Page 1 of 19 AUTHOR SUBMITTED MANUSCRIPT - draft 1 2 3 4 5 6 7 8 9 10 Continuous changes of variables and the Magnus 11 12 expansion 13 14 Fernando Casas,∗ Philippe Chartier,y Ander Muruaz 15 16 July 1, 2019 17 18 19 20 Abstract 21 22 In this paper, we are concerned with a formulation of Magnus and Floquet- 23 Magnus expansions for general nonlinear differential equations. To this aim, we 24 introduce suitable continuous variable transformations generated by operators. 25 As an application of the simple formulas so-obtained, we explicitly compute the 26 first terms of the Floquet-Magnus expansion for the Van der Pol oscillator and 27 the nonlinear Schrodinger¨ equation on the torus. 28 29 30 1 Introduction 31 32 The Magnus expansion constitutes nowadays a standard tool for obtaining both an- 33 alytic and numerical approximations to the solutions of non-autonomous linear dif- 34 ferential equations. In its simplest formulation, the Magnus expansion [28] aims to 35 construct the solution of the linear differential equation 36 37 Y 0(t) = A(t)Y (t);Y (0) = I; (1) 38 39 where A(t) is a n × n matrix, as 40 41 Y (t) = exp Ω(t); (2) 42 43 where Ω is an infinite series 1 44 X 45 Ω(t) = Ωk(t); with Ωk(0) = 0; (3) 46 k=1 47 48 whose terms are increasingly complex expressions involving iterated integrals of nested 49 commutators of the matrix A evaluated at different times. 50 Since the 1960s the Magnus expansion (often with different names) has been used 51 in many different fields, ranging from nuclear, atomic and molecular physics to nu- 52 clear magnetic resonance and quantum electrodynamics, mainly in connection with 53 ∗ 54 Universitat Jaume I, IMAC, Departament de Matematiques,` 12071 Castellon,´ Spain. Email: fer- [email protected] 55 y 56 INRIA, Universite´ de Rennes 1, Campus de Beaulieu, 35042 Rennes, France. Email: [email protected] 57 zKonputazio Zientziak eta A.A. Saila, Infomatika Fakultatea, UPV/EHU, 20018 Donostia-San Se- 58 bastian,´ Spain. Email: [email protected] 59 60 1 AUTHOR SUBMITTED MANUSCRIPT - draft Page 2 of 19 1 2 3 4 5 6 perturbation theory. More recently, it has also been the starting point to construct nu- 7 merical integration methods in the realm of geometric numerical integration (see [7] 8 9 for a review), when preserving the main qualitative features of the exact solution, such 10 as its invariant quantities or the geometric structure is at issue [5, 23]. The convergence 11 of the expansion is also an important feature and several general results are available 12 [6, 10, 31, 27]. 13 Given the favourable properties exhibited by the Magnus expansion in the treat- 14 ment of the linear problem (1), it comes as no surprise that several generalizations 15 have been proposed along the years. We can mention, in particular, equation (1) when 16 the (in general complex) matrix-valued function A(t) is periodic with period T . In that 17 18 case, it is possible to combine the Magnus expansion with the Floquet theorem [16] 19 and construct the solution as 20 21 Y (t) = exp(Λ(t)) exp(tF ); (4) 22 23 where Λ(t + T ) = Λ(t) and both Λ(t) and F are series expansions 24 1 1 25 X X 26 Λ(t) = Λk(t);F = Fk; (5) 27 k=1 k=1 28 with Λ (0) = 0 for all k. This is the so-called Floquet–Magnus expansion [11], 29 k 30 and has been widely used in problems of solid state physics and nuclear magnetic 31 resonance [26, 29]. Notice that, due to the periodicity of Λk, the constant term Fn can 32 be independently obtained as Fk = Ωk(T )=T for all k. n 33 In the general case of a nonlinear ordinary differential equation in R , 34 0 n 35 x = g(x; t); x(0) = x0 2 R ; (6) 36 37 the usual procedure to construct the Magnus expansion requires first to transform (6) 38 into a certain linear equation involving operators [1]. This is done by introducing the 39 Lie derivative associated with g and the family of linear transformations Φt such that 40 Φt[f] = f ◦'t, where 't denotes the exact flow defined by (6) and f is any (infinitely) 41 n 42 differentiable map f : R −! R. The operator Φt obeys a linear differential equation 43 which is then formally solved with the corresponding Magnus expansion [7]. Once 44 the series is truncated, it corresponds to the Lie derivative of some function W (x; t). 45 Finally, the solution at some given time t = T can be approximated by determining 46 the 1-flow of the autonomous differential equation 47 0 48 y = W (y; T ); y(0) = x0 49 50 since, by construction, y(1) ' 'T (x0). Clearly, the whole procedure is different and 51 more involved than in the linear case. It is the purpose of this work to provide a uni- 52 fied framework to derive the Magnus expansion in a simpler way without requiring the 53 54 apparatus of chronological calculus. This will be possible by applying the continuous 55 transformation theory developed by Dewar in perturbation theory in classical mechan- 56 ics [17]. In that context, the Magnus series is just the generator of the continuous 57 transformation sending the original system (6) to the trivial one X0 = 0. Moreover, 58 the same idea can be applied to the Floquet–Magnus expansion, thus establishing a 59 60 2 Page 3 of 19 AUTHOR SUBMITTED MANUSCRIPT - draft 1 2 3 4 5 6 natural connection with the stroboscopic averaging formalism. In the process, the re- 7 lation with pre-Lie algebras and other combinatorial objects will appear in a natural 8 9 way. 10 The plan of the paper is as follows. We review several procedures to derive the 11 Magnus expansion for the linear equation (1) in section 2 and introduce a binary oper- 12 ator that will play an important role in the sequel. In section 3 we consider continuous 13 changes of variables and their generators in the context of general ordinary differen- 14 tial equations, whereas in sections 4 and 5 we apply this formalism for constructing 15 the Magnus and Floquet–Magnus expansions, respectively, in the general nonlinear 16 setting. There, we also show how they reproduce the classical expansions for linear 17 18 differential equations. As a result, both expansions can be considered as the output 19 of appropriately continuous changes of variables rendering the original system into 20 a simpler form. Finally, in section 6 we illustrate the techniques developed here by 21 considering two examples: the Van der Pol oscillator and the nonlinear Schrodinger¨ 22 equation with periodic boundary conditions. 23 24 25 2 The Magnus expansion for linear systems 26 27 There are many ways to get the terms of the Magnus series (3). If we introduce a 28 (dummy) parameter " in eq. (1), i.e., we replace A by "A, then the successive terms in 29 30 2 3 Ω(t) = "Ω1(t) + " Ω2(t) + " Ω3(t) + ··· (7) 31 32 can be determined by inserting Ω(t) into eq. (1) and computing the derivative of the 33 matrix exponential, thus arriving at [24] 34 35 1 X 1 1 1 36 "A = d exp (Ω0) ≡ adk (Ω0) = Ω0 + [Ω; Ω0]+ [Ω; [Ω; Ω0]]+··· (8) Ω (k + 1)! Ω 2 3! 37 k=0 38 39 j j−1 where [A; B] denotes the usual Lie bracket (commutator) and adAB = [A; adA B], 40 ad0 B = 0. At this point it is useful to introduce the linear operator 41 A 42 Z t 43 H(t) = (F £ G)(t) := [F (u);G(t)]du (9) 44 0 45 so that, in terms of 46 47 d 2 3 48 R(t) = Ω(t) = "R1(t) + " R2(t) + " R3(t) + ··· ; (10) dt 49 50 equation (8) can be written as 51 52 1 1 1 "A = R + R £ R + R £ R £ R + R £ R £ R £ R + ··· : (11) 53 2 3! 4! 54 55 Here we have used the notation 56 57 F1 £ F2 £ ··· £ Fm = F1 £ (F2 £ ··· £ Fm): 58 59 60 3 AUTHOR SUBMITTED MANUSCRIPT - draft Page 4 of 19 1 2 3 4 5 6 Now the successive terms Rj(t) can be determined by substitution of (10) into (11) 7 and comparing like powers of ". In particular, this gives 8 9 R = A; 10 1 11 1 1 R2 = − R1 £ R1 = − A £ A; 12 2 2 1 1 13 R = − (R £ R + R £ R ) − R £ R £ R 14 3 2 2 1 1 2 6 1 1 1 15 1 1 = (A £ A) £ A + A £ A £ A 16 4 12 17 18 Of course, equation (8) can be inverted, thus resulting in 19 1 20 X Bk Ω0 = adk ("A(t)); Ω(0) = 0 (12) 21 k! Ω 22 k=0 23 where the Bj are the Bernoulli numbers, that is 24 25 x B B B = 1 + B x + 2 x2 + 4 x4 + 6 x6 + ··· 26 ex − 1 1 2! 4! 6! 27 1 1 1 28 = 1 − x + x2 − x4 + ··· 29 2 12 720 30 In terms of R, equation (12) can be written as 31 32 B B "−1 R = A − B R £ A + 2 R £ R £ A + 4 R £ R £ R £ R £ A + ··· : (13) 33 1 2! 4! 34 35 Substituting (10) in eq. (13) and working out the resulting expression, one arrives to 36 the following recursive procedure allowing to determine the successive terms Rj(t): 37 38 m−j (1) (j) X (j−1) 39 Sm = [Ωm−1;A];Sm = [Ωn;Sm−n ]; 2 ≤ j ≤ m − 1 40 n=1 41 m−1 X Bj 42 R (t) = A(t);R (t) = S(j)(t); m ≥ 2: (14) 1 m j! m 43 j=1 44 45 Notice in particular that 46 47 1 S(1) = A £ A; S(1) = − (A £ A) £ A; S(2) = A £ A £ A: 48 2 3 2 3 49 50 At this point it is worth remarking that any of the above procedures can be used to 51 write each Rj in terms of the binary operation £ and the original time-dependent 52 linear operator A, which gives in general one term per binary tree, as in [25, 24], or 53 equivalently, one term per planar rooted tree.