Symmetry And Group

Lie group analysis. Classical heritage by Ibragimov N.H. (ed.)

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By Ibragimov N.H. (ed.)

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N). Elimination of nm quantities pik from the above q + n equations (if it is possible) yields a system of first-order partial differential equations with the unknown function V. The resulting system Ωk (x1 , . . , xn , z1 , . . , zm , Vx1 , . . , Vxn , Vz1 , . . , Vzm ) = 0 is satisfied by all integral manifolds z1 = ϕ1 , . . , zm = ϕm of the equations F1 = 0, . . , Fq = 0. 53. The question immediately arises of what is the practical significance of this theorem. In order to answer this question we note first of all that if the above elimination is possible and the equations Ωk = 0 are obtained, then we can always find a family of point manifolds generating all integral manifolds.

In other words, we claim that the quantities x, y, z, p, q can have constant values not for every characteristic strip. This immediately follows from the fact that at least some of the five values Vx , Vy , Vz , Vp , Vq appear in Ω = 0. Moreover, the three values x, y, z can also have constant values not for every characteristic strip. This is provided by the fact that the number of existing two-dimensional integral manifolds is at least ∞4 and that the set of all two-dimensional manifolds does not satisfy any first-order partial differential equation other than Ω = 0 so that every characteristic strip belongs at least to one two-dimensional manifold z = f, p = fx , q = fy .

0. * Let a partial differential equation F = 0 of order m with variables z, x1 , x2 , . . , xn (n > 2) admit a two-parameter group of contact transformations generated by two infinitesimal transformations [Wk f ] − Wk ∂f ∂z (k = 1, 2). Then the partial differential equations F = 0, W1 = 0, W2 = 0 have as many common solutions as possible. 72. In general, let a partial differential equation of order m with variables z, x1 , . . , xn admit a q-parameter group of contact transformations [Wk f ] − Wk ∂f ∂z (k = 1, 2, .

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