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A precise definition of reduction of partial differential equations. (English) Zbl 0936.35012
By reduction in the title here is meant Lie symmetry reduction of partial differential equations to ordinary differential equations, more specifically reduction under conditional symmetries. Such types of symmetry came into practical use in the late eighties through the work of {\it P. J. Olver} and {\it P. Rosenau} [Phys. Lett. A114, 107-112 (1986)] and {\it V. I. Fushchich} and {\it I. M. Tsifra }[J. Phys. A 20, L45--L48 (1987; Zbl 0663.35045)] based on an earlier idea of {\it G. W. Bluman} and {\it J. D. Cole} [J. Math. Mech. 18, 1025-1042 (1969; Zbl 0187.03502)]. A partial differential equation, $L=0$ is conditionally invariant under a set of involutive operators $\{Q_a|a=1\ldots n\}$ if the application of their prologations to $L$ vanishes not on the submanifold of jet space defined by the equation but only on its intersection with those associated with the $Q_a$’s. The authors point out that in practice the idea is more easily used for equations with only two independent variables, and their aim is to provide a basis for a less ad hoc treatment of higher dimensional equations. They achieve this by constructing an ansatz adapted to the general solution of the symmetry defining system: $Q_a\equiv 0$. This is a clear, straightforward treatment in which the reduction ansatz implies the presence of a conditional symmetry and vice versa. They apply their results to systems if the form $\square u=F(u)$ where $\square$ is the d’Alembertian in four dimensions, which reduce to d’Alembert-Hamilton systems. They consider special cases and recover some new solutions.

##### MSC:
 35A30 Geometric theory for PDE, characteristics, transformations 58J70 Invariance and symmetry properties
Full Text:
##### References:
 [1] Bluman, G.; Cole, J. D.: The general similarity solution of the heat equation. J. math. Mech. 18, 1025-1042 (1969) · Zbl 0187.03502 [2] Olver, P. J.; Rosenau, P.: The construction of special solutions to partial differential equations. Phys. lett. 114A, 107-112 (1986) · Zbl 0937.35501 [3] Olver, P. J.; Rosenau, P.: Group-invariant solutions of differential equations. SIAM J. Appl. math. 47, 263-278 (1987) · Zbl 0621.35007 [4] Fushchych, W. I.; Tsyfra, I. M.: On a reduction and solutions of nonlinear wave equations with broken symmetry. J. phys. A 20, L45-L48 (1987) · Zbl 0967.81033 [5] Fushchych, W. I.; Zhdanov, R. Z.: Symmetry and exact solutions of nonlinear spinor equations. Phys rep. 172, 123-174 (1989) [6] Clarkson, P.; Kruskal, M. D.: New similarity solutions of the Boussinesq equation. J. math. Phys. 30, 2201-2213 (1989) · Zbl 0698.35137 [7] Levi, D.; Winternitz, P.: Non-classical symmetry reduction: example of the Boussinesq equation. J. phys. A 22, 2915-2924 (1989) · Zbl 0694.35159 [8] Fushchych, W. I.: How to extend symmetry of differential equations?. (1987) [9] Fushchych, W. I.: On symmetry and exact solutions of multi-dimensional nonlinear wave equations. Ukrain. math. J. 39, 116-123 (1987) [10] Fushchych, W. I.; Nikitin, A. G.: Symmetries of Maxwell’s equations. (1987) [11] Fushchych, W. I.; Zhdanov, R. Z.: Symmetry and exact solutions of the nonlinear Dirac equation. Fizika elementar. Chastits i atom. Yadra 19, 1154-1196 (1988) [12] Fushchych, W. I.; Zhdanov, R. Z.: On some new exact solutions of the nonlinear d’Alembert--Hamilton system. Phys. lett. 141A, 113-115 (1989) [13] Fushchych, W. I.; Zhdanov, R. Z.: Non-Lie ansätze and exact solutions of the nonlinear spinor equation. Ukrain. math. J. 42, 958-962 (1990) · Zbl 0711.35102 [14] Fushchych, W. I.; Zhdanov, R. Z.: On non-Lie reduction of the nonlinear Dirac equation. J. math. Phys. 32, 3488-3490 (1991) [15] Fushchych, W. I.; Zhdanov, R. Z.; Yehorchenko, I. A.: On the reduction of the nonlinear multi-dimensional wave equations and compatibility of the d’Alembert--Hamilton system. J. math. Anal. appl. 161, 352-360 (1991) · Zbl 0763.35090 [16] Zhdanov, R. Z.; Fushchych, W. I.: On non-Lie ansatzes and new exact solutions of the classical Yang--Mills equations. J. nonlinear math. Phys. 2, 172-181 (1995) · Zbl 0947.53014 [17] Zhdanov, R. Z.; Fushchych, W. I.: Conditional symmetry and new classical solutions of the Yang-Mills equations. J. phys. A 28, 6253-6263 (1995) · Zbl 0872.35092 [18] Zhdanov, R. Z.: On integration of nonlinear d’Alembert-eikonal system and conditional symmetry of nonlinear wave equations. J. nonlinear math. Phys. 4, 49-62 (1997) · Zbl 0951.35087 [19] Fushchych, W. I.; Zhdanov, R. Z.: Symmetries and exact solutions of nonlinear Dirac equations. (1997) · Zbl 0945.22010 [20] Fushchych, W. I.; Zhdanov, R. Z.: Conditional symmetry and reduction of partial differential equations. Ukrain math. J. 44, 970-982 (1992) [21] Zhdanov, R. Z.: Conditional Lie-Bäcklund symmetry and reduction of evolution equations. J. phys. A 28, 3841-3850 (1995) · Zbl 0859.35115 [22] Olver, P. J.; Vorob’ev, E. M.: Nonclassical and conditional symmetries. (1994) [23] Popovych, R. O.: On reduction and Q-conditional symmetry. (1997) · Zbl 0954.35040 [24] Popovych, R. O.; Korneva, I. P.: On Q-conditional symmetry of the linear n-dimensional heat equation. (1998) · Zbl 0912.35070 [25] Popovych, R. O.: On a class of Q-conditional symmetries and solutions of evolutionary equations. (1998) · Zbl 0913.35145 [26] Fushchych, W. I.: On symmetry and particular solutions of some multi-dimensional mathematical physics equations. (1983) [27] Fushchych, W. I.; Serov, N. I.: The symmetry and some exact solutions of nonlinear many-dimensional Liouville, d’Alembert and eikonal equations. J. phys. A 16, 3645-3656 (1983) · Zbl 0554.35106 [28] Olver, P. J.: Direct reduction and differential constraints. Proc. roy. Soc. London ser. A 444, 509-523 (1994) · Zbl 0814.35003 [29] Fushchych, W. I.; Shtelen, W. M.; Serov, M. I.: Symmetry analysis and exact solutions of equations of non-linear mathematical physics. (1992) [30] Estevez, P. G.; Gordoa, P. R.: Nonclassical symmetries and the singular manifold method--theory and 6 examples. Stud. appl. Math. 95, 517-557 (1995) [31] Zhdanov, R. Z.; Tsyfra, I. M.: Reduction of differential equations and conditional symmetry. Ukrain. math. J. 48, 595-602 (1996) · Zbl 0932.35194 [32] Olver, P. J.: Applications of Lie groups to differential equations. (1989) · Zbl 0743.58003 [33] Schutz, B. F.: Geometrical methods of mathematical physics. (1982) · Zbl 0462.58001 [34] Vorob’ev, E. M.: Partial symmetries of system of differential equations. Soviet math. Dokl. 33, 408-411 (1986) [35] Bateman, H.: Partial differential equations of mathematical physics. (1922) · Zbl 48.0720.16 [36] Smirnov, V. I.; Sobolev, S. L.: New method of solution of the problem of elastic plane vibrations. Proc. seismological inst. Acad. sci. USSR 20, 37-40 (1932) [37] Smirnov, V. I.; Sobolev, S. L.: On application of the new method to study of elastic vibrations in the space with axial symmetry. Proc. seismological inst. Acad. sci. USSR 29, 43-51 (1933) [38] Collins, C. B.: Complex potential equations. I. A technique for solution. Proc. Cambridge philos. Soc. 80, 165-171 (1976) · Zbl 0326.35034 [39] Cartan, E.: Oeuvres completes. (1995) [40] Bateman, H.: Mathematical analysis of electrical and optical wave-motion. (1955) · Zbl 0067.19301 [41] Erugin, N. P.: On functionally-invariant solutions. Proc. acad. Sci. USSR 5, 385-386 (1944) [42] W. I. Fushchych, R. Z. Zhdanov, and, I. V. Revenko, Compatibility and solutions of the nonlinear d’Alembert and Hamilton equations, Preprint No. 90-39, Inst. of Math. Acad. of Sci. Ukraine, Kiev, 1990. · Zbl 0849.35021 [43] Fushchych, W. I.; Zhdanov, R. Z.; Revenko, I. V.: General solutions of the nonlinear wave and eikonal equations. Ukrain. math. J. 43, 1471-1486 (1991) [44] Patera, J.; Sharp, R. T.; Winternitz, P.; Zassenhaus, H.: Subgroups of the Poincaré group and their invariants. J. math phys. 17, 977-985 (1976) · Zbl 0347.20029 [45] Grundland, A.; Harnad, J.; Winternitz, P.: Symmetry reduction for nonlinear relativistically invariant equations. J. math. Phys. 25, 791-806 (1984) · Zbl 0599.70026 [46] Zhdanov, R. Z.; Panchak, O. A.: New conditional symmetries and exact solutions of the nonlinear wave equation. J. phys. A 31, 8727-8734 (1998) · Zbl 0989.35094 [47] Galaktionov, V. A.: On new exact blow-up solutions for nonlinear heat conduction equations with source and applications. Differential integral equations 3, 863-874 (1990) · Zbl 0735.35074 [48] Nucci, M. C.: Iterating the nonclassical symmetries method. Physica D 78, 124-134 (1994) · Zbl 0815.35008