Nonlinear wave interactions for a model of extended thermodynamics with six fields

  • C. Currò
  • N. Manganaro


A nonlinear model of extended thermodynamics with six fields without the near-equilibrium approximation, in one dimensional case, is considered. A class of double wave solutions of the governing system at hand is determined and an exact description of a soliton-like wave interaction is given.


Hyperbolic systems Double wave solutions Nonlinear wave interactions 

Mathematics Subject Classification

35L50 35L60 35N10 



This work was supported by Italian National Group of Mathematical Physics (GNFM-INdAM). The results contained in the present paper have been partially presented in Wascom 2017.


  1. 1.
    De Groot, S.R., Mazur, P.: Non-equilibrium Thermodynamics. Dover, New York (1984)zbMATHGoogle Scholar
  2. 2.
    Landau, L.D., Lifshitz, E.M.: Fluid Mechanichs. Pergamon Press, London (1958)Google Scholar
  3. 3.
    Arima, T., Ruggeri, T., Sugiyama, M., Taniguchi, S.: Extended thermodynamics of dense gases. Contin. Mech. Thermodyn. 24(4), 271–292 (2012)CrossRefzbMATHGoogle Scholar
  4. 4.
    Ruggeri, T., Sugiyama, M.: Rational Extended Thermodynamics Beyond the Momoatomic Gas. Springer, Cham (2015)CrossRefzbMATHGoogle Scholar
  5. 5.
    Arima, T., Ruggeri, T., Sugiyama, M., Taniguchi, S.: Nonlinear extended thermodynamics of real gases with 6 fields. Int. J. Nonlinear Mech. 72, 6–15 (2015)CrossRefzbMATHGoogle Scholar
  6. 6.
    Arima, T., Ruggeri, T., Sugiyama, M., Taniguchi, S.: Overshoot of the non-equilibrium temperature in the shock wave structure of a rarefied polyatomic gas subject to the dynamic pressure. Int. J. Nonlinear Mech. 79, 66–75 (2016)CrossRefGoogle Scholar
  7. 7.
    Arima, T., Ruggeri, T., Sugiyama, M., Taniguchi, S.: Recent results on non-linear extended thermodynamics of real gases with six fields. Part I: general theory. Ric. Mat. 65(1), 263–277 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Müller, I., Ruggeri, T.: Rational Extended Thermodynamics. Springer, New York (1998)CrossRefzbMATHGoogle Scholar
  9. 9.
    Pavić-Čolić, M., Madjaverić, D., Simić, S.: Polyatoic gases with dynamic pressure: kinetic non-linear closure and the shock structure. Int. J. Nonlinear Mech. 92, 160–175 (2017)CrossRefGoogle Scholar
  10. 10.
    Curró, C., Manganaro, N.: Double-wave solutions to quasilinear hyperbolic systems of first order PDEs. ZAMP 68, 103 (2017)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Jeffrey, A.: Quasilinear Hyperbolic Systems and Waves. Research Notes in Mathematics. Pitman Publishing, London (1997)Google Scholar
  12. 12.
    Curró, C., Fusco, D., Manganaro, N.: A reduction procedure for generalized Riemann problems with application to nonlinear transmission lines. J. Phys. A Math. Theor. 44(33), 335205 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Curró, C., Fusco, D., Manganaro, N.: Differential constraints and exact solution to Riemann problems for a traffic flow model. Acta Appl Math. 122(1), 167–178 (2012)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Curró, C., Manganaro, N.: Riemann problems and exact solutions to a traffic flow model. J. Math. Phys. 54(17), 071503 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Curró, C., Manganaro, N.: Generalized Riemann problems and exact solutions for \(p\)-systems with relaxation. Ric. Mat. 65(2), 549–562 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Manganaro, N.: Riemann problems for viscoelastic media. Rend. Lincei Mat. Appl. 28, 479–494 (2017)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Seymour, B.R., Varley, E.: Exact solutions describing soliton-like interactions in a non dispersive medium. SIAM J. Appl. Math. 42, 804–21 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Curró, C., Fusco, D.: On a class of quasilinear hyperbolic reducible systems allowing for special wave interactions. ZAMP J. Appl. Math. Phys. 38, 58094 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Curró, C., Fusco, D., Manganaro, N.: Hodograph transformation and differential constraints for wave solutions to \(2 \times 2\) quasilinear hyperbolic nonhomogeneous systems. J. Phys. A Math. Theor. 45(19), 195207 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Curró, C., Fusco, D., Manganaro, N.: An exact description of nonlinear wave interaction processes ruled by \(2 \times 2\) hyperbolic systems. ZAMP 64(4), 1227–1248 (2013)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Curró, C., Fusco, D.: A reduction method for quasilinear hyperbolic systems of multicomponent field PDEs with application to wave interaction. Int. J. Nonlinear Mech. 37, 281–295 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Curró, C., Fusco, D., Manganaro, N.: Exact description of simple wave interactions in multicomponent chromatography. J. Phys. A Math. Theor. 48(1), 015201 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Curró, C., Manganaro, N., Pavlov, M.V.: Nonlinear wave interaction problems in the three-dimensional case. Nonlinearity 30, 207–224 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Curró, C., Manganaro, N.: Exact solutions and wave interaction for a viscoelastic medium. AAPP Cl. Sci. Fis. Mat. Nat. 96(1), A1 (2018)Google Scholar
  25. 25.
    Yanenko, N.N.: Compatibility theory and methods of integration of systems of nonlinear partial differential equation. In: Proceedings of 4th All-Union Mathematical Congress, Nauka, Leningrad, pp. 247–252 (1964)Google Scholar
  26. 26.
    Manganaro, N., Meleshko, S.: Reduction procedure and generalized simple waves for systems written in the Riemann variables. Nonlinear Dyn. 30(1), 87–102 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Boillat, G.: Non linear hyperbolic fields and wavees. In: Ruggeri, T. (ed.) Recent Mathematical Methods in Nonlinear Wave Propagation. Lecture Notes in Mathematics, vol. 1640. Springer, Berlin (1995)Google Scholar
  28. 28.
    Lax, P.D.: Hyperbolic systems of conservation laws II. Commun. Pure Appl. Math. 10, 537–566 (1957)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Courant, R., Friedrichs, K.O.: Supersonic Flows and Shock Waves. Interscience Publishers, New York (1962)zbMATHGoogle Scholar

Copyright information

© Università degli Studi di Napoli "Federico II" 2018

Authors and Affiliations

  1. 1.MIFTUniversity of MessinaMessinaItaly

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