By Tom Ivie, Tom Tullis
Nicknamed the вЂBluenosed Bastards of BodneyвЂ™ as a result of the garish all-blue noses in their P-51s, the 352nd FG used to be some of the most profitable fighter teams within the 8th Air strength. Credited with destroying virtually 800 enemy plane among 1943 and 1945, the 352nd accomplished fourth within the score of all teams inside of VIII Fighter Command. first and foremost built with P-47s, the gang transitioned to P-51s within the spring of 1944, and it used to be with the Mustang that its pilots loved their maximum luck. quite a few first-hand debts, fifty five newly commissioned works of art and one hundred forty+ images entire this concise heritage of the вЂBluenosersвЂ™.
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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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