New closed form analytic patterns of the quintic Ginzburg-Landau equation in one spatial dimension

Transcription

1 New closed form analytic patterns of the quintic Ginzburg-Landau equation in one spatial dimension R. Conte, Ng Tuen-wai LRC MESO, ENS Cachan et CEA/DAM, Cachan, France Dept of Math, The university of Hong Kong Frontiers in nonlinear waves, Tucson, March 2010

2 Outline Motivation, physical problem, mathematical problem Nonintegrable counting. The general analytic solution Elliptic functions = the simplest nonlinear special functions Existing methods (sufficient) Subequation method (necessary) Examples and results Conclusion & perspectives

3 Problem (math): to find the general analytic solution N:= differential order of the ODE. Integrable ODE (Painlevé property). Pb is: to find the general solution (N arb constants). Discard this case. Partially integrable ODE (fails the P test). The general solution, if it exists, has movable multivaluedness. Definition: general analytic solution : = the largest, M-parameter particular sol without movable critical singularities, M N 1. PROBLEM: to find this general analytic solution in closed form. Equivalent to find the largest subequation with the PP. SOLUTION: IF this solution is elliptic or degenerate of elliptic, method (RC, M. Musette, Stud. Appl. Math. 123 (2009) 63 81) which finds all those solutions in closed form.

4 Integrable case. Example Korteweg-de Vries Skill: Total derivative first integral g 2 : U 6 a UU = 0. (1) U 3 a U2 + ag 2 = 0. (2) Skill: Integrating factor U first integral g 3 : 1 2 U 2 1 a U3 + ag 2 U + 2a 2 g 3 = 0. (3) General solution is elliptic (doubly periodic) U(ξ) = 2a (ξ ξ 0, g 2, g 3 ), arbitrary = ξ 0, g 2, g 3. (4) The above integration relies on skill, but skill is not a method.

5 Partially integrable case. Examples By definition, the equations fail the Painlevé test. Complex Ginzburg-Landau equation (CGL3) ia t + pa xx + q A 2 A iγa = 0, pqγ 0, (A, p, q) C, γ R (a generic amplitude equation, optical fibers, etc). Kuramoto-Sivashinsky equation (KS) u t + νu xxxx + bu xxx + µu xx + uu x = 0, ν 0, (flame on a vertical wall, phase equation of CGL3). Particular travelling waves are known. They are all either elliptic (KS, b 2 = 16µν) or polynomial in tanh(k/2)(ξ ξ 0 ). Q.: Are there other travelling waves? A.: A priori yes. Problem: to find them all.

6 Outline Motivation, physical problem, mathematical problem Nonintegrable counting. The general analytic solution Elliptic functions = the simplest nonlinear special functions Existing methods (sufficient) Subequation method (necessary) Examples and results Conclusion & perspectives

7 General analytic solution, local representation. KS νu + bu + µu + u 2 /2 + A = 0, Fuchs indices = 1, 13 ± i Particular one-parameter sol. = Laurent series (local) χ = ξ ξ 0 u (0) = 120νχ 3 15bχ (16µν b2 ) χ (4µν b2 )b 4 19ν 32 19ν , General three-parameter solution (illusory, irrational Fuchs ind.): u(ξ ξ 0, εc +, εc ) = u (0) +ε[c 1 χ 1 Regular(χ) + c + χ (13+i 71)/2 Regular(χ) + c χ (13 i 71)/2 Regular(χ)] + O(ε 2 )}. c 1 can be set to 0 (Poincaré). Reachable constants=3 2 = 1= the origin ξ 0 of ξ. General analytic solution = closed form expression for u (0) =?

8 Outline Motivation, physical problem, mathematical problem Nonintegrable counting. The general analytic solution Elliptic functions = the simplest nonlinear special functions Existing methods (sufficient) Subequation method (necessary) Examples and results Conclusion & perspectives

9 Elliptic functions: the simplest nonlinear special fns Which functions can be defined by a differential eq. (exclude Γ)? Any linear ODE defines a function (because its general solution is uniformizable) : u (N) = 0 (polynomials, rational functions), u = u (exponential, trigo, hyperbolic trigo), u + p(x)u + q(x)u = 0 (Airy, Bessel, Hermite, Whittaker, Gauss, Heun... ),... Nonlinear (nonlinearizable) ODEs? Much more difficult, but systematic (Painlevé). The result is (Fuchs, Poincaré, Painlevé): Order one F (u, u, x) = 0: one function (modulo some group), the elliptic function. Order two F (u, u, u, x) = 0: the six functions of Painlevé. To summarize: The elliptic function(s) is (are) the simplest nonlinear special function(s).

10 Elliptic functions: their successive degeneracies Good book: Abramowitz and Stegun (1972). Bad book: Byrd and Friedman (1954). Any elliptic f. is a rational f. of, (birational group). Canonical representative = the choice of Weierstrass, 2 = 4 3 g 2 g 3 = 4( e 1 )( e 2 )( e 3 ). Elliptic f ns have two successive degeneracies: rational functions of one exponential e k(z z0) (degeneracy of Weierstrass ODE to constant coeff Riccati), then rational functions of z z 0 : doubly periodic ( cnoidal wave ), e j distinct, (z) = simply periodic = 3q coth 2 ( 3qz) 4q, e 1 = e 2 e 3 rational = 1 z 2, e 1 = e 2 = e 3. Practically. The Weierstrass first order ODE (and its degeneracy the Riccati ODE) are the simplest nonlinear elementary building blocks to build (particular) solutions to higher order autonomous nonlinear ODEs.

11 Elliptic functions, a few patterns Abramowitz and Stegun Fig Figure: The copolar trio (sn, cn, dn) of Jacobi, over one real period, for the value k = 1/3. The curves of sn, cn are close to those of sin, cos, and dn never vanishes. Front = shock = heteroclinic = tanh Pulse = source = homoclinic = sech

12 Outline Motivation, physical problem, mathematical problem Nonintegrable counting. The general analytic solution Elliptic functions = the simplest nonlinear special functions Existing methods (sufficient) Subequation method (necessary) Examples and results Conclusion & perspectives

13 Existing methods (sufficient) Mostly 1983, Weiss, Tabor and Carnevale. 1. Assume a given class of expressions for the general analytic solution, 2. Check whether there are indeed solutions in that class. Various names: truncation method, tanh method, exp method, Jacobi expansion method, new or extended method, etc. Examples of classes: polynomial in tanh (Weiss et al 1983 and followers), polynomial in, (Samsonov, Kudryashov 1989), polynomial in tanh, sech (RC and M. Musette 1992, Pickering 1993). All these methods are sufficient (street lamp), they cannot find any solution outside the given class.

14 Outline Motivation, physical problem, mathematical problem Nonintegrable counting. The general analytic solution Elliptic functions = the simplest nonlinear special functions Existing methods (sufficient) Subequation method (necessary) Examples and results Conclusion & perspectives

15 Two results of Briot and Bouquet Théorie des fonctions elliptiques (Mallet-Bachelier, Paris, 1859). Elliptic order:= nb of poles per period parallelogram. Theorem (BB). Given elliptic functions u, v with same periods, F : m n F (u, v) a j,k u j v k = 0, (22) k=0 j=0 deg(f, u) = order(v) = n, deg(f, v) = order(u) = m. If v = u, m 2m 2k F (u, u ) a j,k u j u k = 0, a0,m 0. (23) k=0 j=0 Theorem (BB, Poincaré, Painlevé). If a first order m-th degree autonomous algebraic ODE has the Painlevé property, it must have the form (23), its general solution u = f (x x 0 ) is either elliptic (two periods) or rational(e kx ) (one period) or rational (no period).

16 General method to find all elliptic solutions M. Musette and RC, Physica D 181 (2003) nlin.ps/ ; RC and M. Musette, The Painlevé handbook (Springer, 2008); RC and M. Musette, Stud. Appl. Math. 123 (2009) arxiv: ingredients: 2 theorems of Briot and Bouquet, Laurent series, algorithm of Poincaré. Input: N-th order (N 2) any degree autonomous algebraic ODE admitting a Laurent series. Output: all its elliptic and degenerate elliptic solutions in closed form.

17 Steps: 1. Find the structure of singularities (e.g., 4 simple poles, 2 double poles). Deduce the elliptic orders m, n of u, u. 2. Compute slightly more than (m + 1) 2 terms in Laurent, J u = χ p u j χ j + O(χ J+p+1 ), χ = ξ ξ 0. (24) j=0 3. Define the first order m-th (or divisor) degree subequation m 2m 2k F (u, u ) a j,k u j u k = 0, a0,m 0, 2m 2k n.(25) k=0 j=0 4. Require each Laurent series (24) to obey F (u, u ) = 0, J F χ m(p 1) F j χ j + O(χ J+1 ), j : F j = 0. (26) j=0 and solve this linear overdetermined system for a j,k. 5. Integrate each resulting ODE F (u, u ) = 0, algorithm of Poincaré. Mark van Hoeij, package algcurves, Maple V

18 Outline Motivation, physical problem, mathematical problem Nonintegrable counting. The general analytic solution Elliptic functions = the simplest nonlinear special functions Existing methods (sufficient) Subequation method (necessary) Examples and results Conclusion & perspectives

19 Tutorial example, KdV U (6/a)UU = 0. (27) One double pole U 2aχ 2, χ = 1. Elliptic orders (U) = 2, (U ) = 3. U = 2aχ 2 + U 4 χ 2 + U 6 χ 4 + U2 4 6a χ6 +..., (9 terms), U 4, U 6 = arbitrary, F U 2 + a0,1 U + a 1,1 UU + a 0,0 + a 1,0 U + a 2,0 U 2 + a 3,0 U 3, a 0,2 = 1, Step 4. Linear overdetermined system (26), F 0 16a 2 a 0,2 + 8a 3 a 3,0 = 0, a 0,2 = 1, F 1 8a 2 a 1,1 = 0, F 2 4a 2 a 2,0 = 0, F 3 4aa 0,1 = 0, F 4 2aa 1,0 16aa 0,2 U a 2 (28) a 3,0 U 4 = 0, F 5 0, F 6 a 0,0 + 4aa 2,0 U 4 32aa 0,2 U a 2 a 3,0 U 6 = 0,...

20 Only one solution U 2 (2/a)U U 4 U + 56aU 6 = 0. (29) (Arbitrary constants U 4, U 6 ) (first integrals). Systematic, no skill required to find first integrals. Step 5 (algorithm of Poincaré). Establish the birational transformation (U, U ) (, ). Maple statements: with(algcurves); genus((29),u, U ); Weierstrassform((29),U, U, wp, wpprime);

21 Example. Complex quintic Ginzburg-Landau ia t + pa xx + q A 2 A + r A 4 A iγa = 0, I(p/r) 0, (p, q, r) C, γ R, A = M(ξ)e i( ωt + ϕ(ξ)), ξ = x ct. Only three solutions were known: 1. Heteroclinic front or shock (van Saarloos, Hohenberg 1992) A = A 0 ((k/2)(tanh kξ/2 + ε)) 1/2 e i[α log cosh kξ/2 + Kξ ωt], ε 2 = 1, 2. Homoclinic source or sink (Marcq, Chaté, RC 1994) (HS 1972 for cosh ka = 0, r 0 = 0; vsh 1992 for r 0 = 0) ( ) k sinh ka 1/2 A = A 0 cosh kξ + cosh ka + r 0 e i[α log(cosh kξ + cosh ka) + Kξ 3. Elliptic solution (Vernov 2007) M = c (ξ) g (ξ) + g 2, ϕ = (c/2)r(1/p) + ( 3g ) 1/2 2, 4 (2ξ) g 3 = 0, g 2 = g r 2 27, c2 0 = 4g r, q = R(r/p) = ci(p) = 0. 3I(r/p)

22 Complex quintic Ginzburg-Landau. Subequation method The ODE system in (M, ϕ ) has third order. Discard ϕ (slave var). M has four simple poles M rχ 1, e 2 i r 4 4e r r 2 3 = 0, e r + ie i = r/p. Ellipt. orders(m, M ) = (4, 8). Table: CGL5. Solutions ordered by nb of Laurent series nb(laurent) genus periods pattern codim Ref front 2 van Saarloos Hoh , r 1 + r 2 = source 3 Marcq, Chaté, RC , r 1 + r 2 = 0 1 2? 3 new 2, r 1 + r 2 0 in progress 3 in progress 4 1 2? 5 Vernov ? 4 new For ci(p) = 0 the last sol reduces to the solution of Vernov 2007: (3M ) 4 M 2 (3e i M 2 4g r ) 3 /e i = 0, 9ψ 2 12ψ 4 g 2 r = 0.

23 Complex quintic Ginzburg-Landau. One new solution q = 0, R(r/p) = 0, g i = (3/16)c 2 i, c i I(c/p), arb.=(g r, c i ) F M 4 2ci MM ci 2 M 2 (3ei M 2 g r ) + 34 ci 8 2e i 2 12 ei 2 + c4 i 2 5 e 2 i ( 2g 2 r + 6e i g r M 2 9e 2 i M 4) e i M 2 ( 3e i M 2 4g r ) 3 = 0. Step 5. (Briot-Bouquet) M = P 4( ) + P 2 ( ) = ugly. P 4 ( ) Maple Weierstrassform command fails to integrate for arbitrary (g r, c i ), it only succeeds for numerical (g r, c i ) = ugly BB form.

24 Results for various partially integrable PDEs KdV : U (6/a)UU = 0, KS : νu + bu + µu + u 2 /2 + A = 0, CGL3 : ia t + pa xx + q A 2 A iγa = 0, qp 0, I(q/p) 0, CGL5 : ia t + pa xx + q A 2 A + r A 4 A iγa = 0, pr 0, I(p/r) Chazy III : u 2uu + 3u 2 = 0. PDE poles m (m + 1) 2 1 a j,k u k solutions Ref KdV 1P ell Kuramoto 1P ell + 4 trig MC 2003 CGL3 2P trig + 1 rat MC 2003; Hone 20 CGL5 4P ell + n trig Vernov CGL3 2P To be done RC, Musette 2000 Chazy III 1P rational Bureau 1P rational

25 Conclusion, perspectives Conclusion All elliptic and degenerate sol (trigo, rational) are found. Includes all truncation, extended, new, etc methods. Makes these methods obsolete. For constant coeff. ODEs, linear complexity. Current and future work Revisit physically interesting PDEs with insufficient solutions. Find general analytic solutions which are not elliptic (e.g. doubly periodic of second or third species, Hermite, Halphen).

26 References C. Briot et J.-C. Bouquet, Théorie des fonctions elliptiques (Mallet-Bachelier, Paris, 1859). Mark van Hoeij, package algcurves, Maple V (1997). hoeij/algcurves.html M. Musette and R. Conte, Physica D 181 (2003) A.N.W. Hone, Physica D 205 (2005) S.Yu. Vernov, J. Phys. A (2007). R. Conte and M. Musette, The Painlevé handbook (Springer, Berlin, 2008). R. Conte and M. Musette, Studies in Applied Mathematics 123 (2009)