Multi-Cuts Solutions of Laplacian Growth

Multi-Cut Solutions of Laplacian Gro wth Ar. Abanov ∗ Dep ar tment of Physics, MS 4242, T exas A&M University, Col le ge Station, TX 77843-4242, USA M. Mineev-W einstein † L os A lamos Nat ional L ab or atory, MS-P365, L os Alamos, NM 8 7545, USA A. Za brodin ‡ Institute of Bio chemic al Physics, Kosygina str. 4, 119334 Mosc ow, Rus sia; also a t ITEP, Bol . Cher emushkinskaya str. 25, 117218 Mosc ow, R ussia (Dated: No vem ber 8, 2018 ) A new class of solutions to Laplacian gro wth ( LG) with zero surface ten si on is presented and sho wn to contain all other known so lutions as sp ecial or limiting cases. These solutions, which are time-dep endent conformal maps with branch cuts inside the unit circle, are gov erned b y a nonlinear integ ral equation and describ e oil fjords with non-parallel wa lls in viscous fingering exp eri ments in Hele-Shaw cells . Integral s of motion for th e m ulti-cut LG solutions in terms of singularities of the Sch w arz function are found , and the dynamics of densities (jump s) on the cuts are d eriv ed. The sub clas s of these solutions with linear Cauc hy den si ties on the cuts of th e S chw arz function is of particular interest, b ecause in this case the integral equation for th e conforma l map becomes linear. These solutions can also b e of physical imp ortance by represen ting oil/air in terfaces , whic h form oil fjords with a constant opening angle, in accordance with recent exp erimen ts in a Hele-shaw cell. P A CS num b ers: 02.30.Ik, 02.30.Zz, 05.45.-a Keyw ords: Laplacian growt h, harm onic momen ts, viscous fing ering domain, in terface, pattern, dynamics Cont ents I. In tro duction 1 I I. The Sc h w arz F unction and Integrals of Motion 2 I II. Ratio na l and Logarithmic Solutions 4 A. Rational So lutio ns 4 B. Logarithmic Solutions 5 C. “Mixed” Rationa l-Logarithmic Solutions 6 IV. Multi-Cut Solutions with Analytic Cauc hy Densities of General T yp e 6 A. Motiv ation 6 B. A Multi-Cut Ansatz for S + ( z ) 7 C. An Integral Equation for the C o nformal Map 8 D. The So lutio n Sc heme 9 V. Multi-Cut Solutions with Linear Cauc hy Densities 9 A. Cauch y Densities with a Pole at Infinity 9 B. Linear Cauch y Densities 10 C. Example: Z N -Symmetric Solutions with N Radial Cuts 10 VI. Discussion and Conclusion 12 ∗ abano v@tam u.edu † mariner@lanl.gov ‡ zabrodin@itep.ru VI I. Ac kno wledgement 12 A. Numerical solution of equation (64) 13 References 13 References 13 I. INTRO DUCTION Backgr ound and motivatio n: The Laplacian g r o wth (LG) is a free b oundary motion governed by gradient of a harmonic field. It describ es numerous physical pr o - cesses [1] with a moving boundary b etw een tw o immis- cible phases (an in terface) far fr o m eq uilibrium and pro- foundly in terconnects v arious branches of mathematica l ph ysics [2, 3]. LG in tw o dimensions is o f special imp or- tance since in the zero surface tension limit it has a re- mark a bly rich integrable structure [4 , 5, 6 ] and an impres- sive list of exact solutions in the form of conformal maps with p ole [7, 8, 9] and loga rithmic [10, 11, 12, 13] time- depe ndent singular ities (many of these solutions cease to exist in a finite time due to the interface instabil- it y .) In addition, there exist a family o f explicit s olu- tions [14, 15, 16] with time-indep enden t branch cuts with fixed endp oint s which describ e g ro wth of self-similar air ‘fingers’ in a wedge geometry (or in the plane with the rotational Z N -symmetry imp osed). Do other solutions exist, a pa rt from those mentioned ab o v e? Answering ‘yes’ to this question, w e hereby present a new family of conformal maps with time- depe ndent cuts, study their dynamics and provide ex- 2 amples. This family is the mos t gene r al solution to the LG problem (in the absence of surfa ce tension) which can b e written in terms of a nalytic functions. All ratio- nal, lo g arithmic and self-similar so lutions known so far (and mentioned ab ov e) are shown to b e sp ecial or limit- ing cases of these newly found ones. In this pap er we consider a simply co nnected exterior 2D LG problem with a sour ce/sink at infinity . In the context of Hele-Shaw flows [17] (the pro to t yp e of all 2D LG pro cesses [18]), it corr esponds to a finite bubble of an inviscid fluid (air) surr ounded b y a visco us fluid (oil) with a source or sink at infinit y . In what follows we ho ld to this hydrodynamic int erpretatio n. The standar d formulation of the exterior LG problem in 2D planar g eometry is as follows:            v = −∇ p ∆ p = 0  in D ( t ) p = co nst V n = − ∂ n p  at Γ( t ) p ( x, y ) = − Q 2 π log R , when R = p x 2 + y 2 → ∞ . (1) Here t is time, Γ( t ) is the b o undary (an oil/air in terface) of an infinite planar domain D ( t ) containing infinity and filled with oil, p is the pres sure field, v is velo cit y of o il, V n is norma l velocity o f the oil/air interface, ∂ n stands for normal deriv ative at the interface, and Q is a pumping rate assumed to be time-indep enden t (p ositiv e for the suction pro blem and negative for the injection one). The first equa tion in the s ystem (1) is the Da r cy law in scaled units, whic h determines the local velocity o f oil in the Hele- Sha w cell. The second eq ua tion follo ws from contin uity and incompressibility of oil, div v = 0. The third equation r esults fro m the fact that in fluids with negligible visc o sit y pressure is the same everywhere in the fluid, p = co nst, so, neglecting surface tension, one can s et p to b e the same constan t alo ng both sides o f the int erface. The fourth equation states that nor mal v elo c- it y o f fluid a t the interface and that of the interface itself coincide (con tin uity). The last equa tion gives the asymp- totic of p far awa y from the in terface in the presence o f a single sink of strength Q pla ced at infinit y . A reformulation o f the LG problem in ter ms of complex v aria bles and analytic functions is particularly ins truc- tive. W e co nclude the introduction by a brief review of the time-dep enden t co nformal map appro a c h. The c onformal map formulation . Let us int ro duce a time-dep e ndent conformal map, z = f ( w, t ), from the exterior of the unit circle, | w | ≥ 1, in an auxilia ry math- ematic al w -plane to the domain D ( t ) o ccupied by oil in the ph ysic al plane z = x + iy . The n the in terface at time t is a closed curv e sw ept b y f ( e iφ , t ) as φ runs from 0 to 2 π . Below we sometimes do not indicate the time depe ndence e x plicitly and write simply f ( w ). It is con- venien t to normalize the confor mal ma p b y the co ndition that ∞ is mapp ed to ∞ a nd the deriv ative a t infinit y is a real p ositive num ber r (the conformal r adius), so that the Laur els expansion at infinity has the form f ( w ) = rw + u 0 + u 1 /w + u 2 /w 2 + . . . (2) (the co efficients u i are in gener al complex n umbers). It is w ell known [19, 20] that the L G dynamics of the interface is equiv a len t to the following equa tion for f ( e iφ , t ): Im( ¯ f t f φ ) = Q 2 π , (3) where subscripts s tand for partial der iv ativ es a nd a bar for complex conjugation. This nonlinear equation is re- mark a ble b ecause it p ossesses an infinite num b er of con- serv a tion la ws [4] and an impres s iv e list of non-trivial so- lutions with moving singularities [7, 8, 9, 10, 11, 12, 13], which are either p oles or logarithmic branch p oin ts ly- ing strictly inside the unit circle . In all these case s the conformal map f ( w ) thus a dmit s an analytic contin ua- tion acr oss the unit circle, so the function f ( w ) is actually analytic no t o nly in its exterior but in some larger infinite domain containing it. Assuming that f ( w , t ) is analyt- ically extendable acr oss the unit cir cle for all t in some time int erv a l, one can analytically contin ue the LG equa- tion (3) itself. Set ¯ f (1 /w, t ) ≡ f (1 / ¯ w, t ), then equation (3) ca n be rewritten a s ∂ w f ( w, t ) ∂ t ¯ f (1 /w, t ) − ∂ t f ( w, t ) ∂ w ¯ f (1 /w, t ) = 1 /w (4) (w e hav e set Q = π that means a rese aling of time units). Under our a ssumption the functions f ( w, t ) and ¯ f (1 /w , t ) hav e a common domain o f analyticity co n ta ining the unit circle | w | = 1, and w in equation (4) is suppo sed to b e- long to this doma in. In fact, for rational and logarithmic solutions, this equation holds everywhere in the w -plane except for p oles and branch cuts of f ( w, t ) and ¯ f (1 /w , t ). W e no te in adv ance that the gener al m ulti-cut so lutio ns constructed b elow hav e the sa me pro perty . Namely , the functions f ( w , t ) and ¯ f (1 /w , t ) hav e a common do main of analyticit y where they actually solve the analytically contin ued LG eq ua tion (4) and no t just (3). As said earlier , there are a lso self-similar Z N - symmetric solutions [3, 14, 15, 16] con taining a time- independent fra ctional p o w er s ing ularit y which corre - sp onds to the interface se lf-intersection p oint at the ori- gin. In this cas e the analytic con tin uation is p ossible through every p oin t on the unit circle except tho se that are ma pped to the o rigin. Again, the functions f ( w , t ) and ¯ f (1 /w , t ) have a non- empt y common domain o f a n- alyticity and equation (4) is v alid everywhere in this do- main. II. THE SCHW ARZ FUNCT ION AND INTEGRALS OF MOTION It is known [2, 4, 5, 10, 12, 21] that the partial dif- ferential e q uation (4) can b e integrated in terms o f the Schwarz function . It is a n analytic function, S , which 3 connects f ( w ) a nd ¯ f (1 / w ) in their common domain of an- alyticity: ¯ f (1 /w ) = S ( f ( w )). Equiv alen tly , the Sch w arz function for a curv e Γ is an analytic function S ( z ) such that ¯ z = S ( z ) for z ∈ Γ [22]. In other words, S ( z ) is an analytic con tin uation o f the function ¯ z awa y from the curve. F or ana ly tic curves this function is known to b e well defined in a strip- lik e neighborho o d of the curve. If the curve dep ends on time, s o do es its Sc h w arz function, S = S ( z , t ). In terms of the Sch warz function the (analytically con- tin ued) LG equation (4) r eads [21] S t = ∂ z log w (5) where w = w ( z , t ) is the function in v erse to the f ( w , t ). Since ∂ z log w ( z ) is analytic everywhere in the oil domain D ( t ), both sides of equation (5) are fr ee of singular ities there. Ther e fo re, all sing ularities of S ( z , t ) located in D ( t ) are time-indep enden t. Equiv a len tly , the function S + ( z ) = I Γ( t ) S ( ζ ) ζ − z d ζ 2 π i (6) defined for z o utside D ( t ) by the integral of Cauch y type (where the in tegration contour Γ has the sta ndard a n ti- clo c kwise orientation) and ana lytically con tin ued to D ( t ) is constant in time: ∂ t S + ( z , t ) = 0 . This implies conser - v ation of harmonic moment s t k [4] which ar e coefficients of the T aylor ex pa nsion of S + ( z ) at z = 0: t k = 1 2 π ik I Γ( t ) ¯ z dz z k = − 1 π k Z D ( t ) dx dy z k , k = 1 , 2 , 3 , . . . , (7) where we assume, without loss of g eneralit y , that the co n- tour Γ( t ) e ncircles the origin. In particular, t 1 = S + (0). Besides, it is easy to see that the ph ysical time t is equal to the area of the air bubble divided by π : t = I Γ( t ) ¯ z dz 2 π i (8) Thu s the LG dynamics is a pro cess such that the area sur- rounded b y the int erface grows linea rly with time while the harmonic moments (7) (or the S + -part of the Sch warz function (6)) a re kept constant. It is impor tan t to note that the function S + (i.e., the infinite set of ha rmonic moments t k ) together with the v ariable t fix the whole Sch warz function S uniquely , at least lo cally in the v a r i- ety of closed analy tic con tours in the plane. In a cer tain sense (made more precise in [23]), these data serv e as lo cal co ordinates in the infinite dimensio nal v ariety of contours. It is also worth while to men tion that the function S − ( z ) = S ( z ) − S + ( z ) is analytic in D ( t ) and v anishes a t infinit y a s O (1 /z ) with the residue res ∞ [ S − ( z ) dz ] = − t (9) (the res idue at infinit y is defined as res ∞ [ dz /z ] = − 1). The decomp osition of the Sc hw a r z function S = S + + S − app ears to b e v ery useful. As it follows from pr operties of Ca uc hy-type in tegrals , the function S − ( z ) is giv en by the same in tegral (6) (with the opp osite sign), where z now belongs to D ( t ). The conformal map and the Sch w arz function are con- nected in the following w a y:  S ( z ) = ¯ f (1 /w ) z = f ( w ) (10) These form ulas mak e it clear that if the functions f ( w ) and ¯ f (1 /w ) ha ve a common do main of analyticity con- taining the unit circle, then the Sc h w arz function is w e ll- defined. They a lso imply the one- to -one corres p ondence betw een singula r ities o f the function f ( w ) inside the unit disk and singularities o f the function S ( z ) in D ( t ) (which, by the pro perties of the Cauch y type integrals, are the same as singularities o f the function S + ( z )). Let us illustrate the one- to -one corr espondence b e- t ween s ingularities of f ( w ) and S + ( z ) by the example of po les in w hich ca se it is esp ecially tra nsparen t. Suppose that the confo r mal map f ( w ) has a p ole of o rder k at a po in t a inside the unit disk, so that the local b eha vior of f nea r a is f ( w ) = A ( w − a ) k + les s singular terms . (11) Then the function S = ¯ f (1 /w ) has a p ole of the s ame order at the p oin t w = 1 / ¯ a : S = ¯ f (1 / w ) = ¯ A ( w − 1 − ¯ a ) k + less singular ter ms . (12) Because the point 1 / ¯ a tog ether with its sma ll neig h bor- ho od lies o utside the unit disk (where f is confor mal), we can linea rize the denominator in (12) near z = f (1 / ¯ a ) and obtain S ( z ) = ¯ A  f ¯ a (1 / ¯ a ) z − f (1 / ¯ a )  k + less singular terms (13) (here f ¯ a (1 / ¯ a ) ≡ ∂ w f (1 /w ) a t w = ¯ a ). Thus S ( z ) neces- sarily has a singula r it y a t z = f (1 / ¯ a ) of the same type as f ( w ) has a t w = a . Now we are r eady to outline the general strategy of int egratio n of the LG problem. T ak en initial data, i.e., a conformal map f ( w ) a t t = t 0 , one finds S + from (6) by rewriting it as integral ov er the unit circle in the mathe- matical plane: S + ( z ) = 1 2 π i I | w | =1 ¯ f (1 /w ) d f ( w ) f ( w ) − z . (14) In or der to obtain the LG dy namics, one then should solve the inv erse problem: g iven the S + ( z ) and time t , to r eco ver the co nformal map f ( w , t ). The last step is a 4 part of the in verse potential problem and thus is hard to implemen t in genera l. F o rmally , this is equiv alen t to a nonlinear integral equation which is easy to derive using the definition of the Sc h warz function in the form f ( ζ ) = ¯ S ( ¯ f (1 /ζ )). In tegrating b oth sides of this eq ualit y with the Cauc hy kernel 1 / ( w − ζ ) o ver the unit cir cle assuming that | w | > 1, and us ing the fact that the S − -part of the Sch warz function do es not contribute to the integral, we get the int egral equation f ( w ) = rw + u 0 + 1 2 π i I | ζ | =1 ¯ S + ( ¯ f (1 /ζ )) dζ w − ζ , | w | > 1 . (15) The same pro cedure at | w | < 1 gives a “complimen tary” equation rw + u 0 = 1 2 π i I | ζ | =1 ¯ S + ( ¯ f (1 /ζ )) dζ ζ − w + ¯ S − ( ¯ f (1 /w )) , | w | < 1 . (16) As their direct consequence, we can write the following useful for m ula e fo r the co efficien ts of the Laur els series (2): u j = 1 2 π i I | ζ | =1 ¯ S + ( ¯ f (1 /ζ )) ζ j − 1 dζ , j ≥ 0 , (17) r = 1 2 π i I | ζ | =1 ¯ S + ( ¯ f (1 /ζ )) ζ − 2 dζ + t r (18) Substituting them bac k in to (2) and using (15), w e get a set o f in tegral equations f ( w ) = rw + d X j =0 u j w − j + 1 2 π i I | ζ | =1 ζ d ¯ S + ( ¯ f (1 / ζ )) dζ w d ( w − ζ ) , | w | > 1 . (19) for any d = − 1 , 0 , 1 , 2 , . . . . Being taken tog ether with (17), all of them ar e equiv alent to the original equation (15) which is repro duced at d = 0. How ever, dep end- ing on the type of singular ities o f the function S + , one or ano ther fo rm may b e mor e co n v enient than other s. The choice d = − 1 g iv es an eq uation wher e no ne of the co efficien ts u j ent er explicitly : f ( w ) = rw + 1 2 π i I | ζ | =1 w ¯ S + ( ¯ f (1 /ζ )) dζ ζ ( w − ζ ) , | w | > 1 . (20) This equation combined with relatio n (18) accomplis he s int egratio n of the LG problem. Let f ( w ) = f ( w | r ) be a solution to equation (20) depending on the par ameter r , then the time dependence is found from (18) whic h determines r as an implicit function of t : t = r 2 − r 2 π i I | ζ | =1 ¯ S + ( ¯ f ( ζ | r )) dζ (21) Alternatively , one may use the r elation t = 1 2 π i I | w | =1 ¯ f (1 /w ) d f ( w ) (22) leading to the sa me results. In principle, this sc heme provides a general solution to the LG problem but in a ra ther implicit form. How ev er, for sev eral imp ortant classes of functions S + ( z ) it can be made more e x plicit. In all effectively so lv able cases, the r.h.s. of the integral equatio n simplifies dra stically after shrinking the integration contour to singula r ities of the function ¯ S + ( ¯ f (1 /ζ )) inside the unit disk. Some exa mples are given in the next section. II I. RA TIONAL AND LOGARITHMIC SOLUTIONS In this sec tio n we a pply the general metho d bas e d on equations (20) , (21) to construction of LG solutions with po les and log arithmic singularities. A. Rational Solutions By rational solutions we mean solutio ns to equa tion (4) whose only singularities are p oles inside the unit disk (not men tioning a simple pole at infinity). The v ery fact that the r a tional ansate is co ns isten t with the LG equation, i.e., that the num b er of p oles is conse rv ed and singulari- ties of other t ypes are not generated, is a consequence of the integral equation (1 5 ). Let us take S + to be a r ational function of a general form S + ( z ) = N X m =1 K m X k =1 T m,k ( z − z m ) k + T 0 , 0 + K 0 X k =1 T 0 ,k z k (23) where all the pole s z m are in D ( t ) and T m,k are arbitra ry complex c onstan ts. Then the function ¯ S + ( ¯ f (1 / ζ )) has a p ole a t 0 of order K 0 and p oles of orders K m at the po in ts a m such that z m = f (1 / ¯ a m ) and do es not have other sing ula rities inside the unit dis k . F or co n venience, we set a 0 = 0. Therefor e , the in tegral in the r.h.s. of (15) is equal to the sum of residues at these p oles. Calculating the residues, we obtain an expressio n for the conformal map in the form of a rational function of w with p oles o f orders not higher than K m at the po in ts a m inside the unit disk: f ( w, t ) = r ( t ) w + u 0 ( t ) + N X m =0 K m X k =1 A m,k ( t ) ( w − a m ( t )) k , (24) The pole at a 0 = 0 is distinguished among the other s bec ause it does not move. All other p oles as w ell as all co efficients dep end o n time. The co efficien ts A m,k are expressed thro ugh deriv a tiv e s of the function ¯ f (1 /w ) 5 at w = a m . The g eneral formulae are quite compli- cated. In fac t what we really need are inv er se formu- lae which expre s s the integrals of motion T m,k through time-dep e ndent co efficients of f ( w , t ). Obviously , T m,k = res z m  ( z − z m ) k − 1 S + ( z ) dz  , m 6 = 0 , and the function S + ( z ) here can b e substituted for the S ( z ) b ecause S − ( z ) is regula r in D ( t ). Passing then to the mathematical plane, w e obtain the full list of con- stants of motion: z m = f (1 / ¯ a m ( t ) , t ) , m 6 = 0 (25) T m 6 =0 ,k = res ¯ a − 1 m ( t )  ( f ( w, t ) − z m ) k − 1 ¯ f ( w − 1 , t ) d f ( w, t )  T 0 ,k = − r es ∞  ( f ( w, t )) − k − 1 ¯ f (1 /w , t ) d f ( w, t )  . W e note that T m,k hav e the meaning of “Whitham times” for a genera l Whitham hierarch y which cov ers the LG problem with zero sur face tensio n. Each Whitham time T m,k is “ coupled” to its own type o f singularity . F ro m this p oin t of view, for m ula e (25) a r e relations of the ho do- graph t yp e which pro vide a solution o f the full genus z e ro Whitham hiera rc h y in an implicit for m [24]. Algebraic equations (25) expre ss time-dep enden t pa- rameters r ( t ), u 0 ( t ), a m ( t ) and A m,k ( t ) implicitly via the constants of motion z m and T m,k . T o o btain a complete set o f relations, we also need the equation (2 2) cont aining time t explicitly . In o ur case the integral is reduced to a sum o f residues: t = − r es ∞ [ ¯ f ( w − 1 , t ) d f ( w, t )] − X m res ¯ a − 1 m ( t ) [ ¯ f ( w − 1 , t ) d f ( w, t )] Comparing w ith (25) at k = 1, we see that the r esidues at 1 / ¯ a m are just constants T m, 1 . Therefore, t = − r es ∞ [ ¯ f (1 / w, t ) d f ( w, t )] − X m T m, 1 . (26) So, we hav e N + K + 1 c o mplex algebraic equations, where K = P N m =0 K m , and one rea l a lgebraic equa tio n for N + K + 1 complex parameters and one r eal parameter which determine them as functions of time and N + K + 1 constants of motion. Given initial conditions in the form (23), a solution to this system allows one to recov er the time-dep e ndent conformal map, acco rding to the stra tegy outlined a t the end o f Section I I. In some impor tan t par ticular cases the general expres- sions giv en abov e b ecome simpler and more explicit. F or example, if no ne o f the p oles a m lies at the orig in (i.e., A 0 ,k = 0 for all k ≥ 1), then T 0 ,k = 0 for k ≥ 1, T 0 , 0 = f (0 , t ) and t = r ( t ) ¯ f ′ (0 , t ) − X m T m, 1 , (27) (here f ′ (0 , t ) is the deriv a tiv e of f ( w , t ) at w = 0). If, moreov er, all p oles in (24) are simple and none of them lies a t the origin ( K m = 1 for m 6 = 0 and K 0 = 0), then the expressions for T m, 1 and t acquire an esp ecially compact and explicit for m: T m, 1 = ¯ A m, 1 ( t ) ∂ w f (1 /w , t )   w = ¯ a m ( t ) = ¯ A m, 1 ( t ) − r ( t ) ¯ a 2 m ( t ) + X l A l, 1 ( t ) (1 − a l ( t )¯ a m ( t )) 2 ! , (2 8 ) t = r ( t ) ¯ f ′ (0 , t ) − X m T m, 1 = r 2 ( t ) + X l,m A l, 1 ( t ) ¯ A m, 1 ( t ) (1 − a l ( t )¯ a m ( t )) 2 . (29) B. Logarithmic Solutions F o r the clas s of logarithmic solutions, the function S + is taken in the form S + ( z ) = t 1 + N X m =1 ¯ A m log  1 − z z m  , (30) where all the bra nc h points z m are in D ( t ) and t 1 , A m are a rbitrary complex constants (note that this t 1 is the first harmo nic momen t from (7)). If A 0 = − N X m =1 A m 6 = 0 , then there is also a branching at infinit y . T his function is m ulti-v alued and one should fix a single - v alued branch. In a neig hbo rhoo d of the o rigin, w e define it b y the con- dition that S + (0) = t 1 . In or der to con tinue it una m- biguously to D ( t ), it is necessar y to intro duce a system of cuts co nnecting a ll the bra nc h p oin ts in suc h a wa y that the domain D ( t ) \ { all cuts } b e s imply connected. There ar e man y wa ys to dr aw the cuts. Although all of them are ultimately eq uiv alen t, it would b e convenien t for us to meet some requir e men ts natural for the g ro wth problem: to resp ect the demo cracy among the br a nc h po in ts z m and to ensure that the cuts alw ays r emain in D ( t ). F or the former , let us fix a po in t q in D ( t ) and make cuts from q to all z m (and, if necessar y , to infin- it y). F or the la tter, it is natural to choos e q = ∞ since ∞ is the only p oin t in D ( t ) which remains there forever irresp ectiv ely of initial conditions. So , we fix a system of (non-intersecting) cuts from the p oint s z m to ∞ . No te that this c hoice of cuts implies that S + ( z ) do e s no t hav e a definite v alue a t z = ∞ , ev en if ∞ is a r egular p oin t, bec ause the limit depe nds on a particular w ay of tending z → ∞ . Reconstructing f from (1 5 ) in a s imilar w ay as for ra- tional solutions, w e introduce the p oint s a m such that f (1 / ¯ a m ) = z m inside the unit disk and notice that the int egral in (15) is shr unk to res idues of the differential d ¯ S + ( ¯ f (1 /ζ )) = N X m =1 A m d log( ¯ f (1 /ζ ) − ¯ f (1 / a m )) 6 which are easy to calculate. In this wa y one o btains f ( w, t ) = r ( t ) w + u 0 ( t ) + N X m =0 A m log( w − a m ( t )) , (3 1) N X m =0 A m = 0 . Again, we distinguish a p o ssible branc h po in t at the or i- gin and set a 0 = 0. By constructio n, the cuts of the function f ( w ) go from 0 to a m inside the unit disk. Sim- ilarly to the Sch warz function, f ( w ) with this choice of cuts do es not hav e a definite v alue at w = 0 , ev en if 0 is a regular p oin t, b ecause the limit dep ends on a par ticular wa y of tending w → 0. The co nstan ts of motion a re given b y t 1 = 1 2 π i I | w | =1 ¯ f (1 /w ) d f ( w ) f ( w ) = ¯ u 0 + N X m =1 ¯ A m log  ¯ a m z m r  , z m = f (1 / ¯ a m ( t ) , t ) = r/ ¯ a m ( t ) + u ( t ) + N X l =1 A l log(1 − a l ( t )¯ a m ( t )) , (32) in ag reemen t with pr evious results [10, 11, 12, 13]. F or- m ulae (21) and (22) allow one to derive few differently lo oking equiv alen t expressio ns for t . The simplest o ne reads t = r 2 + N X m =1 A m  ¯ z m − ¯ u 0 − r a − 1 m  . (33) The o thers can b e also useful in some situations : t = r ∂ w ( ¯ f ( w ) + ¯ A 0 log f (1 / w ))   w =0 + X m ¯ A m z m , (34) t = r 2 + X l,m A l ¯ A m log(1 − a l ¯ a m ) . (35) The solution contains N + 1 complex ( a m and u 0 ) and one real ( r ) time-dep enden t parameter s. They a re to b e determined from the system of N + 1 complex equations (32) and one r eal equation (34). C. “Mixed” Rational-Logarithmic Solutions In a simila r wa y , one can also consider “ mixed” solu- tions with b oth p oles a nd log arithms. In this ca s e the function S + is S + ( z ) = N X m =1 K m X k =1 T m,k ( z − z m ) k + T 0 , 0 + K 0 X k =1 T 0 ,k z k + N X m =1 ¯ A m log  1 − z z m  (36) where a s ingle-v alued branch is fixed by the condition that S + (0) = N X m =1 K m X k =1 T m,k z k m + T 0 , 0 = t 1 . (37) The equation (15) gives f ( w, t ) = r ( t ) w + u 0 ( t ) + N X m =0 K m X k =1 A m,k ( t ) ( w − a m ( t )) k + N X m =0 A m log( w − a m ( t )) , N X m =0 A m = 0 , (38 ) with the same con ven tion a 0 = 0. The constants of mo- tion z m , T m,k are expressed throug h the time-dep enden t parameters of the c o nformal map by means of formulas similar to (25). I n fact one can represent the “ho dogra ph relations” (25) in a for m whic h is suitable for logar ithmic and mixed cases as well: z m = f (1 / ¯ a m ( t ) , t ) T m 6 =0 ,k = − 1 k res 1 / ¯ a m ( t )  ( f ( w, t ) − z m ) k d ¯ f (1 / w, t )  , T 0 ,k 6 =0 = − 1 k res ∞  ( f ( w, t )) − k d ¯ f (1 /w , t )  , (39) The residue a t infinity is well defined since the differen tial d ¯ f (1 /w , t ) is single- v alued. No te that the co efficien t ¯ A m can b e formally under stoo d as T m, 0 : ¯ A m = − k T m,k at k = 0. An expression for T 0 , 0 follows fro m (37) and the int egral for m ula for t 1 in (32) v alid in all cas e s : T 0 , 0 = 1 2 π i I | w | =1 ¯ f (1 / w ) d f ( w ) f ( w ) − N X m =1 K m X k =1 T m,k z k m . (40) Different equiv alent versions of the for m ula for t can b e obtained from (21) or (22). Finally , we note that rational terms in f ( w ) can b e re- garded as a sp ecial (sing ular) limiting case of logarithmic ones with merging bra nc h points. F or instance, lim a 1 → a  rw + A log w − a 1 w − a  = r w + α w − a with A = α/ ( a − a 1 ). All logarithmic, rationa l and “mixed” solutions belong to the same class character- ized b y the proper t y that the first deriv a tiv e of S + is a rational function (or, equiv alently , the fir s t deriv ativ e o f S is a mesomor phic function in D ( t )) . IV. MUL TI-CUT SOLUTIONS WITH ANAL YTIC CAUCHY DENSITIES OF GENERAL TYPE A. Motiv ation It is k no wn that all ratio nal so lutions (except for tho se of the form f ( w , t ) = r ( t ) w + u ( t ) + A ( t ) /w , which de- scrib e a self-s imilar growth of ellipse) cease to ex ist in a 7 finite time, because the dynamics gives rise to a cusp-lik e singularity of the in terface. These ph ysically meaningless singularities just signify tha t the sur face tension effects can not b e negle c ted in a vicinity of highly c urv ed parts of the interface. Nevertheless, a c onsiderable sub class of the purely loga r ithmic solutions is well defined for all po sitiv e times a nd describes a non-singula r in terface dy- namics at zero surface tension [1 0, 11, 12, 13, 25]. F ur - thermore, e a c h loga rithmic ter m in (31) has a clear geo- metric in terpretation of a fjord o f oil with p ar al lel wal ls left b ehind the adv ancing in terface. Its vertex (the stag- nation p oint) is loc a ted at z k − A k log 2, the width is π | A k | , and the angle b e t ween its cen tral line and the r e al axis in the physical plane is a r g A k (for mo re details se e [12, 13, 25]). This interpretation was found to be in an excellent agr eemen t with some exp eriments (see FIG.2 in [26] and numerical work [2 7]) . How ev er, mor e often than not, fjords of oil left behind the moving fr o n ts ha ve non-p ar al lel wal ls . Moreover, their w alls are not alwa ys straight, but more o ft en c ur v ed, and a non-zer o opening angle a lo ng the fjord was observed [28, 29]. Suc h sha pes of fjords can no t be explained by co nformal maps with a finite num ber of logarithmic or ratio na l terms. This was a significant motiv atio n for us to search for a more general class of LG solutions. F rom mathema tical p oin t of view, it also lo oks quite natural to extend the method developed ab o ve to solutions with singula rities of mo re general type than just poles or logar ithmic branch p oints. Below we apply the strategy outlined a t the end of Section II to derive a closed set of equations for the case when S + ( z ) ha s branch cuts of gener al t yp e with analytic Cauch y dens ities outside the interface. W e start from a m ulti-cut ansatz for S + ( z ) and then de r iv e an integral equation for the c o nformal map f ( w ). B. A Multi-Cut Ansatz for S + ( z ) The function S + ( z , t ) for z in D ( t ) is defined as a n analytic contin uation fr om the air do main, where it is given by the in tegral of Cauchy type (6). In the pro - cess o f analytic contin uation one necess a rily encoun ters singularities. T ypical singularities are branc h p oin ts and po les. In order to keep the function S + ( z ) single-v alued one has to introduce a system of branch cuts. The an- alytic contin uation is a chieved by a deformatio n of the int egratio n contour. Moving it towards infinit y as far as po ssible, w e c an write: S + ( z ) = S + (0) + X cuts Z z [ S + ( τ )] cut τ ( τ − z ) dτ 2 π i − X p oles res  z S + ( τ ) dτ τ ( τ − z )  + I | τ | = R,R →∞ z S + ( τ ) τ ( τ − z ) dτ 2 π i where [ S + ( τ )] cut denotes a disco n tinuit y (a jump) of the function [ S + ( τ )] a cross a br anc h cut. Let us deno te branch p oin ts by z m , by ana logy with logarithmic branc h po in ts from the previous section. As in that ca se, and for the same r easons, it is conv enient to make cuts Γ m from ∞ to z m . As b efore, w e assume that the num- ber o f branch p oin ts is finite. Assuming also that S + ( z ) do es not ha v e other singularities (in particular, p oles) in D ( t ), a single-v alued branch of this function can b e rep- resented as a sum of Cauch y t ype integrals with Cauch y densities P m ( τ ) alo ng the cuts plus a complex constant t 1 = S + (0): S + ( z ) = t 1 + X m Z z m ∞ , Γ m z P m ( τ ) τ ( τ − z ) dτ 2 π i . (41) The ansate (41) implies that the jump of S + on the cut Γ m is [ S + ( z )] = S + ( z + ) − S + ( z − ) = P m ( z ) , z ∈ Γ m (42) where z + (resp ectiv ely , z − ) tends to the p oin t z ∈ Γ m from the left (res p ectively , fr om the rig h t) side of the cut oriented from the low er limit of integration to the upp er one. There is a big freedom in how to draw the cuts. How- ever, if the Ca uc h y densities can b e a nalytically contin- ued from the cuts, then differen t c hoices of the cuts with the same endp oin ts ar e equiv ale n t. Indeed, conside r a n- other system o f cuts, ˜ Γ m , connecting the same p oin ts. Let ˜ P m be the Cauch y densities on these cuts. Subtra ct- ing the t wo in tegral represent ations o f the same function S + ( z ) one from the other, w e see that the “old” and “new” cuts combine in to close d loops and the Cauc hy in- tegrals ov er these loops v a nish for all z outside the lo ops. This means that the function ˜ P m is just an ana lytical contin ua tion o f the function P m , so P m ( τ ) should be re- garded as analytic function of the complex v ar ia ble τ defined b y analytic co n tinuation fro m the contour Γ m . Therefore, given analytic functions P m ( τ ) and branch po in ts , the choice of cuts Γ m (if a ll of them lie in D ( t )) is irrelev an t. F or the LG dynamics to b e reconstructed from the m ulti-c ut ansa te (41), all this mea ns that constants of motion a re the functions P m ( z ), the branch points z m and the co nstan t t 1 . Note that ∞ is a branch po in t unless P m P m ( z ) = 0. In this section, we consider only Cauch y densities that are ana ly tically extendable without singular ities from each cut to the whole domain D ( t ) including infinity . F or example, one may keep in mind functions regular in D ( t ) including infinity with fixed singular ities outside D ( t ). If the Cauch y densities are allowed to hav e singular ities in D ( t ), then the analy sis becomes substantially more com- plicated. In the next section, we extend the constr uction to the simplest pos sible singularity of the P m , a simple po le at infinity . This clas s of solutions includes the im- po rtan t case o f line ar densities . It is cle ar that for Cauch y densities which are constant in the ph ysical plane, P m ( z ) = 2 π i ¯ A m , the function S + is a linear combination of logarithms. So, the logarithmic solutions form a subset of the general mult i-cut o nes. On the other hand, the integral representation (41) sug gests that the m ulti-cut case ca n b e formally though t o f as a 8 limit of either rationa l o r (after integrating by parts in (41)) m ulti-logarithmic one, with infinitely many singu- larities forming a dense set of p oin ts co ncen trated alo ng contours. This remar k might be useful for the physical int erpretation of the mult i-cut s olutions. C. An Integral Equation for the Conformal Map The general form of the integral equation ob ey ed by the conformal map f ( w ) is (1 5) (or (20)). Similarly to the rational and logarithmic examples, it c an b e simplified by shrinking the integration contour to the cuts. Namely , plugging ¯ S + ( z ) = ¯ t 1 + X m Z ∞ ¯ z m z ¯ P m ( τ ) τ ( τ − z ) dτ 2 π i int o the r.h.s. of (15), we get f ( w ) = rw + u 0 + 1 2 π i X m I | ζ | =1 dζ w − ζ Z ∞ ¯ z m ¯ f (1 / ζ ) ¯ P m ( τ ) τ ( τ − ¯ f (1 / ζ )) dτ 2 π i = rw + u 0 − 1 (2 π i ) 2 X m Z ∞ ¯ z m dτ ¯ P m ( τ ) τ I | ζ | =1 ¯ f (1 /ζ ) dζ ( w − ζ )( ¯ f (1 /ζ ) − τ ) . Since τ lies in the co nformal reg io n D ( t ), there is a unique ζ ∗ inside the unit disk suc h that ¯ f (1 / ζ ∗ ) = τ . The last int egral can b e calcula ted b y tak ing the re s idue at the ζ ∗ (recall that w is outside). After that it is conv enient to change the in tegration v ariable τ → ξ connected with τ by the relation ¯ f (1 /ξ ) = τ . As a result, we obtain the following in tegral equatio n for the co nformal map [37]: f ( w ) = rw + u 0 − X m Z a m 0 ,γ m ¯ P m ( ¯ f (1 /ξ )) ξ − w dξ 2 π i . (43) Here the po in ts a m are s uc h that z m = f (1 / ¯ a m ). The int egratio n contours γ m are determined b y the change of the integration v aria ble. Specifica lly , consider a contour γ ∗ m from ∞ to 1 / ¯ a m in the mathematical plane, which is the pre-image of Γ m under the map f : Γ m = f ( γ ∗ m ), then γ m is the co n to ur connecting 0 and a m obtained as the inv ersion of the γ ∗ m with re spect to the unit cir- cle (the transfo r mation w → 1 / ¯ w ). All the parameter s in (43) except for co efficients of the functions ¯ P m ( z ) de- pend on t . The integration co ntours dep end on t as well. How ever, as is explained in the previous subsection, the int egral does not dep end on a particular shap e of the contours provided they do not intersect each other and the b oundary of the domain D ( t ). In what follows we do no t indicate the integration contours explicitly . Note that equation (43) can b e a nalytically contin ued inside the unit disk provided w do es not intersect the cuts from 0 to a m . In the logar ithmic case, P m ( z ) = 2 π i ¯ A m , the r.h.s. of equation (43) immediately gives the loga rithmic ansate FIG. 1: The corresp ondence betw een cut s in the mathemati- cal and ph ysical p la ne. for f with moving bra nc h p oin ts a nd constant co effi- cients. In fact it is the integral equation (43) that justifies the log arithmic ansate. Because o f imp ortance of the in tegral equa tion (43) we giv e here an alterna tive deriv ation “by hands” which is long er but less formal and probably mor e instructive. Using the same arg umen ts as in Sec tio n II I, one can see that for ea c h branch p oin t z m of the function S + ( z ) ther e is a corresp onding time-dep enden t br a nc h p oin t a m of the function f ( w ) determined b y the relation z m = f (1 / ¯ a m ). The conforma l map can then b e written a s f ( w ) = rw + u 0 + X m Z a m 0 ,γ m ρ m ( ζ ) ζ − w dζ 2 π i , (44) where ρ m ( ζ ) are time-dependent Cauch y de ns ities on the cuts γ m betw een 0 and a m . Let us fix a path Γ m from ∞ to z m lying en tirely inside the doma in D ( t ) in the physical plane. It is a n image of a path γ ∗ m connecting the p oin ts ∞ a nd 1 / ¯ a m outside the unit disk: Γ m = f ( γ ∗ m ). Let us r e fle c t γ ∗ m with resp ect to the unit circle (i.e., make the transformation w → 1 / ¯ w ). The r e flected path, γ m , connects the p oin ts 0 and a m inside the unit disk. W e take these γ m to b e the int egratio n paths in (44 ). T a k e a po in t w ∈ γ m and consider t wo po ints, w + and w − , whic h are limits to w fro m r espectively le ft and right sides of the branch cut γ m (oriented from 0 to a m ). Similarly to (42), w e hav e f ( w + ) − f ( w − ) = ρ m ( w ). The corres p onding p oin ts in the physical plane, z ± , tend from opp osite sides to the po in t z = f ( w ∗ ) ∈ Γ m , where w ∗ = 1 / ¯ w ∈ γ ∗ m is the image o f w under the inv er sion (see Fig. 1). Since conformal maps pr eserv e orientation, we can write z ± = f ( w ∗ ± ). Howev e r, the in version int erchanges the sides, i.e., w ∗ ± = 1 / ¯ w ∓ , so z ± = f (1 / ¯ w ∓ ). Acco rding to equations (10) r e written as  S = ¯ f ( ¯ w ) z = f (1 / ¯ w ) , (45) the v alues of the Sc hw a rz function at z ± are S ( z ± ) = ¯ f ( ¯ w ∓ ) and the discontin uity of the Sc hw a rz function is th us [ S ( z )] = S ( z + ) − S ( z − ) = ¯ f ( ¯ w − ) − ¯ f ( ¯ w + ) = − ¯ ρ m ( ¯ w ). As singula rities of S + and S in D ( t ) ar e the same, this is exactly the discontin uity of the function S + ( z ). W e thus conclude that o n γ m it holds ρ m ( w ) = − ¯ P m ( z ) , z = ¯ f (1 / w ) (46) 9 F ur ther more, since the r.h.s. is ana ly tically extendable to the whole ¯ D ( t ) and the function ¯ f (1 /w ) se ts up a conformal equiv a lence b e t ween ¯ D ( t ) and the unit disk, the ρ m ( w ) is analytically ex tendable from the cut γ m to the whole unit disk. Therefore, (46) actually holds everywhere in the unit disk. Plugging it into (44), we obtain the non-linear integral equation (43) fo r f ( w, t ). D. The Soluti on Scheme T o summarize, we hav e the following for m ula s which express constants o f motion through the time-dep enden t conformal map: t 1 = 1 2 π i I | w | =1 ¯ f (1 /w ) d f ( w ) f ( w ) , z m = f (1 / ¯ a m ( t ) , t ) . (47) They lo o k exactly like the cor responding for m ulas (32) for the logarithmic c ase. The main difference is that in general any explicit representation of the function f is not a priori av ailable. It is defined implicitly via the integral equation (43) . In principle, these equations s upplemen ted by a for- m ula for t below give a solution to the pr oblem in an im- plicit form, lik e in rational o r logar ithmic cases. Howev e r, the general multi-cut construction is s ubs tan tially more complicated becaus e b efore applying the scheme outlined in section I II one has to find a solution to the integral equation (43) with requir ed a nalytic prop erties. Clearly , such a s olution dep ends on a m , u 0 and r as parameter s: f ( w ) = f ( w | r, u 0 , { a m } ). Then (47) is a sy stem of equa- tions to determine them through the c o nstan ts of motio n. The missing equation which includes t can b e obta ined from (21) or (22 ). One o f its for ms is t = r 2 − 1 2 π i X m Z a m 0 ¯ P m ( ¯ f (1 / ζ )) d ( ¯ f (1 /ζ ) − r / ζ ) , (48) Another form is t = r 2 − X l,m Z a l 0 dζ 2 π i Z a m 0 d ¯ w 2 π i ρ l ( ζ ) ρ m ( w ) (1 − ζ ¯ w ) 2 (49) where ρ m ( ζ ) = −P m ( ¯ f (1 /ζ )). It is an analog of (35) (compare also with (29)). Mean while, a form ula for t 1 similar to (48) also exists: ¯ t 1 = u 0 − 1 2 π i X m Z a m 0 ¯ P m ( ¯ f (1 /ζ )) d log( ζ ¯ f (1 /ζ )) , (50) which follows fr om (17) at j = 0. In the case of con- stant densities, fo r m ulae (48) and (50) immediately g iv e expressions (33) a nd (3 2 ) for t and t 1 from section I II B . V. M UL TI-CUT SOLUTIONS WITH LINEAR CAUCHY DENSITIES The aim of this section is to elabo rate the imp ortant case o f linear Ca uc hy densities of the Sch warz function in the physical plane. This class o f solutions is char- acterized by the prop erty that the s econd deriv ativ e of S + is a ratio nal function (or, equiv a len tly , the second deriv ative of S is a meso morphic function in D ( t )). The int egral equation fo r confor mal map which is in general non-linear, b ecomes linear in this case. In the con text of the universal Whitham hierarch y , so lutio ns o f this t yp e were discussed in [24] (section 7.2) and in [23] (section 3.8). This clas s also includes logarithms and p oles o n the background of the mu lti-cut functions. Ho wev er, the approach of the previous s ection is not directly applica ble bec ause the integrals b ecome divergent. These divergences ar e artificial a nd can b e curb ed for the price of in tro ducing one mo re time-dep enden t pa- rameter. The in tegral e quation should b e mo dified. W e start, in s ubsection V A, with a more gener a l situation when the Cauchy densities P m ( z ) are a nalytic every- where in D ( t ) except for a p ole a t infinit y . In subsec- tion V B we sp ecify the r esults to the mo st imp ortant case of purely linear Cauch y densities. In subsection V C we consider , as a n example, some specia l “finger” pat- terns with Z N rotational symmetry (which a r e equiv a- lent to so lutions in a wedge with angle 2 π / N ). Their asymptotic form at la rge t is given by the known family of Z N -symmetric s elf-similar “fingers” descr ibed by hy- per geometric solutions to the integral equation (43) (see [14, 30, 31, 32, 33, 34, 35]). A. Cauch y Densities w i th a P ol e at Infinit y Let us consider the ca se when P m ( z ) ar e analytic ev- erywhere in D ( t ) except for a simple p ole at infinity . One immediately sees that neither the m ulti-cut ansate (41) nor the integral equation (44) c an be dir e ctly applied to this case because the integrals diverge. Nevertheless, it is po ssible to mo dify these form ulas in suc h a w a y that the divergences disapp ear. T o this end, co nsider a modified m ulti-cut a nsate: S + ( z ) = t 1 + 2 t 2 z + X m Z z m ∞ , Γ m z 2 P m ( τ ) τ 2 ( τ − z ) dτ 2 π i , (51) where t 1 and t 2 are a rbitrary co mplex constants (the first and the s e cond harmo nic mo ments) and we adopt the same con ven tions ab out the cuts Γ m as befor e . A sing le - v alued bra nc h of this function is fix ed by the conditions S + (0) = t 1 , S ′ + (0) = 2 t 2 . Let us plug it into the integral equation (19) with d = 1. A simple calculation similar to the one done in section IV C yields the integral equation f ( w ) = rw + u 0 + u 1 w − X m Z a m 0 ζ ¯ P m ( ¯ f (1 /ζ )) w ( ζ − w ) dζ 2 π i (52) 10 The gener a l s c he me of solution remains the sa me, with the only difference that now we ha ve o ne more time- depe ndent para meter (the co efficient u 1 ). Accor dingly , we need an extra equation connecting it with integrals of motion. The necessary equations, whic h gener alize (33) and (32), can b e o btained from (17), (18). They ar e: 2 ¯ t 2 = u 1 r + 1 2 π i X m Z a m 0 ¯ P m ( ¯ f (1 /ζ )) d  1 ¯ f (1 /ζ ) − ζ r  ¯ t 1 = u 0 − u 1 ¯ u 0 r − 1 2 π i X m Z a m 0 ¯ P m ( ¯ f (1 / ζ )) d  log( ζ ¯ f (1 /ζ )) − ¯ u 0 ζ r  t = r 2 − | u 1 | 2 − 1 2 π i X m Z a m 0 ¯ P m ( ¯ f (1 / ζ )) d  ¯ f (1 / ζ ) − r ζ − ¯ u 1 ζ  . (53) Let us briefly comment on a mor e general case of Cauch y densities with a hig her p ole at infinity (and ana- lytic everywhere else in D ( t )). It should be already clear how to pr oceed. If the leading term of P m ( z ) as z → ∞ is z d , then o ne should extract from S + ( z ) a p olynomial P d +1 k =1 k t k z k − 1 of degr ee d a nd r epresen t the remaining part (whic h is of or der O ( z d +1 ) as z → 0 ) as integrals along the cuts. Plugging this ansa te into (19), one ob- tains an integral equa tio n for the confor mal map co n ta in- ing a Laurels p olynomial rw + P d j =0 u j w − j in the r.h.s . So, apart from positions of branch p oin ts, ther e are d + 2 time-dep e ndent parameters r, u 0 , . . . , u d which are to b e connected with integrals of motion by fo r m ulae similar to (53). B. Linear Cauch y De nsities The case of linear homo geneous Cauch y densities is esp ecially impo rtan t. Set P m ( z ) = 2 π i c m z , (54) where c m are ar bitrary co mplex constants. The explicit form of the function S + is: S + ( z ) = t 1 + 2 t 2 z + z X m c m log  1 − z z m  , (55) It is clear that the integral equation (52) becomes linear: f ( w ) = rw + u 0 + u 1 w + X m ¯ c m Z a m 0 ζ ¯ f (1 /ζ ) dζ w ( ζ − w ) . (56) F o r m ulae (12) rea d: 2 ¯ t 2 = u 1 r − X m ¯ c m Z a m 0 ¯ f (1 /ζ ) d  1 ¯ f (1 / ζ ) − ζ r  ¯ t 1 = u 0 − u 1 ¯ u 0 r + X m ¯ c m Z a m 0 ¯ f (1 /ζ ) d  log( ζ ¯ f (1 /ζ )) − ¯ u 0 ζ r  t = r 2 − | u 1 | 2 + X m ¯ c m Z a m 0 ¯ f (1 /ζ ) d  ¯ f (1 /ζ ) − r ζ − ¯ u 1 ζ  (57) After so me simple transfor mations, they can b e bro ugh t to the form 2 ¯ t 2 = u 1 r + X m ¯ c m  log  a m ¯ z m r  + 1 r Z a m 0  ¯ f (1 /ζ ) − r / ζ  dζ  (58) ¯ t 1 = u 0 − 2 ¯ u 0 ¯ t 2 + X m ¯ c m  ¯ z m + ¯ u 0 log  a m ¯ z m r  − ¯ u 0 − r a m + Z a m 0  ¯ f (1 /ζ ) − r /ζ − ¯ u 0  dζ ζ  (59) t = r 2 − 2 r ¯ u 1 ¯ t 2 + X m ¯ c m  ¯ z 2 m 2 + r ¯ u 1 log  a m ¯ z m r  − ¯ u 2 0 2 − r ¯ u 1 − r 2 2 a 2 m − r ¯ u 0 a m + r Z a m 0  ¯ f (1 /ζ ) − r / ζ − ¯ u 0 − ¯ u 1 ζ  dζ ζ 2  (60) Let us specify the scheme of solution to the cas e of lin- ear dens ities. Suppo se one is able to find a so lution to the linear integral equation (56) dep ending o n the par ame- ters a m , u 0 , u 1 and r : f ( w ) = f ( w | r , u 0 , u 1 , { a m } ) (with fixed c m ). Then, since f (1 / ¯ a m ) = z m are co nstan ts of motion, we get a system o f equations for a m , u 0 , u 1 and r which b ecomes closed after adding equations (58), (59), (60). This system determines the pa rameters as implicit functions o f t . W e conclude the subsection with a r emark that the solutions of the linear integral equation (56) for the con- formal map f ( w , t ), which corresp ond to linear Cauch y density of the Sch warz function defined b y (5 5 ), deser v e a s pecial name in view o f their imp ortance. So, in what follows we will re fer to them a s to hyp er-lo garithmi c so- lutions , since they contain lo garithmic solutions (31) a nd the Gaus s hyp er -geometric function (see (68) b elow) as particular cas es. C. Example : Z N -Symmetric Solutions with N Radial Cuts In this subsection we c onsider p erhaps the simplest non-trivial application of the linear in tegral equation de- 11 rived ab ov e : gr owing patterns with Z N rotational sym- metry whose Sch w arz function has ex actly N radial branch cuts in D ( t ) with linear Cauch y densities. Let ω = e 2 π i/ N be the primitive ro ot o f unit y of degr ee N . The Z N -symmetry implies S ( ω z ) = ω − 1 S ( z ). More- ov er, this relation holds for b oth + and − parts of the Sch warz function sepa rately: S ± ( ω z ) = ω − 1 S ± ( z ). This prompts us to choose the constants o f motion in the for m c m = cω − 2 m +2 , z m = λω m − 1 , m = 1 , 2 , . . . , N , (61 ) with rea l po sitiv e c onstan ts c a nd λ , s o the function S + is S + ( z ) = cz N − 1 X m =0 ω − 2 m log  1 − z ω − m +1 λ  = − cz N − 1 (1 − 2 N ) λ N − 2 2 F 1  1 , 1 − 2 N 2 − 2 N , ( z /λ ) N  , (62) where 2 F 1 is the Gauss hyper geometric function. F o r the confo r mal map the Z N -symmetry mea ns f ( ω k w ) = ω k f ( w ), so the expansio n of the function f ( w ) /w go es in p o w ers of w − N : f ( w ) / ( r w ) = 1 + X k ≥ 1 f k w − kN where f k ≡ u kN /r . The reality of c and λ in (62) implies that the coe fficien ts f k are re a l, i.e., ¯ f ( w ) = f ( w ). I n what follo ws we ass ume that N ≥ 3, so the co efficients u 0 and u 1 in the expansion of f ( w ) (2) are a lways identically zero. W e use the integral equa tion (56). Substituting our data, we get f ( w ) = rw + c N − 1 X m =0 ω 2 m Z aω m 0 ζ f (1 /ζ ) dζ w ( ζ − w ) (63) where a is a p oint betw een 0 and 1 suc h that f (1 /a ) = λ . Cho osing the integration paths to b e straight lines from 0 to aω m and using the Z N -symmetry , w e a r riv e at the equation f ( w ) = rw + cN Z ∞ 1 /a wf ( x ) dx 1 − w N x N , (64) where c is a cons ta n t a nd r , a dep end on time. Let f ( w ) = f ( w | r, a ) b e a solution to this integral eq uation with para meters r , a , then three pa rameters r , a and t are connected by tw o equa tions which determine r , a as implicit functions of t . The first of these equations is obtained from the fact tha t λ = f (1 /a ( t ) , t ) is a constant of mo tio n b y substituting w = 1 /a in to (64): r = λa − cN Z ∞ 1 /a f ( x | r , a ) dx 1 − x N a − N , (65) the second is one or another version of the for m ula for t in terms of f ( w ). FIG. 2: Laplacian Growth in w edge geometry . Air is pushed into oil trough the corner of t h e wedge. β is the fjord angle. W e ha ve obtained a one-parametric family of ex a ct so- lutions to the LG des c ribing growth of N symmetric air fingers o r, wha t is mathematically the same, growth of an air finger in the wedge with interior ang le 2 π / N (Fig. 2). It is not clear at the moment whether the conformal map f ( w ) is a v aila ble in a clo s ed analytic form. How ever, the solution can be ana lyzed numerically (see appendix A). What is the meaning of the par ameter c ? T o answer this question, consider the limit t → ∞ , wherein, as one can easily show, a → 1. Then equation (64) b ecomes ident ical to the linea r integral equation describing self- similar gr o wth of Z N -symmetric fingers (or finge rs in the wedge with interior angle 2 π / N ): f ( w ) = rw + N sin π β π Z ∞ 1 wf ( x ) dx 1 − w N x N (66) with the “b oundary conditio n” f (1) = 0. As shown in our earlier pap er [16], β is the interior ang le of the o il fjord b et ween t w o neighbor ing air fingers. Since c in (64) is a c o nstan t of motio n, w e conclude that se tt ing c = sin π β π (67) we can in terpret β a s the a symptotic fjord a ng le. Meanwhile, an analytic so lutio n of (66 ) in terms of the Gauss hyper geometric function is av aila ble : f ( w ) = rw (1 − w − N ) β 2 F 1  β , β − κ 1 − κ , w − N  . (68) where κ = 2 N . F o r more details, s e e [16], where the self-simila r case was studied in detail. 12 The in tegral equatio n (64) is equiv a len t to an infinite system o f linear equations for the co efficien ts f k . T o r ep- resent it in this for m, let us introduce a function F via f ( w ) = rwF ( w − N ) , F ( w ) = ∞ X k =0 f k w k , f 0 = 1 , then the integral equation (64) b ecomes F ( w ) = 1 + c Z α 0 x − κ F ( x ) dx x − w − 1 , (69) where α = a N . Subs tituting the T aylor expansio n o f F and compar ing the co efficien ts, we get f j = − c ∞ X k =0 α j + k − κ j + k − κ f k , j ≥ 1 . (70 ) (How ever, b ecause the r .h.s. contains f 0 = 1, this is not a n eigenv alue equation but r a ther an inhomog eneous system o f linear equations.) It is also useful to note the formula for t , t = r 2 + 1 2 N cλ 2 + c r 2 X n ≥ 0 α n − κ f n n − κ , (7 1) which follows fr om (60). VI. DISCUSSION AND CONCLUSION In conclusion, let us brie fly comment on the following three ma jor p oin ts of this work: • The inv erse p oten tial problem in tegral equation, (15) (or (20)), • The equations for multi-cut so lutions, (41 ) and (43), • The equations for hyper -logarithmic solutions, (55) and (56). The inverse p otential pr oblem e quation (15): The formu- lation of the LG pr oblem as a linear growth of a domain area with conserved harmonic moment s requires to solve the in verse p otent ial problem, namely to find the confor- mal map f ( w ) from the function S + ( z ) = P ∞ k =1 k t k z k − 1 , whose T aylor co efficien ts ar e conserved harmonic mo- men ts of the growing domain and from the ar ea/time t . While it is straightforw ard to obtain S + ( z ) and t from f ( w ) (the dir ect p oten tial problem), it is a well-known challenge to do it the o ther way around (the inv erse p o- ten tial pr oblem) [3]. The reformulation of the in verse po ten tia l problem as the nonlinear in tegral equation (15) (or (20)) for the function f ( w ), assuming that S + ( z ) is known, app ears to b e ex tremely useful. In particular, it allows us to obtain some important classes of solutions to the LG proble m with prescrib ed integrals of motion (har- monic momen ts), including those with arbitr ary branch cuts as well as those with po le singularities. This inte- gral equation is also exp ected to be o f significant help in a future work on LG and r elated problems of interface dynamics. The e quation for multi-cut solutions, (43): A mathe- matical motiv ation fo r this w ork was a strong feeling that rational and logarithmic conformal maps do not exhaust the list of exact so lutions of LG at zero sur face tensio n. A physical motiv a tion came from a long standing need to describ e most gener a l moving interfaces in Hele-Shaw ex- per imen ts with neg ligible surface tension. As w as alr eady said in section IV A, while finite linear c o m binations of logarithms ca n describ e interface dynamics without finite time singula rities, they fail to desc r ibe an interface with non-para lle l fjords walls. This motiv ated us to lo ok for a mo r e general family of solutions to the LG equation (3). W e exp ect that the multi-cut solutions pre sen ted in this pap er do explain the exp erimen ts [28, 29, 36] where formation and development of oil fjords with non-pa rallel and/or non- s traigh t w alls was observed. Two s pecial kinds of branch p oints and cut s ingular- ities were already considered in e arlier w orks on Lapla- cian growth. F ra c tional time-independent branch p oin ts app eared in studies o f self-similar sing ular interfaces [14, 15, 16], while logar ithmic branch cuts with constant Cauch y densities app eared in [10, 11, 12, 13, 25]. W e hav e sho wn that a ll of them are just v arious sp ecial cases and limits of the presented g eneral construction. In the light of our approach, it also b ecomes c le a r why the p ole and loga rithmic solutions ar e sp ecial: the Cauch y densi- ties are particularly s imple in these cases, so the integral equation can be easily s olv ed. It a lso s ho ws a r o ad to obtain new families o f exact solutions. The e quation for hyp er-lo garithmi c solutions, (56): The ma thematical significance o f the hyper-lo g arithmic solutions is that they corresp ond to the linear integral equation (56), which ought to be mu ch more accessible to analytic tr eatmen t than the g eneral nonlinea r equa - tion (43) for m ulti-cut solutions. The physical impor - tance lies in the b e lief that the hyper -logarithmic solu- tions des c r ibe fjor ds with a constant op ening ang le, in agreement with viscous fingering exper imen ts [29]. The int erface dynamics simulated in a wedge, shown on Fig- ure 3, seems to confirm this belie f, if to co ns ider fjords central lines a s walls of virtual w edges in accorda nce with [28]. A thor o ugh geometric analysis of the corresp onding int erface dyna mics based o n the integral equation (56) will b e published elsewhere . VII. ACKNO WLEDGEMENT Ar.A is grateful to W elch F oundation for the pa r - tial s upport. His work was a lso partially suppo rted by NF PhD-0757 992 grant . All a utho r s gratefully ackno wl- edge a significant help from the pro ject 200704 83ER at 13 the LDRD progr ams of LANL: the w ork of M.M-W. on this problem was fully supp o rted b y this pr o ject, while t wo other a utho r s were partia lly supp orted by the sa me grant during their visits to LANL in 20 0 8. The work of A.Z. was a lso pa rtially supp orted by grants RFBR 08- 02-00 287, RFBR-06-0 1-92054- C E a , Nsh-303 5.2008.2 and NW O 047.01 7.015. A.Z. thanks the Galileo Ga lilei Insti- tute for Theoretical P h y sics for the ho spitalit y and the INFIN for pa r tial supp ort during the completion of this work. APPENDIX A: NUMERICAL SOLUTION OF EQUA T ION (64) Equation (64) with c given by (67) can be rewritten in the following wa y: f ( w, t ) = r ( t ) w − N sin π β π w I | ζ | =1 f ( ζ , t ) log (1 − a ( t ) ζ ) 1 − w N ζ N dζ 2 π i , f (1 /a ( t ) , t ) = λ, (A1) W e are lo oking for a solution cons is ten t with the Z N - symmetry: f ( w, t ) = r ( t ) w + ∞ X k =1 f k ( t ) w − N k +1 (A2) Equation (A1) then takes the form f n ( t ) = − r ( t ) sin π β π α n − κ ( t ) n − κ − N sin π β π ∞ X k =1 f k ( t ) α n + k − κ ( t ) n + k − κ . (A3) where κ = 2 / N , α = a N . Using the relation λ = f (1 /a ( t ) , t ) and re- denoting f k ( α ) ≡ f k ( t ( α )), r ( α ) ≡ r ( t ( α )), we get r ( α ) = α κ/ 2 λ − ∞ X k =1 f k ( α ) α k , (A4) and ∞ X k =1 f k ( α ) k α n + k n + k − κ − π ( n − κ ) sin π β α κ f n ( α ) = λα n + κ/ 2 (A5) This equation is easily solved n umerically for a ny giv en α . The conformal radius r and the time a r e then found from (A4) and (22) (or (71)) r espectively . The solution is shown in figur e 3. 0 0.5 1 1.5 r 0 0.5 1 1.5 2 t α 2/κ =0.999 α 2/κ =0.99 α 2/κ =0.9 α 2/κ =0.7 α 2/κ =0.3 α 2/κ =0.1 α 2/κ =0.01 α 2/κ =0.001 FIG. 3: (color online) The results of the n umerical solution of th e equation (64) for N = 6, β = 0 . 4 / N . The left panel sho ws the dep endence of t he area t on the conformal radius r , the right panel sho ws the grow ing finger at differen t times. REFERENCES [1] K .A .G illo w and S.D.Ho wison, A bibliograph y of free and mo ving boun dary problems for Hele-Sh a w and Stokes flow . http:// www.maths.o x .a c.uk/ ~ how ison/Hele-Shaw/ . [2] I . Kric heve r, M. Mineev-W einstein, P . Wieg- mann, and A. Zabrodin, Physica D 198 , 1 (2004), arXiv:nlin.SI/0311005. [3] B. Gustafsson and A. V asil’ev, Conformal and Poten- tial A nalysis in Hele-Shaw Cel ls (A dvanc es i n Mathemat- ic al Fluid Me chanics) , (Birkh¨ auser, Basel, 2006), 1st ed., ISBN 37643 77038. [4] S . Richardson, J. Fluid Mech., 56 , 609-618 (1972). [5] M. Mineev- W einstein, P . Wiegmann, A. Zabrodin, Ph ys. Rev. Lett., 84 , 5106–5109, (2000). [6] I .Kos tov, I.Krichev er, M.Mineev-W einstein, P .Wiegmann, and A .Za bro din, in R andom Matrix Mo dels and Their Applic ations , ed. by P .Bleher and A.Its, (Cam bridge: Cam bridge Academic Press, 2001), vol .40 of R andom Matric es and Their Applic ations (MSRI publi c ations) , pp. 28 5-299. [7] P . Kufarev, Doklady Ak ademii Nauk S SSR, 60 , 1333 (1948). [8] B. S hraima n, and D. Bensimon, Phys. R ev. A, 30 , 2840 (1984). [9] M. Mineev-W einstein, Physica D, 43 , 288 (199 0). [10] S. D. Howison, Journal of Fluid Mechanics, 167 , 439 (1986). [11] D. Bensimon, and P . Pelc ´ e, Phys. Rev. A, 33 , 4477 (1986). [12] M. B. Mineev-W einstein and S. P . Da wson, Phys. Rev . E 50 ,R24 (1994). [13] S. P . Dawson and M. B. Mineev-W einstein, Physica D, 83 , 383(1994 ). [14] M. Ben Amar, Ph ys. Rev. A , 44 , 3673 (1991). 14 [15] Y. T u, Ph ys. Rev. A, 44 , 1203 (1991). [16] Ar. Abanov, M. Mineev-W einstein, and A. Zabro din, Physica D, 235 , 62 (200 7). [17] H. S. Hele–Shaw , Nature, 58 , 1489 (1898). [18] D. Bensimon, L. P . K adanoff, S . Liang, B. I. Shraiman, and C. T ang, Rev. Mod. Phys., 58 , 97 7 (1986). [19] L. Galin, Dokl. Ak ad. Nauk. S.S.S.R., 47 , 246 (1945). [20] P . Polubarino va– Ko china, Dokl. Ak ad. Nauk. S .S.S.R., 47 , 254 (194 5). [21] S. D. Howis on, European Journal of Applied Mathemat- ics, 3 , 209 (1992). [22] P . J. D a vis, T he Schwar z f unc tion and its applic ations. (The Carus Math. Monographs, No. 17, The Math. As- sociation of America, Buffalo, N.Y., 1974. ) [23] I. Krichev er, A. Marshak o v, and A . Zabro din, Co mmun. Math. Ph ys., 259 , 1 (2005). [24] I. Krichev er, Commun. Pure App l. Math., 437 , 437 (1994). [25] S. P . Dawson and M. Mineev-W einstein, Phys. R ev . E, 57 , 3063 ( 19 98). [26] L. Paterson, Journal of Fluid Mechanics, 113 , 513 (1981). [27] L. M. Sander and P . Ramanlal and E. Ben Jacob, Ph ys. Rev. A, 32 , R 31 60 (1985). [28] E. Lageunesse and Y. Couder, Journal of Fluid Mec han- ics, 419 , 125 (2000). [29] L. Ristroph, M. Thrasher, M. Mineev-W einstein and H. L. Swinney , Physical R eview E, (RC), 74 , 015201-1 (2006). [30] H. Thome, M. Rabaud , V . Hak im, and Y. Couder, Physics of Fluids A: Fluid Dynamics, 1 , 224 (1989). [31] M. Ben Amar, Ph ys. Rev. A , 43 , 5724 (1991). [32] R. Com b esco t, Phys. Rev. A, 45 , 873 (1992). [33] L. Cummings, Euro. J. Appl. Math., 10 , 547 (1999). [34] S. Richardson, Euro. J. Appl. Math., 12 , 665 (2001). [35] I. Markina and A . V asil’ev, Euro. J. Appl. Math., 15 , 781 (2004). [36] O. Praud and H. L. S wi nney , Phys. Rev. E, 72 , 011406 2005). [37] Strictly sp eaking, it is a system of tw o in tegral equations for tw o functions, f ( w ) and ¯ f ( w ).