Physica 23D (1986) 84-90 North-Holland, Amsterdam

SPATIAL STRUCTURE OF TIME-PERIODIC SOLUTIONS OF THE GINZBURG-LANDAU EQUATION*

Philip HOLMES Department of Theoretical and Applied Mechanics, and Mathematics and Center for Applied Mathematics, Cornell Un&ersity, Ithaca, N Y 14853, USA

We study solutions of the form A(x, t) = a(x)e i ' t of the complex Ginzburg-Landau equation iAt + ~ A ~ = flA - ~[AI2A, near those of the integrable limit: the nonlinear SchrSdinger equation iA t + fiA~ = -~,IAI2A, ~, ~' real. We show that spatially periodic and quasiperiodic solutions as well as heteroclinic orbits to them exist, in addition to the spatially decaying (homoclinic) solutions found by Hocking and Stewartson (1972). We speculate briefly on spatially chaotic solutions.

1. Introduction

In this brief note we make some observations on time periodic (harmonic) solutions of the Ginzburg-Landau (G-L) equation

iAt + KAxx= ~A - ¥1AIZA, (1.1)

where A = A (x, t) is a complex function of space and time (x, t) ~ R x R and ~,/3, ~7 are complex and may depend on x. An important limiting case is the integrable nonlinear SchriSdinger (NLS) equation

iA t + aRAxx = --'/RIA[ zA, aR, "/R real constants. (1.2)

Observe that solutions of (1.1) are invariant under translations in space (if 8,/3, ~7 are x-independent) and time (x, t) ~ (x + x 0, t + to) and multiplica- tion by a complex number of unit modulus, A Ae i'ko. This three parameter group of symmetries is reflected in our results below; in particular the latter (phase change) invariance leads to an S 1 symmetry in the dynamical system we study.

Equations (1.1-2) arise in several applications, including fluid mechanics and nonlinear optics [1].

*Partially supported by NSF CME 84-02069 and AFOSR 84-0051.

They typically arise when we assume a field of the form u ( X, T) = A (x, t)e i(K- X- at) + complex con- jugate, where x and t are 'slow' space and time.

Recently, Moon, Huerre and Redekopp [2, 3] and O'Keefe [4] have studied the temporal evolu- tion of spatially periodic solutions of (1.1) using numerical techniques; see their papers and the references cited therein for more information on the derivation of (1.1-2). In addition, Newton and Sirovich [5, 6] and Sirovich and Newton [7, 8] have obtained analytical results on bifurcations of the spatially periodic problem.

In contrast to the rich temporal behavior ex- hibited by the evolution equation studied in the papers mentioned above, we restrict ourselves here to a study of the spatial structure of time periodic (sinusoidal) solutions. Letting A(x, t )= a(x)e i'°t and ()' denotes d( ) /dx , (1.1) becomes

a " - ( a + i B ) a + ( y + i r ) l a l 2 a = O , (1.3)

where

- = + G ) +

+ G ) ) / a ,

= (~R~R + ~I~7~)/a,

and

0167-2789/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

Ph. Holmes / Spatially periodic solutions o f G - L equation 85

For a = a R,/~ = 0, ~ = ~ R , the NLS limit has a = tO/Or R, "y = TR//OtR and t = 8 = 0. We will study the complex Duffing equation (1.3) and thereby obtain results on time periodic solutions of (1.1-2) with particular spatial behaviors. Our methods are essentially an extension of conventional phase- plane techniques, often used for the study of trav- elling waves, into a complex phase space.

Now (1.3) generates a flow on complex 2-space (a, a ' ) ~ C 2 but the flow possesses a n S 1 symme-

try, most easily seen by passing to polar coordi- nates, a = pe i~, in which (1.3) becomes

p,,_ p,,2 = + y 3, (p2,,), = ( B - 8p )p 2

(1.4)

Letting r = 1o 2 = lal 2, v = p ' / p and p2¢, = m (an-

gular momentum), we have

r ' = 2ro,

O t -~ m 2 / r 2 - - o 2 4- ol - - Tr, ( r , v, m) ~ R +X R 2,

( 1 . 5 )

m ' = ( f l - y r ) r .

and m = b e - cd.

Previous work on equations (1.3-6) and a simi- lar system with 0(2) symmetry includes that of Holmes and Wood [9] and Guckenheimer [10]. We will relate our results to theirs as we go along. Both these papers note the fundamental fact that:

Proposition 1.1. Equations (1.3-6) are equivariant under the 1-parameter group of rotations pe i~ ---, pei(~+~). In cartesian coordinates the rotation is

o I y ~ R y ; R = ,,R [s inf f cos~ "

Proof. This is obvious from (1.5), by adjoining the trivial equation ~' = m / r , or by direct calculation using (1.6). []

We describe the phase space of the integrable limit fl = 8 = 0 in the next section. Section 3 contains perturbation computations and we sum- marize our results in theorem 4.1. A brief discus- sion follows in section 5.

The NLS limit fl = 8 = 0 now clearly corresponds to an integrable Hamiltonian system with energy h = r v 2 4- m E / / r - otr + Tr2/2 and momentum m both constant. Note also that (1.5) is a volume preserving flow. Most of the results below will be derived from (1.5), but since the transformation v = p ' / p is singular when p = 0, the representation in R 4 - C 2 is alSO useful:

b' = d, c' = e.

d ' = a b - t i c - ( y b - 8c) (b 2 + c2),

e ' = fib + ac - ( Sb + yc)(b 2 4- c2),

(1.6)

or

y ' = F ( y ) , y - - - ( b , c , d , e ) T,

where a = b + i c . Here for f l = 8 = 0 the invariants are

h = z ~ d 2 + e2 Ix (a2 +2 b2) + T (a2 +4 b2)2

2. T h e i n t e g r a b l e l i m i t

Here we state results on the case fl = 8 = 0 corresponding to the nonlinear Schr6dinger equa- tion. For simplicity we restrict ourselves to the case a, T > 0, although our methods are general.

Proposition 2.1. For fl = y = 0 (1.3) is a com- pletely integrable Hamiltonian system possessing (1) a two dimensional homoclinic manifold h = m = 0, r = lal 2 = ( 2 a / y ) sech2(q~-x), to the hyper- bolic fixed point a = a ' = 0, (2) a one parameter family of periodic solutions a = v~e i~, yr 3 - ar 2

= m 2 parameterized by m, and (3) a two parame- ter family of invariant tori (quasiperiodic solu- tions) parameterized by h and m. All these solutions are invariant under x ~ x + x 0.

Proof. That (1.3) is integrable is a general conse- quence of Proposition (1.1) and a theorem of Marsden and Weinstein [11] (cf. [10]); here one

8 6 Ph. Holmes / Spatially periodic solutions of G- L equation

explicitly checks it from (1.5) or (1.6). The homo- clinic orbits to the hyperbolic saddle are found by linearization of (1.6) and direct substitution of solutions of the form b = B(cos ~ - sin ~), c = B(sin ~k + cos ~k) where B = ~ sech(vrd(x - Xo) ) is a homoclinic orbit for the real-valued Duffing equation B" - aB + ~,B 3 = 0. The peri- odic orbits are fixed points (r, v, m) = (P, 0, rh) with 773_ a~2= j~2 of (1.5), the fourth equation ~ ' = rh/~ = V/, /~-a implying that they are non- trivial periodic solutions except at ~ = a /y , ~h = O. The quasiperiodic solutions are the level sets h = rv 2 + m 2 / r -- a'y + y r 2 / 2 = constant , m =

constant. []

3. Perturbations

We now use the integrable structure of Proposi- tion 2.1 to obtain results on the nearly integrable system (1.3-6) with /3,3 ~: 0, small. Again for simplicity we will take/3, 3 > 0 and constant, but see Remark 3, below. We first note that, in ad- dition to the S 1 symmetry of Proposition 1.1, we also have

Lemma 3.1. Equation (1.5) is equivariant under the transformation (r, v, m, t) ~ (r, - v , - m , - t ) . and possesses the fixed points (r, v, m)_+= ( /3/8,0, + /3 /6 ( /3y /8 - a) 1/2) for /3"l > a& Both these points are hyperbolic spiral saddles, Q ) . having a one dimensional stable and two dimen- sional unstable manifold and ( )_ having a one dimensional unstable and two dimensional stable manifold. For any/3 4:0 the fixed point a = 0 of equation (1.1) persists and is a spiral saddle with two dimensional stable and unstable manifolds.

Proof. Direct computation and linearization at the fixed points. For the final statement we use (1.6).

[]

Remarks. Using a slightly different coordinate system, Holmes and Wood [9] analyze the bifurca- tion poin t /3y = a3 at which the (unique) degener- ate fixed point (r, o, m) = ( a / y =/3 /y ,0 ,0 ) has

eigenvalues (0, _+ i 2 ~ ) . Using averaging they show that, near this bifurcation point, for fly > a3 one expects to find 'almost' invariant tori and spheres of heteroclinic orbits, as in the unfolding of the codimension 2 singularity with linear part

0 (ref. 12; chap. 7.4); see also Broer and 0

Vegter [13] for the volume preserving case (1.5). They argue that Silnikov [14] phenomena (cf. ref. 12, chap. 6.5) are likely to occur and support their arguments with numerical integrations. We will return to this later.

To proceed further we need a modification of the Melnikov (1963) global perturbation method developed by Wiggins and Holmes [15, 16, 17]. Briefly, one computes an approximate Poincar~ map for (1.5) on the section v = 0 by integrating the first variation equations around the closed (and homoclinic) orbits of the limiting f l - -3 = 0 case. Simple computations yield the approximate increments in energy and momentum experienced by solutions due to the perturbation m' =/3r - 3r 2. For/3, 3 4= O, small we obtain

Ah = ~ofT(h" re)h, dx I

= f r (h 'm)E( /3 - - r r ( x ; h , m ) ) m d x ,

Am = foT(h'm)m ' dx j

= foT(h'm)(/3r(x; h, m) - vr2(x; h, m)) dx.

(3.1)

Here T(h, m) denotes the period of the closed orbit r( x; h, m) on the level set h --- rv 2 + m2/r - ar + y r2 /2 = const., m = const. Using r ' = 2rv and h = const, we can rewrite these increments as

A h = ( 2 f l l ° ( h, m) - 81t( h, m ) )m, (3.2)

Am = ½(/311(h, m) - 312(h, m)),

where

i J ( h , m ) = [r2(h,m) r__~Jdr ~rlth, m) Chr - m 2 + ar 2 - ½yr 3 '

T( h, m) = I ° ( h, m) , (3.3)

Ph. Holmes / Spatially periodic solutions of G- L equation 87

and r 2 ___ rl >_ 0 are the non-negative roots of hr - m 2 -4- ar 2 - 13,r3 = 0. Thus the approximate Poincar6 map in energy/momentum coordinates,

P: (h,m)--*(h+Ah(h,m),m+Am(h,m)),

(3.4)

can be evaluated in terms of elliptic integrals. The fixed points Tra(m) - otr2(m) = m 2, v = 0

of the unperturbed problem correspond to the curve h = l (m) = 3yr2(m) /2 - 2ar(m) and the map (3.4) is only defined for h > l(m). For m -- 0 we take the root r (O)=a /T so that h(0)= - a / 2 T 2. The homoclinic orbits of Proposition 2.1 correspond to the point (h, m ) = (0,0) and the perturbed fixed points (r, v, m) ± of Lemma 3.2 to the points,

( h l , - 4 - m l ) = ( f l 2 [ 3 T f l - 4 o t ) , q - f l V f ~ - o t ) 28~ 8

Also note that P is an area preserving map, since it derives from the volume preserving flow of (1.5), and that it inherits the symmetry of (1.5); more precisely: if ((hi , mj)}~_0 is a forward orbit of

m n (3.4) then { h_j, - - j)}j-o is a backward orbit.

Lemma 3.2. Ah = 0 on m = 0. On h --- l(m), Am > 0 for Iml < m t and Am < 0 for Iml > ml. For h sufficiently large, Am < 0 and Ah < 0 (resp. > 0) for m > 0 (resp. < 0). Consequently, P has at least one further fixed point (h 2, 0).

Proof. By inspection of (3.1) for the fixed points ,frS(m) - otr2(m) = m 2 and simple estimates using (3.2-3) for large h. rq

Lemma 3.3. There is a bifurcation set f13' = 8(a + ~ / 3 ) on which equation (1.3) has an S 1 @mmetric two dimensional homoclinic manifold M = W u(0 ) n WS(0); i.e., the integrable homo- clinic structure persists on this set.

the exact solution a = C(sech(~tx)) l+itt, with # =

( ~ + f 1 2 _ a ) / f l , )t 2= (¢a2+ fl 2 + a ) / 2 , C 2=

( ~ + B 2 + 3a)/2V f o r B v = 8 ( a + ~ + B2/3) ,

from the existence of which the result follows. []

Remarks 3.4. 1) The homoclinic solutions corre- spond to 'breathers' (time periodic, spatially de- caying) similar in some respects to the sine-Gordon breathers.

2) Computation of the Melnikov functions A h, Zl m for m =- h = 0 using the unperturbed solu- tion r=(2a /y ) sech(v l -a t ) y i e ld i 0 = oo, i 1 = 4¢ra-/y, 12 = 16av/-~/3~, 2, giving fly = 4a8/3, the leading term in the bifurcation set for fl, 8, small.

3) The S ~ symmetry implies that the homoclinic manifold M is filled with non-transverse homo- clinic orbits, thus we cannot appeal to Silnikov's results [19, 20] on horseshoes near transverse ho- moclinic orbits to four dimensional spiral saddles, cf. Lerman and Umanskii [21], Robinson [22]. In fact, while A m has a simple zero for fixed h, m = 0 as i l l 8 varies, A h is identically zero due to the factor m = 0 in (3.1a). Thus Lerman-Umanskii 's Theorem 1 does not apply.

4) When f l y = 8 ( o t + ot~~-2+f12/3) the homo- clinic orbit corresponds to the additional fixed point of Lemma 3.2 (h2,0) -- (0,0).

Since the fixed points (T, v, m) ± are hyperbolic for (1.5) (with fl), > aS),(h a, + m 1) are also hyper- bolic fixed points for P, on the boundary h = l (m) of its domain. Thus their local stable and unstable manifolds are transverse at (h 1, + m 1) and this, together with the estimates of Lemma 3.2 and the continuity and symmetry of the map, implies that there are curves ~1, ~¢2 connecting (h, + m 1) to (h I, - m 1) on which A m = O and A h = 0 respec- tively, cg2 lies to the right of ~x, both curves intersect m = 0 (the latter at (h2,0)) and

u 1 s 1 Wloc(h , +m 1) and - lie in the Wloc(h, m 1) sector bounded by these curves. See fig. 1, from which it is clear that:

Proof. Hocking and Stewartson [18] first noted Lemma 3.5. For all fl, 3 4:0 small and fly > et8

88 Ph. Holmes / Spatially periodic solutions of G-L equation

m

h=~( / ~'~ ' ' \

) / / . /

/

h

Fig. 1. Qualitative behavior of the approximate Poincar6 map P.

there is a heteroclinic orbit connecting the fixed points (r, v, m) ±.

Remarks 3.6. 1) These methods do not allow us to establish whether the heteroclinic orbits are trans- verse or not, but, taken with the fact that the bounding curve h = l (m) of the domain of P is invariant, Lemma 3.5 does imply the existence of recurrent motions of (1.5) repeatedly passing in the neighborhoods of the two spiral saddle points which, in view of the volume preservation, each satisfy the 'Silnikov condition' that the real eigen- value is larger in magnitude than the real parts of the complex conjugate pair. Thus, by Silnikov's (1965) results (cf. Theorem 6.5.1. of ref. 12), there exist small perturbations of the vector field of (1.5) for f13' - a8 which possess Smale horseshoes. The numerical work of Holmes and Wood [9] suggests that (1.5) possesses horseshoes itself.

2) If /3 and 8 are both negative then results similar to those above also hold, since (1.5) is invariant under the transformation ( r , v, m, t , /3 , 8)

---, (r, o, - j 'n , t, - /3 , - 8 ) . Holmes and Wood [9] consider various possibilities for a,/3, 3', 8 and establish that spatially bounded solutions do not exist for /3/8 < 0 or/33' < etS. Thus the case studied

in this section seems the most relevant (solutions which blow up spatially as x ~ + oo are not nor- mally likely to be of much physical interest).

3) If /3 a n d / o r 8 are spatially periodic (e.g. /3 =/30 +/31(x)) then the perturbed x-flow of (1.5) induces a 3 dimensional Poincar6 map on (r, v, m) space. The Melnikov functions Am and Ah will be periodic in the phase x0 and simple zeros will reveal the presence of transverse S 1 symmetric families of homoclinic orbits or of isolated fami- lies of periodic orbits (cf. ref. 13).

4. Spatial structure of solutions

We now assemble the perturbation results of section 3 to obtain a partial characterization of solutions of the Ginzburg-Landau equation which parallels the description of the integrable limit of Proposition 2.1.

Theorem 4.1. Let a,/3, 3', 8 be defined as in (1.3). Then for those frequencies ~ for which a, 3' > 0 and/3, 8 > 0 are small constants with/33' - a8 > 0, the Ginzburg-Landau equation (1.1) possesses time periodic solutions A( x, t) = a( x )e i~t with the following spatial behaviors: (1) a ( x ) = 0 and, if f ly = 8( a + ~ + /32/3) , a two parameter family of spatially decaying solutions a(x;xo , q~o)= Ce i~v° (sech(•((x - x0)) l+ig, C, X,/1 given in Lemma 3.3.

(2) A two parameter family of periodic or quasi- periodic solutions a(x; Xo, ~Po) ~ ~r(x - Xo) e i~o, where r (x ) is a periodic orbit lying in a level set of ro 2 - ar + 3"r2/2 = const.

(3) A pair of two parameter families of counter rotating periodic solutions

a(X;Xo, o)

and a two parameter family of heteroclinic orbits connecting them.

Ph. Holmes / Spatially periodic solutions of G- L equation 89

There are arbitrarily small perturbations of (1.1) for which spatially chaotic solutions exist in the neighborhood of the homoclinic solutions (1) and in the neighborhood of the periodic solutions (3) and the heteroclinic orbits connecting them.

Proof. Existence of the fixed point and homo- clinic solutions of (1) follows from Lemmas 3.1 and 3.2,3, of the solutions (2) from the 'ad- ditional' fixed point (h2,0) of Lemma 3.2 and of the periodic and heteroclinic solutions of (3) from Lemmas 3.1 and 3.5. To obtain the phase behavior e i~x), we integrate the uncoupled equation ¢p'= re~r, derived from our changes of variables a = pe i~°, r = p2 and m = p2q¢. Silnikov's work [18-20]

implies the final statement. []

' invert ' the evolution equation to obtain

. IAI[0 0 1 - - = 0 j -

_ ~71AI2A , d x fl 1 ~ 0

where B = ~A x and (A, B) ~ HX([0, 2~r/~0]) × L2([0, 2rr/w]) with periodic boundary conditions. A suitable basis for this space of (complex valued) functions is i,,0x ~o {e ) ,=-oo and the analysis above shows that each (complex) 2 dimensional subspace span (ei~ox, e i~'~x) is invariant under the x-flow.

Thus one should be able to push through global perturbation methods for the general problem of nonsinusoidal time-periodic solutions. See ref. 23.

5. D i s c u s s i o n

We have used simple qualitative and perturba- tion techniques on the "phase space" dynamical system for time periodic solutions of the Ginzburg-Landau equation to establish the ex- istence of solutions with specific spatial structures. The main tools used were coordinate changes to effect reduction by the S t (phase) symmetry and the Melnikov global perturbation technique. We cannot ascribe a general physical significance to our results, since in given applications spatial boundary conditions cannot be chosen arbitrarily. However, the existence of spatially periodic solu- tions and the heteroclinic orbits connecting them is likely to be of interest, as are the spatially decaying 'breathers ' of Hocking and Stewartson [18]. A proper study of the approximate Poincar6 map P of (3.4), involving explicit evaluation of the elliptic integrals (3.3) would probably be worthwhile, as would computat ion of the Melnikov functions for specific spatially periodic parameters f l (x) , 8 (x ) (see Remark 3.6 of section 3).

We end with the remark that, in studying gen- eral 2,r/~0 time-periodic solutions of (1.1), one can

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