Spatial dynamics methods for solitary waves on a ferrofluid jet
SSpatial dynamics methods for solitary waveson a ferrofluid jet
M. D. Groves ∗ D. V. Nilsson † Abstract
This paper presents existence theories for several families of axisymmetric solitary waveson the surface of an otherwise cylindrical ferrofluid jet surrounding a stationary metal rod.The ferrofluid, which is governed by a general (nonlinear) magnetisation law, is subject toan azimuthal magnetic field generated by an electric current flowing along the rod.The ferrohydrodynamic problem for axisymmetric travelling waves is formulated as aninfinite-dimensional Hamiltonian system in which the axial direction is the time-like vari-able. A centre-manifold reduction technique is employed to reduce the system to a locallyequivalent Hamiltonian system with a finite number of degrees of freedom, and homo-clinic solutions to the reduced system, which correspond to solitary waves, are detectedby dynamical-systems methods. r = R + η ( z , θ , t ) x y z H n S S Figure 1: Waves on the surface of a ferrofluid jet surrounding a current-carrying wire. ∗ Fachrichtung 6.1 - Mathematik, Universit¨at des Saarlandes, Postfach 151150, 66041 Saarbr¨ucken, Germany;Department of Mathematical Sciences, Loughborough University, Loughborough, LE11 3TU, UK † Centre for Mathematical Sciences, Lund University, PO Box 118, 22100 Lund, Sweden a r X i v : . [ m a t h . A P ] J un e consider an incompressible, inviscid ferrofluid of unit density in the region S := { < r < R + η ( θ, z, t ) } bounded by the free interface { r = R + η ( θ, z, t ) } and a current-carrying wire at { r = 0 } , where ( r, θ, z ) are cylindrical polar coordinates. The fluid is subject to a static magnetic field and thesurrounding region S = { r > R + η ( θ, z, t ) } is a vacuum (see Figure 1). Travelling waves move in the axial direction with constant speed c and without change of shape, so that η ( θ, z, t ) = η ( θ, z − ct ) . We are interested in particularin axisymmetric solitary waves for which η does not depend upon θ and η ( z − ct ) → as z − ct → ±∞ . Waves of this kind for ferrofluids with a linear magnetisation law have beeninvestigated using a weakly nonlinear approximation by Rannacher & Engel [18], experimentallyby Bourdin, Bacri & Falcon [4] and numerically by Blyth & Parau [3]. In this paper we presenta rigorous existence theory for small-amplitude solitary waves and consider fluids with a general(nonlinear) magnetisation law.Our starting point is a formulation of the hydrodynamic problem as a reversible Hamiltoniansystem η z = δHδω , ω z = − δHδη , ˆ φ z = δHδ ˆ ζ , ˆ ζ z = − δHδ ˆ φ (1.1)in which the axial coordinate z plays the role of time, ˆ φ is a variable related to the fluid ve-locity potential φ and ω , ˆ ζ are the momenta associated with the coordinates η , ˆ φ . The spatialHamiltonian system (1.1) is derived from a variational principle for the governing equations inSection 3; it depends upon two dimensionless physical parameters α and β (see equation (2.6)for precise definitions) and the (dimensionless) magnitude m ( | H | ) of the magnetic intensitycorresponding to the magnetic field H in the ferrofluid. Homoclinic solutions of (1.1) (solutions with ( η, ω, ˆ φ, ˆ ζ ) → as z → ±∞ ) are of particu-lar interest since they correspond to solitary waves. We detect such solutions using a techniqueknown as the Kirchg¨assner reduction (Section 4), in which a centre-manifold reduction principleis used to show that all small, globally bounded solutions of a spatial (Hamiltonian) evolution-ary system solve a (Hamiltonian) system of ordinary differential equations, whose solution setcan in principle be determined. In this fashion we reduce (1.1) to a Hamiltonian system withfinitely many degrees of freedom which can be treated by well-developed dynamical-systemsmethods, in particular normal-form theory. We proceed by perturbing the physical parameters β , α around fixed reference values β , α and thus introducing bifurcation parameters ε , ε .The Kirchg¨assner reduction delivers an ε -dependent reduced system which captures the small-amplitude dynamics for small values of these parameters; its dimension is the number of purelyimaginary eigenvalues of the corresponding linearised system at ( ε , ε ) = (0 , . The reductionprocedure is therefore especially helpful in detecting bifurcations which are associated with achange in the number of purely imaginary eigenvalues.Working in the ( β, γ ) parameter plane, where γ = α − β , one finds that there are threecritical curves C , C , C at which the number of purely imaginary eigenvalues changes (seeFigure 2(a)), together with a fourth curve C at which the number of real eigenvalues changes.(In fact C = { ( β,
2) : β < } , C = { ( β,
2) : β > } and explicit formulae for C and2 are given in Section 4.) A similar diagram arises in the study of gravity-capillary travellingwater waves (see Iooss [14], Groves & Wahl´en [13] and the references therein), and there thecurves corresponding to C , C and C are associated with homoclinic bifurcation: homoclinicsolutions of the reduced Hamiltonian system (corresponding to solitary water waves) bifurcatefrom the trivial solution. Figure 2(a) illustrates the parameter regions I, II and III adjacent to C , C and C in which the existence of homoclinic solutions is to be expected. In Section 5we study these regions using the Kirchg¨assner reduction; the basic types of solitary wave foundthere are sketched in Figures 2(b)–(d).In Section 5.1 we examine region I, choosing ( β , γ ) ∈ C , so that α = 2 + β , and writing α = α + µ with < µ (cid:28) . According to the Kirchg¨assner reduction small-amplitude solitarywaves are given by η ( z ) = µ ( β − ) / Q (cid:0) µ / ( β − ) − / z (cid:1) + O ( µ / ) , where ( Q, P ) is a homoclinic solution of the reversible Hamiltonian system ˙ Q = P + O ( µ / ) , (1.2) ˙ P = Q − ˇ c Q + O ( µ / ) (1.3)with ˇ c := ( α m (cid:48) (1) − . This system admits a homoclinic solution which corresponds toa monotonically decaying, symmetric solitary wave of elevation for ˇ c > and depression for ˇ c < . For m (cid:48) (1) close to the critical value α − we write m (cid:48) (1) = α − (6 + 2 µ / ˇ κ ) with < ˇ κ (cid:28) and find that small-amplitude solitary waves are given by η ( z ) = µ / ( β − ) / Q (cid:0) µ / ( β − ) − / z (cid:1) + O ( µ / ) , where ( Q, P ) is a homoclinic solution of the reversible Hamiltonian system ˙ Q = P + O ( µ / ) , (1.4) ˙ P = Q − ˇ κQ − ˇ d Q + O ( µ / ) (1.5)with ˇ d = (12 − α m (cid:48)(cid:48) (1)) . For ˇ d > this system admits a pair of homoclinic solutions whichcorrespond to monotonically decaying, symmetric solitary waves; one is a wave of depression,the other a wave of elevation. Note that in the limit µ = 0 or ( µ, ˇ κ ) = (0 , the variable Q solvesa travelling-wave version of the (generalised) Korteweg-de Vries equation.In Section 5.2 we apply the Kirchg¨assner reduction in region II, finding that small-amplitudesolitary waves are given by η ( z ) = µ P ( µz ) + O ( µ ) , where ( Q, P ) is a homoclinic solution of the reversible Hamiltonian system ˙ Q = − P + (1 + δ ) P + (1 + δ ) P + 3 c P + O ( µ ) , ˙ Q = P + (1 + δ ) P + O ( µ ) , ˙ P = Q + O ( µ ) , ˙ P = Q + (1 + δ ) Q + O ( µ ) IIIII CCC C βγ (a) Bifurcation curves in the ( β, γ ) -plane; the shaded regions indicate the parameter regimes inwhich homoclinic bifurcation is detected.(b) Solitary waves of elevation (left) and depression (right) in region I.(c) Primary solitary waves of elevation (left) and depression (right) in region II.(d) Primary solitary waves of elevation (left) and depression (right) in region III.Figure 2: Summary of the basic types of solitary wave whose existence is established in thepresent paper by the Kirchg¨assner reduction. c = 48 √ m (cid:48) (1) − ; the parameters < µ, δ (cid:28) measure the distance from respec-tively the point ( β , γ ) = ( , and the curve C . This system admits a homoclinic solutionwhich corresponds to a solitary wave of elevation for c > and depression for c < ; the waveis symmetric with an oscillatory decaying tail. For m (cid:48) (1) close to the critical value we write m (cid:48) (1) = (8 + √ ˇ κµ ) with < ˇ κ (cid:28) and find that small-amplitude solitary waves aregiven by η ( z ) = µ P ( µz ) + O ( µ ) , where ( Q, P ) is a homoclinic solution of the reversible Hamiltonian system ˙ Q = − P + (1 + δ ) P + (1 + δ ) P + ˇ κP + 4 d P + O ( µ ) , ˙ Q = P + (1 + δ ) P + O ( µ ) , ˙ P = Q + O ( µ ) , ˙ P = Q + (1 + δ ) Q + O ( µ ) with d = 864 (cid:0) − m (cid:48)(cid:48) (1) (cid:1) . For d > this system admits a a pair of homoclinic solutionswhich correspond to symmetric solitary waves with oscillatory decaying tails; one is a wave ofdepression, the other a wave of elevation. Note that in the limit µ = 0 or ( µ, ˇ κ ) = (0 , thevariable P solves a travelling-wave version of the (generalised) Kawahara equation.It is instructive to interpret the above results for two well-studied magnetic intensities. (i) The linear magnetisation law m ( s ) = s. In region I we find that ˇ c < for α < (solitary waves of depression) and ˇ c > for α > (solitary waves of elevation); furthermore ˇ d = 2 , so that both types of waves exist for α near . This region has also been studied by Rannacher & Engel [18] using a weakly nonlinear ap-proximation. In terms of the magnetic Bond number B = α /β (corresponding to B < ) theyderived a Korteweg-de Vries equation equivalent to (1.2), (1.3) and found solitary waves of de-pression for < B < (that is, α < ) and of elevation for < B < (that is, α > ),in agreement with our results. (Continuing their weakly nonlinear analysis to the next order inthis region would lead to a cubic Korteweg-de Vries equation equivalent to (1.4), (1.5) and theprediction of both types of waves for B near ). In region II we find that c = − √ (solitarywaves of depression). (ii) The Langevin magnetisation law m ( s ) = coth( λs ) − ( λs ) − coth λ − λ − , where λ > is a dimensionless parameter. In Region I we find that ˇ c < for α < and α > , λ ∈ ( λ (cid:63) ( α ) , ∞ ) (solitary waves of depression), while ˇ c < for α > , λ ∈ (0 , λ (cid:63) ( α )) (solitary waves of elevation), where λ (cid:63) ( α ) is the unique solution of the equation λ − − λ cosech λ coth λ − λ − = 6 α − (so that λ (cid:63) (6) = 0 ). Furthermore ˇ d > , so that both types of waves exist for ( λ, α ) near ( λ (cid:63) , α ( λ (cid:63) )) (with α (0) = 6 ). In region II we find that c < (solitary waves of depression).5n Section 5.3 we turn to region III. Introducing a bifurcation parameter µ so that positivevalues of µ correspond to points on the ‘complex’ side of C , one obtains the reduced (reversible)Hamiltonian system ˙ A = ∂ ˜ H µ ∂ ¯ B , ˙ B = − ∂ ˜ H µ ∂ ¯ A , ˜ H µ = i s ( A ¯ B − ¯ AB ) + | B | + ˜ H ( | A | , i( A ¯ B − ¯ AB ) , µ ) + O ( | ( A, B ) | | ( µ, A, B ) | n ) , where ˜ H is a real polynomial which satisfies ˜ H = 0 ; it contains the terms of order , . . . , n +1 in the Taylor expansion of ˜ H µ . The substitution A ( z ) = e i sz a ( z ) , B ( z ) = e i sz b ( z ) convertsthe ‘truncated normal form’ obtained by neglecting the remainder term into the system ˙ a = b + ∂ b ˜ H ( | a | , i( a ¯ b − ¯ ab ) , µ ) , ˙ b = − ∂ ¯ a ˜ H ( | a | , i( a ¯ b − ¯ ab ) , µ ) (which, as evidenced by the scaling z (cid:55)→ µ / z , ( a, b ) (cid:55)→ ( µ / a, µb ) , is at leading order equiv-alent to the nonlinear Schr¨odinger equation). Supposing that the coefficients of certain terms in ˜ H have the correct sign, one finds that the latter system admits a circle of homoclinic solutions,two of which are real. The corresponding pair of homoclinic solutions to the original ‘truncatednormal form’ are reversible and persist when the remainder terms are reinstated (see Iooss &P´erou`eme [15]). They generate symmetric solitary waves which take the form of periodic wavetrains modulated by exponentially decaying envelopes; one is a wave of depression, the other awave of elevation.Each of the basic types of solitary waves in regions II and III is the primary member of an in-finite family of multipulse solitary waves which resemble multiple copies of the primary. Thesewaves are generated by corresponding multipulse homoclinic solutions which make several largeexcursions away from the origin in their four-dimensional phase space. A more precise descrip-tion of the multipulse waves, together with a discussion of the relevant existence theories (whichare based on variational and dynamical-systems arguments) is given in Sections 5.2 and 5.3.Although the techniques used in the present paper are generalisations of those developed forthe water-wave problem (see Iooss [14], Groves & Wahl´en [13] and the references therein), weemploy different methods to compute the reduced Hamiltonian systems. The spatial Hamiltoniansystem (1.1) is invariant under the transformation ˆ φ (cid:55)→ ˆ φ + c , c ∈ R (‘variation of potential base-level’), and the quantity (cid:82) r ˆ ζ d r is conserved. In many hydrodynamic problems it is possible toeliminate a symmetry of this kind before applying the Kirchg¨assner reduction (see e.g. Groves,Lloyd & Stylianou [11, § before lowering the order of the system since itcan be ‘absorbed’ into the changes of variable associated with the Kirchg¨assner reduction; thisprocedure greatly simplifies our later calculations. We present a general result for this purpose(Theorem 4.4), whose proof is based upon the method given by Bridges & Mielke [5, Theorem4.3] and which may also be helpful in other applications.6 The ferrohydrodynamic problem
We consider an incompressible, inviscid ferrofluid of unit density in the region S := { < r < R + η ( θ, z, t ) } bounded by the free interface { r = R + η ( θ, z, t ) } and a current-carrying wire at { r = 0 } , where ( r, θ, z ) are cylindrical polar coordinates. The fluid is subject to a static magnetic field and thesurrounding region S = { r > R + η ( θ, z, t ) } is a vacuum (see Figure 1).We denote the magnetic and induction fields in the fluid and vacuum by respectively H , B and H , B , and suppose that the relationships between them are given by the identities B = µ ( H + M ( H )) , B = µ H , where µ is the magnetic permeability of free space and M is the (prescribed) magnetic intensityof the ferrofluid. We suppose that M ( H ) = m ( | H | ) H | H | where m is a (prescribed) nonnegative function, so that in particular M and H are collinear.According to Maxwell’s equations the magnetic and induction fields are respectively irrota-tional and solenoidal, and introducing magnetic potential functions ψ , ψ with H = −∇ ψ , H = −∇ ψ , we therefore find that ∇ · ( µ ( |∇ ψ | ) ∇ ψ ) = 0 in S , ∆ ψ = 0 in S , in which µ ( s ) = 1 + m ( s ) s is the magnetic permeability of the ferrofluid relative to that of free space. We suppose thatthe ferrofluid flow is irrotational, so that its velocity field v is the gradient of a scalar velocitypotential φ . The Euler equation for the ferrofluid is given by v t + ( v . ∇ ) v = −∇ p (cid:63) + µ ( M · ∇ ) H (Rosensweig [19, § p (cid:63) is its composite pressure, and the calculations ( M · ∇ ) H = | M |∇ ( | H | ) = ∇ (cid:32)(cid:90) | H | m ( t ) d t (cid:33) , ( v . ∇ ) v = ∇ (cid:18) | v | (cid:19) show that this equation is equivalent to φ t + 12 |∇ φ | − µ (cid:90) | H | m ( t ) d t + p (cid:63) = c , (2.1)7here c is a constant.Next we turn to the boundary conditions at { r = R + η ( θ, z, t ) } . The magnetic boundaryconditions are H · t = H · t , B · n = B · n , where t and n denote tangent and normal vectors to the free surface; it follows that ψ − ψ (cid:12)(cid:12)(cid:12) r = R + η ( θ,z,t ) = 0 , ψ n − µ ( |∇ ψ | ) ψ n (cid:12)(cid:12)(cid:12) r = R + η ( θ,z,t ) = 0 . The (hydro-)dynamical boundary condition is given by p (cid:63) + µ M · n ) = 2 σκ, (Rosensweig [19, § σ > is the coefficient of surface tension and κ = − η θ − ( R + η ) (1 + η z ) + ( R + η ) η zz + ( R + η ) η θ η zz − R + η ) η θ η z η θz + ( R + η )(1 + η z ) η θθ (( R + η )(1 + η z ) + η θ ) / is the mean curvature of the interface; using (2.1), we find that φ t + 12 |∇ φ | − µ ν ( |∇ ψ | ) + 2 σκ − µ µ ( |∇ ψ | ) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r = R + η ( θ,z,t ) = c , where ν ( s ) = (cid:90) s m ( t ) d t. Finally, the (hydro-)kinematic boundary condition is ( ∂ t + v . ∇ )( r − R − η ( θ, z, t )) = 0 , that is − η t + φ r − r φ θ η θ − φ z η z = 0 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r = R + η ( θ,z,t ) . The relevant conditions at r = 0 and in the far field are v . e r , B . e r → as r → , so that φ r , ψ r → as r → , and B . e r → as r → ∞ , so that ψ r → as r → ∞ .The constant c is selected so that H = J πr e θ , H = J πr e θ , v = , η = 0 (that is ψ = ψ = − J θ/ π , φ = 0 , η = 0 ) is a solution to the above equations (correspondingto a uniform magnetic field and a circular cylindrical jet with radius R ); we therefore set c = − µ ν ( J/ πr ) + σ/R . Seeking axisymmetric waves for which η and φ are independent of θ , onefinds that ψ = ψ = − J θ/ π , so that the hydrodynamic problem decouples from the magneticproblem and is given by φ rr + 1 r φ r + φ zz = 0 , < r < R + η ( z, t ) ,φ r = 0 , r = 0 − η t + φ r − φ z η z = 0 ,φ t + 12 ( φ r + φ z ) − µ ν (cid:18) J π ( R + η ) (cid:19) + µ ν (cid:18) J πR (cid:19) + σ ( R + η )(1 + η z ) / − ση zz (1 + η z ) / − σR = 0 for r = R + η ( z, t ) .The next step is to seek travelling wave solutions for which η and φ depend upon z and t onlythrough the combination z − ct , and to introduce dimensionless variables (ˆ z, ˆ r ) := 1 R ( z − ct, r ) , ˆ φ := 1 cR φ, ˆ η := 1 R η. and functions ˆ m ( s ) := 2 πRJ χ m (cid:18) J πR s (cid:19) , ˆ ν ( s ) := 4 π R J χ ν (cid:18) J πR s (cid:19) , where χ = 2 πRJ m (cid:18) J πR (cid:19) (note that ˆ m (1) = ˆ ν (cid:48) (1) = 1 ). Dropping the hats for notational simplicity, we find that φ rr + 1 r φ r + φ zz = 0 , < r < η ( z, t ) , (2.2) φ r = 0 , r = 0 (2.3)and η z + φ r − φ z η z = 0 , (2.4) − φ z + 12 ( φ r + φ z ) − α T (cid:48) ( η )1 + η + β (cid:18) η )(1 + η z ) / − η zz (1 + η z ) / − (cid:19) = 0 (2.5)for r = 1 + η ( z, t ) , where T ( η ) = (cid:90) η (cid:18) ν (cid:18)
11 + s (cid:19) − ν (1) (cid:19) (1 + s ) d s and α = µ J χ π R c , β = σc R (2.6)are dimensionless parameters. Solitary waves are nontrivial solutions of (2.2)–(2.5) with η ( z ) , φ ( r, z ) → as z → ±∞ . Finally, note that equations (2.2), (2.4) and (2.5) follow from theformal variational principle δ (cid:90) (cid:26)(cid:90) η (cid:18) rφ r + 12 rφ z − rφ z (cid:19) d r − αT ( η ) + β (1 + η )(1 + η z ) / − β (1 + η ) (cid:27) d z = 0 , where the variations are taken with respect to η and φ .9 Spatial dynamics
The first step is to use the ‘flattening’ transformation ˆ r = r η to map the variable domain { < r < η } into a fixed strip (0 , × R and the free interface { r = 1 + η ( z ) } into { ˆ r = 1 } . Dropping the hat for notational simplicity, we find that thecorresponding ‘flattened’ variable ˆ φ (ˆ r, z ) = φ ( r, z ) satisfies the equations ( rφ r ) r (1 + η ) + r (cid:18) φ z − rη z φ r η (cid:19) z − r η z η (cid:18) φ z − rη z φ r η (cid:19) r = 0 , < r < (3.1)with boundary conditions φ r | r =0 = 0 (3.2)and η z + φ r η − (cid:18) φ z − rη z φ r η (cid:19) η z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r =1 = 0 , (3.3) − (cid:18) φ z − rη z φ r η (cid:19) + 12(1 + η ) φ r + 12 (cid:18) φ z − rη z φ r η (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r =1 − α T (cid:48) ( η )1 + η + β (cid:18) η )(1 + η z ) / − η zz (1 + η z ) / − (cid:19) = 0 . (3.4)Observe that equations (3.1), (3.3) and (3.4) follow from the new variational principle δ L = 0 ,where L ( η, φ ) := (cid:90) (cid:40) (cid:90) (cid:40) (cid:32) rφ r + (cid:18) φ z − rη z φ r η (cid:19) (1 + η ) r (cid:33) − (cid:18) φ z − rη z φ r η (cid:19) (1 + η ) r (cid:41) d r − αT ( η ) + β (1 + η )(1 + η z ) / − β (1 + η ) (cid:41) d z (3.5)and the variations are taken in η and φ (the functional L is obtained from the variational func-tional for (2.2), (2.4) and (2.5) by ‘flattening’).We exploit this variational principle by regarding L as an action functional of the form L = (cid:90) L ( η, φ, η z , φ z ) d z,
10n which L is the integrand on the right-hand side of equation (3.5), and deriving a canonicalHamiltonian formulation of (3.1)–(3.4) by means of the Legendre transform. To this end, let usintroduce new variables ω and ξ by the formulae ω = δLδη z = (cid:90) (cid:26) − (cid:18) φ z − rη z φ r η (cid:19) (1 + η ) r φ r + (1 + η ) r φ r (cid:27) d r + β (1 + η ) η z (1 + η z ) / ,ξ = δLδφ z = (cid:18) φ z − rη z φ r η (cid:19) (1 + η ) − (1 + η ) and define the Hamiltonian function by H ( η, ω, φ, ξ )= η z ω + (cid:90) rφ z ξ d r − L ( η, φ, η z , φ z )= (cid:90) (cid:40) (cid:18) ξ (1 + η ) + 1 (cid:19) (1 + η ) r − rφ r (cid:41) d r + αT ( η ) − (1 + η ) (cid:112) β − W + 12 β (1 + η ) , (3.6)in which W = 11 + η (cid:18) ω + 11 + η (cid:90) r φ r ξ d r (cid:19) . Writing ( β, α ) = ( β + ε , α + ε ) , where ( β , α ) are fixed, and ξ = ζ − (since ( η, ω, φ, ξ ) = (0 , , − , is the ‘trivial’ solution of Hamilton’s equations), we find that Hamil-ton’s equations are given explicitly by η z = δH ε δω = W (cid:112) ( β + ε ) − W , (3.7) ω z = − δH ε δη = (cid:90) (cid:40)(cid:18) ( ζ − (1 + η ) − (cid:19) (1 + η ) r + W r φ r ( ζ − η ) (cid:112) ( β + ε ) − W (cid:41) d r − ( α + ε ) T (cid:48) ( η ) + ( β + ε ) (cid:112) ( β + ε ) − W − ( β + ε )(1 + η ) , (3.8) φ z = δH ε δζ = (cid:18) ( ζ − η ) + 1 (cid:19) + W (cid:112) ( β + ε ) − W rφ r η , (3.9) ζ z = − δH ε δφ = − r ( rφ r ) r + W (cid:112) ( β + ε ) − W r ( r ( ζ − r η , (3.10)where the superscript denotes the dependence upon ε , with boundary condition rφ r − W (cid:112) ( β + ε ) − W r ( ζ − η (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r =1 = 0 , (3.11)the second of which arises from the integration by parts necessary to compute (3.10) Notethat our equations are reversible, that is invariant under the transformation ( η, ω, φ, ζ )( z ) (cid:55)→ S ( η, ω, φ, ζ )( − z ) , where the reverser is defined by S ( η, ω, φ, ζ ) = ( η, − ω, − φ, ζ ) .11o make this construction rigorous we recall the differential-geometric definitions of a Hamil-tonian system and Hamilton’s equations for its associated vector field. Definition 3.1.
A Hamiltonian system consists of a triple ( M, Ω , H ) , where M is a manifold, Ω :
T M × T M → R is a closed, weakly nondegenerate bilinear form (the symplectic -form)and the Hamiltonian H : N → R is a smooth function on a manifold domain N of M (thatis, a manifold N which is smoothly embedded in M and has the property that T N | n is denselyembedded in T M | n for each n ∈ N ).Its Hamiltonian vector field v H with domain D ( v H ) ⊆ N is defined as follows. The point n ∈ N belongs to D ( v H ) with v H | n := w ∈ T M | n if and only if Ω | n ( w, v ) = d H | n ( v ) for all tangent vectors v ∈ T M | n (by construction d H | n ∈ T ∗ N | n admits a unique extension d H | n ∈ T (cid:63) M | n ). Hamilton’s equations for ( M, Ω , H ) are the differential equations ˙ u = v H | u which determine the trajectories u ∈ C ( R , M ) ∩ C ( R , N ) of its Hamiltonian vector field. Definition 3.1 applies to the above formulation. Note that the identity mapping is(up to the scaling factor √ π ) an isometry ˇ L ( B (0)) → L r (0 , , ˇ H ( B (0)) → H r (0 , and ˇ H ( B (0)) → { φ ∈ H r (0 ,
1) : φ r ∈ L r − (0 , } , where B (0) is the unit ball in R and ˇ H s ( B (0)) denotes the closed subspace of H s ( B (0)) consisting of axisymmetric functions (seeBernardi, Dauge & Maday [2, Theorem II.2.1]). We therefore let M be a neighbourhood of theorigin in X := { ( η, ω, φ, ζ ) ∈ R × R × H r (0 , × L r (0 , } and N = Y ∩ M with Y := { ( η, ω, φ, ζ ) ∈ R × R × H r (0 , × H r (0 ,
1) : φ r ∈ L r − (0 , } , so that elements ( η, ω, φ, ζ ) ∈ Y satisfy φ r | r =0 = 0 (see Bernardi, Dauge & Maday [2, RemarkII.1.1]). We consider values of ( ε , ε ) in a neighbourhood Λ of the origin in R and choose M and Λ small enough so that | ε | < β , η > − > − , | W | < β < β + ε . The formula
Ω(( η , ω , φ , ζ ) , ( η , ω , φ , ζ )) = ω η − η ω + (cid:90) r ( ζ φ − φ ζ ) d r defines a weakly nondegenerate bilinear form M × M → R and hence a constant symplectic -form T M × T M → R (its closure follows from the fact that it is constant), and the function H ε given by (3.6) belongs to C ∞ ( N, R ) , so that the triple ( M, Ω , H ) is a Hamiltonian system.Applying the criterion in the definition, one finds that D ( v H ε ) = (cid:40) ( η, ω, φ, ζ ) ∈ N : rφ r − W (cid:112) ( β + ε ) − W r ( ζ − η (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r =1 = 0 (cid:41) ( M, Ω , H ε ) and a solution to the ‘flattened’ hydrodynamic problem (3.1)–(3.4). Suppose that ( η, ω, φ, ζ ) is a smooth solution of Hamilton’s equations. An explicit calculation shows that thevariables ˜ η , ˜ φ given by ˜ η ( z ) = η ( z ) , ˜ φ ( r, z ) = φ ( z )( r ) solve (3.1)–(3.4) (see Groves & Toland[12, pp. 212-214] for a discussion of this procedure in the context of water waves). Our strategy in finding solutions to Hamilton’s equations (3.7)–(3.10) for ( M, Ω , H ε ) consistsin applying a reduction principle which asserts that ( M, Ω , H ε ) is locally equivalent to a finite-dimensional Hamiltonian system. The key result is the following theorem, which is a parametrised,Hamiltonian version of a reduction principle for quasilinear evolutionary equations presented byMielke [17, Theorem 4.1] (see Buffoni, Groves & Toland [8, Theorem 4.1]). Theorem 4.1.
Consider the differential equation ˙ u = L u + N ( u ; λ ) , (4.1) which represents Hamilton’s equations for the reversible Hamiltonian system ( M, Ω λ , H λ ) . Here u belongs to a Hilbert space X , λ ∈ R (cid:96) is a parameter and L : D ( L ) ⊂ X → X is a denselydefined, closed linear operator. Regarding D ( L ) as a Hilbert space equipped with the graphnorm, suppose that is an equilibrium point of (4.1) when λ = 0 and that(H1) The part of the spectrum σ ( L ) of L which lies on the imaginary axis consists of a finitenumber of eigenvalues of finite multiplicity and is separated from the rest of σ ( L ) in thesense of Kato, so that X admits the decomposition X = X ⊕ X , where X = P ( X ) , X = ( I − P )( X ) and P is the spectral projection corresponding the purely imaginarypart of σ ( L ) .(H2) The operator L = L| X satisfies the estimate (cid:107) ( L − i sI ) − (cid:107) X →X ≤ C | s | , s ∈ R , for some constant C that is independent of s .(H3) There exists a natural number k and neighbourhoods Λ ⊂ R (cid:96) of and U ⊂ D ( L ) of such that N is ( k + 1) times continuously differentiable on U × Λ , its derivatives arebounded and uniformly continuous on U × Λ and N (0 ,
0) = 0 , d N [0 ,
0] = 0 .Under these hypotheses there exist neighbourhoods ˜Λ ⊂ Λ of and ˜ U ⊂ U ∩X , ˜ U ⊂ U ∩X of and a reduction function r : ˜ U × ˜Λ → ˜ U with the following properties. The reduction function r is k times continuously differentiable on ˜ U × ˜Λ , its derivatives are bounded and uniformlycontinuous on ˜ U × ˜Λ and r (0; 0) = 0 , d r [0; 0] = 0 . The graph ˜ M λ = { u + r ( u ; λ ) ∈X ⊕ X : u ∈ ˜ U } is a Hamiltonian centre manifold for (4.1) , so that i) ˜ M λ is a locally invariant manifold of (4.1) : through every point in ˜ M λ there passes aunique solution of (4.1) that remains on ˜ M λ as long as it remains in ˜ U × ˜ U .(ii) Every small bounded solution u ( x ) , x ∈ R of (4.1) that satisfies ( u ( x ) , u ( x )) ∈ ˜ U × ˜ U lies completely in ˜ M λ .(iii) Every solution u : ( x , x ) → ˜ U of the reduced equation ˙ u = L u + PN ( u + r ( u ; λ ); λ ) (4.2) generates a solution u ( x ) = u ( x ) + r ( u ( x ); λ ) (4.3) of the full equation (4.1) .(iv) ˜ M λ is a symplectic submanifold of M and the flow determined by the Hamiltoniansystem ( ˜ M λ , ˜Ω λ , ˜ H λ ) , where the tilde denotes restriction to ˜ M λ , coincides with the flow on ˜ M λ determined by ( M, Ω λ , H λ ) . The reduced equation (4.2) is reversible and representsHamilton’s equations for ( ˜ M λ , ˜Ω λ , ˜ H λ ) . Mielke’s theorem cannot be applied directly to (3.7)–(3.10) because of the nonlinear bound-ary condition (3.11) in the domain of the Hamiltonian vector field v H ε (the right-hand sides of(3.7)–(3.10) define a smooth mapping g ε : Y → X with v H ε | u = g ε ( u ) for any u ∈ D ( v H ε ) ).We overcome this difficulty using the change of variable G ε : ( η, ω, φ, ζ ) (cid:55)→ ( η, ˆ ω, ˆ φ, ζ ) , where ˆ ω = (cid:90) r φ r d r, ˆ φ = φ − W (cid:112) ( β + ε ) − W
11 + η (cid:90) r s ( ζ −
1) d s, (4.4)which transforms the boundary condition in D ( v H ε ) into r ˆ φ r | r =1 = 0 . Lemma 4.2.
For each ε ∈ Λ the mapping G ε is a smooth diffeomorphism from the neighbour-hood M of the origin in X onto a neighbourhood ˆ M of the origin in X , and from N = M ∩ Y onto ˆ N = ˆ M ∩ Y . The diffeomorphisms and their inverses depend smoothly upon ε ∈ Λ . Proof.
These results follow from the explicit formulae (4.4) and ω = ( β + ε )Γ(1 + η ) √ −
11 + η (cid:90) r ˆ φ r ( ζ −
1) d r − Γ(1 + η ) (cid:90) r ( ζ − d r,φ = ˆ φ + Γ1 + η (cid:90) r s ( ζ −
1) d s, where Γ = (1 + η ) (cid:18) ˆ ω − (cid:90) r ˆ φ r d r (cid:19)(cid:30)(cid:90) r ( ζ −
1) d r, for G ε and its inverse ( G ε ) − : ( η, ˆ ω, ˆ φ, ζ ) (cid:55)→ ( η, ω, φ, ζ ) .14 simple calculation shows that the diffeomorphism G transforms u z = g ε ( u ) into u z = ˆ g ε ( u ) , (4.5)where ˆ g ε : Y → X is the smooth vector field defined by ˆ g ε ( u ) = d G ε [( G ε ) − ( u )] ( g ε (( G ε ) − ( u ))) . Formula (4.5) represents Hamilton’s equations for the Hamiltonian system ( ˆ
M , Υ ε , ˆ H ε ) , where Υ ε (cid:12)(cid:12) m ( v , v ) = Ω(d G ε [( G ε ) − ( m )] − ( v ) , d G ε [( G ε ) − ( m )] − ( v )) , m ∈ ˆ M , v , v ∈ T ˆ M | m , and ˆ H ε ( n ) = H ε (( G ε ) − ( n )) , n ∈ ˆ N .
The domain of the Hamiltonian vector field v ˆ H ε is D ( v ˆ H ε ) = { ( η, ˆ ω, ˆ φ, ζ ) ∈ ˆ N : r ˆ φ r | r =1 = 0 } and v ˆ H ε | n = ˆ g ε ( n ) for any n ∈ D ( v ˆ H ε ) .The next step is to verify that (4.5) satisfies the hypotheses of Theorem 4.1 (with X = X ), sothat we obtain a finite-dimensional reduced Hamiltonian system ( ˜ M ε , ˜Γ ε , ˜ H ε ) . We write (4.5) as u z = Lu + N ε ( u ) , in which L = d v ˆ H [0] and verify the spectral hypotheses on L by considering the operator K : D ( K ) ⊆ X → X , where K ηωφζ = β (cid:18) ω − (cid:90) r φ r d r (cid:19) − (cid:90) rζ d r − η + ( α − β ) ηζ + 2 η − r ( rφ r ) r − β (cid:18) ω − (cid:90) r φ r d r (cid:19) (4.6)and D ( K ) = (cid:40) ( η, ω, φ, ζ ) ∈ Y : − rφ r − r β (cid:18) ω − (cid:90) r φ r d r (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r =1 (cid:41) (the formal linearisation of v H at the origin); the formula K = d G [0] − L d G [0] shows that thespectral properties of K and L are identical. It follows from Lemma 4.3 below that L satisfieshypotheses (H1) and (H2); hypothesis (H3) is clearly satisfied for an arbitrary value of k . Part(i) of Lemma 4.3 is proved using the elementary theory of ordinary differential equations, whilepart (ii) is established using arguments similar to those employed for other problems treated usingcentre-manifold reduction (e.g. see Buffoni, Groves & Toland [8, Proposition 3.2] or Groves &Wahl´en [13, Lemma 3.4]). 15 emma 4.3. (i) The spectrum σ ( L ) of L consists entirely of isolated eigenvalues of finite algebraic mul-tiplicity. A complex number λ is an eigenvalue of L if and only if λJ ( λ ) = ( γ − β λ ) J ( λ ) , where γ = α − β . (In particular, is an eigenvalue of L and σ ( L ) ∩ i R is a finite set.)(ii) There exist real constants C , s > such that (cid:107) ( L − i sI ) − (cid:107) L ( X,X ) ≤ C | s | for each real number s with | s | > s . According to Lemma 4.3(i), a purely imaginary number λ = i s is an eigenvalue of L if andonly if sI ( s ) = ( γ + β s ) I ( s ) . Straightforward computations show that there are three critical curves C = (cid:26) ( β , γ ) = (cid:18) (cid:18) − I ( s ) I ( s ) I ( s ) (cid:19) , s (cid:18) − I ( s ) I ( s ) (cid:19)(cid:19) : s ∈ (0 , ∞ ) (cid:27) and C = { ( β , γ ) : β < , γ = 2 } , C = { ( β , γ ) : β > , γ = 2 } in the ( β , γ ) parameter plane at which purely imaginary eigenvalues of L collide, together witha fourth curve C = (cid:26) ( β , γ ) = (cid:18) (cid:18) − J ( k ) J ( k ) J ( k ) (cid:19) , k ( J ( k ) + J ( k ) )2 J ( k ) (cid:19) : k ∈ (0 , j , ) (cid:27) at which real eigenvalues collide (see Figure 3). Here J , J , . . . and I , I , . . . denote respec-tively the Bessel functions and modified Bessel functions of the first kind, and j , > is thesmallest zero of J . Furthermore, L has a geometrically simple zero eigenvalue whose algebraicmultiplicity is two for γ (cid:54) = 2 , four for γ = 2 , β (cid:54) = and and six for ( β , γ ) = ( , .The centre manifold ˜ M ε is equipped with the single coordinate chart ˜ U ⊂ X and coordinatemap π : ˜ M ε → ˜ U defined by π − ( u ) = u + r ( u ; ε ) . It is however more convenient to use analternative coordinate map for calculations. We define the function ˜ r : ˜ W × ˜Λ → ˜ U × ˜ U with ˜ W = P ( G ε ) − ( ˜ U × ˜ U ) (which in general has components in X and X ) by the formula w + ˜ r ( w ; ε ) = ( G ε ) − (cid:0) w + r ( w ; ε ) (cid:1) , (4.7)where ˜ r (0; 0) = 0 , d ˜ r [0; 0] = 0 , and equip ˜ M ε with the coordinate map ˆ π : ˜ M ε → ˜ W given by ˆ π − ( w ) = w + ˜ r ( w ; ε ) , so that ˜ H ε ( w ) = H ε ( w + ˜ r ( w ; ε )) , ˜Ω ε | w ( v , v ) = Ω( v + d ˜ r [ w ; ε ]( v ) , v + d ˜ r [ w ; ε ]( v ))= Ω( v , v ) + O ( | ( ε, w ) | ) (4.8)as ( ε, w ) → . Furthermore, using a parameter-dependent version of Darboux’s theorem (e.g.see Buffoni & Groves [7, Theorem 4]), we may assume that the remainder term in (4.8) vanishesidentically. 16 CCC C βγ Figure 3: Eigenvalues of L ; solid and hollow dots denote respectively algebraically simple andmultiple eigenvalues. The curves C j , j = 1 , . . . , consist of points in ( β , γ ) parameter spaceat which the qualitative nature of the eigenvalue picture changes. We proceed by choosing a symplectic basis { f , . . . , f n , f , . . . , f n } for the centre subspaceof K (so that Ω( f i , f i ) = 1 for i = 0 , . . . , n and the symplectic product of any other combinationof these vectors is zero); here either f or f is the eigenvector (0 , , , T corresponding to thezero eigenvalue of K . Using coordinates q , . . . , q n , p , . . . , p n , where w = q f + q f + · · · + q n f n + p f + p f + · · · + p n f n , we find that ˜Ω ε is the canonical -form. Note that equations (3.7)–(3.11) are invariant under thetransformation φ (cid:55)→ φ + c , c ∈ R , and the quantity (cid:82) rζ d r is conserved. This symmetry isinherited by the reduced system: one of the variables q , p is cyclic (that is, ˜ r and ˜ H ε do notdepend upon it), so that the other is conserved.According to the classical theory, the next step is to lower the dimension of the reduced sys-tem by two by setting the conserved variable to zero, solving the resulting decoupled system for q , . . . , q n , p , . . . p n , and recovering the cyclic variable by quadrature; the lower-order system istypically studied using a canonical change of variables which simplifies its Hamiltonian ˜ H ε | q =0 (a ‘normal-form’ transformation). For our purposes it is convenient to use a normal-form trans-formation before lowering the order of the system since it can be ‘absorbed’ into ˜ r in the sameway as the Darboux transformation; this procedure greatly simplifies our later calculations. Thefollowing general result (whose proof is based upon the method given by Bridges & Mielke [5,Theorem 4.3]) shows that this procedure is possible; we assume for definiteness that p is cyclicand use the construction by Elphick [10] as our ‘usual’ normal form. The result is applied tothe specific parameter regimes shown in Figure 2(a) in Section 5 below, where we denote thenonlinear part of the reduced Hamiltonian vector field v ˜ H ε by P ε ( w ) .17 heorem 4.4. Consider the ( n + 1) -degree-of-freedom Hamiltonian system ˙ q i = ∂ ˜ H ε ∂p i , ˙ p i = − ∂ ˜ H ε ∂q i , i = 1 , . . . , n, (4.9) ˙ q = ∂ ˜ H ε ∂p , ˙ p = − ∂ ˜ H ε ∂q , (4.10) where ˜ H ε ( q, p, q ) = O ( | ( ε, q , q, p ) || ( q , q, p ) | ) and p is cyclic (so that q is conserved).There exists a near-identity canonical change of variables ( q, p, q , p ) (cid:55)→ ( Q, P, Q , P ) with the properties that P is cyclic, Q = q and the lower-order Hamiltonian system ˙ Q i = ∂ ˜ H ε ∂P i ( Q, P, , ˙ P i = − ∂ ˜ H ε ∂Q i ( Q, P, , i = 1 , . . . , n, adopts its usual normal form. (Here, with a slight abuse of notation, we denote the transformedHamiltonian by ˜ H ε ( Q, P, Q ) .) Proof.
Consider the n -degree of freedom Hamiltonian system ˙ q i = ∂ ˜ H ε ∂p i , ˙ p i = − ∂ ˜ H ε ∂q i , i = 1 , . . . , n, (4.11)in which q and ε are parameters. The standard theory asserts the existence of a canonical changeof variables Q = q + h ε ( q, p, q ) ,P = p + h ε ( q, p, q ) with h εj ( q, p, q ) = O ( | ( ε, q , q, p ) || ( q, p ) | ) , j = 1 , , which converts (4.11) into its parameter-dependent normal form; note that M T1 J M = J , where M = (cid:18) Q q Q p P q P q (cid:19) = (cid:18) I + ∂ q h ε ∂ p h ε ∂ q h ε I + ∂ p h ε (cid:19) , J = (cid:18) I − I (cid:19) , and this condition may also be written as ( I + ∂ q h ε )( I + ∂ p h ε ) − ∂qh ε ∂ p h ε = I. (4.12)We seek a change of variable for (4.9), (4.10) of the form Q = q + h ε ( q, p, q ) ,P = p + h ε ( q, p, q ) ,Q = q ,P = p + h ε ( q, p, q ); h ε ( q, p, q ) = O ( | ( ε, q , q, p ) || ( q, p ) | ) is subject to the requirement that M T2 J M = J , where M = Q q Q p Q q Q p P q P p P q P p Q q Q p Q q Q p P q P p P q P p = I + ∂ q h ε ∂ p h ε ∂ q h ε ∂ q h ε I + ∂ p h ε ∂ q h ε
00 0 1 0 ∂ q h ε I + ∂ p h ε ∂ q h ε ,J = I − I − , and this condition may be written as ( I + ∂ q h ε )( I + ∂ p h ε ) − ∂ q h ε ∂ p h ε = I, (4.13) ( I + ∂ q h ε ) ∂ q h ε − ∂ q h ε ∂ q h ε = ∂ q h ε , (4.14) ∂ p h ε ∂ q h ε − ( I + ∂ p h ε ) ∂ q h ε = ∂ p h ε . (4.15)It is possible to find h ε satisfying these conditions since the compatibility condition for (4.14),(4.15) is the derivative of (4.13) with respect to q , and (4.13) is automatically satisfied becauseof (4.12). C At each point of the curve C in Figure 3 two real eigenvalues become purely imaginary bycolliding at the origin and increasing the algebraic multiplicity of the zero eigenvalue from twoto four. This resonance is associated with the bifurcation of a branch of homoclinic solutions intothe region with real eigenvalues (the parameter regime marked I in Figure 2. Let us thereforefix reference values ( β , γ ) ∈ C , so that β > , α = 2 + β , and introduce a bifurcationparameter by choosing ( ε , ε ) = (0 , µ ) , where < µ (cid:28) .The four-dimensional centre subspace of K is spanned by the generalised eigenvectors e = , e = , e = ( β − ) − r + A , e = ( β − ) + A − r − ( β − ) , where A = − ( β − ) − (cid:0) + β ( β − (cid:1) has been chosen so that Ke = 0 , Ke j = e j − for j = 2 , , , Ω( e , e ) = − (cid:0) β − (cid:1) , Ω( e , e ) = (cid:0) β − (cid:1) e , . . . e is zero. Writing w = q f + p f + qf + pf , f i := (cid:0) β − (cid:1) − / e i , we therefore find that q , q , p and p are canonical coordinates for the reduced Hamiltoniansystem, which has the cyclic variable p and reverser S : ( q , q, p , p ) (cid:55)→ ( q , q, − p , − p ) ; with aslight abuse of notation we abbreviate ˜ H ε | ( ε ,ε )=(0 ,µ ) to ˜ H µ .The usual normal-form theory for the two-dimensional system with Hamiltonian ˜ H µ ( q, p, asserts that, after a canonical change of variables, ˜ H µ ( q, p,
0) = p + ˜ H ( q, µ ) + O ( | ( q, p ) | | ( µ, q, p ) | n ) , where ˜ H ( q, µ ) is a polynomial of order n + 1 in ( q, µ ) with ˜ H ( q, µ ) = O ( | q | | ( µ, q ) | ) . It follows that, after a canonical change of variables, ˜ H µ ( q, p, q ) = p − qq + ˜ H µ nl ( q, p, q ) with ˜ H µ nl ( q, p, q ) = ˜ H NF ( q, q , µ ) + ˜ H r ( q, q , µ ) + O ( | ( q, p, q ) | | ( µ, q, p, q ) | n ); here ˜ H NF ( q, q , µ ) is a polynomial of order n + 1 with ˜ H NF ( q, q , µ ) = O ( | q | | ( µ, q, q ) | ) and ˜ H NF ( q, , µ ) = ˜ H ( q, µ ) , and ˜ H r ( q, q , µ ) is an affine function of its first argument whichsatisfies ˜ H r ( q, q , µ ) = O ( | ( q, q ) || q || ( µ, q, q ) | ) . Note that P µ ( q, p, q ) = − ∂ q ˜ H µ nl ( q, p, q ) f − ∂ q ˜ H µ nl ( q, p, q ) f . Writing ˜ H ( q, p, q ) = c q + c q q + c qq + c q , ˜ H ( q, p, q ) = c q + c qq + c q , where µ j ˜ H jk ( q, p, q ) denotes the part of the Taylor expansion of ˜ H µ ( q, p, q ) which is homoge-neous of order j in µ and k in ( q, p, q ) , one finds that c = (cid:0) β − (cid:1) − / ( α m (cid:48) (1) − , c = − (cid:0) β − (cid:1) − (see Appendix (i)). Setting q = 0 and introducing scaled variables Z = µ / ( β − ) − / z, q ( z ) = µ ( β − ) / Q ( Z ) , p ( z ) = µ / P ( Z ) , yields ˜ H µ ( q, p,
0) = µ (cid:2) P − Q + ˇ c Q (cid:3) + O ( µ / ) , ˇ c = ( α m (cid:48) (1) − , and the lower-order Hamiltonian system ˙ Q = P + O ( µ / ) , (5.1) ˙ P = Q − ˇ c Q + O ( µ / ) , (5.2)which is reversible with reverser S : ( Q, P ) (cid:55)→ ( Q, − P ) . Suppose ˇ c (cid:54) = 0 . In the limit µ = 0 equations (5.1), (5.2) are equivalent to the single equation ∂ Z u − u + u = 0 for the variable u = ˇ c Q .Let us now suppose that m (cid:48) (1) is close to the critical value α − and introduce a secondbifurcation parameter κ by setting m (cid:48) (1) = α − (6 + κ ) and observing that ˜ r ( q, p, q ; µ, κ ) = O ( | ( q, p, q ) || ( µ, q, p, q ) | ) + O ( | κ || ( q, p, q ) | ) , ˜ H µ,κ ( q, p, q ) = O ( | ( q, p, q ) | | ( µ, q, p, q ) | ) + O ( | κ || ( q, p, q ) | ) (with a slight change of notation). Writing ˜ H , ( q, p, q ) = d q + d q q + d q q + d qq + d q , where µ i κ j ˜ H i,jk ( q, p, q ) denotes the part of the Taylor expansion of ˜ H µ,κ ( q, p, q ) which is ho-mogeneous of order i in µ , j in κ and k in ( q, p, q ) , one finds that d = ( β − ) − (12 − α m (cid:48)(cid:48) (1)) (see Appendix (ii)). Setting q = 0 , introducing scaled variables Z = µ / ( β − ) − / z, q ( z ) = µ / ( β − ) / Q ( Z ) , p ( z ) = µP ( Z ) and writing κ = 2 µ / ˇ κ, thus yields ˜ H µ,κ ( q, p,
0) = µ (cid:2) P − Q + ˇ κQ + ˇ d Q (cid:3) + O ( µ / ) , where ˇ d = (12 − α m (cid:48)(cid:48) (1)) , and the lower-order Hamiltonian system ˙ Q = P + O ( µ / ) , (5.3) ˙ P = Q − ˇ κQ − ˇ d Q + O ( µ / ) , (5.4)21hich is of course reversible with reverser S : ( Q, P ) (cid:55)→ ( Q, − P ) . Suppose that ˇ d > . In thelimit ( µ, ˇ κ ) = 0 equations (5.3), (5.4) are equivalent to the single equation ∂ Z u − u + u = 0 for the variable u = ˇ d / Q .The phase portrait of the equation ¨ u − u + u m = 0 (5.5)for a fixed natural number m (which is a travelling-wave version of the generalised Korteweg-deVries equation) is readily obtained by elementary calculations and is sketched in Figure 4; thehomoclinic orbits are of particular interest. Lemma 5.1. (i) Suppose that m is even. Equation (5.5) has precisely one homoclinic solution h (up totranslations). This solution is positive and symmetric, and monotone increasing to the left,monotone decreasing to the right of its point of symmetry.(ii) Suppose that m is odd. Equation (5.5) has precisely two homoclinic solutions ± h , where h is symmetric, and monotone increasing to the left, monotone decreasing to the right ofits point of symmetry.In both cases the homoclinic solutions intersect the symmetric section { ˙ u = 0 } in the two-dimensional phase space { ( u, ˙ u ) ∈ R } transversally. A familiar argument shows that Lemma 5.1(i) also applies to (5.1), (5.2) for small, positivevalues of µ , while Lemma 5.1(ii) applies to (5.3), (5.4) for small, positive values of µ and small,values of ˇ κ (that is, small values of κ ); the qualitative statements apply to the variable ˇ c Q or ˇ d / Q . The homoclinic orbits at µ = 0 (and ˇ κ = 0 ) intersect the symmetric section Fix R = { P = 0 } transversally, and these orbits therefore persist (as small, uniform perturbations of theirlimits) for small, positive values of µ (and small values of ˇ κ ).Altogether we have established the existence of a symmetric, monotonically decaying soli-tary wave of depression for m (cid:48) (1) < α − and elevation for m (cid:48) (1) > α − ; the correspondingferrofluid surface { r = 1 + η ( z ) } is obtained from the homoclinic solution of (5.1), (5.2) by theformula η ( z ) = µ ( β − ) / Q (cid:0) µ / ( β − ) − / z (cid:1) + O ( µ / ) . Furthermore, a pair of symmetric, monotonically decaying solitary waves exists for small valuesof m (cid:48) (1) − α − provided that m (cid:48)(cid:48) (1) < α − ; one is a wave of depression, the other a wave ofelevation. The corresponding ferrofluid surface { r = 1 + η ( z ) } is obtained from a homoclinicsolution of (5.3), (5.4) by the formula η ( z ) = µ / ( β − ) / Q (cid:0) µ / ( β − ) − / z (cid:1) + O ( µ / ) . (A more detailed analysis of a codimension-two bifurcation of this kind was given by Kirrmann[16, §
4] in the context of two-layer fluid flow.) Figure 5 shows a sketch of the ferrofluid surfacecorresponding to solitary waves of the present type.22 u. u u.
Figure 4: Phase portrait of equation (5.5) for even (left) and odd (right) values of m z Figure 5: A solitary wave of elevation (left) and depression (right) generated by a homoclinicsolution (top) in region I C At each point of the curve C in Figure 3 two pairs of real eigenvalues become complex bycolliding at non-zero points on the real axis. Of particular interest here is the local part of C near the point ( β , γ ) = ( , (which is given by β = + µ + O ( µ ) , γ = 2+ µ + O ( µ ) for < µ (cid:28) ) since we can access this curve using the centre-manifold technique. To this endwe choose β = , α = and ε = µ , ε = µ + µ , µ = (1 + δ ) µ , µ = µ . (5.6)Notice that µ indicates the distance in ( β , γ ) parameter space from the point ( , , while δ plays the role of a bifurcation parameter (varying δ through zero from above we cross the criticalcurve C from above); the parameter regime marked II in Figure 2 corresponds to small, positivevalues of δ and µ . 23he six-dimensional centre subspace of K is spanned by the generalised eigenvectors e = , e = , e = − r , e = − − r ,e = − − r + r , e = − − r + r where Ke = 0 , Ke j = e j − for j = 2 , . . . , , Ω( e , e ) = , Ω( e , e ) = − , Ω( e , e ) = and the symplectic product of any other combination of the vectors e , . . . e is zero. Writing w = q f + p f + q f + p f + q f + p f , f i := 8 √ e i , we therefore find that q , q , q , p , p and p are canonical coordinates for the reduced Hamilto-nian system, which has the cyclic variable q and reverser S : ( q , q , q , p , p , p ) (cid:55)→ ( − q , − q , − q , p , p , p ) ; with a slight abuse of notation we abbreviate ˜ H ε | ( ε ,ε )=( µ ,µ + µ ) to ˜ H µ ,µ .The usual normal-form theory for the four-dimensional system with Hamiltonian ˜ H µ ,µ ( q, p, ,where q = ( q , q ) , p = ( p , p ) asserts that, after a canonical change of variables, ˜ H µ ,µ ( q, p,
0) = p − q q + ˜ H ( q, p, µ , µ ) + O ( | ( q, p ) | | ( µ , µ , q, p ) | n ) , where ˜ H ( q, p, µ , µ ) is a polynomial of order n +1 which depends upon q , q , p , p throughthe combinations p , q − p p , q + 3 p q − p p q , − p p + 3 p q − p q − q q + 18 p p q q and satisfies ˜ H ( q, p, µ , µ ) = O ( | ( q, p ) | | ( µ , µ , q, p ) | ) . It follows that, after a canonical change of variables, ˜ H µ ,µ ( q, p, p ) = p − q q + p p + ˜ H µ ,µ nl ( q, p, p ) with ˜ H µ ,µ nl ( q, p, p ) = ˜ H NF ( q, p, p , µ , µ )+ ˜ H r ( q, p, p , µ , µ )+ O ( | ( q, p, p ) | | ( µ , µ , q, p, p ) | n ); here ˜ H NF ( q, p, p , µ , µ ) is a polynomial of order n + 1 which depends upon q , q , p , p through the above combinations and satisfies ˜ H NF ( q, p, q , µ , µ ) = O ( | ( q, p ) | | ( µ , µ , q, p, p ) | ) , ˜ H NF ( q, p, , µ , µ ) = ˜ H ( q, p, µ , µ ) , and ˜ H r ( q, p, p , µ , µ ) is an affine function of itsfirst two arguments which satisfies ˜ H r ( q, p, p , µ , µ ) = O ( | ( q, p, p ) || p || ( µ , µ , q, p, p ) | ) . Note that P µ ,µ ( q, p, q ) = ∂ p ˜ H µ ,µ nl ( q, p, p ) f + ∂ p ˜ H µ ,µ nl ( q, p, p ) f − ∂ q ˜ H µ ,µ nl ( q, p, p ) f − ∂ q ˜ H µ ,µ nl ( q, p, p ) f + ∂ p ˜ H µ ,µ nl ( q, p, p ) f . Writing ˜ H , ( q, p, p ) = c p + c p p + c p p + c p + c p ( q − p p ) + c p ( q − p p ) + c p p , ˜ H , ( q, p, p ) = c , p + c , p p + c , p + c , ( q − p p ) + c , p p , ˜ H , ( q, p, p ) = c , p + c , p p + c , p + c , ( q − p p ) + c , p p , ˜ H , ( q, p, p ) = c , p + c , p p + c , p + c , ( q − p p ) + c , p p , where µ i µ j ˜ H i,jk ( q, p, p ) denotes the part of the Taylor expansion of ˜ H µ ,µ ( q, p, q ) which ishomogeneous of order i in µ , j in µ and k in ( q, p, p ) , one finds that c = 48 √ m (cid:48) (1) − , c , = 0 , c , = − , c , = 512 , c , = − (see Appendix (iii)). Setting p = 0 , choosing µ , µ according to (5.6) and introducing thescaled variables Z = µz, q ( z ) = µ Q ( Z ) , q ( z ) = µ Q ( Z ) , p ( z ) = µ P ( Z ) , p ( z ) = µ P ( Z ) , thus yields ˜ H µ ,µ ( q, p, µ (cid:2) P − P − Q Q − (1 + δ )( Q − P P ) + (1 + δ ) P + c P (cid:3) + O ( µ ) and the lower-order Hamiltonian system Q Z = − P + (1 + δ ) P + (1 + δ ) P + 3 c P + O ( µ ) , (5.7) Q Z = P + (1 + δ ) P + O ( µ ) , (5.8) P Z = Q + O ( µ ) , (5.9) P Z = Q + (1 + δ ) Q + O ( µ ) , (5.10)which is reversible with reverser S : ( Q, P ) (cid:55)→ ( − Q, P ) . Suppose c (cid:54) = 0 . In the limit µ = 0 equations (5.7)–(5.10) are equivalent to the single fourth-order ordinary differential equation ∂ Z u − δ ) ∂ Z u + u − u = 0 for the variable u = 3 c P . 25et us now suppose that m (cid:48) (1) is close to the critical value and introduce a further bifur-cation parameter κ by setting m (cid:48) (1) = (8 + κ ) and observing that ˜ r ( q, p, p ; µ , µ , κ ) = O ( | ( q, p, p ) || ( µ , µ , q, p, p ) | ) + O ( | κ | ( q, p, p ) | ) , ˜ H µ ,µ ,κ ( q, p, p ) = O ( | ( q, p, p ) | | ( µ , µ , q, p, p ) | ) + O ( | κ || ( q, p, p ) | ) (with a slight change of notation). Writing ˜ H , , ( q, p, p ) = d p + d p p + d p p + d p p + d p + d ( q − p p ) + d p ( q − p p ) + d p ( q − p p ) + d p p ( q − p p )+ d ( − p p + 3 p q − p q − q q + 18 p p q q ) + d p p where µ i µ j κ k ˜ H i,j(cid:96) ( q, p, q ) denotes the part of the Taylor expansion of ˜ H µ ,µ ( q, p, q ) which ishomogeneous of order i in µ , j in µ , k in κ and (cid:96) in ( q, p, q ) , one finds that d = 864 (cid:16) − m (cid:48)(cid:48) (1) (cid:17) (see Appendix (iv)). Setting p = 0 , choosing µ , µ according to (5.6), introducing the scaledvariables Z = µz, q ( z ) = µ Q ( Z ) , q ( z ) = µ Q ( Z ) , p ( z ) = µ P ( Z ) , p ( z ) = µ P ( Z ) , and writing κ = √ ˇ κµ thus yields ˜ H µ ,µ ,κ ( q, p, µ (cid:2) P − P − Q Q − (1 + δ )( Q − P P ) + (1 + δ ) P + ˇ κP + d P (cid:3) + O ( µ ) and the lower-order Hamiltonian system Q Z = − P + (1 + δ ) P + (1 + δ ) P + ˇ κP + 4 d P + O ( µ ) , (5.11) Q Z = P + (1 + δ ) P + O ( µ ) , (5.12) P Z = Q + O ( µ ) , (5.13) P Z = Q + (1 + δ ) Q + O ( µ ) , (5.14)which is of course reversible with reverser S : ( Q, P ) (cid:55)→ ( − Q, P ) . Suppose d > . In the limit ( µ, ˇ κ ) → equations (5.11)–(5.14) are equivalent to the single fourth-order ordinary differentialequation ∂ Z u − δ ) ∂ Z u + u − u = 0 for the variable u = 2 d / P . 26xistence theories for homoclinic solutions to the equation .... u − δ )¨ u + u − u m = 0 . (5.15)for a fixed natural number m ≥ (which is a travelling-wave version of the generalised Kawa-hara equation) are given in Theorems 5.2 and 5.3 below. These theorems are generalisations ofresults given by Buffoni, Champneys & Toland [6] (see also Devaney [9]) for the special case m = 2 ; a full discussion of their generalisation to m ≥ is given by Ahmad [1]. Theorem 5.2.
Suppose that δ ≥ .(i) Suppose that m is even. Equation (5.15) has precisely one homoclinic solution h (up totranslations). This solution is positive and symmetric, and monotone increasing to the left,monotone decreasing to the right of its point of symmetry.(ii) Suppose that m is odd. Equation (5.15) has precisely two homoclinic solutions ± h , where h is symmetric, and monotone increasing to the left, monotone decreasing to the right ofits point of symmetry.In both cases the homoclinic solutions are transverse, that is, the stable and unstable mani-folds of the zero equilibrium intersect transversally with respect to the zero level surface of theHamiltonian at their point of symmetry. Theorem 5.3.
The primary homoclinic solutions found in the previous theorem persist (as small,uniform perturbations of their limits at δ = 0 ) for small, negative values of δ .Furthermore, each primary homoclinic solution h in the region δ < generates a family oftransverse multipulse homoclinic solutions which resemble multiple copies of h ‘glued’ togetherwith small oscillations in between. More precisely, for each all natural numbers (cid:96) , . . . , (cid:96) n − with n = 1 , , . . . there exists a homoclinic solution n ( (cid:96) , . . . , (cid:96) n − ) associated with h which(i) has n local extrema at t , . . . , t n ,(ii) oscillates (cid:4) (cid:96) k (cid:5) times and has (cid:4) (cid:96) k − (cid:5) extrema in each interval ( t k , t k +1 ) ,(iii) oscillates infinitely often in the intervals ( −∞ , t ) and ( t n , ∞ ) . Theorem 5.2(i) also applies to (5.7)–(5.10) for small, positive values of µ , while Theorem5.2(ii) applies to (5.11)–(5.14) for small, positive values of µ and small, values of ˇ κ (that is, smallvalues of κ ); the qualitative statements apply to the variable ˇ c P or ˇ d / P . The homoclinicorbits at µ = 0 (and ˇ κ = 0 ) are transverse and therefore persist (as small, uniform perturbationsof their limits) for small, positive values of µ (and small values of ˇ κ ). Similarly, Theorem 5.3applies to any of these persistent primary homoclinic orbits.Altogether we have established the existence of a primary and accompanying multipulsefamily of solitary waves of depression for m (cid:48) (1) < and elevation for m (cid:48) (1) > ; the corre-sponding ferrofluid surface { r = 1 + η ( z ) } is obtained from the homoclinic solution of (5.1),(5.2) by the formula η ( z ) = µ P ( µz ) + O ( µ ) . Figure 6: A solitary wave of elevation (left) and depression (right) generated by a ‘primary’homoclinic solution (top) in region II z Figure 7: A solitary wave of elevation (left) and depression (right) generated by a ‘ (2) ’ homo-clinic solution (top) in region II m (cid:48) (1) − provided that m (cid:48)(cid:48) (1) (cid:54) = ; one consists of waves of depression, the other of waves of elevation.The corresponding ferrofluid surface { r = 1 + η ( z ) } is obtained from a homoclinic solution of(5.3), (5.4) by the formula η ( z ) = µ P ( µz ) + O ( µ ) . C At each point of the curve C in Figure 3 two pairs of purely imaginary eigenvalues becomecomplex by colliding at non-zero points ± i s on the imaginary axis and forming two Jordanchains of length 2. This resonance is associated with the bifurcation of a branch of homoclinicsolutions into the region with complex eigenvalues (the parameter regime marked III in Figure2). Let us therefore choose β = 12 (cid:18) − I ( s ) I ( s ) I ( s ) (cid:19) , γ = 12 s (cid:18) − I ( s ) I ( s ) (cid:19) (so that α = γ − β ) and introduce a bifurcation parameter µ by writing ( ε , ε ) = (0 , µ ) ,where < µ (cid:28) .The six-dimensional centre subspace of K is spanned by the generalised eigenvectors e = , e = γ − − γ − , e, ¯ e, f, ¯ f , where e = I ( s )i sβ I ( s ) − i I ( s ) − i I ( sr ) sI ( sr ) − I ( s ) , f = − i I ( s ) + i s I ( s ) β I ( s ) − s I ( s ) − I ( s ) − rI ( sr ) − i I ( sr ) − i rsI ( sr ) + 2 iI ( s ) − s I ( s ) − i τ τ e and τ = 2 I ( s ) − s I ( s ) I ( s ) + sI ( s ) I ( s ) − I ( s ) ,τ = − (cid:18) s ( − s ) I ( s ) − s I ( s ) I ( s ) + 9 I ( s ) I ( s ) − I ( s ) I ( s ) + 1 s (5 + s ) I ( s ) (cid:19) ; note that Ke = 0 , Ke = e , ( K − i sI ) e = 0 , ( K − i sI ) f = e , Ω( e , e ) = − γ − , Ω( e, ¯ f ) = τ , Ω(¯ e, f ) = τ and the symplectic product of any other combination of the vectors e e , e , f , ¯ e , ¯ f is zero.Writing w = q f + p f + AE + BF + ¯ A ¯ E + ¯ B ¯ F , f = ( − γ − / ) − e , f = ( − γ − ) − / e , E = τ − / e, F = τ − / f, we therefore find that q , p , A and B are canonical coordinates for the reduced Hamiltoniansystem, which has the cyclic variable q and reverser S : ( q , p , A, B ) (cid:55)→ ( − q , p , ¯ A, − ¯ B ) ;with a slight abuse of notation we abbreviate ˜ H ε | ( ε ,ε )=(0 ,µ ) to ˜ H µ .The usual normal-form theory for the two-dimensional system with Hamiltonian ˜ H µ ( A, B, ¯ A, ¯ B, asserts that, after a canonical change of variables, ˜ H µ ( A, B, ¯ A, ¯ B,
0) = i s ( A ¯ B − ¯ AB )+ | B | + ˜ H ( | A | , i( A ¯ B − ¯ AB ) , µ )+ O ( | ( A, B ) | | ( µ, A, B ) | n ) , where ˜ H is a real polynomial function of its arguments which satisfies ˜ H ( | A | , i( A ¯ B − ¯ AB ) , µ ) = O ( | ( A, B ) | | ( µ, A, B ) | ) . It follows that, after a canonical change of variables, ˜ H µ ( A, B, ¯ A, ¯ B, p ) = i s ( A ¯ B − ¯ AB ) + | B | + p + ˜ H µ nl ( A, B, ¯ A, ¯ B, p ) with ˜ H µ nl ( A, B, ¯ A, ¯ B, p ) = ˜ H NF ( | A | , i( A ¯ B − ¯ AB ) , p , µ )+ ˜ H r ( A, B, ¯ A, ¯ B, p , µ ) + O ( | ( A, B, p ) | | ( µ, A, B, p ) | n ); here ˜ H NF is a real polynomial function of its arguments which satisfies ˜ H NF ( | A | , i( A ¯ B − ¯ AB ) , p , µ ) = O ( | ( A, B ) | | ( µ, A, B, p ) | ) and ˜ H NF ( | A | , i( A ¯ B − ¯ AB ) , , µ ) = ˜ H ( | A | , i( A ¯ B − ¯ AB ) , µ ) , and ˜ H r ( A, B, ¯ A, ¯ B, p , µ ) isan affine function of its first four arguments which satisfies ˜ H r ( | A | , i( A ¯ B − ¯ AB ) , p , µ ) = O ( | ( A, B, p ) || p || ( µ, A, B, p ) | ) Note that P µ ( A, B, ¯ A, ¯ B, p ) = ∂ ¯ B ˜ H µ nl ( A, B, ¯ A, ¯ B, p ) E + ∂ B ˜ H µ nl ( A, B, ¯ A, ¯ B, p ) ¯ E − ∂ ¯ A ˜ H µ nl ( A, B, ¯ A, ¯ B, p ) F − ∂ A ˜ H µ nl ( A, B, ¯ A, ¯ B, p ) ¯ F + ∂ p ˜ H µ nl ( A, B, ¯ A, ¯ B, p ) f . Writing ˜ H ( A, B, p ) = c p + c | A | + c i( A ¯ B − ¯ AB ) + c p A + ¯ c p ¯ A + c p B + ¯ c p ¯ B, ˜ H ( A, B, p ) = c p + c p | A | + c i p ( A ¯ B − ¯ AB ) + c p A + ¯ c p ¯ A + c p B + ¯ c p ¯ B ˜ H ( A, B, p ) = d p + d p | A | + d i p ( A ¯ B − ¯ AB ) + d | A | + d i( A ¯ B − ¯ AB ) | A | − d ( A ¯ B − ¯ AB ) + d p A + ¯ d p ¯ A + d p B + ¯ d p ¯ B, µ j ˜ H jk ( A, B, p ) denotes the part of the Taylor expansion of ˜ H µ ( A, B, p ) which is homo-geneous of order j in µ and k in ( A, B, p ) , one finds that d = I ( s ) τ (cid:32) ( − s + s β − sT + 4 s ST − α m (cid:48) (1))( − s − s β − sS + 4 s ST − α m (cid:48) (1))2( γ + 4 s β − sT ) − ( s β − sS + 2 + α m (cid:48) (1))(3 sS − α m (cid:48) (1)) γ −
2+ 7 s − s β + s β + 6 sS − s S + 4 s S T − s ST − α m (cid:48) (1) − α m (cid:48)(cid:48) (1) (cid:33) ,c = − I ( s ) τ , (5.16) where S = I ( s ) I ( s ) , T = I (2 s ) I (2 s ) (see Appendix (v)).The lower-order Hamiltonian system A Z = ∂ ¯ B ˜ H µ ( A, B, ¯ A, ¯ B, , , (5.17) B Z = − ∂ ¯ A ˜ H µ ( A, B, ¯ A, ¯ B, (5.18)has been examined in detail by Iooss & P´erou`eme [15]. The ‘truncated normal form’ obtained byignoring the remainder terms in ˜ H µ ( A, B, ¯ A, ¯ B, is conveniently handled using the substitution A ( z ) = e i sz a ( z ) , B ( z ) = e i sz b ( z ) , which converts it into the system ˙ a = b + ∂ b ˜ H ( | a | , i( a ¯ b − ¯ ab ) , µ ) , (5.19) ˙ b = − ∂ ¯ a ˜ H ( | a | , i( a ¯ b − ¯ ab ) , µ ) . (5.20)Supposing that the coefficients c and d are respectively negative and positive, one finds that(5.19), (5.20) admits a real, reversible homoclinic solution ( a h , b h ) , which evidently generatesa circle { e i θ ( a h , b h ) : θ ∈ [0 , π ) } of further homoclinic solutions, two of which (those with θ = 0 and θ = π ) are reversible. The corresponding pair of homoclinic solutions to the original‘truncated normal form’ are reversible and persist when the remainder terms are reinstated. Atheory of multipulse homoclinic solutions to (5.17), (5.18) has also been given by Buffoni &Groves [7] (under the same hypotheses on the normal-form coefficients). Theorem 5.4. (i) (Iooss & P´erou`eme) For each sufficiently small, positive value of µ the two-degree-of-freedom Hamiltonian system (5.17) , (5.18) has two distinct symmetric homoclinic solu-tions.(ii) (Buffoni & Groves) For each sufficiently small, positive value of µ the two-degree-of-freedom Hamiltonian system (5.17) , (5.18) has an infinite number of geometrically distincthomoclinic solutions which generically resemble multiple copies of one of the homoclinicsolutions in part (i). Figure 8: Sketches of η = η ( z ) and the corresponding symmetric unipulse (left) and multipulsesolitary waves (right) generated by homoclinic solutions in region IIIThe homoclinic solutions identified above correspond to envelope solitary waves whose ampli-tude is O ( µ / ) and which decay exponentially as z → ±∞ ; they are sketched in Figure 8. Appendix: Calculation of the normal-form coefficients
The coefficients in the reduced Hamiltonian ˜ H ε ( w ) are determined using the equations K ˜ r ( w ; ε ) − d ˜ r [ w ; ε ]( Kw ) = P ε ( w ) + d ˜ r [ w ; ε ]( P ε ( w )) − g ε nl ( w + ˜ r ( w ; ε )) , (A.1) B l ˜ r ( w ; ε ) = − B ε nl ( w + ˜ r ( w ; ε )) (A.2)to compute the Taylor series of ˜ H ε ( w ) and ˜ r ( w ; ε ) systematically in powers of ( q, p, q ) or ( q, p, p ) . Here K = d g [0] and g ε nl = g ε − K are the linear and nonlinear parts of g ε (with thisslight abuse of notation K is given by the explicit formula (4.6)), and B l , B ε nl are the linear andnonlinear parts of the boundary-value operator B ε : N → R defined by the left-hand side of(3.11). Throughout these calculations we also make use of the identity Ω( Ku + g ε nl ( u ) , v ) + ( B l ( u ) + B ε nl ( u )) φ v | r =1 = d H ε [ u ]( v ) , in which v = ( η v , ω v , φ v , ζ v ) . We denote the parts of H ε ( w ) , B ε nl ( w ) , g ε nl ( w ) which are homoge-neous of order m in ε and n in w by ε m H mn ( w ) , ε m B m nl ,n ( w ) , ε m g m nl ,n ( w ) , and the part of ˜ r ( w ; ε ) which is homogeneous of order m in ε and n in w by ˜ r mn ( w ; ε ) ; the notation is modified inthe natural fashion when ε is replaced by a more specific parameterisation. Finally, arbitraryconstants arising from solving differential equations are denoted by a i .32 omoclinic bifurcation at C (i) Write ˜ r nm ( w ; µ ) = (cid:88) h + i + j = m µ n ˜ r nhij q h p i q j and consider the q and µq components of (A.1), (A.2), namely q : (cid:40) K ˜ r = − c f − c f − g , ( f , f ) ,B l ˜ r = − B , ( f , f ) , (A.3) µq : (cid:40) K ˜ r = − c f − c f − g , ( f ) ,B l ˜ r = 0 . Using these equations we find that c = H ( f , f , f ) + 2 H (˜ r , f )= H ( f , f , f ) + Ω( K ˜ r , f ) + B l ˜ r φ f | r =1 = H ( f , f , f ) + 3 c − Ω( g , ( f , f ) , f )= − H ( f , f , f ) + 3 c , which implies that c = H ( f , f , f ) = (cid:0) β − (cid:1) − / ( α m (cid:48) (1) − , and c = H ( f , f ) + 2 H (˜ r , f ) = H ( f , f ) + 2 c − Ω( g , ( f ) , f ) = − H ( f , f ) + 2 c , which implies that c = H ( f , f ) = − (cid:0) β − (cid:1) − . (ii) Write ˜ r , m ( w ; µ, κ ) = (cid:88) h + i + j = m ˜ r , hij q h p i q j and consider the q component of (A.1), (A.2), namely q : (cid:40) K ˜ r , = − e f − e f − g , , ( f , f , f ) − g , , ( f , ˜ r , ) ,B l ˜ r , = − B , , ( f , f , f ) − B , , ( f , ˜ r , ) . (A.4)The coefficient d can be expressed as d = H , ( f , f , f , f ) + 3 H , ( f , f , ˜ r , ) + 2 H , ( f , ˜ r , ) + H , (˜ r , , ˜ r , ) , (A.5)33here H , ( f , ˜ r , ) = Ω( K ˜ r , , f ) + B l ˜ r , φ f | r =1 = 4 d − Ω( g , , ( f , f , f ) , f ) − g , , (˜ r , , f ) , f ) + B l ˜ r , φ f | r =1 = 4 d − H , ( f , f , f , f ) − H , ( f , f , ˜ r , )+ (cid:0) B l ˜ r , + 2 B , , ( f , ˜ r , ) + B nl , ( f , f , f ) (cid:1) φ f | r =1 , and we find from the boundary condition in (A.4) that the sum inside the parentheses vanishes.From (A.3) we find that ˜ r , = (cid:0) β − ) − / − c (cid:1) f + a f , and it follows from (A.5) that d = H , ( f , f , f , f )+ H , ( f , f , ˜ r , ) − H , (˜ r , , ˜ r , ) = ( β − ) − (12 − α m (cid:48)(cid:48) (1)) . Homoclinic bifurcation at C (iii) Write ˜ r n ,n m ( w ; µ , µ ) = (cid:88) h + i + j + k + l = m µ n µ n ˜ r m ,m hijk l q h p i q j p k p l and consider the p , µ p and µ p components of (A.1), (A.2), namely p : (cid:40) K ˜ r , = 3 c f − c f + c f − g , , ( f , f ) ,B l ˜ r , = − B , , ( f , f ) , (A.6) µ p : (cid:40) K ˜ r , = 2 c , f − c , f + c , f − g , , ( f ) ,B l ˜ r , = 0 ,µ p : (cid:40) K ˜ r , = 2 c , f − c , f + c , f − g , , ( f ) ,B l ˜ r , = − B , , ( f ) . (A.7)Using the method described in part (i) above, we find from these equations that c = H , ( f , f , f ) = 48 √ m (cid:48) (1) − ,c , = H , ( f , f ) = − ,c , = H , ( f , f ) = 0 . Combining µ q : (cid:40) K ˜ r , − ˜ r , = − c , f − g , , ( f ) ,B l ˜ r , = − B , , ( f ) , (A.8)with ˜ r , = − c , f + c , f + a f , c , = 13 H , ( f , f ) = − . Similarly, combining µ p : (cid:40) K ˜ r , = 2 c , f − c , f + c , f − c , ˜ r , ,B l ˜ r , = 0 (A.9)with ˜ r , = − c , f + c , f + a f + a f + √ , which is obtained from (A.8), yields c , = H , ( f , f ) + 2 H , (˜ r , , f ) + 2 H , (˜ r , , f ) + H , (˜ r , , ˜ r , )= 2 c , − , so that c , = 512 .(iv) Write ˜ r , , m ( w ; µ , µ , κ ) = (cid:88) h + i + j + k + l = m ˜ r , , hijik l q h p i q j p k p l and note that d = H , , ( f , f , f , f )+3 H , , ( f , f , ˜ r , , )+2 H , , ( f , ˜ r , , )+ H , , (˜ r , , , ˜ r , , ) . Since H , , ( f , ˜ r , , ) = 4 d − H , , ( f , f , f , f ) − H , , ( f , f , ˜ r , , ) − c Ω(˜ r , , , f ) , where we have used p : (cid:40) K ˜ r , , = 4 d f + d f − d f − g , , , ( f , f , f ) − g , , , ( f , ˜ r , , ) − c ˜ r , , ,B l ˜ r , , = − B , , , ( f , f , f ) − B , , , ( f , ˜ r , , ) , it follows that d = 3 H , , ( f , f , f , f )+3 H , , ( f , f , ˜ r , , ) − H , , (˜ r , , , ˜ r , , )+2 c Ω(˜ r , , , f ) . (A.10)In order to compute d it is therefore necessary to compute ˜ r , , , ˜ r , , and c .From (A.6) one finds that ˜ r , , = − c f + ( c + 6 √ f + a f , and p q : (cid:40) K ˜ r , , − r , , = − c f − g , , , ( f , f ) ,B l ˜ r , , = − B , , , ( f , f ) ˜ r , , = − c f + (2 c + 12 √ f + a f + a f + r . Furthermore, q : (cid:40) K ˜ r , , − ˜ r , , = c f + c f − g , , , ( f , f ) ,B l ˜ r , , = − B , , , ( f , f ) , and p p : (cid:40) K ˜ r , , − ˜ r , , = − c f − c − g , , , ( f , f ) ,B l ˜ r , , = − B , , , ( f , f ) yield ˜ r = − c f + (2 c + 12 √ f + c f + a f + a f + a f + −
108 + 144 r , ˜ r , , = − c f + (2 c + 12 √ f − c f + a f + a f + a f + − r ) , and using these results we find that the solvability condition for q p : (cid:40) K ˜ r , , − r , , − ˜ r , , = − g , , , ( f , f ) ,B l ˜ r , , = − B , , , ( f , f ) is c = − √ . Inserting these expressions for ˜ r , , , ˜ r , , and c into (A.10), we obtain d = 864 (cid:18) − m (cid:48) (1)) (cid:19) . Homoclinic bifurcation at C (v) Here we write ˜ r nm ( w ; µ ) = (cid:88) h + i + j + k + l = m ˜ r nhijk l µ n A h B i ¯ A j ¯ B k p l . The coefficient c is found from µA : (cid:40) ( K − i sI )˜ r = c i E − c F + c f − g , ( E ) ,B l ˜ r = 0 . (A.11)36oting that Ω( K ˜ r , ¯ E ) = 2 H (˜ r , ¯ E ) = Ω( K ¯ E, ˜ r ) , we find from (A.11) that c = − Ω(˜ r , ( K + i sI ) ¯ E ) + Ω( g , ( E ) , ¯ E ) = 2 H ( E, ¯ E ) = − I ( s ) τ . Finally, to compute d we consider A | A | : (cid:40) ( K − i sI )˜ r = i d E − d F − g , ( E, E, ¯ E ) − g , ( ¯ E, ˜ r ) − g , ( E, ˜ r ) ,B l ˜ r = − B , ( E, E, ¯ E ) − B , ( ¯ E, ˜ r ) − B , ( E, ˜ r ) . Taking the symplectic product with ¯ E and simplifying in the usual fashion, we find that d = 6 H ( E, E, ¯ E, ¯ E ) + 3 H (˜ r , ¯ E, ¯ E ) + 3 H (˜ r , E, ¯ E ) , (A.12)where ˜ r and ˜ r are obtained from A : (cid:40) ( K − sI )˜ r = − g , ( E, E ) ,B l ˜ r = − B , ( E, E ) and | A | : (cid:40) K (˜ r − c f ) = − g , ( E, ¯ E ) ,B l (˜ r − c f ) = 0 , where c = 6 H ( E, ¯ E, f ) because of p A : (cid:40) ( K − i sI )˜ r = i c E − c F + 2 c f − g , ( E, f ) ,B l ˜ r = − B , ( E, f ) (note that ˜ r is determined up to addition of a f ). Altogether (A.12) shows that d = 6 H ( E, E, ¯ E, ¯ E ) + 3 H (˜ r , ¯ E, ¯ E ) + 3 H (˜ r − c f , E, ¯ E ) + 18 H ( E, ¯ E, f ) , and the result of this calculation is given in equation (5.16). References [1] A
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