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Symbolic Software for Lie Symmetry Computations

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Symbolic Software for Lie Symmetry Computations Invited Lecture 1 Symbolic Software for Lie Symmetry Computations Willy Hereman Dept. Mathematical and Computer Sciences Colorado School of Mines Golden, CO 80401-1887 U.S.A. ISLC Workshop Nordfjordeid, Norway Tuesday, June 18, 1996 16:00 I. INTRODUCTION Symbolic Software • Solitons via Hirota’s method (Macsyma & Mathematica) • Painlev´etest for ODEs or PDEs (Macsyma & Mathematica) • Conservation laws of PDEs (Mathematica) • Lie symmetries for ODEs and PDEs (Macsyma) Purpose of the programs • Study of integrability of nonlinear PDEs • Exact solutions as bench mark for numerical algorithms • Classification of nonlinear PDEs • Lie symmetries −→ solutions via reductions • Work in collaboration with Unal¨ G¨okta¸s Chris Elmer Wuning Zhuang Ameina Nuseir Mark Coffey Erik van den Bulck Tony Miller Tracy Otto Symbolic Software by Willy Hereman and Collaborators Software is freely available from anonymous FTP site: mines.edu Change to subdirectory: pub/papers/math cs dept/software Subdirectory Structure: – symmetry (Macsyma) systems of ODEs and PDEs systems of difference-differential equations – hirota (Macsyma) – painleve (Macsyma) single system – condens (Mathematica) – painmath (Mathematica) single system – hiromath (Mathematica) Computation of Lie-point Symmetries – System of m differential equations of order k ∆i(x, u(k)) = 0, i = 1, 2, ..., m with p independent and q dependent variables p x = (x1, x2, ..., xp) ∈ IR u = (u1, u2, ..., uq) ∈ IRq – The group transformations have the form x˜ = Λgroup(x, u), u˜ = Ωgroup(x, u) where the functions Λgroup and Ωgroup are to be deter- mined – Look for the Lie algebra L realized by the vector field p q X i ∂ X ∂ α = η (x, u) + ϕl(x, u) l i=1 ∂xi l=1 ∂u Procedure for finding the coefficients – Construct the kth prolongation pr(k)α of the vector field α – Apply it to the system of equations – Request that the resulting expression vanishes on the solution set of the given system (k) i pr α∆ |∆j=0 i, j = 1, ..., m – This results in a system of linear homogeneous PDEs i for η and ϕl, with independent variables x and u (determining equations) – Procedure thus consists of two major steps: deriving the determining equations solving the determining equations Procedure to Compute Determining Equations p – Use multi-index notation J = (j1, j2, ..., jp) ∈ IN , to denote partial derivatives of ul |J| l l ∂ u uJ ≡ j j j , ∂x1 1∂x2 2...∂xp p where |J| = j1 + j2 + ... + jp – u(k) denotes a vector whose components are all the partial derivatives of order 0 up to k of all the ul – Steps: (1) Construct the kth prolongation of the vector field q (k) X X J (k) ∂ pr α = α + ψl (x, u ) l , 1 ≤ |J| ≤ k l=1 J ∂uJ J The coefficients ψl of the first prolongation are: p Ji X l j ψl = Diϕl(x, u) − uJ Diη (x, u), j=1 j th where Ji is a p−tuple with 1 on the i position and zeros elsewhere Di is the total derivative operator ∂ q ∂ D = + X X ul , 0 ≤ |J| ≤ k i J+Ji l ∂xi l=1 J ∂uJ Higher order prolongations are defined recursively: p J+Ji J X l j ψl = Diψl − uJ+J Diη (x, u), |J| ≥ 1 j=1 j (2) Apply the prolonged operator pr(k)α to each equation ∆i(x, u(k)) = 0 Require that pr(k)α vanishes on the solution set of the system (k) i pr α ∆ |∆j=0 = 0 i, j = 1, ..., m (3) Choose m components of the vector u(k), say v1, ..., vm, such that: (a) Each vi is equal to a derivative of a ul (l = 1, ..., q) with respect to at least one variable xi (i = 1, ..., p). (b) None of the vi is the derivative of another one in the set. (c) The system can be solved algebraically for the vi in terms of the remaining components of u(k), which we denoted by w: vi = Si(x, w), i = 1, ..., m. (d) The derivatives of vi, i i vJ = DJS (x, w), j1 j2 jp where DJ ≡ D1 D2 ...Dp , can all be expressed in terms of the components of w and their derivatives, without ever reintroducing the vi or their derivatives. For instance, for a system of evolution equations i i (k) ut(x1, ..., xp−1, t) = F (x1, ..., xp−1, t, u ), i = 1, ..., m, where u(k) involves derivatives with respect to the vari- i i ables xi but not t, choose v = ut. (4) Eliminate all vi and their derivatives from the ex- pression prolonged vector field, so that all the remaining variables are independent (5) Obtain the determining equations for ηi(x, u) and ϕl(x, u) by equating to zero the coefficients of the re- l maining independent derivatives uJ. Reducing & Solving Determining Equations – Reduce the Determining Equations in Standard Form ∗ Riquier-Janet-Thomas theory ∗ Differential Gr¨obnerbasis – Solve the Determining Equations ∗ Standard integration techniques ∗ Heuristic rules Solving the Determining Equations No algorithms available to solve any system of linear homogeneous PDEs The following heuristic rules can be used: (1) Integrate single term equations of the form |I| ∂ f(x1, x2, ..., xn) i i i = 0 ∂x1 1∂x2 2...∂xn n where |I| = i1 + i2 + ... + in, to obtain the solution n ik−1 X X j f(x1, x2, ..., xn) = hkj(x1, x2, ..., xk−1, xk+1, ..., xn)(xk) k=1 j=0 Thus introducing functions hkj with fewer variables (2) Replace equations of type n X j fj(x1, x2, ..., xk−1, xk+1, ..., xn)(xk) = 0 j=0 by fj = 0 (j = 0, 1, ..., n) Splitting equations (via polynomial decomposition) into a set of smaller equations is also allowed when fj are differen- tial equations themselves, provided the variable xk is missing (3) Integrate linear differential equations of first and second order with constant coefficients Integrate first order equations with variable coefficients via the integrating factor technique, provided the resulting in- tegrals can be computed in closed form (4) Integrate higher-order equations of type n ∂ f(x1, x2, ..., xn) n = g(x1, x2, ..., xk−1, xk+1, ..., xn) ∂xk n successive times to obtain (x )n f(x , x , ..., x ) = k g(x , x , ..., x , x , ..., x )(1) 1 2 n n! 1 2 k−1 k+1 n xn−1 + k h(x , x , ..., x , x , ..., x ) (n − 1)! 1 2 k−1 k+1 n + ... + r(x1, x2, ..., xk−1, xk+1, ..., xn) where h, ..., r are arbitrary functions (5) Solve any simple equation (without derivatives) for a function (or a derivative of a function) provided both (i) it occurs linearly and only once (ii) it depends on all the variables which occur as arguments in the remaining terms (6) Explicitly integrate exact equations (7) Substitute the solutions obtained above in all the equa- tions (8) Add differences, sums or other linear combinations of equations (with similar terms) to the system, provided these combinations are shorter than the original equations Beyond Lie Symmetries – Contact and generalized symmetries i The η and φl depend on a finite number of derivatives of u, i.e. p q X i (k) ∂ X (k) ∂ α = η (x, u ) + ϕl(x, u ) l i=1 ∂xi l=1 ∂u Case k = 0, with u(0) = u: point symmetries Case k = 1: classical contact symmetry – Nonclassical or conditional symmetries Add q invariant surface conditions p l l (1) X i ∂u Q (x, u ) = η (x, u) −ϕl(x, u) = 0, l = 1, ..., q i=1 ∂xi and their differential consequences, to the given system • Review Papers on Lie Symmetry Software: – W. Hereman Symbolic Software for Lie Symmetry Analysis In: CRC Handbook of Lie Group Analysis of Differential Equations Volume 3: New Trends in Theoretical Developments and Computational Methods Chapter 13, Ed.: N.H. Ibragimov, CRC Press, Boca Ra- ton, Florida (1995) pp. 367-413 – W. Hereman Review of symbolic software for the computation of Lie symmetries of differential equations Euromath Bulletin, vol. 2, no. 1 (1894) pp. 45-82 – W. Hereman Review of Symbolic Software for Lie Symmetry Anal- ysis Mathematical and Computer Modeling Special issue on Algorithms for Nonlinear Systems, vol. 21 Eds: W. Oevel and B. Fuchssteiner (1996) in press • Papers on SYMMGRP.MAX: – B. Champagne, W. Hereman and P. Winternitz The computer calculation of Lie point symmetries of large systems of differential equations Computer Physics Communications, vol. 66, pp. 319- 340 (1991) – W. Hereman SYMMGRP.MAX and other symbolic program for sym- metry analysis of partial differential equations in: Exploiting Symmetry in Applied and Numerical Anal- ysis Lect. in Appl. Math. 29, Eds.: E. Allgower, K. Georg and R. Miranda Proceedings of the AMS-SIAM Summer Seminar, Fort Collins July 26-August 1, 1992 American Mathematical Society, Providence, Rhode Is- land pp. 241-257 (1993) Lie-point & Lie-B¨acklund (generalized) Symmetries • LIE by Eliseev, Fedorova & Kornyak (Reduce, 1985) • SPDE by Schwarz (Reduce, Scratchpad, 1986) • LIEDF/INFSYM by Kersten & Gragert (Reduce, 1987) • Lie-B¨acklund symmetries by Fedorova, Kornyak & Fushchich (Reduce, 1987) • Crackstar by Wolf (Formac, 1987) • Lie-point symmetries by Schwarzmeier & Rosenau (Macsyma, 1988) • Hereditary symmetries by Fuchssteiner & Oevel (Reduce, 1988) • Special symmetries by Mikhailov (Pascal, 1988) • Higher Symmetries by Mikhailov et al. (muMATH, 1990) • CRACK by Wolf (Reduce, 1990) • LIE by Head (muMath, 1990) • NUSY by Nucci (Reduce, 1990) • PDELIE by Vafeades (Macsyma, 1990-1992) • DEliA by Bocharov (Pascal, 1990-1993) • SYM DE by Steinberg (Macsyma, 1990) • SYMCAL by Reid & Wittkopf (Maple, Macsyma, 1990) • SYMMGRP.MAX by Champagne, Hereman & Winternitz (Macsyma, 1990) • Liesymm by Carminati, Devitt & Fee (Maple, 1992) • SYMSIZE by Schwarz (Reduce, 1992) • Standard Form Package by Reid and Wittkopf (Maple, 1992) • DIMSYM by Sherring & Prince (Reduce, 1992) • Symgroup.c by B´erub´eand de Montigny (Mathematica, 1992) • Adjoint Symmetries by Sarlet & Vanden Bonne (Reduce, 1992) • LIEPDE by Wolf (Reduce, 1993) • Symmgroup.m by Coult (Mathematica, 1992) • Lie & LieBaecklund by Baumann (Mathematica, 1993) • MathSymm by Herod (Mathematica, 1993) • DIFFGROB2 by Mansfield (Reduce, 1993) • Symmetries & Gr¨obner basis by Gerdt (Reduce, 1993) • JET by Seiler et al.
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