Abstract
We discuss here the notion of conservation laws and briefly the theory of Abelian coverings over infinitely prolonged equations. Computation of conservation laws is also closely related to that of cosymmetries , and we shall continue this discussion in Chap. 4 below. In this chapter we give the solution to Problems 1.7, 1.13, and 1.15 posed in Chap. 1.
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References
Baran, H., Krasil′shchik, I.S., Morozov, O.I., Vojčák, P.: Nonlocal symmetries of Lax integrable equations: a comparative study. Submitted to Theor. Math. Phys. (2016). arXiv:1611.04938
Ferapontov, E.V.: Surfaces in 3-space possessing nontrivial deformations which preserve the shape operator. In: Integrable Systems in Differential Geometry, Tokyo, July 2000, Contemporary Mathematics, vol. 508, pp. 145–159. AMS, Providence (2002). arXiv:math/0107122
Gibbons, J., Tsarev, S.P.: Reductions of the Benney equations. Phys. Lett. A 211, 19–24 (1996)
Göktas, Ü., Hereman, W.: Symbolic computation of conserved densities for systems of nonlinear evolution equations. J. Symb. Comput. 24(5), 591–621 (1997)
Golovko, V.A., Kersten, P.H.M., Krasil′shchik, I.S., Verbovetsky, A.M.: On integrability of the Camassa-Holm equation and its invariants. Acta Appl. Math. 101, 59–83 (2008)
Hereman, W., Adams, P.J., Eklund, H.L., Hickman, M.S., Herbst, B.M.: Direct methods and symbolic software for conservation laws of nonlinear equations. In: Yan, Z. (ed.) Advances in Nonlinear Waves and Symbolic Computation, chap. 2, pp. 19–79. Nova Science Publishers, New York (2009)
Kersten, P.: Supersymmetries and recursion operator for N = 2 supersymmetric KdV-equation. Sūrikaisekikenkyūsho Kōkyūroku 1150, 153–161 (2000)
Kersten, P., Krasil′shchik, I., Verbovetsky, A.: (Non)local Hamiltonian and symplectic structures, recursions and hierarchies: a new approach and applications to the N = 1 supersymmetric KdV equation. J. Phys. A 37, 5003–5019 (2004)
Kersten, P., Krasil′shchik, I., Verbovetsky, A.: A geometric study of the dispersionless Boussinesq type equation. Acta Appl. Math. 90, 143–178 (2006)
Kersten, P., Krasil′shchik, J.: Complete integrability of the coupled KdV-mKdV system. In: Morimoto, T., Sato, H., Yamaguchi, K. (eds.) Lie Groups, Geometric Structures and Differential Equations—One Hundred Years After Sophus Lie. Advanced Studies in Pure Mathematics, vol. 37, pp. 151–171. Mathematical Society of Japan, Tokyo (2002)
Krasil′shchik, I.: A natural geometric construction underlying a class of Lax pairs. Lobachevskii J. Math. 37(1), 60–65 (2016). arXiv:1401.0612
Krasil′shchik, I.S., Kersten, P.H.M.: Deformations and recursion operators for evolution equations. In: Prastaro, A., Rassias, T.M. (eds.) Geometry in Partial Differential Equations, pp. 114–154. World Scientific, Singapore (1994)
Krasil′shchik, I.S., Kersten, P.H.M.: Symmetries and Recursion Operators for Classical and Supersymmetric Differential Equations. Kluwer, Dordrecht/Boston (2000)
Krasil′shchik, I.S., Morozov, O.I., Sergyeyev, A.: Infinitely many nonlocal conservation laws for the ABC equation with A + B + C ≠ 0. Calc. Var. Partial Differ. Equ. 55(5), 1–12 (2016). arXiv:1511.09430
Krasil′shchik, I.S., Sergyeyev, A.: Integrability of S-deformable surfaces: conservation laws, Hamiltonian structures and more. J. Geom. Phys. 97, 266–278 (2015). arXiv:1501.07171
Morozov, O.I., Sergyeyev, A.: The four-dimensional Martínez Alonso–Shabat equation: reductions and nonlocal symmetries. J. Geom. Phys. 85(11), 40–45 (2014)
Pavlov, M.V., Chang, J.H., Chen, Y.T.: Integrability of the Manakov–Santini hierarchy. arXiv:0910.2400
Poole, D., Hereman, W.: Symbolic computation of conservation laws for nonlinear partial differential equations in multiple space dimensions. J. Symb. Comput 46(12), 1355–1377 (2011)
Popovych, R.O., Samoilenko, A.M.: Local conservation laws of second-order evolution equations. J. Phys. A 41, 362002 (2008). arXiv:0806.2765
Popovych, R.O., Sergyeyev, A.: Conservation laws and normal forms of evolution equations. Phys. Lett. A 374, 2210–2217 (2010). arXiv:1003.1648
Sergyeyev, A.: New integrable (3 + 1)-dimensional systems and contact geometry. Lett. Math. Phys. (2017). https://doi.org/10.1007/s11005-017-1013-4
Wolf, T.: An efficiency improved program LIEPDE for determining Lie-symmetries of PDEs. In: Proceedings of Modern Group Analysis: Advanced Analytical and Computational Methods in Mathematical Physics. Kluwer (1993)
Wolf, T.: A comparison of four approaches to the calculation of conservation laws. Eur. J. Appl. Math. 13(2), 129–152 (2002)
Wolf, T., Brand, A.: CRACK, user guide, examples and documentation. http://lie.math.brocku.ca/Crack_demo.html
Wolf, T., Brand, A.: Investigating des with crack and related programs. SIGSAM Bull. Spec. Issue 1–8 (1995)
Zabolotskaya, E.A., Khokhlov, R.V.: Quasi-plane waves in the nonlinear acoustics of confined beams. Sov. Phys. Acoust. 15, 35–40 (1969)
Zakharevich, I.: Nonlinear wave equation, nonlinear Riemann problem, and the twistor transform of Veronese webs. arXiv:math-ph/0006001
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Krasil’shchik, J., Verbovetsky, A., Vitolo, R. (2017). Conservation Laws and Nonlocal Variables. In: The Symbolic Computation of Integrability Structures for Partial Differential Equations. Texts & Monographs in Symbolic Computation. Springer, Cham. https://doi.org/10.1007/978-3-319-71655-8_3
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