Superintegrability? Hidden linearity? Classical quantization? Symmetries and more symmetries!

Superintegrability? Hidden linearity? Classical quantization? Symmetries and more symmetries! Maria Clara Nucci University of Perugia & INFN-Perugia, Italy Conference on Nonlinear Mathematical Physics: Twenty Years of JNMP, June 4-4, 203 Nordfjordeid, Norway

The effectiveness of Lie symmetries Eugene P. Wigner, The unreasonable effectiveness of Mathematics in the Natural Sciences,Comm. Pure Appl. Math 3 960-4 Paraphrasing Eugene Paul Wigner: Lie symmetries turn up in entirely unexpected connection e.g., hidden linearity, nonlocal symmetries, calculus of variations Lie symmetries permit unexpectedly accurate description of the phenomena in this connection e.g., conservation laws, blow-up solutions A theory formulated in terms of Lie symmetries maybe uniquely appropriate. e.g., quantization

[Jacobi, 842-45] i= Jacobi last multiplier Af = n i= a i f x i = 0 dx = dx 2 =... = dx n. a a 2 a n f, ω, ω 2,..., ω n = MAf x, x 2,..., x n ω i, i =,..., n solutions of ie first integrals of n Ma i = 0 d logm n a i = x i x i IMPORTANT PROPERTY: M M 2 = First Integral NB: ratio may be quite trivial. i=

Enter Lie [S. Lie, 874] dx a = dx 2 a 2 =... = dx n a n. If there exist n symmetries of, say Γ i = ξ ij xj, i =, n then JLM is given by M =, provided that 0, where a a n ξ, ξ,n = det.. ξ n, ξ n,n Corollary: if M=const, then is a first integral.

Volterra s last paper Calculus of Variations and the Logistic Curve, Human Biology, 939 I have been able to show that the equations of the struggle for existence depend on a question of Calculus of Variations In order to obtain this result, I have replaced the notion of population by that of quantity of life. In this manner I have also obtained some results by which dynamics is brought into relation to problems of the struggle for existence. The quantity of life X and the population N of a species are connected by the relation N = dx. Thus Volterra takes a system of first-order equations and transform it into a system of second-order equations.

Volterra-Verhulst-Pearl equation One of the equations Volterra considered is that through becomes d 2 X 2 dn = Nε λn 2 = dx ε λ dx. 3

Volterra-Verhulst-Pearl equation One of the equations Volterra considered is that through becomes d 2 X 2 dn = Nε λn 2 = dx ε λ dx. 3 [Nucci & Tamizhmani, J. Nonlinear Math. Phys. 202]

Volterra-Verhulst-Pearl equation One of the equations Volterra considered is that through becomes d 2 X 2 dn = Nε λn 2 = dx ε λ dx. 3 [Nucci & Tamizhmani, J. Nonlinear Math. Phys. 202] Equation 3 admits an eight-dimensional Lie symmetry algebra generated by the following operators: Γ = expλx εt t, Γ 2 = expλx t + ε λ X, Γ 3 = exp λx + εt X, Γ 4 = exp λx X, Γ 5 = expεt λ ε t + X, Γ6 = X, Γ 7 = exp εt t, Γ 8 = t. Therefore the equation is linearizable y = exp εt, u = λ expλx εt d2 u dy 2 = 0

JLM = 2 L [Jacobi, 842] Ẋ 2 Ten Lagrangians

Ten Lagrangians JLM = 2 L Ẋ 2 [Jacobi, 842] JLM/n = 2 L x n 2 [Jacobi,845]

Ten Lagrangians JLM = 2 L Ẋ [Jacobi, 842] 2 JLM/n = 2 L x n 2 [Jacobi,845] + X, Lag 4 = expεt λ log dx Lag 5 = exp λx ε Lag 7 = 2λ dx dx log dx exp2εt λx, + ε log λ dx ε dx + λ, Lag 8 = expεt λx λ dx ε 2 ε log dx ε log λ dx ε, Lag 23 = λ exp εx log ε λ dx + λx, Lag 25 = ε exp εt λx 2λεt λ dx, Lag 34 = 2ε exp εt + 2λX dx Lag 36 = exp εt + λx λ dx λ 2 ε log ε λ dx Lag 37 = ε expλx dx log dx dx + λ ε, Lag 68 = dx ε log dx + ελ ε λ dx log ε λ dx + X 2, λ dx,

Conservation laws For example the Lagrangian Lag 34 yields the following five Noether symmetries and corresponding first integrals of equation 3 Γ 3 = Int 3 = expλx ε + λ dx, Γ 4 = Int 4 = exp εt + λx dx, Γ 5 = Int 5 = exp2λx ε λ dx 2, Γ 6 + 2 λ ε Γ 8 = Int 6 = exp εt + 2λX dx dx Γ 7 = Int 7 = exp 2εt + 2λX ε λ dx 2.,

Superintegrability and symmetries [Post & Winternitz, J. Phys. A 20] This Hamiltonian which does not admit separation of variables H = 2 p2 + p 2 2 + αu 2 u 2/3 yields a superintegrable maximally integrable system u = p, u 2 = p 2, ṗ = 2αu 2, ṗ 2 = α 3u 5/3 since there exist two independent integrals of motion, i.e.: I = 3p 2 p 2 + 2p 3 2 + 9αu /3 p + 6αu 2 p 2 /u 2/3, u 2/3 I 2 = p 4 + 4αu 2 p 2 /u 2/3 2u /3 αp p 2 2α 2 9u 2 2u 2 2/u 4/3.

The corresponding Lagrangian equations are ü = 2αu 2, ü 2 = α 3u 5/3 u 2/3 They admits a 2-dim Lie symmetry algebra A 2 generated by Γ = t, Γ 2 = 5 6 t t + u u + u 2 u2..

The corresponding Lagrangian equations are ü = 2αu 2, ü 2 = α 3u 5/3 u 2/3 They admits a 2-dim Lie symmetry algebra A 2 generated by Γ = t, Γ 2 = 5 6 t t + u u + u 2 u2. [Nucci & Post, J.Phys.A 202].

The corresponding Lagrangian equations are ü = 2αu 2, ü 2 = α 3u 5/3 u 2/3 They admits a 2-dim Lie symmetry algebra A 2 generated by Γ = t, Γ 2 = 5 6 t t + u u + u 2 u2. [Nucci & Post, J.Phys.A 202] Since system u = p, u 2 = p 2, ṗ = 2αu 2, ṗ 2 = α 3u 5/3. u 2/3 is autonomous we can choose one of the dependent variables as new independent variable, namely u = y. Then u 2 = p 2 p, p = 2αu 2 3y 5/3 p, p 2 = α y 2/3 p,

Eliminating one of the dependent variables, e.g. from the second equation u 2 = 3y 5/3 p p /2α and then the system becomes: p = 2y /3 αp 2 5yp 2p 3y 2 p p 2 3y 2 p 2, p 2 = α y 2/3. 4 p It admits a 0-dim Lie symmetry algebra isomorphic to the de Sitter algebra o3, 2 generated by: X = 972α 2 y 2/3 3yp 8 2 324α2 y 5/3 p 4 2 + 8748α4 y 7/3 972α 2 y 5/3 p 2 p2 2 y X 2 = X 3 = X 4 = X 5 = +944α 3 yp 2 2 p 08α 2 y 2/3 p 5 2 + 5832α4 y 4/3 p 2 p2 + 296α 4 y 4/3 p p p 8 2 +296α 3 yp 3 2 24αy /3 p 7 2 + 432α2 y 2/3 p p 4 2 + 324α2 y 2/3 p 3 p2 2 + 944α3 p 2 p 2 y p 486α 2 y 2/3 62α 2 y 5/3 p 3 2 3yp7 2 + 486α2 y 5/3 p 2 p 2 y +8α 2 y 2/3 p 4 2 458α4 y 4/3 972α 3 yp p 2 p2 + 2αy /3 p 6 2 486α3 yp 2 2 486α 3 yp 2 270α2 y 2/3 p p 3 2 62α2 y 2/3 p 2 p 3 + p p 7 2 p, 243α 2 y 2/3 243α 2 y 5/3 p 2 3yp6 2 y + 54α2 y 2/3 p 3 2 486α3 yp p2 +p p 6 2 8α2 y 2/3 p 3 + 8αy /3 p 5 2 62α2 y 2/3 p p 2 2 p, 972α 2 y 2/3 3yp 4 2 + 62y 5/3 α 2 y + 54α 2 y 2/3 p 2y /3 αp 3 2 p p 4 2 p, 27α 2 y 2/3 3yp 3 2 y + 27α2 y 2/3 p2 + 9p 2 2 y /3 α + p p 3 2 p,,

X 6 = X 7 = X 8 = X 9 = X 0 = 36α 2 y 2/3 54y 5/3 α 2 3yp2 4 y + 36y 2/3 α 2 p 2 p2 +p2p 4 8y 2/3 α 2 p + 2y /3 αp2 3 p, 54α 2 y 2/3 62y 5/3 α 2 p 2 3yp2 5 y + 54p2y 2 2/3 α 2 p2 +p p2 5 + 5y /3 αp2 4 62α 3 y 54y 2/3 α 2 p p 2 p, 3y 2/3 3y y p p, 3y 2/3 3yp 2 y 3y /3 α + p p 2 p, p 2 3y 2/3 3yp 2 y 6y /3 α + p p 2 p, which implies that system 4 is

Here is system 2 again: = 2y /3 αp 2 5yp 2p 3y 2 p p 2 3y 2 p 2, p 2 = α y 2/3. p p From the second equation p = α/y 2/3 p 2 yields p 2 = 9α 2 p 2 y 2 6y /3 p 2 p 2 5 y 2 + 4α 2 p 2 2 + 9α 2 p 2p 2y + 27α 2 p 2 2 y 2. This equation admits a 7-dim Lie symmetry algebra generated by Y = y/3 p 2 3 9α 2 y + p2, Y 2 = y/3 p 2 5 +54α2 yp 2 54α 2 y + p2 2 3 p2, Y 3 = y/3 p 2 4 +54α2 y 54α 2 y, Y 4 = y/3 p 4 2 9α 2 y + p 2 p2, Y 5 = y /3 y, Y 6 = y /3 p 2 y, Y 7 = y /3 p 2 2 y. Thus it is linearizable 0-dim Lie algebra are its contact symm. ỹ = p 2, p 2 = 3 2 y 2/3 + 36α 2 p 4 2

Since y = u and u 2 = 3y 5/3 p p /2α then the 0-dim Lie symmetry algebra is obtained: Γ = V t + Γ 2 = V 2 t + 972α 3 u 4/3 324u 7/3 α 3 p 4 2 3u5/3 αp 8 2 8748u3 α5 + 972u 7/3 α 3 p 2 2 p2 u +5832u 8/3 p 2 2 α4 252α 2 u 2 p6 2 5832α5 u 2 u 2 u 4/3 p 8 2 p2 24αu5/3 p 7 2 p 2αu 2/3 p 8 2 u 2 3888u 5/3 p 2 α 4 u 2 p 296u 2/3 p 2 2 α4 u 2 2 648u4/3 p 2 2 α3 u 2 p 2 648u 4/3 p 4 2 α3 u 2 + 944u 7/3 p 3 2 α3 p u2 +24u α 2 p 7 2 296u2 α5 p + u 2/3 αp 8 2 p 296u 5/3 α 4 p 3 2 432u 4/3 α 3 p 4 2 p 324u 4/3 α 3 p 2 2 p3 944u5/3 α 4 p 2 p 2 p +08p 5 2 u4/3 α 3 5832p 2 u 2 α5 944p 2 2 u5/3 α 4 p p2, 486α 3 u 4/3 3u 5/3 p 7 2 α + 62u7/3 p 3 2 α3 + 486u 7/3 p 2 α 3 p 2 u +972u 7/3 p 2 2 α3 p 2αu 5/3 p 6 2 p 2αu 2/3 p 7 2 u 2 + 458u 8/3 α 4 p 2 972u 5/3 α 4 u 2 p 648u 2/3 α 4 u 2 2 p 2 89p 5 2 α2 u 2 u4/3 p 7 2 p2 324u 4/3 p 2 α 3 u 2 p 2 432u4/3 p 3 2 α3 u 2 u2 +u 2/3 αp 7 2 p + 2u α 2 p 6 2 486u5/3 α 4 p 2 2 486u5/3 α 4 p 2 270u4/3 α 3 p 3 2 p 62u 4/3 α 3 p 2 p 3 p + 8u4/3 α 3 p 4 2 458u2 α5 972u 5/3 α 4 p 2 p p2,

Γ 3 = V 3 t + Γ 4 = V 4 t + 243α 3 u 4/3 243u 7/3 α 3 p 2 3u5/3 αp 6 2 u 8αu 5/3 p 5 2 p + 2αu 2/3 p 6 2 u 2 + 324u 2/3 u 2 2 α4 + 35p 4 2 α2 u 2 + u4/3 p 6 2 p2 +62u 4/3 α 3 u 2 p 2 + 324u4/3 α 3 u 2 p 2 2 486u7/3 α 3 p p 2 u2 +u 2/3 αp 6 2 p 8u 4/3 α 3 p 3 + 8u α 2 p 5 2 62u4/3 α 3 p 2 2 p p +54u 4/3 α 3 p 3 2 486u5/3 α 4 p p2, 972α 3 u 4/3 3u 5/3 αp 4 2 + 62u7/3 α 3 u +2αu 5/3 p 3 2 p + 2αu 2/3 p 4 2 u 2 + 54p 2 2 α2 u 2 + u4/3 +54u 4/3 α 3 p u 2/3 αp 4 2 p 2u α 2 p 3 2 p Γ 5 = V 5 t u/3 p2 3 9α 2 u Γ 6 = V 6 t + Γ 7 = V 7 t + 27α 3 u 4/3 p 4 2 p2 + 26u4/3 α 3 u 2 u2, 9p 2 2 αu5/3 p + 2p 3 2 αu2/3 u 2 +27p 2 α 2 u 2 + p3 2 u4/3 p 2 u 2 p3 2 u2/3 αp + 9p 2 2 u α 2 p + p2, 36α 3 u 4/3 54u 7/3 α 3 3u 5/3 αp 4 2 u 2αu 5/3 p 3 2 p + 2αu 2/3 p 4 2 u 2 + 54p 2 2 α2 u 2 + u4/3 +u 2/3 αp 4 2 p 8u 4/3 α 3 p + 2u α 2 p 3 2 p 54α 3 u 4/3 62u 7/3 p 2 α 3 3u 5/3 p 5 2 α u p 4 2 p2 p 2 + p 2 p2, 5p 4 2 αu5/3 p + 2p 5 2 αu2/3 u 2 + 90p 3 2 α2 u 2 + u4/3 p 5 2 p2 u 2 +u 2/3 αp 5 2 p + 5u α 2 p 4 2 62u5/3 α 4 54u 4/3 α 3 p 2 p p + p 2 2 p 2,

Γ 8 = V 8 t + Γ 9 = V 9 t + 3αu 2/3 3αu 2/3 αp 2 p + 3u /3 α 2 p, Γ 0 = V 0 t + u /3 p 2 2 u + 3αu 4/3 3αu u + 2u 2 α + u 2/3 p 2 u 2 αp p, 3αu p 2 u + 3αp u + 2αu 2 p 2 + u 2/3 p 2 p 2 u 2 6αp p 2 u 5/3 + 2αp 2 2 u2/3 u 2 +9α 2 u 2 + p2 p2 2 u4/3 u2 αp 2 6αu + p p 2 u 2/3 p. Each V k = V k t, u, u 2, p, p 2, k =,..., 0, satisfies: 486u 5/3 α 2 V y + 486u 5/3 α 2 p V u + 486u 5/3 α 2 p 2 V u2 + 324α 3 u 2 V p 486α 3 u V p2 +c u p 8 2 648u α 3 u 2 p 2 2 + 5832u7/3 α 4 648u 5/3 α 2 p 2 2 p2 378u5/3 α 2 p 4 2 c 2 648u α 3 u 2 p 2 + 2u p 7 2 + 648u5/3 α 2 p 2 p 2 + 432u5/3 α 2 p 3 2 +c 3 4u p 6 2 648u α 3 u 2 648u 5/3 α 2 p 2 324u5/3 α 2 p 2 2 c 4 u p 4 2 + 54u5/3 α 2 + 36u p 3 2 c 5 + c 6 27u p 4 2 972u5/3 α 2 +c 7 8u p 5 2 944u5/3 α 2 p 2 324u α 2 c 8 324α 2 p 2 u c 9 324u α 2 p 2 2 c 0 = 0 with c k = and all the other c j k = 0, j =,..., 0.

We have 330 matrices M n,m,k,j = det p p 2 2αu 2 3u 5/3 α u 2/3 V n G,n G 2,n G 3,n G 4,n V m G,m G 2,m G 3,m G 4,m V k G,k G 2,k G 3,k G 4,k V j G,j G 2,j G 3,j G 4,j, where n, m, k, j =,..., 0. Since we are interested in autonomous first integrals we only need to consider the determinant of any the 20 matrices M n,m,k = det 2αu 2 p p 2 3u 5/3 α u 2/3 G,n G 2,n G 3,n G 4,n G,m G 2,m G 3,m G 4,m G,k G 2,k G 3,k G 4,k, where n, m, k =,..., 0 are all the possible combinations without repetition of the three indices and the generators of each Lie symmetry.

We present some of those first integrals, e.g.: M 5,8,0 = 2α 2 p2 + p2 2 + αu 2 = 2αH u 2/3 that is the Hamiltonian H;

We present some of those first integrals, e.g.: M 5,8,0 = 2α 2 p2 + p2 2 + αu 2 = 2αH u 2/3 that is the Hamiltonian H; M 5,6,8 = 2 3α 2 p2 + p 2 2 + αu 2 u 2/3 2 = 2 3α H2 that is the square of the Hamiltonian H and a polynomial of order 4 in p and p 2. Also the two known independent first integrals can be obtained: M 7,8,9 = α 3 3p 2p 2 + 2p2 3 + 9αu/3 p + 6α u2p2 = α u 2/3 3 I, M 7,8,0 = α 2 2u /3 αp p 2 + 8α 2 u 2/3 + p2 4 + 2p2 3 p2 2 + 4α u2p2 2 = α 2 I 2 4H 2 u 2/3

Since Γ contains powers of p up to order 3 and powers of p 2 up to order 8, Γ 2 up to order 3 and 7, respectively, and Γ 3 up to order 3 and 6, respectively, then the following first integral

Since Γ contains powers of p up to order 3 and powers of p 2 up to order 8, Γ 2 up to order 3 and 7, respectively, and Γ 3 up to order 3 and 6, respectively, then the following first integral M,2,3 = 4776α 3 u 8/3 78732u 4 α4 p 0 2 + 3402224u8/3 α 9 u 3 2 + u8/3 p 8 2 +425528u 4/3 α 6 p 6 2 + 27p6 u8/3 p 2 2 + 27u8/3 p 4 2 p4 + 486u0/3 p 4 2 α2 + 9u 8/3 p 6 2 p2 +08p αu 3 p5 2 + 648p3 αu3 p3 2 + 866052p2 α4 u 4 p8 2 + 972p5 αu3 p 2 + 8u2 p6 2 αu 2 +2960u 2 p0 2 α4 u 2 2 + 889568u2 α7 p 4 2 u3 2 + 2550968u4 α7 p 4 2 u 2 + 26u 2/3 p 2 2 α3 u 3 2 +6804u 8/3 p 2 2 α3 u 2 + 34992u /3 p 2 α3 p + 2204496u 8/3 p 6 2 α6 u 2 2 +6804u 0/3 p 2 α2 p 2 2 + 5632u0/3 p 2 α2 p 2 2 + 5632u/3 p 3 α3 p 9 2 +6038u 0/3 p 4 α2 p 0 2 + 2834352p α 5 u 3/3 p 7 2 + 08u4/3 p 4 2 α2 u 2 2 +34992u 4/3 p 8 2 u3 2 α5 + 787320u 0/3 α 5 p 8 2 u 2 + 508336u 0/3 α 8 u 2 2 p2 2 +279936p α 4 u 3 u 2p 9 2 + 337408p α 7 u 3 u2 2 p3 2 + 209952p3 α4 u 3 u 2p 7 2 +08p 2 αu2 p4 2 u 2 + 62p 4 αu2 p2 2 u 2 + 34992p 2 α4 u 2 p8 2 u2 2 +324p 2 α2 u 4/3 p 2 2 u2 2 + 49904p α 5 u 7/3 u 2 2 p7 2 + 3888p α 3 u 5/3 p 2 u2 2 +3888p α 3 u 5/3 p 2 α5 u 7/3 u 2 2 p7 2 + 337408p α 6 u /3 u 2 p 5 2 +3888p 3 α2 u 7/3 p 2 u 2 + 46656p 2 α3 u 8/3 p 0 2 u 2 + 2204496p 2 α5 u 0/3 p 6 2 u 2 +296p α 2 u 7/3 p 3 2 u 2 + 944784u 8/3 α 6 p 4 2 p2 u2 2 + 8748u8/3 α 3 p 8 2 p4 u 2 is polynomial of order 6 and 8, respectively, in the components p and p 2 of the momenta. Many other polynomials first integrals of lesser order in p, p 2 can be derived, e.g. M,2,4 is a polynomial of degree 6 and 6 in p, p 2, respectively; M,2,5 is a polynomial of degree 6 and 5; M,2,6 and M,3,7 are also polynomials of degree 6 and 6; M,3,5, M,4,7 and M 2,3,6 are polynomials of degree 6 and 4; and so on.

λ-symmetry and JLM [Nucci & Levi,Nonlinear Anal. 202]

λ-symmetry and JLM [Nucci & Levi,Nonlinear Anal. 202] Painlevé-Ince XV equation: y = y 2 y + y y + rty 2 y d r t 5 rt is known to possess a first integral, i.e. r t 2 y 2 rt y + y + 2 rty + rt = a. 6 The divergence of Painlevé-Ince XV equation 5 yields one λ-symmetry λ J = y + D t log y 2 = X λ = y 2 t r ty + rt rty 2 y. The first prolongation of X λ yields invariants ỹ = r t rt + y +, t = rty + rt y that substituted into 5 generate dỹ d t = ỹ = ỹ 2 2 t = a = 6.

Quantizing with symmetries Problem of a charged particle in a uniform magnetic field in the plane: its classical Lagrangian is L = 2 ẋ 2 + ẏ 2 + ωyẋ xẏ 7 and consequently the Lagrangian equations are ẍ = ωẏ, ÿ = ωẋ. These equations admit a 5-dim Lie symmetry algebra, and the Lagrangian 7 admits 8 Noether symmetries generated by X = cosωt t ω 2 sinωtx + cosωty x + ω cosωtx sinωty y, 2 X 2 = sinωt t 2 cosωtωx sinωtωy x sinωtωx + cosωtωy y, 2 X 3 = t, X 4 = y x + x y, X 5 = sinωt x + cosωt y, X 6 = cosωt x sinωt y, X 7 = y, X 8 = x.

The Schrödinger equation was determined by Darwin 927 to be 2iψ t + ψ xx + ψ yy iωyψ x xψ y ω2 4 x 2 + y 2 ψ = 0. 8 It admits an infinite Lie symmetry algebra generated by the operator αt, x, y ψ, where α is any solution of the equation itself, and also a finite dimensional Lie symmetry algebra generated by the following operators: Y = X + 4 Y 2 = X 2 + 4 Y 3 = X 3, Y 4 = X 4, 2 sinωtω i cosωtω 2 x 2 + y 2 ψ, 2 cosωtω + i sinωtω 2 x 2 + y 2 ψ, Y 5 = X 5 2 ω x cosωt + y sinωt ψ, Y 6 = X 6 + 2 ω x sinωt y cosωt ψ, Y 7 = X 7 + i 2 ωx ψ, Y 8 = X 8 i 2 ωy ψ, Y 9 = ψ ψ. 9

Evolution equation: u t = H Lie symmetry operator: Classical symmetries t, x, u, u x, u xx,..., uxx }{{} n Γ = V t, x, u t + V 2 t, x, u x + Gt, x, u u Determining equation: Invariant surface: Determining equation: Γ u t H = 0 n {u t H=0} V u t + V 2 u x G = 0 Nonclassical symmetries Bluman & Cole, 969 Γ u t H n { u t H = 0 V u t + V 2 u x G = 0 } = 0

Iterating NCSM V t, x, uu t + V 2 t, x, uu x = Gt, x, u V = 0, V 2 = u x = Gt, x, u G equation ξ t, x, u, GG t + ξ 2 t, x, u, GG x + ξ 3 t, x, u, GG u = ηt, x, u, G ξ = 0, ξ 2 =, ξ 3 = G G x + GG u = u xx = ηt, x, u, G η equation η x + Gη u + ηη G = u xxx = Ωt, x, u, G, η Ω equation Ω x + GΩ u + ΩΩ G + ΩΩ η = u xxxx = ρt, x, u, G, η, Ω ρ equation. Heir equations

[MCN, Physica D 994], [MCN, J.Phys.A 996], Heir-equations Definition: Hierarchy of equations which admit the same Lie symmetry algebra heirs as the original one. Each equation has one additional independent variable than the previous equation in the hierarchy, and thus WE CAN GET MORE SOLUTIONS FROM THE SAME SYMMETRY. CLASSICAL vs. NONCLASSICAL BOTH SYMMETRIES ARE PARTICULAR SOLUTIONS OF THE SAME HEIR-EQUATION [MCN, J.Math.Anal.Appl. 2003]

Second Order: Invariant surface condition: Outline of the method u t = u xx + H t, x, u, u x V t, x, uu t + V 2 t, x, uu x = F t, x, u V = u t + V 2 t, x, uu x = F t, x, u u xx + H t, x, u, u x + V 2 t, x, uu x = F t, x, u i.e. η = F t, x, u V 2 t, x, ug H t, x, u, G Generate the η-equation and search for the particular solution. Example: u t = u xx + uu x yields η = F t, x, u V 2 t, x, ug ug.

A class of reaction-diffusion equations [Hashemi & Nucci, J.Nonlinear Math.Phys 203] u t = u xx + cu x 2 x 2 u 3 + 3u 2 + 2 c2 u ut, x = c 2 c e c2 t+ cx 2 c 2 + cxe cx 2 c 2 e c2 t 4 + c cx 2e c2 t+ cx 2 + 0 + 5cx + c 2 x 2 e cx 2

A class of reaction-diffusion equations [Hashemi & Nucci, J.Nonlinear Math.Phys 203] u t = u xx + cu x 2 x 2 u 3 + 3u 2 + 2 c2 u c 2 c e c2 t+ cx 2 c 2 + cxe cx 2 ut, x = c 2 e c2 t 4 + c cx 2e c2 t+ cx 2 + 0 + 5cx + c 2 x 2 e cx 2 c = 0., c = 2/0 5, c 2 = 0]

ut, x = u t = u xx + cu x 2 ecx u 3 + c2 4 u + e cx 2 3 2 2 R 2 t sin 3 2 3x/4 + cos 3 2 3x/4 e cx/2 [ 3 +R 2 t sin 2 3x/4 3 3 cos 2 3x/4 3 3 ] 3 sin 2 3x/4 cos 2 3x/4 R 2 t = tan 3 3/2 2 7/3 t

ut, x = u t = u xx + cu x 2 ecx u 3 + c2 4 u + e cx 2 3 2 2 R 2 t sin 3 2 3x/4 + cos 3 2 3x/4 e cx/2 [ 3 +R 2 t sin 2 3x/4 3 3 cos 2 3x/4 3 3 ] 3 sin 2 3x/4 cos 2 3x/4 R 2 t = tan 3 3/2 2 7/3 t [c = 2]

Conclusions Paraphrasing Eugene Paul Wigner 960: Why DOES the success of Lie symmetries appear so baffling? The enormous usefulness of Lie symmetries in the natural sciences is NOT something bordering on the mysterious and THERE IS an obvious rational explanation for it. The appropriateness of the language of Lie symmetries for the formulation of the laws of physics is a wonderful gift, which SOME PHYSICISTS STILL neither understand nor deserve.