On solutions of the shape equation for membranes and strings

20 November

1997

PHYSICS

ELSEVIER

LETTERS

B

Physics Letters B 414 (1997158-64

On solutions of the shape equation for membranes and strings B .G. Konopelchenko Dipartimento di Fisica. dell’Unicersitci di Lrcce. e Sezione INFN, 73100 Lecce, Italy and Budker Institute of Nuclear Physics, Novosibirsk-90. Russiu

Received 16 June 1997 Editor: L. Alvarez-GaumC

Abstract Classical configurations

of strings and shapes of membranes

in equilibrium

are defined by a nonlinear equation.

It is

shown that this equation has a simple form in terms of the inverse mean curvature and density of squared mean curvature. Broad variety of its solutions (in particular. kink, vortex and solitons) and corresponding possible shapes are given. A new type of degeneracy of membrane shapes and string configurations via the integrable Veselov-Novikov equation is discussed. 0 1997 Elsevier Science B.V.

Modern string theory and theory of biological membranes are, perhaps, the two opposite extreme points in a wide variety of nonline~ phenomena in which surfaces play a key role (see e.g. [l-4]). These two theories deal with the objects of considerably different natures. But, amazingly certain phenomena in these fields are associated, basically, with the same functional over surfaces (extrinsic Polyakov action and bending energy). A simple model, generally accepted now, regards fluid membranes and lipid vesicles (closed membranes) as the two-dimensional surfaces embedded in three-dimensional space with no elastic energy associated with displacement within the surface (see [1,3,4]). The corresponding free energy has the form [5,6] F=K//(H-H,T)ZdS+ZjjKdS+AjjdS+ApjdV

(1)

where dS and dV are surface area and volume elements. Here H and K are mean curvature and Gaussian curvature of the surface, respectively. H, is a spontaneous mean curvature, K and 7i are bending and Gaussian rigidity, Ap is the osmotic pressure difference and A is a tensile stress. Depending on situation the third and fourth terms either take into account of the contraints of constant total area and volume or represent actual work. Respectively, A and .dp serve as the Lagrange multipliers or they are prescribed experimentially. In the string theory the functional (I) (without the last term) represents an action in the Polyakov’s integral over random surfaces (see [1,2,4]). The third term is an old Nambu-Goto action while the first term (with H, = 0) has been introduced recently in [7]. The presence of this extrinsic Polyakov action is crucial for the infrared and critical properties of random surfaces [7]. The second term in (2) (total Gaussian curvature) is relevant only for surface fluctuations which change topology and usually can be omitted. Note that the first term in (1) (Canham-HeIfrich bending energy or Polyakov extrinsic action) has been known for a long time as the Willmore functional in the differential geometry of surfaces (see [8]). 037~~693/97/$17.~ 0 1997 Elsevier Science B.V. All rights reserved PII s0370-~49~~97~01137-4

13.G.

Konopelchenko/

Physics

Letters

3 414

(1997)

58-64

59

Theoretical explanation of a broad variety of shapes of membranes observed ex~~rnent~ly (see e.g. f 1,3,41) is an important problem of the theory of membranes. Within the theory governed by the energy (1) shapes of membranes in equi~b~um and classical string conjurations are defined by the co~esponding Euler-Lagr~ge equation (shape equation). It is of the form [9,lO] ~~A~~2~~~2-~~~Z~~~~

+2(&-Ah)&Ap=O

(21

when: A is the two-dimensional Laplace-Beltrami operator. In the case H, = h = Ap = 0 see [8]. Various equilibrium shapes have been investigated (see e.g. [9-191). There are shapes with rotational and azimuthal symmetries [9,11,10], spherical, cylindrical, toroidal shapes and also surfaces with higher genus [17] among them. Certain types of shapes were predicted first theoretically and then have been observed experimentally (for instance, the Clifford torus shape (see [ 13,141)). So an analysis of solutions of the shape equation (21, especially its exact solutions is of the great importance. In the string theory few exact solutions of Eq. (2) have been found analytically fsee e.g. [20,21,2]) so far. Another irn~~nt problem is a possible degeneracy of equ~b~um shapes. Wi~in the difference geometry it was shown that the Willmore functional (the functions (1) with H, = h = Ap = 0) is inv~ant under the conformal ~~sfo~ations in the three-dimensional Euclidean space (see [8]), This property gives rise to the conformal degeneracy of the equil~b~um shapes [17]. In this Letter we discuss both above problems. First, we present Eq. (2) in a simpIe form choosing the conformai metric on a surface and introducing the inverse mean curvature and squared mean curvature density as the new dependent variables, This new form of the shape equation is easily emenable to an analysis which provides us a wide variety of exact solutions. Among them one has the solutions of kink, vortex and soliton types which are well-known for other nonlinear phenomena. They give rise to a new types of possible ~uilib~nm shapes of membranes, For the theory with H, = h = Ap = 0 we demons~ate that the functions (1) and the co~esponding shape equation are invariant under in~nite-p~~e~c group of transforma~ons defined by the infinite hier~chy of Veselov-Novikov integrable equations. This Sykes provides us a new possible integrable degeneracy of the eq~~b~~ shapes and co~esponding integrable diffusion. So we first choose metric on a surface in the conformal form dP = 4U2fdr2 + dy2) = 4u2&&

(3)

where x, y are real local coordinates on the surface, z = x f iy, bar denotes the complex conjugation, U(z, 2) is a scalar real-valued function. For any surface one can choose coordinates x and y in such a way that the line element will has the form (3) (see e.g. [8,22]). Then we introduce the variables cp= (H - &)-I and 4 = ucp-‘. In these variables dl= = 4q2rp”dz&

(41

and the functional (1) with Z = 0 and Eq. (2) have the form (AH = 7, F=k

~~q~~dy

+ 4~jjq~~2~dy

+ ApjdV,

‘P;~+2((logq)zr+q2)(C+q2(~2(P2+(yj(p3+C11q(P4)=0 where o’2=6Hs,

a,=8Hs”--2-

h

K= -u-~(logulzT):

2h and LY~=~H,~- -H,

(6) 1 - - Ap. Thus, the function q is nothing but the

squared mean curvature density &d the unction tp dzes not Tontribute to the first term (- jH2ds> in the functional (1). The facto~~ation (4) of the metric gives rise not only to the simple formulae (5) and (6). It allows us also to prescribe q and cp separately including the ~eatment of their ~guments as slow or fast variables.

B.G. Konopelchenko/ Physics Letters B 414 (1997) 58-64

60

We will consider here three main particular cases: 1) constant mean curvature, 2) constant squared mean curvature density and 3) the case H, = h = Ap = 0. The simplest solution of Eq. (6) corresponds to the constant mean curvature, i.e. to rp = qc, = const. In this case Eq. (61 is reduced to the Liouville equation for @= 210gq: 8,1+pe@=o

(7)

where j3 = 2 ‘p. + CQ4002 + CX~ 9; + 9:. The general solution of Eq. (7) is of the form (see e.g. [22j) A& expe=

p12

+

p/2)2

=

q2

while for /3 = 0 one has q2 = A(z)/%?) where A(z) is an arbitrary analytic function. Consequently, at p > 0 one has spherical equilibrium shapes f K = const > 0, H = const) and at p < 0 they are pseudospherical surfaces (K = const < 0, H = const). At p = 0 one has K = 0 and the co~esponding shapes are of the cylin~cal type (K = 0, H = const). In general /3 may be equal to zero for one ore three real 40, (except ‘pO= 0). Hence, there may be one or three points (in H) of transition (depending on the values of H,, K, h and Ap) between spherical and pseudospherical shapes. General expressions for the values of these points are straightfo~~d but cumbersome. In the particular case h = Ap = 0 the equation j3 = 0 is reduced to 1 + 35 + 45 ’ + 2 5 s = 0 where 5 = H, 4po. The single real solution of this equation is 5 = - 1. The co~es~nding mean curvature is zero. Hence, the transition shape is a plane. Since then the sign of j3 coincides with the sign of H one has two phases: spherical shapes with positive mean curvatures and pseudospherical shapes with negative mean curvatures. In the case H, = Ap = 0 one has ~3= 2 ‘ps(l - 1 9:). So there are two cylindrical shapes ( p = 0) with H,, = & m. At H > fi forH<-mandO
and fi K

< H < 0 ke shapes are spherical ( j3 > 0 and, hence, K > 0) while one has pseudosphe~cal shapes ( /3 < 0 and K < 0). Note the discontinu-

Ap Another simple case is H,?= 0, h = 0, Ap f 0. One has J3 = ZC,Q(1 - - cpi>. So the cylindrical shapes K

( fl = 0) with the single value of H = shapes(~>Oand

K>O)have

1/3 exist for both signs of Ap. For positive Ap the spherical

H>

one has spherical shapes with H > the discontinuity at H = 0 as in the previous case. Now we consider the second particular case q = q,, = const. The shape equation is now

where pi = 2q,2, pi = qzcci (i = 2,3,4) are constants. In the simplest case H,?= h = Ap = 0 we have the linear equation Ppx*+ ‘pyy+ S&P = 0.

(10)

General solution and classes of particular solutions of this equation are easily available. For the square domain 0 ( n, y < 7~and vanishing 9 at the boundary, one has the well-known family of solutions 4p,= A,,sinnxsinmy where n,m= 1,2,3,. . ., A,_,, are arbitrary constants and 89; = n2 + m2. The corresponding closed surfaces

61

B.G. Konopelchenko/ Physics Letters B 414 (1997) 58-64 (r?+m”)

have dl’ = =

+ dy*), the mean curvature

A,.,sin*nxsin*my(uIx* 2

Qr2A2 and the total area S,, = L(n2

7+2+Wz2)

H, # 0, A = 4xH”,

At the:ase

‘pzz+ 2q;( cp- 3H,(p’)

Ap = 4x:,;

J/H2dS

H, = (A,,,sinnxsinmyI-‘,

+ m2>. Eq. (9) is of the form

= 0.

This equation possesses the vortex solution [23] intensively studied in plasma physics. The corresponding also has the vortex structure. The next simple example arises in the case Hs = Ap = 0 for which A K

cpzi+2q,zq-2qq,?Eq. (11) is well-known cp=

(p3=o

shape

(11)

in various fields of physics. It has the famous kink (antikink)

solution

(see e.g. [24])

th [ qO( einz + emfaZ + b)]

+ -

where a and b are arbitrary real constants. The mean curvature H = cp-’ of the surface has the singularity along the line 2cosa . x - 2sina . y + b = 0. At last, in general case solutions of Eq. (9) which depend only on x(z + Z) satisfy the equation 1 q~:+

--p,‘p’+

P? 3(p3+

P3 4(p4+

P4

jcp’=const.

Solutions of this equation with ad = 0 are given by standard elliptic functions while in the case (Ye# 0 solution cp is expressed via hyperelliptic functions. Eq. (9) arises, also in a more general case when (logq)l, = 0. Since then q2 = A(.z where A(z) is an arbitrary analytic function, Eq. (6) is reduced to (9) after the change of variable z + A( z>. Note that the solutions given above can be considered as approximate solutions of the basic equation (6) when the mean curvature (i.e the function cp) has a slow dependence on x, y with respect to the function q (first case) and when the function q (squared mean curvature density) has slow dependence on x, y with respect to cp (second case). ALA2 Among the cases when both q # const and cpf const we mention one with q* = 2 where A is an (A+x)2 arbitrary analytic function. Changing z + 5 = A(z), one gets from (6) the equation (~+~)2~~E+6~+20(zcp2+2cu3cp3+2~4~4=0. In the linear case (Y?= cz3 = a4 = 0 (H, = h ;Ap = 0) the general solution of the equation (5-t $)2~tg + 3 6~ = 0 is of the form (see [25]) p = 9 - B( 5 I + C.C. where B( ,$I is an arbitrary analytic function and t+5 1 i cc. means complex conjugation. Among the different possible contraints on q and cp there is one of importance in string theory. It is the gauge of constant mean curvature density H& = 1 in which the Virasoro symmetry is easily revealed (see e.g. [2]). In our variables it is the constraint

q2q = 1. In this gauge one has (at Ap = 0) F = 4//

in the case H,r = Ap = 0 the corresponding

shape equation

is of the form

(12)

In the one-dimensional case (say p,, = 0) this equation is reduced to

K

where v, = z..P,

that is the standard equation for the Weiersbass elliptic function with the invariants g2 = I+,

h

g, = 0 (see e.g. [22]1. The shape equation (12) is also solvable by the elliptic change of the dependent varia$e rp and then by the use of the method of characte~stics. Note also that in terms of the variables 8 = In7 and li”o &o 6 = 2,/a, pcLo z Eq. (12) looks like 0[ (9,= she. At H,V# 0, Ap + 0 the corresponding shape equation contains the fourth order term and it is solvable in terms of elliptic functions. Now we will consider our third particular case H,Y= A = Ap = 0, when Eq. (9) is reduced to the linear equation

where hi= 2(logq)z, + 2q2. For the periodic functions 4 we have W = ~~~~~~ = 2~~~~~~. So the energy (51 in this case is completely defined by the potential L: in the linear equation (13). Even the simplest case II= 0 is rather nontrivial. The inverse mean curvature is harmonic function + IB(Z) +Bfi) and the function q is given by the formula (8) with j3 = 2. Surfaces with the harmonic i ( inverse mean curvature have been studied recently in [26J. The results of 1261provide us the variety of shapes of vesicles (closed surfaces) with the harmonic inverse mean curvature and the metrics (3) with u = (B+ii) AJZ (IA’+ I)” For u = const, i.e. for Eq. (10) we have infinite family of solutions. To define 4 one has to solve the equation O,, + 2exp6 = const where q’ = exp@. The existence of the wide class of periodic solutions of this equation was claimed in [Z’]. Using those results and solution of (10) presented above, one can construct the variety of cfosed surfaces (vesicle shapes) with the same value of energy F = 2rr% ~1. In the one-dimensional case 9 = f;c(x), L:= u(x) one can use the known results about the one-dimensional Schrodinger equation - qX, + Q( X)Q = Ecp (see e.g. [28]). There are several solvable cases. One of them is 2V5 2 provided by the potential Q = -sinZx ~Su~erland model [29]). Taking the periodic wavefunction gc= sinx sin x 2 Calculating q. one gets q = associated with the energy E = 1, one has 4 L’ = 1 - sin2x ’ 2&(&L - sinx) ’ sin and u = (6 - sinx)-“. The corresponding surface is nothing but the Clifford torus (see [S]). Thus H= 2\/2 It has been already discussed within the shape problem in [13-151 in different par~et~zation. Here we would like to emphasize its connection with the solvable ~Sutherland~ case of Eq. (13). Other solvable potentials in (13) may yde remarkable surfaces too. The reality of the c~~~~~ding 9 is the problem. For yihy for &== - ~h2~ {Bargmann potential) and E = 1, one has H = x

and pure imaginary q =

2JZ(Y5 4 chx) * Consequently, the metric is negatively defined a2 = -tfi +-chxf-‘. A variety of exact solutions for the two-dimensional Schrijdinger equation and, hence, Eq, (131 is available

B.G. Konopefchen~/

63

Physics Letters B 414 (1997) 58-64

too. They have been constructed by the inverse spectral transform method (see e.g. the review in [301). We presented here only periodic, finite-gap, solutions (see 131,321):

8(&(x+$

+&? [

-t&(X-iy)

-r)(p)

-e/n>

e(v,(x+i,l)+v,(x-i;v)-e/n)

xexp[(x+iY)G(p)

W/IQ @(rl( P) -e/n>

+(x-+)Wp)I

I (14)

where 8 is the Prym &function, V,, V, are matrices of periods, LZ,(p), fi,( p), q(p) are certain Abel’s integrals. These solutions provides us the variety of compact surfaces, which are possible shapes of vesicles in equilibrium. Finally, we proceed to the problem of degeneracy of possible shapes. The existence of infinite family of shapes with the same energy in the case H, = h = Ap = 0 is quite obvious from the shape equation (13). For fixed q the energy F is fixed while choosing different solutions 40 of (13) one gets different shapes. For 2 in addition to the solution discussed above one has also the instance, for the Sutherland model Q = 7 sm2x 2& solutions cp= + ffM-& - cosx) where a(t) is an arbitrary function of the parameter t. sin x There is another type of possible deformations of surfaces which preserve the functional /SH2dS. Let the potential v and the function c;oin (13) depend on the parameter rn in such a way that n-1 (or,=

a.?“+‘+$2”+‘+

i

B,( m=O c

n-1

Z,Z)az2m+'+ c &#*+~ m=O

p I

where B, are functions and n is positive integer. In the simplest case iz = 1 the compability condition of Eq. (131 and (15) is equivalent to the Veselov-Novikov equation 1321 Ur,+l’,:~+L’Eil+3(0(~‘L:;))r+3((1(a-’uL))~=0.

(16)

This equation is integrable by the inverse spectral ~ansfo~ method [32]. Choosing p1= 2,3,4,..., one gets an infinite hierarchy of integrable equations. Equations integrable by the inverse spectral transform method have a number of remarkable properties (see e.g. [30]). Finite-gap periodic solutions of the Veselov-Novikov equation (16) and the whole hierarchy are given by (14) with the substitution V, + V2 -+ V, i- V, + hzn+‘tn 1321. The Veselov-Novikov equation (16) and the whole hierarchy have an infinite set of integrals of motion. The a simplest of them is C = ~vdxdy (i.e. c C = 0). So the hierarchy of the Veselov-Novikov equations (in particular, their solutions (14)) provides “us the infinite-parametric family of shapes with the same energy F = Ic/lH’dS = K~v~dy (see also [33]). This type of possible degeneracy of shapes is essentially different from those discussed previously. Note that in the one-dimensional limit U, = 0 Eq. (16) is nothing but the famous Korteweg-de Vries equation v,, + a~,,,~ + ~LIU,= 0 (see e.g. [30]).

References It1 D. Nelson,‘I’.Piran,S. Weinberg,

Eds., Statistical mechanics of membmnes and surfaces (World Scientific, Singapore, 1989). [21 D.J. Gross, T. Piran, S. Weinberg, Eds., Two dimensional quantum gravity and random surfaces (World Scientific, Singapore, 1992). [31 R. Lipowsky, E. Sackman, Eds., Structure and dynamics of membranes (Elsevier Science. Amsterdam, 1994). [41 P. David, P. Ginsparg, Y. Zinn-Yustin, Eds., Fluctuating geometries in statistical mechanics and field theory (Elsevier Science, Amsterdam, 1996).

64

B.C. Konopelchenko/~l~~sics Letters B 414 (IY9?158-64

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