Duke Mathematical Journal

Generalized Laplace invariants and the method of Darboux

Martin Juráš and Ian M. Anderson

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Article information

Source
Duke Math. J. Volume 89, Number 2 (1997), 351-375.

Dates
First available in Project Euclid: 19 February 2004

Permanent link to this document
http://projecteuclid.org/euclid.dmj/1077241021

Mathematical Reviews number (MathSciNet)
MR1460626

Zentralblatt MATH identifier
0885.35075

Digital Object Identifier
doi:10.1215/S0012-7094-97-08916-X

Subjects
Primary: 58A17: Pfaffian systems
Secondary: 35A30: Geometric theory, characteristics, transformations [See also 58J70, 58J72] 35L10: Second-order hyperbolic equations

Citation

Juráš, Martin; Anderson, Ian M. Generalized Laplace invariants and the method of Darboux. Duke Math. J. 89 (1997), no. 2, 351--375. doi:10.1215/S0012-7094-97-08916-X. http://projecteuclid.org/euclid.dmj/1077241021.


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References

  • [1] I. M. Anderson and N. Kamran, The variational bicomplex for hyperbolic second-order scalar partial differential equations in the plane, Duke Math. J. 87 (1997), no. 2, 265–319.
  • [2] R. Bryant, P. A. Griffiths, and L. Hsu, Hyperbolic exterior differential systems and their conservation laws. I, Selecta Math. (N.S.) 1 (1995), no. 1, 21–112.
  • [3] F. Calogero, A solvable nonlinear wave equation, Stud. Appl. Math. 70 (1984), no. 3, 189–199.
  • [4] G. Darboux, Leçons sur la théorie générale des surfaces et les applications géométriques du calcul infinitésimal, Gauthier-Villars, Paris, 1896.
  • [5] E. D. Fackerell, D. Hartley, and R. Tucker, An obstruction to the integrability of a class of non-linear wave equations by $1$-stable Cartan characteristics, J. Differential Equations 115 (1995), no. 1, 153–165.
  • [6] A. Forsyth, Theory of differential equations. 1. Exact equations and Pfaff's problem; 2, 3. Ordinary equations, not linear; 4. Ordinary linear equations; 5, 6. Partial differential equations, Six volumes bound as three, Dover Publications Inc., New York, 1959.
  • [7] R. B. Gardner and N. Kamran, Characteristics and the geometry of hyperbolic equations in the plane, J. Differential Equations 104 (1993), no. 1, 60–116.
  • [8] E. Goursat, Leçon sur l'intégration des équations aux dérivées partielles du second ordre á deux variables indépendantes, Tome 1, Hermann, Paris, 1896.
  • [9] E. Goursat, Leçon sur l'intégration des équations aux dérivées partielles du second ordre á deux variables indépendantes, Tome 2, Hermann, Paris, 1896.
  • [10] W. F. Ames, R. L. Anderson, V. A. Dorodnitsyn, E. V. Ferapontov, R. K. Gazizov, N. H. Ibragimov, and S. R. Svirshchevskiĭ, CRC handbook of Lie group analysis of differential equations. Vol. 1, CRC Press, Boca Raton, FL, 1994.
  • [11] M. Juráš, Geometric aspects of second-order scalar hyperbolic partial differential equations in the plane, Ph.D. thesis, Utah State University, in preparation.
  • [12] M. Juráš, Generalized Laplace invariants and classical integration methods for second order scalar hyperbolic partial differential equations in the plane, Differential geometry and applications (Brno, 1995), Masaryk Univ., Brno, 1996, pp. 275–284.
  • [13] A. Kumpera and D. Spencer, Lie equations. Vol. I: General Theory, Ann.Math.Stud., vol. 73, Princeton Univ. Press, Princeton, N.J., 1972.
  • [14] A. V. Mikhaĭ lov, A. B. Shabat, and V. V. Sokolov, The symmetry approach to classification of integrable equations, What is integrability? ed. V. E. Zakharov, Springer Ser. Nonlinear Dynam., Springer, Berlin, 1991, pp. 115–184.
  • [15] V. V. Sokolov and A. V. Zhiber, On the Darboux integrable hyperbolic equations, Phys. Lett. A 208 (1995), no. 4-6, 303–308.
  • [16] T. Tsujishita, On variation bicomplexes associated to differential equations, Osaka J. Math. 19 (1982), no. 2, 311–363.
  • [17] P. J. Vassiliou, Geometry and the method of Darboux, Lie theory, differential equations and representation theory (Montreal, PQ, 1989) ed. V. Hussin, Univ. Montréal, Montreal, QC, 1990, pp. 395–404.
  • [18] E. Vessiot, Sur les équations aux dérivées partielles du second ordre $F(x, y, z, p, q, r, s, t)=0$, intégrables par la méthode de Darboux, J. Math. Pures Appl. (9) 18 (1939), 1–61.
  • [19] E. Vessiot, Sur les équations aux dérivées partielles du second ordre, $F(x,y,z,p,q,r,s,t)=0$, intégrables par la méthode de Darboux, J. Math. Pures Appl. (9) 21 (1942), 1–66.