Paris, 297I (1983) pp. 141–143.
G. Boillat, Evolution des chocs caractéristiques dans les champs dérivant d’un principe variationnel, J. Maths. Pures et Appl., 56 (1977) pp. 137–147.
E. Hölder, Historischer Überblick zur mathematischen Theorie von Unstetigkeitswellen seit Riemann und Christoffel, in E. B. Christoffel: The Influence of his Work on Mathematics and the Physical Sciences, Internat. Symp., Aachen, 1979 (1981).
P. Germain, Schock waves, jump relations and structure, Adv. in Appl. Mech. Ed. by Chia-Shun Yih, Vol. 12, Academic Press, New York (1972) pp. 131–144.
P. D. Lax, Hyperbolic systems of conservation laws and the mathematical theory of shock waves, Regional Conf. Series in Appl. Math., No 11, SIAM, Philadelphia (1973).
C. M. Dafermos and R. J. Diperna, The Riemann problem for certain classes of hyperbolic systems of conservation laws, J. Diff. Eqs., 20 (1976) pp. 90–114.
M. Berger and M. Berger, Perspectives in Nonlinearity, W.A. Benjamin Inc., New York (1968) p.137.
G. Boillat and T. Ruggeri, Limite de la vitesse des chocs dans les champs à densité d’énergie convexe, C. R. Acad. Sci. Paris, 289 A (1979) pp. 257–258.
G. Boillat and A. Strumia, Limitation des vitesses d’onde et de choc quand la densité relativiste d’entropie (ou d’énergie) est convexe, Ibid., 307 I (1988) pp. 111–114.
B. L. Keyfitz and H. C. Kranzer, A system of non-strictly hyperbolic conservation laws arising in elasticity theory, Arch. Ratl. Mech. Analys., 72 (1980) pp. 219–241.
C. M. Dafermos, Admissible wave fans in nonlinear hyperbolic systems, Ibid., 106 (1989) pp. 243–260.
T.-P. Liu, Admissible solutions of hyperbolic conservation laws, Memoirs Am. Math. Soc., 240 (1981) pp. 1–78.
G. Boillat, Hamilton-Jacobi equation for shocks with a convex entropy density, in 7th Conf. on Waves and Stability in Continuous Media, Bologna, 1993. Ed. by S. Rionero & T. Ruggeri. Quaderno CNR. Series in Adv. in Math. & Appl. Sciences, Vol. 23, World Scientific, Singapore, New Jersey, London, Hong-Kong (1994) pp. 22–25.
G. Boillat, Chocs avec contraintes et densité d’énergie convexe, C. R. Acad. Sci. Paris, 295 I (1982) pp. 747–750.
G. Boillat, De la nature des chocs, in Onde e stabilità nei mezzi continui, Cosenza, giugno 1983. Quaderni del CNR, Catania (1986) pp. 35–39.
G. Boillat, De la vitesse de choc considérée comme valuer propre, C. R. Acad. Sci. Paris, 302 I (1986) pp. 555–557.
P. D. Lax, Shock waves and entropy in Contribution to non linear functional analysis. Ed. by Zarantonello, Academic Press, New York (1971).
C. Dafermos, Generalized characteristics in hyperbolic systems of conservation laws, Arch. Ratl. Mech. Analys., 107 (1989) p. 127.
D. Fusco, Alcune considerazioni sulle onde d’urto in fluidodinamica, Atti Sem. Mat. Fis. Univ. Modena, 28 (1979) p. 223.
N. Virgopia and F. Ferraioli, On the shock wave generating function in a single mixture of gases, N. Cim., D 7 (1988) p. 151.
A. Strumia, A detailed study of entropy jump across shock waves in relativistic fluid dynamics, N. Cim., B 92 (1986) p. 91.
W. Dreyer and S. Seelecke, Entropy and causality as criteria for the existence of shock wave in low temperature ideal conduction, Cont. Mech. Thermodyn., 4 (1992) pp. 23–36.
L. Seccia, Shock wave propagation and admissibility criteria in a non linear dielectric medium, Ibid.,, 7 (1995) pp. 277–296.
N. Manganaro and G. Valenti, On shock propagation for a compressible fluid supplemented by a generalized von Kármán law, Atti. Sem. Mat. Fis. Univ. Modena, XXXVIII (1990) pp. 109–122.
G. Boillat and A. Muracchini, The structure of the characteristic shocks in constrained symmetric systems with applications to magnetohydrodynamics, Wave Motion, 11 (1989) pp. 297–307.
G. Boillat, Chocs caractéristiques et ondes simples exceptionnelles pour les systèmes conservatifs à intégrale d’énergie; forme explicite de la solution, C. R. Acad. Sci. Paris, 280 A (1975) pp. 1325–1328.
T. Ruggeri, On same properties of electromagnetic shock waves in isotropic nonlinear materials, Boll. Un. Mat. Ital., 9 (1974) pp. 513–522.
G. Boillat, Expression explicite des chocs caractéristiques de croisement, C. R. Acad. Sci. Paris, 312 I (1991) pp. 653–656.
A. Jeffrey, Magnetohydrodynamics, Oliver & Boyd, London (1966).
H. Cabannes, Theoretical magnetofluid-dynamics, Academic Press, New York (1970).
G. Boillat and A. Muracchini, Characteristic shocks of crossing velocities in magnetohydrodynamics, NoDEA, 3, (1996), pp. 217–230.
G. Boillat and T. Ruggeri, Characteristic shocks: completely and strictly exceptional systems, Boll. Un. Mat. Ital., 15 A (1978) pp.197–204.
P. D. Lax, Development of singularities of solutions of nonlinear hyperbolic partial differential equations, J. Math. Phys., 5 (1964) pp.611–613.
G. Boillat, Shock relations in nonlinear electrodynamics, Phys. Lett., 40 A (1972) pp.9–10.
G. Boillat and T. Ruggeri, Euler equations for nonlinear relativistic strings. Introduction of a new Lagrangian, Progr. Theor. Phys., 60 (1978) pp.1928–1929.
A. Greco, On the strict exceptionality for a subsonic flow, in 2o Congresso Naz. AIMETA, Università di Napoli, Facoltà di Ingegneria, 4 (1974) pp.127–134.
G. Boillat, A relativistic fluid in which shock fronts are also wave surfaces, Phys. Lett., 50 A (1974) pp.357–358.
L. Brun, Ondes de choc finies dans les solides élastiques, in Mechanical waves in solids. Ed. by J. Mandel & L. Brun, Springer, Vienna (1975).
A. Morro, Atti Accad. Naz. Lincei Rend., 64 (1978) p.177; Acta Mech., 38 (1981) p.241; On stablity in the interaction between shock waves and acoustic waves, in Onde e stabilità nei mezzi continui, Catania, Nov.1981. Quaderni del CNR, Catania (1982) pp.18–47.
A. Strumia, Transmission and reflexion of a discontinuity wave through a characteristic shock in nonlinear optics, Riv. Mat. Univ. Parma, 4 (1978) p.15.
G. Boillat and T. Ruggeri, Reflection and transmission of discontinuity waves through a shock wave. General theory including also the case of characteristic shocks, Proc. Roy. Soc. Edinburgh, 83 A (1979) pp. 17–24.
A. Donato and D. Fusco, Nonlinear wave propagation in a layered halfspace, Int. J. Nonlinear Mechanics, 15 (1980) pp.497–503.
A. M. Anile and S. Pennisi, On the mathematical structure of the relativistic magnetofluid-dynamics, Ann. Inst. Henri Poincaré, 46 (1987) p.27.
A. M. Anile, Relativistic fluids and magneto-fluids, Cambridge University Press, 1989.
G. Boillat and T. Ruggeri, Wave and shock velocities in relativistic magnetohydrodynamics compared with the speed of light, Continuum Mech. Thermodyn., 1 (1989) pp.47–52.
A. Lichnerowicz, Ondes de choc, ondes infinitésimales et rayons en hydrodynamique et magnétohydrodynamique relativistes, in Relativistic Fluid Dynamics, Corso CIME, Bressanone, 1970. Ed. by C. Cattaneo, Cremonese, Roma (1971); Shock waves in relativistic magnetohydrodynamics under general assumptions, J. Math. Phys., 17 (1976) p.2125.
W. Israel, Relativistic theory of shock waves, Proc. Roy. Soc., 259 A (1960) p.129.
A. Lichnerowicz, Magnetohydrodynamics: waves and shock waves in curved space-time. Mathematical Studies, 14. Kluwer Academic Publ. Group, Dordrecht (1994).
Y. Choquet-Bruhat, Fluides chargés non abéliens de conductivité infinie, C. R. Acad. Sci. Paris, 314 I (1992) pp.87–91; Hydrodynamics and magnetohydrodynamics of Yang-Mills fluids, in 7th Conf. on Waves and Stability in continuous media, Bologna (1993) (op. cit.).
S. Pennisi, A covariant and extended model for relativistic magnetofluiddynamics, Ann. Inst. Henri Poincaré, 58 (1993) pp.343–361.
G. Boillat, Sur l’élimination des contraintes involutives, C. R. Acad. Sci. Paris, 318 I (1994) pp.1053–1058.
C. Cattaneo, Sulla conduzione del calore, Atti. Sem. Mat. Fis. Univ. Modena, 3 (1948) p.1.
T. Ruggeri, Struttura dei sistemi alle derivate parziali compatibili con un principio di entropia e termodinamica estesa, Suppl. Boll. Un. Mat. Ital., 4 (1985) p.261; Thermodynamics and symmetric hyperbolic systems, Rend. Sem. Mat. Univ. Torino, special issue: Hyperbolic equations, 167 (1987).
A. Morro and T. Ruggeri, Non-equilibrium properties of solids obtained from second-sound measurements, J. Phys. C: Solid State Phys., 21 (1988) p.1743.
F. Franchi and A. Morro, Global existence and asymptotic stability in nonlinear heat conduction, J. Math. Analys. Appls., 188 (1994) pp.590–609.
G. B. Nagy, O. E. Ortiz and O. A. Reula, The behavior of hyperbolic heat equations’ solutions near their parabolic limits, J. Math. Phys., 35 (1994) pp.4334–4356.
A. M. Anile, S. Pennisi and M. Sammartino, A thermodynamical approach to Eddington factors, J. Math. Physics, 32 (1991) pp.544–550.
G. Boillat, Convexité et hyperbolicité en électrodynamique non linéaire, C. R. Acad. Sci. Paris, 290 A (1980) pp.259–261.
G. Boillat and A. Giannone, A constraint on wave, radial and shock velocities in non-linear electromagnetic materials, ZAMP, 40 (1989) pp.285–289.
Editor information
Rights and permissions
Copyright information
© 1996 Springer-Verlag
About this chapter
Cite this chapter
Boillat, G. (1996). Non linear hyperbolic fields and waves. In: Ruggeri, T. (eds) Recent Mathematical Methods in Nonlinear Wave Propagation. Lecture Notes in Mathematics, vol 1640. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0093705
Download citation
DOI: https://doi.org/10.1007/BFb0093705
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-61907-9
Online ISBN: 978-3-540-49565-9
eBook Packages: Springer Book Archive