dmitry.lyakhov@kaust.edu.sa Symbolic Computations May 4, 2018 1 / 25 Algorithmic Lie Symmetry Analysis and Group Classication for Ordinary Dierential Equations Dmitry A. Lyakhov 1 1 Computational Sciences Group, Visual Computing Center King Abdullah University of Science and Technology, Jeddah, Saudi Arabia Kolchin Seminar in Dierential Algebra
Introduction Symmetry group analysis is classical and the most fundamental method for solving dierential equations and constructing solutions with particular properties. Modern Lie theory has a lot of applications in natural sciences: automatic (analytical) solvers of ordinary dierential equations in computer algebra systems construction of conservation laws, equivalence mappings for partial dierential systems numerical (Lie) integrators (which preserve symmetries) reveal much better numerical behavior pattern recognition in computer vision (based on group equivalence of boundary curves) dmitry.lyakhov@kaust.edu.sa Symbolic Computations May 4, 2018 2 / 25
dmitry.lyakhov@kaust.edu.sa Symbolic Computations May 4, 2018 3 / 25 Two equations 1. Constant coecient equation 2. Euler equation d n y dx n + a d n 1 y n 1 dx n 1 +... + a dy 1 dx + a 0y = 0. d n y dx n + a n 1x n 1 d n 1 y dx n 1 +... + a 1x dy dx + a 0y = 0.
dmitry.lyakhov@kaust.edu.sa Symbolic Computations May 4, 2018 4 / 25 Two equations 1. Constant coecient equation d n y dx n + a d n 1 y n 1 dx n 1 +... + a dy 1 dx + a 0y = 0 admits shifts for x x = x + a and dilations for y ȳ = C y
dmitry.lyakhov@kaust.edu.sa Symbolic Computations May 4, 2018 5 / 25 Two equations 2. Euler equation admits dilations for x and dilations for y d n y dx n + a n 1x n 1 d n 1 y dx n 1 +... + a 1x dy dx + a 0y = 0 x = A x ȳ = C y
Two equations Transformation of independent variable, which maps dilation into shift x = A x ln( x) = ln(a) + ln(x) denes change of variables t = exp(x), u = y which maps Euler equation into Constant coecient equation. Remark. Two equivalent equations have isomorphic symmetry groups. dmitry.lyakhov@kaust.edu.sa Symbolic Computations May 4, 2018 6 / 25
dmitry.lyakhov@kaust.edu.sa Symbolic Computations May 4, 2018 7 / 25 Symmetry Consider dierential system of general form (x, y (n) ) = 0 (1) where x = (x 1,..., x p ) X and y = (y 1,..., y q ) Y are vectors of independent and dependent variables correspondingly. Denition. The point transformation g : (x, y) ( x, ȳ) of phase space E = X Y is called symmetry if it transforms any solution y(x) of (1) into new function dened by ȳ( x), which is also solution.
dmitry.lyakhov@kaust.edu.sa Symbolic Computations May 4, 2018 8 / 25 Symmetry (example) Trivial second-order ODE y (x) = 0
dmitry.lyakhov@kaust.edu.sa Symbolic Computations May 4, 2018 9 / 25 Symmetry (example) General solution: y = C 1 x + C 2 Symmetry condition means to transform solutions into solutions, which implies to map straight lines straight lines on plane.
dmitry.lyakhov@kaust.edu.sa Symbolic Computations May 4, 2018 10 / 25 Symmetry (example, continue) Trivial second-order ODE y (x) = 0 by substitution [u = f (x, y), t = g(x, y)] is transformed into u (t) + A 3 (u ) 3 + A 2 (u ) 2 + A 1 u + A 0 = 0. Symmetry condition implies A 3 = 0, A 2 = 0, A 1 = 0, A 0 = 0.
dmitry.lyakhov@kaust.edu.sa Symbolic Computations May 4, 2018 11 / 25 Symmetry (example, continue) A 3 = 2 g y 2 f y + 2 f y 2 g y = 0, A 2 = 2 g f y 2 x + 2 f g x y y + 2 2 f g y y x 2 2 g f x y y = 0, A 1 = 2 g f x 2 y + 2 f g x 2 y + 2 2 f g x y x 2 2 g f x y x = 0, A 0 = 2 g x 2 f x + 2 f x 2 g x = 0.
Lie symmetry One of the prominent idea of Sophus Lie is to study symmetry properties under one-parameter group of transformation. Denition. Set of transformation g a : (x, y) ( x, ȳ) is called one-parameter group of transformation of dierential system (x, y (n) ) = 0 if 1) g a is symmetry transformation for a, 2) it is a local group: g a g b = g a+b, where (a group parameter). Typically it leads to overdetermined system of linear partial dierential equations of nite type, which could be eciently analyzed symbolically by means of computer algebra for further reduction and explicit solving. dmitry.lyakhov@kaust.edu.sa Symbolic Computations May 4, 2018 12 / 25
dmitry.lyakhov@kaust.edu.sa Symbolic Computations May 4, 2018 13 / 25 Lie symmetry (examples) 1. Shift 2. Dilation x = x + a, ȳ = y x = e a x, ȳ = y 3. Rotation ] [ x = ȳ 4. Shear 5. Projection x = [ ] [ ] cos(a) sin(a) x sin(a) cos(a) y x = x + ay, ȳ = y x 1 ax, ȳ = y 1 ax
dmitry.lyakhov@kaust.edu.sa Symbolic Computations May 4, 2018 14 / 25 Innitesimal generators The important role plays the rst term in Taylor expansion of one-parameter group of transformation x = x + ε ξ(x, y) + O(ε 2 ), ȳ = y + ε η(x, y) + O(ε 2 ). It is equivalent to And ξ(x, y) = x(x, y, a) a, η(x, y) = a=0 ȳ(x, y, a) a a=0 x (a + b) = Thus assuming b = 0: x(a + b) b = x( x(x, y, a), ȳ(x, y, a), b), ȳ (a + b) =... a x (a) = ξ( x, ȳ), ȳ (a) = η( x, ȳ).
dmitry.lyakhov@kaust.edu.sa Symbolic Computations May 4, 2018 15 / 25 Examples 1. Shift 2. Dilation 3. Rotation 4. Shear 5. Projection [ x = ỹ] x = x = x + a, ỹ = y ξ = 1, η = 0 x = e a x, ỹ = y ξ = x, η = 0 [ [ ] cos(a) sin(a) x ξ = y, η = x sin(a) cos(a)] y x = x + ay, ỹ = y ξ = y, η = 0 x 1 ax, ỹ = y 1 ax ξ = x 2, η = xy
dmitry.lyakhov@kaust.edu.sa Symbolic Computations May 4, 2018 16 / 25 Symmetry Analysis (example, continue) Substitution to symmetry condition implies General solution g(x, y) := x + a ξ(x, y) + o(a), f (x, y) := y + a η(x, y) + o(a). A 3 = 0, A 2 = 0, A 1 = 0, A 0 = 0 2 η x 2 = 0, 2 ξ x 2 + 2 2 η x y = 0, 2 ξ y 2 = 0, 2 η y 2 + 2 2 ξ x y = 0. ξ(x, y) = (C 7 x + C 8 ) y + C 5 x 2 + C 3 x + C 4, η(x, y) = (C 5 y + C 6 ) x + C 7 y 2 + C 1 y + C 2.
dmitry.lyakhov@kaust.edu.sa Symbolic Computations May 4, 2018 17 / 25 Consider ODE (n 2) solved with respect to the highest order derivative where f is rational function. y (n) + f (x, y, y,..., y (n 1) ) = 0, y (k) := d k y dx k (2)
dmitry.lyakhov@kaust.edu.sa Symbolic Computations May 4, 2018 18 / 25 Algorithms What can we do algorithmically? generation of determining equations reduction by integrability conditions dimension of solution space structure constants of Lie algebra Initial value problem are given by nite number of values: { l 1 +m 1ξ x l 1 y m 1 (x 0, y 0 ), l 2+m 2η x l 2 y m 2 (x 0, y 0 ) }, [ ξ η ] = n i,j=0 [ ai,j b i,j ] x i y j Since Lie algebra is closed under Lie bracket, then m X = truncated Taylor series [X i, X j ] = Ci,jX k k, 1 i < j m. Remark. Beautiful point of construction that you actually need only truncated Taylor series (approximate solution) to obtain exact values of structure constant. k=1
Linearizability Theorem. Eq. (2) with n 2 is linearizable by a point transformation if and only if one of the following conditions is fullled: 1 n = 2, m = 8; 2 n 3, m = n + 4; 3 n 3, m {n + 1, n + 2} and derived algebra is abelian and has dimension n. Proof. According to group classication of linear equations: 1. Trivial equations y (n) (x) = 0 have maximal possible dimension n + 4 2. Constant coecients equations have Lie algebra span by operators { f 1 (x) y, f 2(x) y,..., f n(x) y, x, y } y 3. Generic case corresponds to Lie algebra { f 1 (x) y, f 2(x) y,..., f n(x) y, y } y dmitry.lyakhov@kaust.edu.sa Symbolic Computations May 4, 2018 19 / 25
dmitry.lyakhov@kaust.edu.sa Symbolic Computations May 4, 2018 20 / 25 Group classication The problem of group classication of dierential equations was rst posed by Sophus Lie He also began to solve the problem of group classication of the second-order ordinary equation of general form y + f (x, y, y ) = 0
dmitry.lyakhov@kaust.edu.sa Symbolic Computations May 4, 2018 21 / 25 Group classication Lev Ovsyannikov considered simpler case of y + f (x, y) = 0 and solved the problem of group classication by admissible operators f X 1 X 2 X 3 f (y) x 0 0 e y x x x 2 y 0 y k, k 3 x (k 1)x x 2y y 0 ±y 3 x 2x x + y y x 2 x + xy y x 2 g(y) x x 0 0
dmitry.lyakhov@kaust.edu.sa Symbolic Computations May 4, 2018 22 / 25 Group classication 2 η x 2 f x ξ f y η + f ( η y 2 ξ x ) = 0, 2 ξ x 2 + 2 2 η ξ 3f x y y = 0, 2 ξ y 2 = 0, 2 η y 2 + 2 2 ξ x y = 0.
dmitry.lyakhov@kaust.edu.sa Symbolic Computations May 4, 2018 23 / 25 Group classication (invariant form) The form of second-order ODE y + f (x, y, y ) = 0 given by f = F 3 (x, y)(y ) 3 + F 2 (x, y)(y ) 2 + F 1 (x, y) y + F 0 (x, y). (3) is invariant under point transformation.
Group classication (linearizable branch) Sophus Lie showed that only equations of the following form are linearizable by point transformations if and only if Theorem. 3(F 3 ) xx 2(F 2 ) xy + (F 1 ) yy 3F 1 (F 3 ) x + 2F 2 (F 2 ) x 3F 3 (F 1 ) x + 3F 0 (F 3 ) y + 6F 3 (F 0 ) y F 2 (F 1 ) y = 0, (4) (F 2 ) xx 2(F 1 ) xy + 3(F 0 ) yy 6F 0 (F 3 ) x + F 1 (F 2 ) x 3F 3 (F 0 ) x + 3F 0 (F 2 ) y + 3F 2 (F 0 ) y 2F 1 (F 1 ) y = 0. dmitry.lyakhov@kaust.edu.sa Symbolic Computations May 4, 2018 24 / 25
Thank you! dmitry.lyakhov@kaust.edu.sa Symbolic Computations May 4, 2018 25 / 25