Travelling waves for a thin liquid film with surfactant on an inclined plane

IOP PUBLISHING Nonlinearity (009) 85 NONLINEARITY doi:0.088/095-775///006 Travelling waves for a tin liquid film wit surfactant on an inclined plane Vaagn Manukian and Stepen Scecter Matematics Department, Nort Carolina State University, Box 805, Raleig, NC 7695 USA E-mail: vemanuki@ncsu.edu and scecter@mat.ncsu.edu Received 7 April 008, in final form September 008 Publised 5 December 008 Online at stacks.iop.org/non//85 Recommended by B Eckardt Abstract We sow te existence of travelling wave solutions for a lubrication model of surfactant-driven flow of a tin liquid film down an inclined plane, in various parameter regimes. Our arguments use geometric singular perturbation teory. Matematics Subject Classification: 76D08, 34B6, 34B40, 34C37 (Some figures in tis article are in colour only in te electronic version). Introduction We study te flow of a tin liquid film down an inclined plane in te presence of a surfactant tat is insoluble and terefore remains on te liquid surface. Te governing system of partial differential equations includes an equation for te fluid eigt and an equation for te surfactant concentration. After an initial normalization, te system contains tree parameters, wic are proportional to deviation of te inclined plane from te perpendicular, te capillary number of te liquid and te diffusion constant of te surfactant. Te model includes te Marangoni force, a tangential force at te liquid surface due to spatial variation in te surfactant concentration. Te reader wo is unfamiliar wit te Marangoni force can observe its effect by placing a small piece of paper in a pan of water and adding a drop of diswasing liquid. Te system of partial differential equations in one space dimension is t ( ) Ɣ x x + α ( 3 ) 3 x = C ( 3 ) xxx 3 x + β ( 3 ) x 3 x, (.) Ɣ t (ƔƔ x ) x + α ( Ɣ ) x = C ( xxx Ɣ ) x + β ( x Ɣ ) x + DƔ xx. (.) In tese equations, x increases as one descends te inclined plane; is te eigt of te liquid film; Ɣ is te concentration of te surfactant on te liquid surface; α (respectively, β) 095-775/09/00085+38$30.00 009 IOP Publising Ltd and London Matematical Society Printed in te UK 85

86 V Manukian and S Scecter is proportional to te sine (respectively, cosine) of te smaller angle tat te inclined plane makes wit te orizontal; C is proportional to te capillary number of te liquid and D is proportional to te diffusion constant of te surfactant. We assume α and are positive, so te plane is not orizontal and is not dry anywere; Ɣ, β, C and D are nonnegative. After an initial normalization, α can be made equal to. Our work is motivated by te papers [0, ], were te reader will find discussion of te pysical background and references to te literature. Te focus of tese papers is te existence of travelling wave solutions ((ξ), Ɣ(ξ)), ξ = x st, for wic ( ) = L > R = ( ) are given and Ɣ(ξ) dξ is finite. Te latter condition means tat te total amount of surfactant is finite; te integral is an invariant of te system. For certain values of te parameters, L, and R, te autors of tese papers find tat tere is a one-parameter family of suc travelling waves, parametrized by Ɣ(ξ) dξ. In oter words, given te amount of surfactant, tere is a travelling wave. Te autors of [0, ] consider cases in wic at least one of te parameters C, D and β is zero. Our interest in tis paper is te case in wic all tree of tese parameters are positive. Anoter contrast wit [0, ] is tat we make extensive use of geometric singular perturbation teory [6 8]. For te same range of L and R considered in [0, ] we sow, in certain parts of parameter space, te existence of a one-parameter family of travelling waves, and describe te outstanding features of tese waves. In oter parts of parameter space we elucidate te underlying geometry but are not able to provide existence proofs. We note tat te paper [] includes a numerical simulation of te partial differential equation (.) (.) wit C, D and β all positive in wic a travelling wave clearly develops (figures 7 and 8 of []). Tus tere is numerical evidence tat te travelling waves we study are asymptotically stable, at least in part of parameter space. As far as we know, tere as not yet been a systematic numerical investigation of parameter space to determine weter stability persists, nor is tere yet a proof of stability for any parameter values. Te plan of te paper is as follows. We begin, in section, by deriving tree normalizations of system (.) (.), and we describe six regions of parameter space tat we will study in some detail; see figure. In section 3 we derive te travelling wave system as a first-order system in R 4. In section 6 we examine equilibria and invariant spaces for te travelling wave system in R 4. Te travelling waves we seek correspond to solutions of te travelling wave system tat connect two equilibria. Te equilibria ave unstable and stable manifolds of dimensions 3 and 3, respectively, so if tey meet transversally, tey do so in a two-dimensional manifold, wic represents a one-parameter family of travelling waves. In section 4, we sow numerical computations of some travelling waves, in order to familiarize te reader wit teir typical features. Tese features include, for certain values of te parameters and amount of surfactant: () nonmonotonicity of (ξ), leading to a capillary ridge; () a four-step appearance to te grap of (ξ) and (3) bot presence and absence of oscillations in te grap of (ξ), depending on te parameter values. In section 5 we derive some preliminary results. One is a compactification of te travelling wave system at Ɣ = tat is convenient for studying travelling waves wit large Ɣ, i.e. large concentrations of surfactant. Two oter preliminary results give existence and transversality of connecting orbits for a class of first-order systems in R 3. Tese systems correspond to te tird-order equation... Bḣ + F () = 0 wit B 0, F () = 0 at two points, and F () < 0 if and only if is between tose points. Tere is a literature on existence of connecting orbits

Travelling waves for a tin liquid film 87 for systems in tis class, wic we review. Te transversality result seems to be new. It states tat if F as a unique critical point and, along te connecting orbit, (t) passes troug tat point only once, ten te connecting orbit represents a transversal intersection of te unstable and stable manifolds of two equilibria. In section 7 we sow tat for any positive parameter values, connecting orbits of te travelling wave system wit Ɣ small exist, provided te transversality condition just mentioned olds witin te tree-dimensional invariant space Ɣ = 0. In sections 8 3 we investigate te six regions of parameter space in some detail. For eac of te six regions, we work wit one of te tree normalizations of (.) (.). We fix L and R in te range considered in [0, ] and study te existence of connecting orbits. In eac region at least one normalized parameter is small. Our metod is always to allow te small normalized parameters to approac 0, possibly wit some rescalings, and look at te limit. In region we explain some of te geometry but are not able to prove anyting. In regions and 4 tere is a two-dimensional normally yperbolic invariant manifold on wic connecting orbits wit max Ɣ arbitrary exist. In region 3, if a transversality condition olds, ten again connecting orbits wit max Ɣ arbitrary exist, but tey do not stay in a twodimensional normally yperbolic invariant manifold. Regions 5 and 6 are te most interesting from te point of view of geometric singular perturbation teory. In bot regions a two-dimensional invariant manifold loses normal yperbolicity wen Ɣ and a parameter become 0. In tis situation Ɣ and te parameter must be simultaneously rescaled. In geometric singular perturbation teory tis procedure is called blowing-up [5, 9]. Typically one obtains a system on a neigbourood of a spere, wic must be studied in several coordinate patces. In our situations, owever, we are able to coose coordinates in wic te connecting orbits lie in a single coordinate patc, wic we believe makes te analysis easier to follow. In regions 5 and 6, we do not sow tat for fixed parameter values, connecting orbits wit max Ɣ arbitrary exist. Instead, in region 5 (respectively, region 6), connecting orbits wit max Ɣ η (respectively, η max Ɣ η ) are proved to exist, were η 0 as a parameter goes to 0. In region 5 te connecting orbits again lie in a two-dimensional invariant manifold, but in te limit tis manifold becomes two planes at rigt angles to eac oter, wit normally yperbolic equilibria along te line of intersection. Tracking solutions as tey pass near suc a line of equilibria requires wat te second autor as called a corner lemma [6]. We prove suc a lemma appropriate to te present situation in section.3. In region 6 te connecting orbits begin and end in te tree-dimensional unstable and stable manifolds of te equilibria, and teir middle portions lie near (not in) two-dimensional invariant manifolds like tose for region 5. However, te line of equilibria in te limit is not normally yperbolic. Tus two additional complications are introduced. A corner lemma appropriate to tis situation is proved in section 5. Te proofs of te corner lemmas follow an outline based on Deng s lemma tat was recently used to prove a general excange lemma [7]. Complex eigenvalues at an equilibrium of te travelling wave equation correspond to travelling waves tat oscillate at te end tat corresponds to tat equilibrium. Tey occur in regions and 3. Complex eigenvalues are discussed in section 6. We conclude te paper by discussing in section 6 te resemblance between te travelling waves in te various regions and te travelling waves, described in [], in nearby parts of parameter space were one parameter is 0.

88 V Manukian and S Scecter ~ C (N3) ~ C 3 ~ β 3 ~ β (N) 6 ~ D 5 ~ C ~ D 3 ~ β 4 (N) ~ D Figure. Te equivalence classes include lines troug te origin in D β-space; curves C = constant D 3 in C D-space and curves C = constant β 3 in C β-space. Te normalized coordinate systems (N), (N) and (N3) are sown, as are regions 6 described below. Eac region is coloured in te coordinate system in wic it is analysed.. Partial differential equations and normalizations Te scaling x = αx, t = α t (.) converts (.) (.) to te following equivalent system, in wic we ave dropped te tildes over te scaled variables: t ( Ɣ x ) x + 3 Ɣ t (ƔƔ x ) x + ( 3 ) x = C ( 3 xxx 3 )x + β ( 3 ) x 3 x, (.) ( Ɣ ) x = C ( xxx Ɣ ) x + β ( x Ɣ ) x + DƔ xx. (.3) In (.) (.3), C = α C, D = D and β = β. Note tat α as become. In (.) (.3), te system wit parameters ( C, D, β) is considered equivalent to te system wit parameters (Ĉ, ˆD, ˆβ) if te first can be converted to te second by scaling te variables x, t and Ɣ; we avoid scaling in order to keep te values of L and R fixed. Te scaling is necessarily of te form ˆx = kx, ˆt = kt, ˆƔ = kɣ, (.4) wit k>0. Ten Ĉ = k 3 C, ˆD = k D and ˆβ = k β. Te equivalence class of eac point oter tan te origin in {( C, D, β) : C 0, D 0, β 0} is a curve. See figure. By an appropriate coice of k we can make C, D or β equal to. We tus obtain tree two-parameter normalizations of (.) (.). Tey are system (.) (.3) wit te parameters (N) C =, D = α 3 C 3 D and β = α 3 C 3 β or (N) C = α CD 3, D =, β = D β or (N3) C 3 = α Cβ 3, D 3 = Dβ and β 3 =. Eac two-parameter normal form can be tougt of as a local cross-section to te equivalence classes in C D β-space. See figure.

Travelling waves for a tin liquid film 89 Te tree normal forms are related as follows: C = D 3, β = D β ; D = C 3, β = C 3 β ; (.5) C 3 = β 3 C, D 3 = β ; C = C 3 D 3 3, β = D 3 ; (.6) C 3 = β 3, D 3 = β D ; D = C 3 3 D 3, β = C 3 3. (.7) Our goal is to study te existence and structure of travelling waves of (.) (.) of te form ((ξ), Ɣ(ξ)), ξ = x st, ( ) = L, ( ) = R, Ɣ(± ) = 0. (.8) We ave replaced te condition tat Ɣ(ξ) dξ be finite wit te condition Ɣ(± ) = 0; te latter implies te former if Ɣ(ξ) approaces 0 exponentially as ξ ±, wic will be te case. We are concerned only wit C, D and β all positive. However, we are especially interested in te limits C 0, D 0 and β 0, as well as allowing more tan one of tese parameters to approac 0 simultaneously. We sall study te cases in wic one or two parameters approac 0 by allowing one or two normalized parameters to approac 0 in system (.) (.3) wit parameters (N), (N) or (N3). By reinterpreting te results one of course obtains considerable information about te limit in wic all tree of C, D, β approac 0. We sall study te following six regions in some detail. We describe eac region in only te system of normalized parameters in wic we sall study it. Formulae (.5) (.7) and figure can be used to find te equivalent regions in te oter systems. In te description of eac region, η i > 0 and δ i > 0 are small; if bot are used, te coice of δ i depends on η i. Region : C =, 0 < D δ,0< β η. Region : 0 < C 3 δ, η D 3 η, β 3 =. Region 3: η 3 β 3 C η 3 β 3, D =, 0 < β δ 3. Region 4: η 4 β 5 C η 4 β 5, D =, 0 < β δ 4. Region 5: 0 < C 3 δ 5 D 3,0< D 3 δ 5, β 3 =. Region 6: 0 < C 3 δ 6,0< D 3 δ 6 C 3, β 3 =. Te numbers η i and δ i are cosen separately for eac region, so, depending on tese coices, many of te regions may overlap. Figure sows te regions for one way of coosing te numbers η i and δ i. Regions and 5 can overlap, as can regions and 6; tese pairs of regions are sown as adjacent. 3. Travelling wave system In te normalized system (.) (.3), we look for a solution of te form (.8). After integration we obtain te system of ODEs were s Ɣ + 3 3 = C 3 3 + β 3 3 + K, (3.) sɣ ƔƔ + Ɣ = C Ɣ + β Ɣ + DƔ, (3.) s = 3 ( L + L R + R ), K = 3 L R ( L + R ).

90 V Manukian and S Scecter We now rewrite (3.) and (3.) as a first-order system. We first rewrite (3.) and (3.) as β C = ( 3 3s 3K 3 3 ) Ɣ, (3.3) β C = ( ) Ɣ Ɣ sɣ ƔƔ DƔ. (3.4) We equate te rigt-and sides of (3.3) and (3.4), and solve for Ɣ : Ɣ = Ɣ s +3K Ɣ +4 D. (3.5) Next we substitute (3.5) into (3.3): β C = ( 3 3s 3K 3 3Ɣ s +3K ). (3.6) Ɣ +4 D Writing (3.6), (3.5) as a first-order system, we obtain = u, (3.7) u = v, (3.8) Cv = βu ( 3 3s 3K 3 3Ɣ s +3K ), (3.9) Ɣ +4 D Ɣ = Ɣ s +3K Ɣ +4 D. (3.0) To avoid te possibility of dividing by zero in a limit, we rescale by multiplying te rigt-and side of (3.7) (3.0) by Ɣ +4 D. Let H () = ( 3 ) 3s 3K 3, Q() = (s +3K ), P () = H () 3 Q() = 3 ( 3 6s K ). We obtain te travelling wave system ḣ = (Ɣ +4 D)u, (3.) u = (Ɣ +4 D)v, (3.) ( ) ( ) C v = Ɣ βu P () +4 D βu H (), (3.3) Ɣ = ƔQ(). (3.4) Note tat s and K ave been absorbed into P, Q and H. Tus L and R ave been fixed; owever, C, D and β are parameters. We consider tis system wit >0, u and v arbitrary, Ɣ 0. We sall assume 0 < R < L < ( 3 ) R. (3.5) Ten by [] tere are numbers, and, wit suc tat See figure. 0 < R < < < L <, H () = 3 ( L)( R )( + L + R ), P () = 3 ( )( )( + + ). Q() = s ( ),

Travelling waves for a tin liquid film 9 R Q() H() P() * L Figure. Te functions Q(), H () and P (). We are interested in solutions of te travelling wave system (3.) (3.4) tat satisfy te boundary conditions (, u, v, Ɣ)( ) = ( L, 0, 0, 0), (, u, v, Ɣ)( ) = ( R, 0, 0, 0). (3.6) Tese solutions correspond to travelling wave solutions of te system of PDEs (.) (.3) of te form (.8). 4. Numerical results We consider te travelling wave system (3.) (3.4), wit te boundary conditions (3.6). We sall typically refer to te rescaled independent variable in tis and oter differential equations as time t, wit te ope tat tis will not cause confusion. To satisfy te boundary conditions, we want solutions tat lie in te unstable manifold of ( L, 0, 0, 0) and te stable manifold of ( R, 0, 0, 0). Denote te linear approximations of tese manifolds by E u ( L, 0, 0, 0) and E s ( R, 0, 0, 0); teir equations are readily computable. In order to approximate solutions numerically, we coose times T <T and searc for solutions tat lie in E u ( L, 0, 0, 0) at time T and in E s ( R, 0, 0, 0) at time T. Te error due to tis approximation decays exponentially as T and T increase; see [] for te metod of analysis. Te numbers C, D, β, L, R, T T Ɣ(t) dt, T and T are regarded as parameters. We set C = D = β =, L = 4, R = and T T Ɣ(t) dt = 0. Te last coice implies tat Ɣ 0, so equation (3.) can temporarily be dropped. Using xpp (available from ttp://www.mat.pitt.edu/ bard/xpp/xpp.tml) we found pictorially a solution of (3.) (3.4) wit Ɣ = 0 tat approximately connects (4, 0, 0) to (, 0, 0). Te time interval on wic te solution is defined gives T and T. We now ave an approximate solution of our boundary value problem for certain values of te parameters. Using AUTO (available from ttp://indy.cs.concordia.ca/auto) we can improve tis solution and, in principle, continue it as any of te parameters vary. Figure 3 sows tree computed travelling waves wit L = 4, R = and max Ɣ, wic is related to T T Ɣ(t) dt, between.06 and.7; te waves are sown as parametrized curves in uvɣ-space projected into te Ɣ-plane. In figure 3(a) te parameters are in or close to region. Te solution oscillates near (, u, v, Ɣ) = (, 0, 0, 0). Tis can be seen more clearly wen te solution is projected onto u-space instead of Ɣ-space; see figure 4. In figure 3(b) te parameters are in or close to region. Te projected solution approaces (4, 0) along te Ɣ-axis but approaces (, 0) from an oblique direction; tere is no oscillation. In figure 3(c) te parameters are in or close to region 5. Te projected solution approaces bot (4, 0) and (, 0) along te Ɣ-axis; tere is no oscillation.

9 V Manukian and S Scecter.5 (a).5 (b) Gamma.5 0.5 Gamma.5 0.5 0 0.5.5.5 3.5 4.5.5 Gamma (c).5 0.5 0 0.5.5.5 3.5 4.5 0 0.5.5.5 3.5 4.5 Figure 3. Tree computed travelling waves wit L = 4, R = and max Ɣ between.06 and.7; te waves are sown as curves in te Ɣ-plane tat connect (4, 0) to (, 0). (a) ( C, D, β) = (.0, 0.36 644,.0). (b) ( C, D, β) = (0.000 7 946,.0,.0). (c) ( C, D, β) = (0.000 067 47 7, 0.07 638,.0). We refer to te portions of tese curves tat are near and approximately parallel to te Ɣ- axis as feet. Figures 3(a) and (c) ave two feet; figure 3(b) as one. If we vary te parameters in figure 3(b) a little, te projected solution continues to approac (, 0) obliquely, so tere is still only one foot. Figure 5 sows te effect of canging max Ɣ. Te parameter values are tose of figure 3(c), in region 5. In figure 5(a), for wic max Ɣ is small, bot feet point in. In tis case, (t) is a decreasing function, i.e. te eigt of te travelling wave decreases from = 4 at te left to = at te rigt. In figure 5(a), for wic max Ɣ is large, te foot near (, u) = (4, 0) points out. In tis case te eigt of te travelling wave increases from = 4 at te left to above 5 before decreasing to = at te rigt. Suc an increase in is called a capillary ridge. It is also present in figures 3(a) (c). For max Ɣ large two portions of te curve are approximately vertical. Tey are near = and = defined earlier. For te travelling wave sown in figure 5(b). Figure 6 sows and Ɣ as functions of normalized time. (AUTO normalizes te time interval [T,T ] to [0, ].) Te increase in from 4 to above 5 is te capillary ridge. In addition, te -component of te wave exibits a four-level structure tat is missing or less apparent wen max Ɣ is small; te steps are at = L,, and = R Virtually te entire cange in Ɣ occurs close te time period wen falls from near to near. For more plots of travelling waves see [].

Travelling waves for a tin liquid film 93 0 (a) 0.0-0. 0.005 0 u -0.4 u (b) -0.005-0.6-0.8 0 3 4 5-4 x 0 5 (c) -0.0-0.05 0.96 0.98.0.04.06 0 u 5 0-5 0.997 0.998 0.999.00 Figure 4. (a) Te solution sown in figure 3(a) projected onto u-space instead of Ɣ-space. (b) Zoom into a small rectangle around (, u) = (, 0). (c) Zoom into a smaller rectangle around (, u) = (, 0). 0.06 (a) 60 50 (b) Gamma 0.04 0.0 Gamma 40 30 0 0 0 0.5.5.5 3.5 4.5 0 3 4 5 6 Figure 5. Travelling waves sown as curves in te Ɣ-plane. Te parameter values are tose of figure 3(c). (a) Solution wit max Ɣ approximately 0.057. (b) Solution wit max Ɣ approximately 56.4. 5. Preliminaries 5.. Compactification of te Ɣ-interval We wis to consider travelling waves of all sizes, so tat Ɣ may be large. In order to deal conveniently wit te interval 0 Ɣ <, in (3.) (3.4) we make te cange of variables Ɣ = γ γ, (5.)

94 V Manukian and S Scecter 6 5 (a) 60 50 (b) 4 40 3 0 0 0. 0.4 0.6 0.8 normalized time Gamma 30 0 0 0 0 0. 0.4 0.6 0.8 normalized time Figure 6. Te travelling wave of figure 5(b); and Ɣ are sown as functions of normalized time. (AUTO normalizes te time interval in a boundary value problem to [0, ].) so tat te interval 0 Ɣ < corresponds to 0 γ <. After multiplying by γ, we obtain ḣ = (γ +4 D( γ ))u, (5.) u = (γ +4 D( γ ))v, (5.3) ( ( ) C v = γ βu P ()) +4 D( γ ) βu H (), (5.4) γ = γ ( γ ) Q(). (5.5) We consider tis system on >0, u and v arbitrary, 0 γ. Note tat te infinite Ɣ-interval as been compactified. 5.. Linear combinations of P and H Te following proposition will be useful. Most of it is proved in []. Proposition 5.. Let F (a, b, ) = ap () + bh (). If a 0, b 0, and (a, b) (0, 0), ten: () For 0 << R (respectively, << L, respectively, << ), F (a, b, ) is positive (respectively, negative, respectively, positive). () Te equation F (a, b, ) = 0 as exactly two solutions in > 0, wic we denote (a, b) < (a, b). We ave R (a, b) and L (a, b). (3) (0,b)= R, (a, 0) =, (0,b)= L and (a, 0) =. (4) F (a, b, (a, b)) < 0 and F (a, b, (a, b)) > 0. (5) (a,b) 0 F (a, b, ) d >0 and (a,b) (6) Te equation F F (a, b, ) d >0. (a, b, ) = 0 as a unique solution in >0. Proof. () is a consequence of te signs of P and H on te given intervals; see figure. For b = 0 or a = 0, () (6) follow from te formulae for P and H. We terefore assume a>0 and b>0. Write F (a, b, ) = a 3 ( 4 +a 3 3 +a +a +a 0 ) = a 3 ( µ )( µ )( µ 3 )( µ 4 ). For a>0 and b>0, we see from te formulae for P and H tat a 3 = (µ +µ +µ 3 +µ 4 )>0

Travelling waves for a tin liquid film 95 and a 0 = µ µ µ 3 µ 4 > 0. By () te equation F (a, b, ) = 0 as at least two solutions in >0, so two µ i are distinct positive numbers. It follows tat te oter two ave negative real part, so we ave (). (4) and (5) follow easily. To prove (6), we calculate tat F = a ( 4 + b 4 3 3 + b + b + b 0 ) wit b 3 = 0, b > 0 and b 0 < 0. Ten F = 0 if and only if 4 + b + b 0 = b. Te grap of y = 4 + b + b 0 is convex wit a minimum at (0,b 0 ), b 0 < 0. It follows easily tat it meets te grap of y = b in two points, one wit >0, one wit <0. 5.3. Connection teorem Let I = (, + ) be an open interval, wit = and + = allowed. Let F : I R be a C function tat satisfies te following conditions. (C) Te equation F () = 0 as exactly two solutions in I. We denote tem a and b, wit b < a. (C) F ( b )<0and F ( a )>0. (C3) a F () d >0and + b F () d >0. See figure 7(a). For example, te tird assumption is satisfied if te integrals are. Consider te system ḣ = u, (5.6) u = v, (5.7) v = Bu F (), (5.8) wit B 0. Tis system as exactly two equilibria, at ( a, 0, 0) and ( b, 0, 0). One easily cecks tat bot are yperbolic, W u ( a, 0, 0) as dimension two, and W s ( b, 0, 0) as dimension two. Teorem 5. (Connection teorem). Assume (C) (C3). Ten W u ( a, 0, 0) and W s ( b, 0, 0) ave nonempty intersection. System (5.6) (5.8) is equivalent to te tird-order equation... Bḣ + F () = 0. Tere is a literature on connecting orbits for tis equation tat is larger tan one migt expect. For B = 0 and F () =, existence of a connecting orbit was proved by Kopell and Howard [3] using sooting, and later by Conley [] using te Conley Index. Troy [8] gave a sooting argument for existence for B = 0 and a different F. For B = 0 and general F, Mock [4] gave a nice sooting argument, but is proof tat a certain function is continuous (top of p 387) is not correct. Micelson [] gave a Conley index proof completely different from tat in [] for B = 0 and general F ; in fact, e allows a more general differential operator of odd order at least 3 in place of.... Renardy [5], motivated by a problem involving surfactants, treated a closely related situation. His proof would apply to B 0 and general F. However, an inequality at te top of p 9 cannot be used on an infinite interval. We follow Micelson s argument, wic easily generalizes to te case B>0. Te proof uses a Lyapunov function and te Conley Index [], and is given in nine steps. In step we give te Lyapunov function. In steps 5 we find bounds on te set of bounded solutions of

96 V Manukian and S Scecter F() G() b a * b * a a * b * b a (a) (b) Figure 7. Te functions F () and G() wit = 0 and + =. (5.6) (5.8). In steps 6 9 we deform te set of bounded solutions to te empty set by perturbing (5.6) (5.8), and use te Conley Index conclude te result. Proof.. Let G() = F () d, and let L(, u, v) = uv + G(). See figure 7(b). Ten L = (Bu + v ), so L is decreasing along solutions of (5.6) (5.8).. Let ((t), u(t), v(t)) be a bounded solution of (5.6) (5.8). Ten its α- and ω-limit sets are invariant sets contained in {(, u, v) : L = 0}. It follows easily tat eac is an equilibrium, so lim t ((t), u(t), v(t)) and lim t ((t), u(t), v(t)) are bot equilibria. 3. Using assumption (C3) we see tat tere exist a, < a < b, suc tat G( a ) > G( a) and b, a < b < +, suc tat G( b ) > G( b ). See figure 7(b). 4. Let ((t), u(t), v(t)) be a bounded solution of (5.6) (5.8). We claim tat a < (t) < b for all t. Te proof is by contradiction. Suppose max (t) = (t ) b. Ten L((t ), u(t ), v(t )) = G((t )) because u(t ) = ḣ(t ) = 0. Since (t ) b, G((t )) G( b ) < G( b), and of course G( b ) < G( a ). But L( b, 0, 0) = G( b ) and L( a, 0, 0) = G( a ). Since L is decreasing on solutions, ((t), u(t), v(t)) cannot approac an equilibrium as t. Tis is a contradiction. Similarly, if min (t) a, ten ((t), u(t), v(t)) cannot approac an equilibrium as t. 5. By step 4, for any bounded solution ((t), u(t), v(t)) of (5.6) (5.8), < b. We claim tat if M 0 is cosen sufficiently large, we ave ḣ = u < M 0 and ḧ = v <M 0. Te proof relies on Nirenberg s inequality in R, wic states tat if ( ) p = j + a r m + ( a) q ten tere is a constant C suc tat and j a, (5.9) m w (j) p C w (m) a r w a q (5.0) wenever bot sides of te inequality make sense [3]. Let ((t), u(t), v(t)) be a bounded solution of (5.6) (5.8). From steps and, B(ḣ(t)) + (ḧ(t)) dt = (L( ) L( )) G( a ) G( b ). Terefore tere is a uniform bound on ḧ. By Nirenberg s inequality wit j =, p =, r =, m =, q = and a = 3, ḣ C ḧ 3 3.

Travelling waves for a tin liquid film 97 Terefore tere is a uniform bound on ḣ. Let w = v = ḧ. By Nirenberg s inequality wit j = 0, p =, r =, m =, q = and a = 3, ḧ = w C ẇ 3 w 3 = C... 3 ḧ 3. Now te formula... = Bḣ F (), togeter wit our uniform bounds on and ḣ, yields a uniform bound on.... Tis togeter wit our uniform bound on ḧ yields a uniform bound on ḧ. 6. Consider te one-parameter family of differential equations ḣ = u, (5.) u = v, (5.) v = Bu (F () + µ). (5.3) Let µ = min F (). For 0 µ µ, tis system as one or more equilibria on te -axis between ( b, 0, 0) and ( a, 0, 0), and no oter equilibria; for µ>µ, it as no equilibria. 7. For eac µ wit 0 µ µ, let E µ denote te set of points (u, v, w) tat lie in bounded solutions of (5.6) (5.8). From steps 4 and 5, E 0 lies in te interior of te closure of te open set U 0 ={(, u, v) : a < (t) < b, u <M 0, v <M 0 }. E 0 is te largest invariant set contained in U 0, and U 0 is an isolating neigbourood of E 0. By te same argument wit minor modifications, for eac µ wit 0 µ µ, tere is a number M µ > 0 suc tat E µ lies in te interior of te closure of an open set U µ defined like U 0 except tat M 0 is replaced by M µ. 8. It is easy to see tat M µ can be cosen independently of µ for 0 µ µ. Tus we can find a single open set U tat serves as an isolating neigbourood for all te E µ, 0 µ µ. 9. For µ>µ, te system as no bounded solutions. It follows tat te Conley index of E 0 equals te Conley index of te empty set. Since te Conley index of two yperbolic fixed points does not equal te Conley index of te empty set, E 0 must contain a nontrivial solution connecting two equilibria. Because of te Lyapunov function L, it must connect ( a, 0, 0) to ( b, 0, 0). 5.4. Transversality teorem In addition to te ypoteses of te Connection teorem, assume: (C4) B>0. (C5) F as exactly one critical point c. Of course, b < c < a. If (C5) olds, we will call a solution of (5.6) (5.8) tat lies in W u ( a, 0, 0) W s ( b, 0, 0) a simple connection from ( a, 0, 0) to ( b, 0, 0) if tere is exactly one time t = t c at wic (t) = c. Teorem 5.3 (Transversality teorem). Assume (C) (C5). Let ( (t), u (t), v (t)) be a simple connection from ( a, 0, 0) to ( b, 0, 0). Ten W u ( a, 0, 0) and W s ( b, 0, 0) meet transversally along ( (t), u (t), v (t)).

98 V Manukian and S Scecter We note tat Kopell and Howard [3] prove a transversality result for te case B = 0 and F () =, but teir metod does not appear to generalize. Proof. Linearizing (5.6) (5.8) along ( (t), u (t), v (t)), we obtain =ū, ū = v, v = Bū F ( (t)). (5.4) Note tat (t) approaces a (respectively, b ) as t (respectively, t ). Since (5.4) wit (t) replaced by a (respectively, b as two eigenvalues wit positive (respectively, negative) real part, tis linear system as a two-dimensional space of solutions tat approac 0 exponentially as t, and a two-dimensional space of solutions tat approac 0 exponentially as t. We must sow: (I) tese two-dimensional spaces meet transversally. Given a real linear differential equation ẋ = L(t)x, te adjoint system is ψ = ψl(t) Te adjoint system for (5.4) is ψ = F ( (t))ψ 3, ψ = ψ Bψ 3, ψ 3 = ψ. (5.5) Tis linear system as a one-dimensional space of solutions tat approac 0 exponentially as t, and a one-dimensional space of solutions tat approac 0 exponentially as t. Equivalently to (I), we must sow tat (II) tese one-dimensional spaces ave trivial intersection. Equation (5.5), like any nonautonomous linear differential equation, defines a nonautonomous differential equation ṗ = f (t, p) on RP, te space of lines troug te origin in R 3. A point of RP represents an equivalence class of vectors in R 3 \{0}; we write p = [ψ], wit ψ ψ is tere is a number s 0 suc tat ψ = s ψ. Let p a (respectively, p b ) denote te one-dimensional eigenspace for te unique positive (respectively, negative) eigenvalue of (5.5) at t = (respectively, t = ), i.e. te positive (respectively, negative) eigenvalue of (5.5) wen (t) is replaced by a (respectively, b ). Equivalently to (II), we must sow tat (III) no solution of ṗ = f (t, p) approaces p a as t and p b as t. RP is covered by tree coordinates systems; tese coordinates, and te differential equation ṗ = f (t, p) in eac, are ψ 3 ; φ = F ( (t)) + φ φ, φ = φ B + φ. () η = (η, η 3 ), η = ψ ψ, η 3 = ψ3 ψ ; η = F ( (t))η 3 + η + Bη η 3, η 3 = +η η 3 + Bη3. () φ = (φ, φ ), φ = ψ ψ 3, φ = ψ (3) χ = (χ, χ 3 ), χ = ψ ψ, χ 3 = ψ3 ψ, χ = Bχ 3 F ( (t))χ χ 3, χ 3 = χ F ( (t))χ 3. Let U a = {p RP : p = [ψ] wit ψ > 0, ψ < 0, and ψ 3 > 0}, and let U b ={p RP : p = [ψ] wit ψ > 0, ψ > 0, and ψ 3 > 0}. U a and U b are disjoint and open in RP, and it is straigtforward to ceck tat p a U a and p b U b. More precisely, consider te system of equations F () + φ φ = 0, φ B + φ = 0, (5.6) derived by setting te rigt-and side of te system in φ-coordinates equal to 0. For eac c, tere is a unique solution φ() of (5.6) wit φ > 0. (Te system as 0, or solutions wit φ < 0.) Ten φ () as sign opposite tat of F (). We ave p a = [(φ ( a ), φ ( a ), )] and p b = [(φ ( b ), φ ( b ), )]. We claim tat (A) tere is no solution of ṗ = f (t, p) tat lies in U a for large negative t and in U b for large positive t. Tis implies (III). To prove (A), we will sow te te following. Suppose p(t) is a solution of ṗ = f (t, p) tat lies in U a for large negative t and q(t) is a solution of ṗ = f (t, p) tat lies in U b for large positive t. Ten (a) p(t) U a for t<t c, (b) q(t) U b for t>t c and (c) p(t c ) q(t c ).

Travelling waves for a tin liquid film 99 Let us first sow (a). Suppose p(t) is a solution of ṗ = f (t, p) tat lies in U a for large negative t. Suppose p(t) first reaces te boundary of U a at time t < t c. Since t < t c, assumption (4) implies tat (t) > c, so F ( (t)) > 0. We consider eac coordinate patc. In φ-coordinates, U a corresponds to {(φ, φ ) : φ > 0 and φ < 0}. On te boundary of tis set: If t<t c, φ 0 and φ = 0, ten φ < 0. If t<t c, φ = 0 and φ < 0, ten φ > 0. Terefore, p(t) does not leave U a troug its boundary in te φ-coordinate patc at a time t<t c. A similar argument sows tat p(t) does not leave U a troug its boundary in te oter coordinate patces at a time t<t c. Tis sows (a). A similar argument sows (b), i.e we sow tat q(t) does not leave U b troug its boundary in backward time at any time t>t c. It remains to sow (c). Suppose p(t) first leaves U a at time t = t c and q(t) first leaves U b in backward time at time t = t c. Suppose tat p(t c ) equals q(t c ). We consider te different coordinate patces. Suppose p(t c ) = q(t c ) is a point of te φ-coordinate patc. In φ-coordinates, U a corresponds to {(φ, φ ) : φ > 0 and φ < 0} and U b corresponds to {(φ, φ ) : φ > 0 and φ > 0}. Teir common boundary is te ray {(φ, φ ) : φ 0 and φ = 0}. On tis ray, for t = t c, φ < 0. (At te origin we need B>0at tis point.) But ten neiter p(t c ) nor q(t c ) can belong to tis ray, so tey are certainly not equal. A similar argument applies to te oter coordinate patces. 6. Invariant spaces and equilibria For system (5.) (5.5), te tree-dimensional spaces γ = 0 (no surfactant) and γ = (infinite surfactant) are invariant. For every C 0, D 0 and β 0, tere are equilibria of (5.) (5.5) at (, u, v, γ ) = ( L, 0, 0, 0), ( R, 0, 0, 0), (, 0, 0, ) and (, 0, 0, ). If D >0, tese are te only equilibria. Travelling waves (.8) of (.) (.3) correspond to solutions of (5.) (5.5) tat go from ( L, 0, 0, 0) to ( R, 0, 0, 0). For C >0, te linearization of (5.) (5.5) at ( L, 0, 0, 0) or ( R, 0, 0, 0) as te matrix 0 4 D 0 0 0 0 4 D 0. 4 C DH () 4 C D β 0 C (6.) P () 0 0 0 Q() Caracteristic equation: (λ Q())(λ 3 6 C D βλ + 64 C D 3 H ()) = 0. (6.) Let p(λ) = λ 3 6 C D βλ+ 64 C D 3 H () ave roots λ, λ, λ 3. Ten λ + λ + λ 3 = 0, and, for C >0 and D >0, te sign of λ λ λ 3 is opposite to tat of H (). Conclusions for C >0 and D >0: () At ( L, 0, 0, 0) tere are tree eigenvalues wit positive real part and one wit negative real part. () At ( R, 0, 0, 0) tere are one eigenvalue wit positive real part and tree wit negative real part. Tus for C >0 and D >0, W u ( L, 0, 0, 0) and W s ( R, 0, 0, 0) are tree dimensional. If tey are transverse, te intersection is two dimensional. We also note tat

00 V Manukian and S Scecter () At ( L, 0, 0, 0), two of te eigenvalues wit positive real part are complex if 4 β 3 < 7 CH ( L ). () At ( L, 0, 0, 0), two of te eigenvalues wit negative real part are complex if 4 β 3 < 7 CH ( R ). In region, for fixed L (respectively, R ), complex eigenvalues occur for β less tan a value of order one. In region 3, for fixed L (respectively, R ), complex eigenvalues occur for C greater tan a number of order one times β 3. In te oter regions, for fixed L (respectively, R ), complex eigenvalues do not occur provided te constants in te definition of te region are taken sufficiently small. 7. Connecting orbits wit Γ small We consider te travelling wave system (3.) (3.4) wit te normalization (N). Hence C =, β = β 0, and we assume D = D > 0: ḣ = (Ɣ +4 D )u, (7.) u = (Ɣ +4 D )v, (7.) ( ) ( ) v = Ɣ β u P () +4 D β u H (), (7.3) Ɣ = ƔQ(). (7.4) Te space Ɣ = 0 is invariant. Restricting (7.) (7.4) to Ɣ = 0 and dividing by 4 D > 0, we obtain ḣ = u, (7.5) u = v, (7.6) v = β u H (). (7.7) Te only equilibria of (7.5) (7.7) are ( L, 0, 0) and ( R, 0, 0). Botareyperbolic.Teformer as two-dimensional unstable manifold; te latter as two-dimensional stable manifold. By te connection teorem, W u ( L, 0, 0) W s ( R, 0, 0) is nonempty. Let C be one of te curves in te intersection. Suppose W u ( L, 0, 0) and W s ( R, 0, 0) meet transversally along C. (According to te transversality teorem, tis is te case if β > 0 and C is a simple connection from ( L, 0, 0) to ( R, 0, 0).) Ten for system (7.) (7.4), te tree-dimensional manifolds W u ( L, 0, 0, 0) and W s ( R, 0, 0, 0) meet transversally in a two-dimensional manifold of connecting orbits tat includes C. Tus (3.) (3.4) as a one-parameter family of connecting orbits from ( L, 0, 0, 0) to ( R, 0, 0, 0). Note tat tis argument proves te existence only of connecting orbits wit small Ɣ. Moreover, if D approaces 0, some eigenvalues at te equilibria, wic satisfy (6.), approac 0. Terefore te size of te Ɣ-interval for wic connections are guaranteed to exist by te argument of tis section goes to 0. 8. Region We consider te normalized travelling wave system (7.) (7.4) of te previous section. In region, β lies in a given bounded interval 0 < β η, and D is small.

Travelling waves for a tin liquid film 0 For D = 0, (7.) (7.4) becomes ḣ = Ɣu, (8.) u = Ɣv, (8.) ( ) v = Ɣ β u P (), (8.3) Ɣ = ƔQ(). (8.4) Te space Ɣ = 0 consists of equilibria of (8.) (8.4). It is normally yperbolic for away from. Tus for small D, system (7.) (7.4), altoug not a slow-fast system, as te essential structure of one: for D = 0 it as a large manifold of equilibria. In (7.) (7.4), Ɣ = 0 remains invariant for D > 0. Restricting (7.) (7.4) to Ɣ = 0 and dividing by 4 D, we obtain (7.5) (7.7). Tis system plays te role of te slow system. Te fast system is (8.) (8.4). Let Wloc u ( L, 0, 0) denote an open subset of te two-dimensional unstable manifold of ( L, 0, 0) wose closure is contained in {(, u, v) : > }. Similarly, let Wloc s ( R, 0, 0) denote an open subset of te two-dimensional stable manifold of ( R, 0, 0) wose closure is contained in {(, u, v) : < }. For D = 0 (i.e. for (8.) (8.4)), Wloc u ( L, 0, 0) {0} (respectively, Wloc s ( R, 0, 0) {0}) is a two-dimensional manifold of equilibria tat as a tree-dimensional unstable (respectively, stable) manifold in uvɣ-space. For D > 0 te manifolds Wloc u ( L, 0, 0) {0} and Wloc s ( R, 0, 0) {0} remain invariant because system (7.) (7.4), after restriction to Ɣ = 0 and division by D, is (7.5) (7.7) independent of D. Te unstable manifold of Wloc u ( L, 0, 0) {0} and te stable manifold of W s loc ( R, 0, 0) {0}, respectively, perturb sligtly. Terefore, if tese tree-dimensional manifolds intersect transversally for D = 0 (i.e. for (8.) (8.4)), ten tey do so for small D > 0, and we ave a two-dimensional manifold of connecting orbits from ( L, 0, 0, 0) to ( R, 0, 0, 0). In tis case connecting orbits for D > 0 are close to te following singular connecting orbits for D = 0: () a solution of te slow system (7.5) (7.7) from ( L, 0, 0) to a point ( 0,u 0,v 0 ) in W u loc ( L, 0, 0), te unstable manifold of ( L, 0, 0) for te slow system; () a connecting orbit of te fast system (8.) (8.4) from ( 0,u 0,v 0, 0) to (,u,v, 0), were (,u,v ) W s loc ( R, 0, 0), te stable manifold of ( R, 0, 0) for te slow system; (3) a solution of te slow system from (,u,v ) to ( R, 0, 0). One way to study connecting orbits of (8.) (8.4) from Ɣ = 0 to itself is to first divide (8.) (8.4) by Ɣ: ḣ = u, (8.5) u = v, (8.6) v = β u P (), (8.7) Ɣ = Q(). (8.8) System (8.5) (8.8) is solved wit initial condition (, u, v, Ɣ)(0) = ( 0,u 0,v 0, 0) and 0 >. (Actually, te system for (, u, v) is independent of Ɣ.) Suppose tere is a t > 0 suc tat Ɣ(t ) = 0. Let (,u,v ) = (, u, v)(t ). Ten tere is a connecting orbit of (8.) (8.4) from ( 0,u 0,v 0, 0) to (,u,v, 0). We define π( 0,u 0,v 0 ) = (,u,v ). If te two-dimensional manifold π(wloc u ( L, 0, 0)) is transverse to te two-dimensional manifold Wloc s ( R, 0, 0)) in uv-space, ten te desired transversality condition olds. See figure 8. However, we do not see ow to sow tat tis is te case.

0 V Manukian and S Scecter π v u R L Figure 8. Mapping π for system (8.) (8.4). 9. Region Teorem 9.. Let L and R satisfying (3.5) be given. Consider te travelling wave system (5.) (5.5) wit te normalization (N3): C = C 3 0, D = D 3 0 and β =. Ten for eac η wit 0 < η <, tere exists δ > 0 suc tat, if region is defined using tese values, ten for eac parameter triple ( C 3, D 3, ) in region, tere exists a one-parameter family of travelling waves connecting L to R. Moreover, max γ ranges from 0 to. Proof. In te travelling wave system (5.) (5.5), let c = C, ˆv = cv. After dropping te at and multiplying by c, we obtain te system ḣ = c(γ +4 D( γ ))u, (9.) u = (γ +4 D( γ ))v, (9.) ( ( ) v = γ βu P ()) +4 D( γ ) βu H (), (9.3) γ = cγ ( γ ) Q(). (9.4) Using te normalized parameters (N3), (9.) (9.4) becomes ḣ = c(γ +4 D 3 ( γ ))u, (9.5) u = (γ +4 D 3 ( γ ))v, (9.6) v = γ (u P ()) +4 D 3 ( γ ) (u H ()), (9.7) γ = cγ ( γ ) Q(), (9.8) wit c = C 3. To study region, given η wit 0 < η <, we assume η D 3 η and c is small. Since c is small, we ave a slow-fast system wit two slow variables ( and γ ) and two fast variables (u and v). Set c = 0 in (9.5) (9.8): ḣ = 0, (9.9) u = (γ +4 D 3 ( γ ))v, (9.0) v = γ (u P ()) +4 D 3 ( γ ) (u H ()), (9.) γ = 0. (9.)

Travelling waves for a tin liquid film 03 Te set u = γ P () +4 D 3 ( γ )H (), v = 0, > 0, 0 γ (9.3) γ +4 D 3 ( γ ) is a normally yperbolic manifold of equilibria of dimension. (One positive eigenvalue, one negative eigenvalue.) Perturbed normally yperbolic invariant manifold for small c>0: u = K(, γ,c, D 3 ), K(, Ɣ, 0, D 3 ) = γ P () +4 D 3 ( γ )H (), (9.4) γ +4 D 3 ( γ ) v = L(, γ,c, D 3 ), L(, γ, 0, D 3 ) = 0. (9.5) Since (, u, v, γ ) = ( L, 0, 0, 0) and ( R, 0, 0, 0) are equilibria for all (c, D 3 ), K( L, 0,c, D 3 ) = L( L, 0,c, D 3 ) = K( R, 0,c, D 3 ) = L( R, 0,c, D 3 ) = 0. Since (, u, v, γ ) = (, 0, 0, ) and (, 0, 0, ) are equilibria for all (c, D 3 ), K(,,c, D 3 ) = L(,,c, D 3 ) = K(,,c, D 3 ) = L(,,c, D 3 ) = 0. Te vector field on te perturbed normally yperbolic invariant manifold is ḣ = c(γ +4 D 3 ( γ ))K(, Ɣ,c, D 3 ) In slow time (i.e. divide by c): = c ( γ P () +4 D 3 ( γ )H () + O(c) ), (9.6) γ = cγ ( γ ) Q(). (9.7) = γ P () +4 D 3 ( γ )H () + O(c), (9.8) γ = γ ( γ ) Q(). (9.9) For all (c, D 3 ) tere are equilibria of (9.8) (9.9) at (, Ɣ) = ( L, 0), ( R, 0), (, ) and (, ). Te first is a repeller, te second is an attractor and te last two are semiyperbolic. For c = 0, te flow can be drawn wit te elp of proposition 5.. In te notation of tat teorem, te nullclines are te curves = i (γ, 4 D 3 ( γ )), i =, and = ; (γ, 4 D 3 ( γ )) < < (γ, 4 D 3 ( γ )). In particular: For 0 << (γ, 4 D 3 ( γ )), ḣ>0 and γ < 0. For (γ, 4 D 3 ( γ )) < <, ḣ<0 and γ < 0. For << (γ, 4 D 3 ( γ )), ḣ<0 and γ > 0. For (γ, 4 D 3 ( γ )) <, ḣ>0 and γ > 0. Te centre manifold of (, ) (respectively, (, )), wic is a separatrix, connects to ( L, 0) (respectively, ( R, 0)). Te flow is sown in figure 9: tere is a one-parameter family of connections from ( L, 0) to ( R, 0) tat fills te region (γ, 4 D 3 ( γ )) < < (γ, 4 D 3 ( γ )),0< γ <. Te flow for c small is exactly te same. For c = 0, te eigenvalues of te linearization of (9.8) (9.9) at ( L, 0) and ( R, 0) are 4 D 3 H () and Q(). Bot are positive at L ;botarenegativeat R. Teir relative size affects te sape of solution curves in figure 9: () If 0 < 4 D 3 H ( L ) < Q( L ), all integral curves in >0 but one tat approac ( L, 0) as t do so tangent to te -axis. () If 0 < Q( L ) < 4 D 3 H ( L ), all integral curves in > 0 tat approac ( L, 0) as t do so tangent te eigendirection for te eigenvalue Q( L ), wic is ( L P ( L ), 4 D 3 H ( L ) Q( L )). Bot components are positive.

04 V Manukian and S Scecter γ.... R * L Figure 9. Flow of (9.8) (9.9) for small c 0. Nullclines are dased. Centre manifolds of te equilibria at (, 0) and (, 0) are sown. (3) If Q( R )<4 D 3 H ( R )<0, all solutions in >0but one tat approac ( R, 0) as t do so tangent to te -axis. (4) If 4 D 3 H ( R ) < Q( R ) < 0, all solutions in > 0 tat approac ( R, 0) as t do so tangent to te eigendirection for te eigenvalue Q( R ), wic is ( R P ( R ), Q( R ) 4 D 3 H ( R )). Bot components are positive. Te first and tird cases, wic occur for small D 3, give rise to feet in te solutions. Te second and fourt cases, wic occur for larger D 3, do not. However, in te second and fourt cases, te eigenvector as positive slope; wen tis occurs at ( L, 0), te result is a rise in te travelling wave above L (capillary ridge). 0. Region 3 In te travelling wave system (5.) (5.5), let ( ) C ɛ =, ˆv = ɛv. β 3 After dropping te at and multiplying by ɛ, we obtain te system ḣ = ɛ(γ +4 D( γ ))u, (0.) u = (γ +4 D( γ ))v, (0.) v = γ (u P ()) +4 D( γ ) (u H ()), (0.3) γ = βɛγ( γ ) Q(). (0.4) We use te normalized parameters (N): C = C 0, D = and β = β 0. Tus (0.) (0.4) becomes ḣ = ɛ(γ +4( γ ))u, (0.5) u = (γ +4( γ ))v, (0.6) v = γ (u P ()) +4( γ ) (u H ()), (0.7) γ = β ɛγ( γ ) Q(), (0.8)

Travelling waves for a tin liquid film 05 γ C C R L (u,v) Figure 0. Flow of (0.9) (0.). wit ɛ = ( C / β 3 ). To study region 3, given η wit 0 < η <, we assume η C / β 3 η and β is small. Terefore η ɛ η. Hence we ave a slowfast system wit one slow variable (γ ) and tree fast variables (, u and v). Set β = 0: ḣ = ɛ(γ +4( γ ))u, (0.9) u = (γ +4( γ ))v, (0.0) v = γ (u P ()) +4( γ ) (u H ()), (0.) γ = 0. (0.) Eac plane γ = constant is invariant and as equilibria at points (, u, v) wit γ P () +4( γ )H () = 0 and u = v = 0. By proposition 5., te equilibria constitute two one-dimensional normally yperbolic invariant manifolds: C i ={(, u, v, γ ) : = i (γ, 4( γ )), u = 0, v= 0, 0 γ }, i =,. See figure 0. Equilibria on C ave two eigenvalues wit positive real part and one wit negative real part; equilibria on C ave one eigenvalue wit positive real part and two wit negative real part. On te plane γ = constant, after division by γ +4( γ )>0 te system becomes ḣ = ɛu, (0.3) u = v, (0.4) γ v = γ +4( γ ) (u P ()) + 4( γ ) (u H ()). (0.5) γ +4( γ ) By a simple cange of variables we can convert ɛ in (0.3) to. Ten by te connection teorem, wose ypoteses are verified by proposition 5., te two-dimensional unstable manifold of eac equilibrium on C meets te two-dimensional stable manifold of te equilibrium on C wit te same value of γ. Teorem 0.. Let L and R satisfying (3.5) be given. Consider te travelling wave system (5.) (5.5) wit te normalization (N). Assume tere is a γ 0, 0 γ 0, suc tat for (0.3) (0.5) wit γ = γ 0, te two-dimensional unstable manifold of te equilibrium on C meets te two-dimensional stable manifold of te equilibrium on C transversally. Ten for eac η 3 wit 0 < η 3 <, tere exists δ 3 > 0 suc tat, if region 3 is defined using tese values, ten for eac parameter triple ( C,, β ) in region 3, tere exists a one-parameter family of travelling waves connecting L to R wit max γ near γ 0.

06 V Manukian and S Scecter We note tat according to te transversality teorem, te transversality condition witin te plane γ = γ 0 is satisfied if te connection tere is simple. Proof. Te assumptions imply tat for (0.3) (0.5), te tree-dimensional unstable manifold of C meets te tree-dimensional stable manifold of C transversally near γ = γ 0 in a two-dimensional family of fast connecting orbits. For small β > 0, C (respectively, C ) perturbs to a normally yperbolic invariant curve connecting ( L, 0, 0) to (, 0, 0) (respectively, ( R, 0, 0) to (, 0, 0)). On te perturbed normally yperbolic invariant manifolds corresponding to C i, we ave γ = β ɛγ( γ ) Q( i (γ, 4( γ )) + O( β )), wic is positive on te interior of C and negative on te interior of C. Tere is a repeller at L and an attractor at R. Singular solutions ( β = 0) from ( L, 0, 0, 0) to ( R, 0, 0, 0) consist of a slow solution on C from ( L, 0, 0, 0) to level γ,0< γ < ; a fast connection from C to C at level γ, and a slow solution on C to ( R, 0, 0, 0). Because of te transversality, for γ near γ 0 tey persist for small β > 0. Note tat at ( L, 0), C as slope 4H ( L )/ L P ( L )>0; at ( R, 0), C as slope 4H ( R )/ R P ( R )>0. As in te previous section, te positive slope at ( L, 0) produces a rise in te travelling wave above L (capillary ridge).. Region 4 Teorem.. Let L and R satisfying (3.5) be given. Consider te travelling wave system (5.) (5.5) wit te normalization (N): C = C 0, D = and β = β 0. Ten for eac η 4 wit 0 < η 4 <, tere exists δ 4 > 0 suc tat, if region 4 is defined using tese values, ten for eac parameter triple ( C,, β ) in region 4, tere exists a one-parameter family of travelling waves connecting L to R. Moreover, max γ ranges from 0 to. Proof. In te travelling wave system (0.5) (0.8), wit ɛ = ( C / β 3), let β = β ( ) β 5 ɛ =. C Te system becomes ḣ = ɛ(γ +4( γ ))u, (.) u = (γ +4( γ ))v, (.) v = γ (u P ()) +4( γ ) (u H ()), (.3) γ = β ɛ γ ( γ ) Q(). (.4) Given η 4 wit 0 < η 4 <, we assume η 4 C / β 5 η 4 and β is small. Terefore η 4 β η 4, and ɛ = β /β is small. Hence (.) (.4) is a slow-fast system wit two slow variables ( and Ɣ) and two fast variables (u and v). For ɛ = 0, te set u = γ P () +4( γ )H (), v = 0 (.5) γ +4( γ )

Travelling waves for a tin liquid film 07 is a normally yperbolic manifold of equilibria of dimension (one positive eigenvalue, one negative eigenvalue); compare (9.3). For small ɛ > 0, te perturbed normally yperbolic invariant manifold is given by γ P () +4( γ )H () u = K(, γ, ɛ, β ), K(, γ, 0, β ) =, (.6) γ +4( γ ) v = L(, γ, ɛ, β ), L(, γ, 0, β ) = 0; (.7) compare (9.4) (9.5). Te vector field on te perturbed normally yperbolic invariant manifold is ḣ = ɛ ( γ P () +4( γ )H () + O(ɛ) ), (.8) In slow time (i.e. divide by ɛ): γ = β ɛ γ ( γ ) Q(). (.9) = γ P () +4( γ )H () + O(ɛ), (.0) γ = β ɛγ( γ ) Q(). (.) In tese coordinates, for ɛ = 0, eac line γ = constant is invariant and as equilibria at points wit γ P () + 4( γ )H () = 0. By proposition 5., te equilibria constitute two one-dimensional normally yperbolic invariant manifolds C i ={(, γ ) : = i (γ, 4( γ )), 0 γ }, i =,. See figure. Equilibria on C are repellers; equilibria on C are attractors. For small ɛ > 0, C (respectively, C )perturbstoanormallyyperbolicinvariantcurve connecting ( L, 0, 0) to (, 0, 0) (respectively, ( R, 0, 0) to (, 0, 0)). On te perturbed normally yperbolic invariant manifolds corresponding to C i, we ave γ = β ɛγ( γ ) Q( i (γ, 4( γ )) + O(β )), wic is positive on te interior of C and negative on te interior of C. Tere is a repeller at L and an attractor at R. Singular solutions (ɛ = 0) from ( L, 0) to ( R, 0) consist of a slow solution on C from ( L, 0) to level γ,0< γ < ; a fast connection from C to C at level γ ; and a slow solution on C to ( R, 0). Tey persist for small ɛ > 0. See figure. As in region 3, te slopes of C and C are positive were tey meet ( L, 0) and ( R, 0), respectively. Te positive slope at ( L, 0) produces a rise in te travelling wave above L (capillary ridge).. Region 5 Teorem.. Let L and R satisfying (3.5) be given. Consider te travelling wave system (3.) (3.4) wit te normalization (N3): C = C 3 0, D = D 3 0 and β =. Ten for eac small η > 0, tere exists δ 5 > 0 suc tat, if region 5 is defined using tis value, ten for eac parameter triple ( C 3, D 3, ) in region 5, tere exists a one-parameter family of travelling waves connecting L to R. Moreover, max Ɣ ranges from 0 to η. If we decrease η, ten apparently δ 5 must srink. Tus, unlike in regions and 4, we are not able to define a single region 5 in wic connecting orbits of all sizes exist. To prove te teorem, in te travelling wave system (3.) (3.4) let ( ) C ɛ =, û = βu, ˆv = β ɛv. β 3