Abstract
A test in terms of invariants for the existence of a factorization of a bivariate, non-hyperbolic third-order Linear Partial Differential Operator (LPDO) which has a given factorization of its principal symbol is found. The invariants that are used are with respect to known gauge transformations, which is together with constructive factorization itself are essentially involved in modern exact integration algorithms. The invariants, and even a generating system of those were found in previous paper using Moving Frames methods.
In order to find the expressions in terms of invariants that guarantee the existence of a factorization of a certain type, we show that the operation of taking the formal adjoint can be also defined in terms of invariants, that is for equivalence classes of LPDOs, and explicit formulae defining this operation in the space invariants are obtained. The operation of formal adjoint is highly interesting for factorization of LPDOs for if the initial operator has a factorization, its adjoint has also one, and they are related. (informally, the factorization types are symmetric).
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References
Tsarev, S., Shemyakova, E.: Differential transformations of parabolic second-order operators in the plane (submitted, 2008), http://arxiv.org/abs/0811.1492
Tsarev, S.: Generalized laplace transformations and integration of hyperbolic systems of linear partial differential equations. In: ISSAC 2005: Proceedings of the 2005 international symposium on Symbolic and algebraic computation, pp. 325–331. ACM Press, New York (2005)
Tsarev, S.: Factorization of linear partial differential operators and darboux’ method for integrating nonlinear partial differential equations. Theo. Math. Phys. 122, 121–133 (2000)
Anderson, I., Juras, M.: Generalized Laplace invariants and the method of Darboux. Duke J. Math. 89, 351–375 (1997)
Anderson, I., Kamran, N.: The variational bicomplex for hyperbolic second-order scalar partial differential equations in the plane. Duke J. Math. 87, 265–319 (1997)
Athorne, C.: A z ×r toda system. Phys. Lett. A. 206, 162–166 (1995)
Zhiber, A.V.: Integrals, solutions and existence of the laplace transformations for a linear hyperbolic system of equations. Math. Notes 74(6), 848–857 (2003)
Startsev, S.: Cascade method of laplace integration for linear hyperbolic systems of equations. Mathematical Notes 83 (2008)
Grigoriev, D., Schwarz, F.: Factoring and solving linear partial differential equations. Computing 73(2), 179–197 (2004)
Grigoriev, D., Schwarz, F.: Generalized loewy-decomposition of d-modules. In: ISSAC 2005: Proceedings of the 2005 international symposium on Symbolic and algebraic computation, pp. 163–170. ACM, New York (2005)
Grigoriev, D., Schwarz, F.: Loewy decomposition of third-order linear pde’s in the plane. In: ISSAC 2008: Proceedings of the 2005 international symposium on Symbolic and algebraic computation, pp. 277–286. ACM, New York (2008)
Shemyakova, E., Mansfield, E.: Moving frames for laplace invariants. In: Proceedings of ISSAC 2008 (The International Symposium on Symbolic and Algebraic Computation), pp. 295–302 (2008)
Fels, M., Olver, P.J.: Moving coframes. I. A practical algorithm. Acta Appl. Math. 51(2), 161–213 (1998)
Fels, M., Olver, P.J.: Moving coframes. II. Regularization and theoretical foundations. Acta Appl. Math. 55(2), 127–208 (1999)
Olver, P., Pohjanpelto, J.: Pseudo-groups, moving frames, and differential invariants. In: Eastwood, M., Miller, W.J. (eds.) Symmetries and Overdetermined Systems of Partial Differential Equations. IMA Volumes in Mathematics and Its Applications, vol. 144, pp. 127–149. Springer, New York (2007)
Olver, P., Pohjanpelto, J.: Moving frames for Lie pseudo-groups. Canadian J. Math (to appear) (2007)
Darboux, G.: Leçons sur la théorie générale des surfaces et les applications géométriques du calcul infinitésimal, vol. 2. Gauthier-Villars (1889)
Shemyakova, E., Winkler, F.: A full system of invariants for third-order linear partial differential operators in general form. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2007. LNCS, vol. 4770, pp. 360–369. Springer, Heidelberg (2007)
Shemyakova, E., Winkler, F.: On the invariant properties of hyperbolic bivariate third-order linear partial differential operators. In: Kapur, D. (ed.) ASCM 2007. LNCS (LNAI), vol. 5081. Springer, Heidelberg (2008)
Loewy, A.: Ueber reduzible lineare homogene differentialgleichungen. Math. Annalen 56, 549–584 (1903)
Blumberg, H.: Über algebraische Eigenschaften von linearen homogenen Differentialausdrücken. PhD thesis, Göttingen (1912)
Shemyakova, E.: The Parametric Factorizations of Second-, Third- and Fourth-Order Linear Partial Differential Operators on the Plane. Mathematics in Computer Science 1(2), 225–237 (2007)
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Shemyakova, E. (2009). Invariant Properties of Third-Order Non-hyperbolic Linear Partial Differential Operators. In: Carette, J., Dixon, L., Coen, C.S., Watt, S.M. (eds) Intelligent Computer Mathematics. CICM 2009. Lecture Notes in Computer Science(), vol 5625. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02614-0_16
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DOI: https://doi.org/10.1007/978-3-642-02614-0_16
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