## Duke Mathematical Journal

### The variational bicomplex for hyperbolic second-order scalar partial differential equations in the plane

#### Article information

Source
Duke Math. J. Volume 87, Number 2 (1997), 265-319.

Dates
First available in Project Euclid: 19 February 2004

http://projecteuclid.org/euclid.dmj/1077242147

Digital Object Identifier
doi:10.1215/S0012-7094-97-08711-1

Mathematical Reviews number (MathSciNet)
MR1443529

Zentralblatt MATH identifier
0881.35069

#### Citation

Anderson, Ian M.; Kamran, Niky. The variational bicomplex for hyperbolic second-order scalar partial differential equations in the plane. Duke Math. J. 87 (1997), no. 2, 265--319. doi:10.1215/S0012-7094-97-08711-1. http://projecteuclid.org/euclid.dmj/1077242147.

#### References

• [1] I. M. Anderson, The Variational Bicomplex, forthcoming.
• [2] I. M. Anderson and T. Duchamp, On the existence of global variational principles, Amer. J. Math. 102 (1980), no. 5, 781–868.
• [3] I. M. Anderson and M. Fels, Variational operators for ordinary differential equations, Hamiltonian operators, and the method of equivalence, in preparation.
• [4] I. M. Anderson and M. Juráš, Generalized Laplace invariants and the method of Darboux, to appear in Duke Math. J.
• [5] I. M. Anderson and N. Kamran, The variational bicomplex for second-order scalar partial differential equations in the plane, Department of Mathematics, Utah State University, Septembre 1994.
• [6] I. M. Anderson and N. Kamran, La cohomologie du complexe bi-gradué variationnel pour les équations paraboliques du deuxième ordre dans le plan, C. R. Acad. Sci. Paris Sér. I Math. 321 (1995), no. 9, 1213–1217.
• [7] I. M. Anderson and G. Thompson, The inverse problem of the calculus of variations for ordinary differential equations, Mem. Amer. Math. Soc. 98 (1992), no. 473, vi+110.
• [8] I. M. Anderson and C. G. Torre, Symmetries of the Einstein equations, Phys. Rev. Lett. 70 (1993), no. 23, 3525–3529.
• [9] I. M. Anderson and C. G. Torre, Aysmptotic conservation laws in field theory, Phys. Rev. Lett. 77 (1996), 4109–4133.
• [10] I. M. Anderson and C. G. Torre, Lower degree conservation laws in field theory, in preparation.
• [11]1 G. Barnich, F. Brandt, and M. Henneaux, Local BRST cohomology in the antifield formalism. I. General theorems, Comm. Math. Phys. 174 (1995), no. 1, 57–91.
• [11]2 G. Barnich, F. Brandt, and M. Henneaux, Local BRST cohomology in the antifield formalism. II. Application to Yang-Mills theory, Comm. Math. Phys. 174 (1995), no. 1, 93–116.
• [12] R. L. Bryant, S. S. Chern, R. B. Gardner, H. L. Goldschmidt, and P. A. Griffiths, Exterior differential systems, Mathematical Sciences Research Institute Publications, vol. 18, Springer-Verlag, New York, 1991.
• [13] R. L. Bryant and P. A. Griffiths, Characteristic cohomology of differential systems. I. General theory, J. Amer. Math. Soc. 8 (1995), no. 3, 507–596.
• [14] R. L. Bryant and P. A. Griffiths, Characteristic cohomology of differential systems. II. Conservation laws for a class of parabolic equations, Duke Math. J. 78 (1995), no. 3, 531–676.
• [15]1 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.
• [15]2 R. Bryant, P. A. Griffiths, and L. Hsu, Hyperbolic exterior differential systems and their conservation laws. II, Selecta Math. (N.S.) 1 (1995), no. 2, 265–323.
• [16] G. Darboux, Leçons sur la théorie générale des surfaces et les applications géométriques du calcul infinitésimal, Gauthier-Villars, Paris, 1896.
• [17] R. P. Delong, Killing Tensors and the Hamilton-Jacobi Equation, PH.D. dissertation, University of Minnesota, 1982.
• [18] S. V. Duzhin and T. Tsujishita, Conservation laws of the BBM equation, J. Phys. A 17 (1984), no. 16, 3267–3276.
• [19] A. Forsyth, Theory of Differential Equations, Vol. 6, Dover, New York, 1959.
• [20] R. B. Gardner, A differential geometric generalization of characteristics, Comm. Pure Appl. Math. 22 (1969), 597–626.
• [21] 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.
• [22] E. Goursat, Leçon sur l'intégration des équations aux dériées partielles du second ordre à deux variables indépendantes, Tome 1, Hermann, Paris, 1896.
• [23] E. Goursat, Leçon sur l'intégration des équations aux dériées partielles du second ordre à deux variables indépendantes, Tome 2, Hermann, Paris, 1896.
• [24] 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.
• [25] M. Juráš Ph.D. thesis, Utah State University, Logan, Utah, in preparation.
• [26] N. G. Khorkova, On the $\scr C$-spectral sequence of differential equations, Differential Geom. Appl. 3 (1993), no. 3, 219–243.
• [27] I. S. Krasilśhchik, V. V. Lychagin, and A. M. Vinogradov, Geometry of jet spaces and nonlinear partial differential equations, Advanced Studies in Contemporary Mathematics, vol. 1, Gordon and Breach Science Publishers, New York, 1986.
• [28] A. V. Mikhailov, A. B. Shabat, and V. V. Sokolov, The symmetry approach to classification of integrable equations, What is integrability?, Springer Ser. Nonlinear Dynam., Springer, Berlin, 1991, pp. 115–184.
• [29] P. J. Olver, Euler operators and conservation laws of the BBM equation, Math. Proc. Cambridge Philos. Soc. 85 (1979), no. 1, 143–160.
• [30] P. J. Olver, Conservation laws in elasticity. II. Linear homogeneous isotropic elastostatics, Arch. Rational Mech. Anal. 85 (1984), no. 2, 131–160.
• [31] P. J. Olver, Conservation laws in elasticity. I. General results, Arch. Rational Mech. Anal. 85 (1984), no. 2, 111–129.
• [32] P. J. Olver, Applications of Lie groups to differential equations, Graduate Texts in Mathematics, vol. 107, Springer-Verlag, New York, 1986.
• [33] P. J. Olver, Darboux's theorem for Hamiltonian differential operators, J. Differential Equations 71 (1988), no. 1, 10–33.
• [34] A. V. Shapovalov and I. V. Shirokov, On the symmetry algebra of a linear differential equation, Theoret. and Math. Phys. 92 (1992), 697–703.
• [35] V. V. Sokolov and A. V. Zhiber, On the Darboux integrable hyperbolic equations, Phys. Lett. A 208 (1995), no. 4-6, 303–308.
• [36] T. Tsujishita, On variation bicomplexes associated to differential equations, Osaka J. Math. 19 (1982), no. 2, 311–363.
• [37] T. Tsujishita, Formal geometry of systems of differential equations, Sugaku Exposition 2 (1989), 1–40.
• [38] T. Tsujishita, Homological method of computing invariants of systems of differential equations, Differential Geom. Appl. 1 (1991), no. 1, 3–34.
• [39] W. M. Tulczyjew, The Euler-Lagrange resolution, Differential geometrical methods in mathematical physics (Proc. Conf., Aix-en-Provence/Salamanca, 1979), Lecture Notes in Math., vol. 836, Springer, Berlin, 1980, pp. 22–48.
• [40] 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.
• [41] 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.
• [42]1 A. M. Vinogradov, The $\cal C$-spectral sequence, Lagrangian formalism, and conservation laws. I. The linear theory, J. Math. Anal. Appl. 100 (1984), no. 1, 1–40.
• [42]2 A. M. Vinogradov, The $\cal C$-spectral sequence, Lagrangian formalism, and conservation laws. II. The nonlinear theory, J. Math. Anal. Appl. 100 (1984), no. 1, 41–129.
• [43] V. V. Zharinov, Differential algebras and low-dimension conservation laws, Math. USSR Sb. 71 (1992), 319–329.
• [44] A. V. Zhiber and A. B. Shabat, Klein-Gordon equation with nontrivial group, Soviet Math. Dokl. 24 (1979), 607–609.