By Liman F.N.
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Additional resources for 2-Groups with normal noncyclic subgroups
N generate the Galilean transformations t = t, xa = xa + va t, U = U exp which are essentially diﬀerent from (12). i va va xa va + t 2 2 (15) Unique symmetry of two nonlinear generalizations of the Schr¨odinger equation 33 So, equation (5) for arbitrary λ1 and b2 = 0, which is a particular case of equation (4), is invariant with respect to the algebra AG(1, n), I , but in the case b2 = 0 (see equation (6)) it has the AG2 (1, n)-symmetry with the additional unit operator I. Statement 4. Equation (5) for λ1 = 1 and b2 = 0 (see equation (7)) is invariant with respect to the Lie algebra with the basic operators (9), (13) and U U Pa + xa Pt , D1 = −i ln ∗ Q + xa Pa , U∗ U 2 U U U Π1 = − ln ∗ Q − 2i ln ∗ xa Pa + |x|2 Pt + in ln ∗ I, U U U |x|2 U U n ixa + it ln ∗ Pa + xa xb Pb − xa I − ln ∗ Q.
7) is invariant under the linear transformations of dependent variables 2 uj → uj = (8) αjk uk + βj , k=1 where αjk , βj , j = 1, 2 are arbitrary constants with det αjk = 0. That is why we carry out symmetry classiﬁcation of Eqs. (7) within the equivalence transformations (8). Theorem 1. Z. I. V. Marko (iii) (iv) (v) (vi) a−2 u2 F1 (ω), b 2 ✷u2 = exp − u2 F2 (ω), ω = au2 − b ln u1 ; b a−2 u1 ✷u1 = (u21 + u22 )−1/2 exp arctan u2 F1 (ω) + u1 F2 (ω) , b u2 a−2 u1 u2 F2 (ω) − u1 F1 (ω) , ✷u2 = (u21 + u22 )−1/2 exp arctan b u2 u1 ω = b ln(u21 + u22 ) − 2a arctan ; u2 2 F1 (ω) + u2 F2 (ω) , ✷u1 = exp − u2 b 2 ✷u2 = b exp − u2 F2 (ω), ω = 2bu1 − u22 ; b ✷u1 = 0, ✷u2 = 0; ✷u1 = exp (9) where F1 , F2 are arbitrary smooth functions, a, b are arbitrary constants.
System of PDEs (7) is invariant under the conformal group C(1, 3) iﬀ it is equivalent to the following system: ✷uj = u31 Fj u1 u1 u1 , , u2 u3 u4 , j = 1, 2, 3, 4. g. [2, 5, 6]). Here we present the scheme of the proof of Theorem 1 only. Within the framework of the Lie method, a symmetry operator for system of PDEs (7) is looked for in the form X = ξµ (x, u)∂µ + ηj (x, u)∂uj , (10) j = 1, . . , 4, where ξµ (x, u), ηj (x, u) are some smooth functions. ,4 = 0, (11) j = 1, . . , 4, where X stands for the second prolongation of the operator X.
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