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Regularization of Hele-Shaw flows, multiscaling expansions

and the Painlevé I equation ∗

L. Mart́ınez Alonso1 and E. Medina2

1 Departamento de F́ısica Teórica II, Universidad Complutense

E28040 Madrid, Spain
2 Departamento de Matemáticas, Universidad de Cádiz

E11510 Puerto Real, Cádiz, Spain

Abstract

Critical processes of ideal integrable models of Hele-Shaw flows are considered. A regularization
method based on multiscaling expansions of solutions of the KdV and Toda hierarchies character-
ized by string equations is proposed. Examples are exhibited in which the tritronquée solution of
the Painlevé-I equation turns out to provide the leading term of the regularization

Key words: Hele-Shaw flows, integrable hierarchies, multiscale expan-
sions, Painlevé-I equation.
1991 MSC: 58B20.

∗Partially supported by MEC (Ministerio de Educación y Ciencia) project FIS2005-00319 and ESF (European Science
Foundation) programme MISGAM

1

http://arxiv.org/abs/0710.3731v1


1 Introduction

A Hele-Shaw cell is a narrow gap between two plates filled with two fluids: say oil surrounding
one or several bubbles of air. Several dispersionless (quasiclassical) limits of integrable systems
have been found [1]-[6] which provide ideal models of Hele-Shaw flows in the absence of surface
tension. Moreover, the same integrable structures also arise in random matrix models of two-
dimensional quantum gravity [7]. Integrable systems of dispersionless type are solved by means
of hodograph equations, so that generic initial conditions reach points of gradient catastrophe in
a finite time. In the ideal Hele-Shaw models this feature gives rise to the cusp formation in the
motion of the interface, while in random matrix models it manifests itself as the critical points of
the asymptotic expansions for large matrix dimension N . In both cases a regularization mechanism
of the underlying integrable models is required. As it is well known in random matrix theory [7]-
[10], the double scaling limit method provides a regularization scheme of the large N expansions
which leads to models of two-dimensional quantum gravity. On the other hand, recent work [4]-[6]
suggests the use of methods of asymptotic solutions of integrable systems [11]-[13] to cure the
singularities in Hele-Shaw flows.

The present work is concerned with the regularization of a family of critical Hele-Shaw processes
[4]-[6]. We mainly consider the case in which the interface develops an isolated finger which is close
to becoming a cusp [4]-[5]. Then at a small scale the boundary of the finger tip is described by a
curve

Y (X) = P (X)
√

X − v(x, t1),

where (X,Y ) are Cartesian coordinates, P (X) is a given polynomial and v(x, t1) is a particular
solution of the dispersionless KdV (Hopf) equation 2∂t1v = 3 v vx. Here x ∼ Q (t − tc), where t is
the physical time, tc is the critical time and t1 is a deformation parameter. The starting point of
our analysis is the fact that v satisfies the dispersionless limit of the string equations [14] which
characterize the one-matrix models of topological and two-dimensional quantum gravity [15]-[17].
Then, from the quasi-triviality property [18] of the KdV hierarchy, there is a unique solution u of
the dispersionful KdV equation

∂t1u =
1

4

(
ǫ2 uxxx + 6uux

)
, u =

∑

k≥0

ǫ2k u(k), (1)

and its associated hierarchy such that v = u(0). More precisely [14], the τ -function associated with
u is the large N limit of the Kontsevich integral over hermitian matrices

τN (t) : =
AN (Θ)

BN (Θ)
, Θ := diag(θ1, . . . , θN ), k tk = ǫTr(−Θ)k,

AN (Θ) : =

∫
X. exp Tr(X2 Θ − 1

3
X3), BN (Θ) :=

∫
X. exp Tr(X2 Θ).

In this KdV picture, singular Hele-Shaw processes correspond to critical points of the expansion of
u, which in turn are associated to points of gradient catastrophe for u(0). In our work we follow the
same regularization procedure that is used in the one-matrix model description of two-dimensional
quantum gravity [8]-[10] : we apply a multiscaling limit method to obtain the leading term of the
regularization of u near critical points. Then we use it to continue the Hele-Shaw flow on critical

2


regions. We notice that according to recent work [19, 20], the multiscale regularization of KdV
solutions gives a correct asymptotic approximation near the edges of the oscillatory zone (Whitham
zone) which emerges at points of gradient catastrophe. We illustrate our strategy by regularizing
the (2, 5) critical finger studied in [5].

We also consider in this paper the critical processes of break-off and merging of Hele-Shaw
bubbles [6]. The same analysis as in the critical finger case applies, but now the ruling integrable
structure is supplied by the solution of the Toda hierarchy which underlies the large N limit of the
partition function of the Hermitian matrix model

τN :=

∫
H. exp

(
tr(

∑

k≥1

tk Hk)
)
. (2)

In this case the method is illustrated with an example of regularization for the merging of two
bubbles.

Our analysis makes use of the method developed by Takasaki and Takebe [21] for determining
solutions of integrable systems by means of string equations (see also [22]-[24]). In the examples
considered in both cases, KdV and Toda structures, the leading term of the regularization turns
out to be provided by a particular solution of the Painlevé-I equation (P-I)

Wξξ = 6W 2 − ξ, (3)

the so called tritronquée solution discovered by P. Boutroux [25]. This is the same function which
appears in the study of some critical processes in plasma [26] as well as in the analysis of the
critical behavior of solutions to the focusing nonlinear Schrödinger equation [27]. As it is known, a
different solution of P-I emerges in the random matrix models of two-dimensional quantum gravity
[7].

2 Hele-Shaw flows and the KdV hierarchy

In the set-up considered in [4, 5] the cell is permeable to air but not oil. When air is injected the
bubble develops a finger whose tip is pushed away and may become a cusp. By assuming that the
finger is symmetric with respect to the X-axis and that the cusp is formed at the origin, then near
the origin the finger turns to be described by a curve of the form

Y (z) :=
(∑l+1

k=1(k +
1

2
) tk z2k−1

√
z2 − v

)
⊕

√
z2 − v, X = z2. (4)

Here the subscript ⊕ denotes the projection of z-series on the positive powers, and tk, k ≥ 1 are
deformation parameters. The function v stands for the distance between the tip and the origin
and it is determined by imposing the asymptotic behaviour

Y (z) =

l+1∑

k=1

(k +
1

2
) tk z2k−1 +

x

2 z
+ O(z−3), z → ∞, (5)

where the coefficient x is proportional to time x ∼ Q (t − tc). The resulting equation for v is the
hodograph equation

H(t, v) :=
∑

k≥1

(2 k + 1) tk rk(v) + x = 0, (6)

3


where t := (x, t1, t2, . . .) with tk = 0, k > l+1, and rj are the coefficients of the generating function

r :=
z√

z2 − v
=

∑

k≥0

rk(v)

z2k
, r0 = 1.

The hodograph equation (6) is the basic piece to solve the system of string equations

(z2)− = 0, (m z−1)− = 0. (7)

for the Lax-Orlov functions of the dispersionless KP (dKP) hierarchy

z = p + u0 + O(p−1), m =

∞∑

k=1

(2 k + 1) tk z2k + x + O(z−2), (8)

{z,m} := zp mx − zx mp = 1. (9)

Here and henceforth the subscripts − and + denote the projection of p-series on the strictly negative
and positive powers, respectively. According to Theorem 1.5.1 of [21], if (z,m) are solutions of (7)
satisfying the asymptotic forms (8), then they verify the dKP hierarchy. More concretely, as the
first string equation means that z2 = p2 + v, then the function v verifies the dispersionless KdV
(dKdV) hierarchy ∂tjv = 2 ∂x rj+1.

To solve the second string equation (m z−1)− = 0 and satisfy the asymptotic behaviour (8), one
sets m z−1 =

∑∞
k=1(2 k + 1) tk(z2k−1)+. Thus by taking into account that (z2k−1)+ = (z2k−2 r)⊕ p,

it follows that (7) reduces to (6). In particular from (4) and (5) we may identify

Y (z) =
m(z)

2 z
,

so that the dynamics of the curve Y = Y (X) with respect to t is governed by the dispersionless
KdV hierarchy. The problem is that near critical points (tc, vc)

∂H

∂v

∣∣∣
tc,vc

= . . . =
∂m−1H

∂vm−1

∣∣∣
tc,vc

= 0,
∂mH

∂vm

∣∣∣
tc,vc

6= 0, m ≥ 2, (10)

the solutions v of (6) are multivalued and have singular derivatives (gradient catastroph). These
situations correspond to the critical regimes of the Hele-Shaw fingers described by (4).

In order to find regularizations of these Hele-Shaw flows we consider solutions of the disper-
sionful version of the KdV equation

∂t1u =
1

4

(
ǫ2 uxxx + 6uux

)
. (11)

and the higher members of its hierarchy ∂tj u = 2∂xRj+1(u). Here Rj are the Gel’fand-Dikii
polynomials determined by

ǫ2 (R Rxx − 1

2
R 2

x ) − 2 (z2 − v)R2 + 2 z2 = 0, R =
∑

n≥0

Rn(u)

z2n
, (12)

or by the third-order differential equation

∂x Rn+1 =
(1

4
ǫ2 ∂3

x + v ∂x +
1

2
ux

)
Rn. (13)

4


Our first observation is that there is solvable dispersionful version of the string equations (7)
given by

(L2)− = 0,
1

2
(M L−1 − ǫ

2
L−2)− = 0. (14)

Here L and M are Lax-Orlov operators ([L,M ] = ǫ)

L = ǫ ∂x + u1 ∂−1
x + · · · , M =

∑

j≥1

(2 j + 1) tj L2 j + t0 + O(L−2), (15)

of the dispersionful KP hierarchy. Now, the ± parts of a pseudo-differential operator denote the
truncations of ∂x-series in the positive and strictly negative power terms, respectively. According
to Proposition 1.7.11 of [21], given a solution (L,M) of (14) satisfying (15) and [L,M ] = ǫ, then
they are Lax-Orlov operators of the dispersionful KP hierarchy. The first string equation in (14)
constitutes the KdV reduction condition and leads to a Lax operator of the form

L = (ǫ2 ∂2
x + u)

1

2 .

The second string equation together with the asymptotic condition on M and [L,M ] = ǫ can be
satisfied by setting

M L−1 − ǫ

2
L−2 =

∞∑

j=1

(2j + 1) tj (L2j−1)+.

Thus, by taking into account the identity [L2, (L2j+1)+] = −2ǫ ∂x Rj+1, the problem reduces to
finding solutions of the form

u =
∑

k≥0

ǫ2k u(k), (16)

verifying ∑

k≥1

(2k + 1) tkRk + x = 0, (17)

or equivalently ∮

γ

dz

2πi z
Vz R + x = 0, (18)

where V denotes the function V :=
∑

j≥1 z2j+1 tj and γ is a large positively oriented closed path.
To solve this equation for u one may use (12) to determine the coefficients of the ǫ-expansion

R =
∑

n≥0

ǫ2n R(n), R(0) =
z

(z2 − u(0))
1

2

, (19)

and substitute them in (18) to get the system

∮

γ

dz

2πi z
Vz(z)R(l)(z) + δl0 x = 0. (20)

From (12), (19) and (20) an iterative scheme for obtaining the expansion (16) of u follows.
In particular, for l = 0 it reduces to the hodograph equation (6) for u(0). This means that the
expansion (16) is not valid near critical points (10) so that a different expansion must be used to
construct a regularization of the solutions of (6) on critical regions.

5


3 Multiscaling expansions and asymptotic matching

Given a m-th order critical point (tc, vc) (10) of (6), let us introduce a new small parameter ǫ̃ and
new variables t̃j and x̃ given by

ǫ̃ := ǫ
2

2m+1 , tj = tc,j + ǫ̃m t̃j, x = xc + ǫ̃m x̃. (21)

Let us now look for solutions to (18) of the form

u = vc +
∑

n≥1

ǫ̃n ũ(n), (22)

where now R is expanded in the form R =
∑

n≥0 ǫ̃n R̃(n). To determine the coefficients of R, we

first observe that ǫ ∂x = ǫ̃1/2 ∂x̃, so that (12) can be rewritten as

ǫ̃ (R Rx̃x̃ − 1

2
R2

x̃) − 2 (z2 − u)R2 + 2 z2 = 0. (23)

From this equation and taking into account that vc,x̃ ≡ 0 one deduces a recursion relation for the

coefficients R̃(n) and that they can be expressed in the form

R̃(n) = R̃(0)
n∑

r=1

G̃n,r

(z2 − vc)r
, R̃(0) =

z

(z2 − vc)
1

2

, (24)

where the functions G̃n,r are differential polynomials in ũ(k), 0 ≤ k ≤ n − r + 1. In particular,

from (23) it follows that the leading coefficients Gn := G̃n,n, G0 = 1 satisfy the same recursion
relation as that arising from (12) (with ǫ ≡ 1) for the Gel’fand-Dikii polynomials Rn. In other
words

Gn = Rn(ũ(1)).

If we now substitute (21)-(22) in (18) and identify coefficients of ǫ̃-powers we get the system of
equations





∮

γ

dz

2πi z
Vz(tc) R̃(k) + δk0 xc = 0, k = 0, . . . ,m − 1,

∮

γ

dz

2πi z
Vz(tc) R̃(k) +

∮

γ

dz

2πi z
Vz(t̃) R̃(k−m) + δkm x̃ = 0, k ≥ m.

(25)

Since vc is a m-th order critical point of (10) we have that

∮

γ
dz

Vz(tc)

(z2 − vc)
2k+1

2

+ δk0 xc = 0, k = 0, . . . ,m − 1.

Hence, in view of (24) the first m equations of the system are identically satisfied, while the
remaining ones determine recursively the coefficients ũ(n) for n ≥ 1. In particular for k = m we
get the following differential equation for the leading contribution ũ(1)

(∑

j≥1

cjm(vc) tc,j

)
Rm(ũ(1)) +

∑

j≥1

cj0(vc) t̃j + x̃ = 0, (26)

6


where Rm(ũ(1)) is the m-th Gelfand-Dikii polynomial in ũ(1) and

cjr(vc) = (2j + 1)

∮

γ

dz

2πi

z2j

(z2 − vc)
2r+1

2

From (26) it follows that ṽ(1) depends on the rescaled variables t̃ through the linear combination∑
j≥1 cj0(vc) t̃j + x̃, so that ∂t̃j

ũ(1) = cj0(vc) ũ
(1)
x̃ .

Having in mind the applications to Hele-Shaw flows, we must match the solutions (16) and (22)
for tk = tc,k, ∀k ≥ 1 and x varying on some overlap interval such us as ǫ → 0+ we have x−xc → 0−

and x̃ → −∞. To this end we observe that since u(0) satisfies the hodograph equation (6), then
near an m-th order critical point (tc, vc) it behaves as

u(0)(x, tc,1, tc,2, . . .) ∼ vc + m
√

c (x − xc), c := −m!
(∂mH

∂um
(tc, vc)

)−1
. (27)

Hence, the solutions u = u(0) +O(ǫ) and u = vc + ǫ̃ ũ(1) +O(ǫ̃2) match to first order in ǫ̃ provided
ũ(1) is a solution of the differential equation (26) such that

ũ(1) ∼ m
√

c x̃, x̃ → −∞. (28)

Let us illustrate our analysis by studying the (2,5) critical finger considered in [5]. If we set
tj = 0 for all j ≥ 2 except t3 = 2/7, the hodograph equation (6) for u = u(0) becomes

5

8
u3 +

3

2
t1 u + x = 0, (29)

so that we have

u(0) =

(
2

5

)2/3 (√
5(4t13 + 5x2) − 5x

) 1

3

−
2 3

√
2
5 t1

(√
5(4t13 + 5x2) − 5x

) 1

3

. (30)

We consider the case t1 < 0 which leads to a cusp of the type Y 2 ∼ X3 which can not be continued
[5]. The hodograph equation (29) has a 2-nd order critical point at

xc = −tc,1

√
−4

5
tc,1, vc =

√
−4

5
tc,1,

so that the solution behaves as

u ∼ vc +

√
− 8

15 vc
(x − xc), x → x−

c .

If we take t1 = t1,c (i.e. t̃1 = 0) equation (26) becomes

ũ
(1)
x̃x̃ + 3 ũ(1)2 = − 8

5 vc
x̃, (31)

and the matching condition requires

ũ(1) ∼
√

− 8

15 vc
x̃. (32)

7


By introducing the rescalings

W := −1

2

(5 vc

4

)2/5
ũ(1), ξ := −

( 4

5 vc

)1/5
x̃,

we have that the differential equation (31) reduces to the Painlevé I (P-I) equation (3), while the
matching condition reads

W ∼ −
√

ξ/6, ξ → ∞. (33)

As it is known [28], the tritronquée solution discovered by P. Boutroux [25] is the unique solution
of P-I having no poles in the sector |arg ξ| < 4π/5 for sufficiently large |ξ|. Moreover, it has no
poles on the positive real axis [29].

One may now proceed [27, 29] by taking a numerical approximation to the tritronquée solution
for large positive values of ξ and use it to supply initial data for the P-I equation. To this end we
have taken the approximation provided by Eq.(3.3) of [28], and have set t1 = tc,1 = −4

5 so that
xc = 16

25 = 0.64 and uc = 4
5 . Thus, with the aid of the ODE solver of Mathematica we construct

a numerical solution that matches with the dispersionless approximation (30). The numerical
analysis shows that for ǫ = 10−5 we have that |u(0) − (vc + ǫ̃ũ(1))| < 5 · 10−4 in the x-interval
(0.6365, 0.6395). As both functions take values larger than 0.8, the relative error on this interval
is smaller than 0.000625. The matching between both solutions on the x-intervals [0.58, 0.64022]
and [0.6, 0.6403] can be observed in Fig.1.

Figure 1: u
(0) (solid line) and uc + ǫ̃ũ

(1) (dashed line)

In this way, near the critical point the regularized Hele-Shaw dynamics associated to our asymp-
totic approximation is given by the curve

Y (X) =

(
X2 +

u

2
X +

3

8
u2 − 6

5

)√
X − u. (34)

The function u is given by u(0)(x) for x < 0.638, and by vc + ǫ̃ũ(1) for x ≥ 0.638. It is defined
until a certain x∗ which corresponds to the first pole of the tritronquée solution on the negative
ξ-axis. Figures 2 and 3 exhibit a sequence between x = 0.6 and x = 0.6402302. Notice that u
is a decreasing function of x and that a cusp is formed when one of the roots of the polynomial
P (X) = X2 + u

2X + 3
8u2 − 6

5 coalesces with u. Thus, for x < 0.64 the tip of the finger moves to
the left, and it starts forming a cusp as x is close to a certain value slightly higher that 0.64 (u
coincides with the largest root of P ). Next a bubble appears at the tip of the finger. Subsequently
a new cusp forms in this bubble (u coincides with the smallest root of P ), and a second bubble
grows from this cusp while the first bubble declines until it annihilates. Finally, the remaining
bubble is absorbed by the finger (when both roots of P coincide).

8


Figure 2: Cusp formation and creation of a bubble which develops a new cusp

Figure 3: Emergence, annihilation and absorbtion of bubbles

3.1 Hele-Shaw flows and the Toda hierarchy

In the Hele-Shaw set-up considered in [6] air is injected in two fixed points of a simply-connected
air bubble making the bubble break into two emergent bubbles. Before the break-off the interface
oil-air remains free of cusp-like singularities and develops a smooth neck. The reversed evolution
describes the merging of two bubbles.

The analysis of [6] concludes that after the break-off the local structure of a small part of the
interface containing the tips of the bubbles falls into universal classes characterized by two even
integers (4n, 2), n ≥ 1, and a finite number 2n of real deformation parameters tk. By assuming
symmetry of the curve with respect to the X-axis, the general solution for the curve and the

9


potential in the (4n, 2) class are

Y (z) :=
(∑2n

k=1(k + 1) tk+1 zk

√
(z − a)(z − b)

)
⊕

√
(z − a)(z − b), X = z. (35)

where a and b are the positions of the bubbles tips. Due to the physical assumptions of the problem,
the expansion

Y (z) =

2n∑

k=1

(k + 1) tk+1 zk +

∞∑

k=0

Yn

zn
, z → ∞, (36)

of the function Y must satisfy two conditions Y0 = t (physical time) and Y1 = 0 which determine
the positions a, b of the tips. As it was shown in [6], imposing these two conditions on (36) leads
to a pair of hodograph equations

∞∑

k=1

k tkrk−1(u, v) = 0,
∞∑

k=1

k tk rk(u, v) + 2x = 0, (37)

where tk = 0 for k > 2n and

r : =
z√

(z − u)2 − 4v
=

∑

k≥0

rk(u, v)

zk
,

(38)

a : = u − 2
√

v, b = u + 2
√

v.

These equations arise in the dispersionless AKNS hierarcy [6] . However, it is straightforward to
see [22] that by setting

Y = 2m −
2n∑

k=1

(k + 1) tk+1 zk, x = 0, t = t1, tj = 0, ∀j ≥ 2n + 2, (39)

they also coincide with the hodograph equations which solve the string equations

z̄ = z, m = m, (40)

for the Lax-Orlov functions

z =p + u0 + O(p−1), m =

∞∑

k=1

k tkz
k−1 +

x

z
+ O(z−2),

z̄ =
v0

p
+ v1 + O(p), m =

∞∑

=k1

j t̄kz̄
k−1 − x

z̄
+ O(z̄−2),

of the dispersionless 2-Toda (d2-Toda) hierarchy. In particular the first string equation represents
the 1-Toda reduction z = z̄ = p + u + v p−1.

In order to regularize the critical points of the hodograph equations (37) one may consider
appropriate solutions of the dispersionful version of the Toda hierarchy . The natural candidates
are provided by the string equations

L = L̄, M = M, (41)

10


for the Lax-Orlov operators [L,M ] = [L̄,M ] = ǫ of the dispersionful 2-Toda hierarchy [23, 24].
The first equation determines the 1-Toda reduction L = L̄ = Λ + u + v Λ−1, (Λ := exp (ǫ ∂x)).
The system (41) characterizes the partition function of the hermitian matrix model in the large-N
limit . In this way the well-known double-scaling limit method for this matrix model can be used
to regularize the critical points of (37).

As an example let us analyze the critical process of a merging of two bubbles studied in section
VII of [6] . Thus we set t2 = 0, tn = 0, n > 3, t := t1 so that (37) reduces to

t + 3t3 (u2 + 2v) = 0, 6t3v u + x = 0. (42)

There is a 2nd-order critical point: vc = u2
c , with uc satisfying

tc + 9t3u
2
c = 0, 6t3u

3
c + xc = 0, (43)

and consequently
4t3c + 81t3x

2
c = 0.

On the other hand, the system of string equations (41) of the dispersionful 2-Toda reduces to

t + 3t3(u
2 + v + v(x + ǫ)) = 0, 3t3 (u + u(x − ǫ)) v + x = 0. (44)

where (u, v) are characterized by expansions of the form

u =
∑

k≥0

ǫk u(k), v =
∑

k≥0

ǫ2k v(2k). (45)

Obviously, from (44) it follows that the leading terms (u(0), v(0)) satisfy the hodograph equations
(37). To deal with solutions of (44) near critical points (tc, u

(0), v(0)) = (tc, uc, vc) of (37), one
introduces

ǫ̃ := ǫ
1

5 , x = xc + ǫ̃4x̃, t = tc + ǫ̃4t̃. (46)

It can be proved that (44) admits solutions of the form

u = uc +

∞∑

j=2

ǫ̃jU (j), v = vc +

∞∑

j=1

ǫ̃2jV (2j). (47)

Thus, by equating the coefficients of ǫ̃j for j = 2, 3 and 4 in (44) we get

U (2) = − 1

uc
V (2), U (3) = − 1

2uc
V

(2)
x̃ ,

2(V (4) + ucU
(4)) = − t̃

3t3
− (U (2))2 − 1

2
V

(2)
x̃x̃ , (48)

1

2
∂2

x̃(V (2) − ucU
(2)) + ucU

(3)
x̃ − 2

uc
U (2)V (2) + (U (2))2 =

1

3t3uc
(x̃ − uct̃).

This provides us with an expression of U (4) in terms of (V (2), V (4)) and implies that V (2) verifies

V
(2)
x̃x̃ +

6

u2
c

(V (2))2 =
2

3t3uc
(x̃ − uct̃). (49)

11


Near the critical point the solution of (42) behaves as

u ∼ uc − 1

3

√
1

t3
(tc − t), v ∼ vc +

uc

3

√
1

t3
(tc − t) as t → t−c

so that matching requires a solution of (49) satisfying

V (2) ∼ uc

3

√

− t̃

t3
, as t̃ → −∞. (50)

Now, if we set x = xc = 1 and introduce the change of variables

W = −
(

2u2
c

3t3

)−2/5

V (2), ξ = −
(

2u2
c

3t3

)1/5

t̃,

it follows that W must satisfy the P-I equation (3) and the asymptotic condition W ∼ −
√

ξ
6 as

ξ → ∞, so that it must be the tritronquée solution of P-I.
Thus near the critical point the regularized Hele-Shaw dynamics of this example is characterized

by the curve
Y (X) = 3t3(X + u)

√
(X − u)2 − 4v,

where

u = uc −
ǫ̃2

uc
V (2), v = vc + ǫ̃2V (2),

The resulting process represents the merging of the tips of two bubbles. It turns out that the right
bubble develops a cusp, then a new bubble appears at this cusp and it grows until it merges with
the tip of the left bubble.

Acknowledgements

The authors wish to thank the Spanish Ministerio de Educación y Ciencia (research project
FIS2005-00319) and the European Science Foundation (MISGAM programme) for their support.

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13

http://arxiv.org/abs/math/0108160
http://arxiv.org/abs/math-ph/0511011
http://arxiv.org/abs/math-ph/0702038
http://arxiv.org/abs/0706.0592
http://arxiv.org/abs/0710.2798
http://arxiv.org/abs/0704.0501

	Introduction
	 Hele-Shaw flows and the KdV hierarchy
	Multiscaling expansions and asymptotic matching 
	Hele-Shaw flows and the Toda hierarchy