Spatial dynamics and solitary hydroelastic surface waves

This paper presents an existence theory for solitary waves at the interface between a thin ice sheet (modelled using the Cosserat theory of hyperelastic shells) and an ideal ﬂuid (of ﬁnite depth and in irrotational motion). The theory takes the form of a review of the Kirchg¨assner reduction to a ﬁnite-dimensional Hamiltonian system, highlighting the reﬁne-ments in the theory over the years and presenting some novel aspects including the use of a higher-order Legendre transformation to formulate the problem as a spatial Hamiltonian sys-tem, and a Riesz basis for the phase space to complete the analogy with a dynamical system. The reduced system is to leading order given by the focussing cubic nonlinear Schr¨odinger equation, agreeing with the result of formal weakly nonlinear theory (which is included for completeness). We give a precise proof of the persistence of two of its homoclinic solutions as solutions to the unapproximated reduced system which correspond to symmetric hydroeleastic solitary waves.


The main result
In this article we examine the propagation of solitary waves on the surface of an ocean under ice, regarding the water as a perfect fluid in irrotational flow and the ice sheet as an elastic shell which bends with the surface without stretching and without friction or cavitation between it and the fluid beneath.For this purpose we consider the model derived by Plotnikov and Toland [33] using the Euler equations for inviscid fluid flow and the Cosserat theory of hyperelastic shells.
We suppose that the fluid occupies the region bounded below by a rigid horizontal bottom {y = 0} and above by the free surface {y = h + η(x, t)}, where h is the depth of the water in its undisturbed state.Travelling waves move in the x-direction with constant speed c and without change of shape, so that η(x, t) = η(x − ct), and solitary waves are localised travelling waves, so that η(x − ct) → 0 as x − ct → ±∞.In terms of an Eulerian velocity potential φ, the governing equations for the hydrodynamic problem in dimensionless coordinates and in a coordinate system moving with the wave are with boundary conditions and asymptotic conditions η → 0, (φ x , φ y ) → (0, 0) as x → ±∞ (see Guyenne & Parau [16]).
The dimensionless parameters α and β are given by where D is the coefficient of flexural rigidity of the ice sheet, g is the acceleration due to gravity, c is the wave speed and ρ is the constant water density.
Suppose that b 2 > 0, that is, choose s sufficiently small, or equivalently β 0 sufficiently large (corresponding to sufficiently shallow water in physical variables).Equation (1.10) admits the family e iθ , θ ∈ [0, 2π) of homoclinic solutions (solutions which decay to zero as x → ±∞), which correspond to the solitary waves These waves take the form of periodic wave trains modulated by exponentially decaying envelopes; the wave with θ = 0 is a symmetric wave of elevation while the wave with θ = π is a symmetric wave of depression (see Figure 3).In this article we confirm the predictions of the weakly nonlinear theory and prove the following theorem.

Spatial dynamics and the Kirchgässner reduction
We prove Theorem 1.1 using the Kirchgässner reduction: the hydrodynamic problem is formulated as a spatial Hamiltonian system and reduced to a locally equivalent Hamiltonian system with finitely many degrees of freedom; homoclinic solutions of the reduced system correspond to solitary waves.The method was introduced by Kirchgässner [21] and has been used for many problems in fluid mechanics, in particular for water waves (see Dias & Iooss [6] for a review), and more recently for water waves with vorticity (Groves & Wahlén [14,15], Kozlov & Lokharu [23], Kozlov, Kuznetsov & Lokharu [22]) and ferrofluids (Groves, Lloyd & Stylianou [11], Groves & Nilsson [12]).In this paper we review the method as it applies to hydroelastic solitary waves, presenting various refinements and new features.Our starting point in Section 2 is the observation that the equations (1.1)-(1.4)follow from the formal variational principle in which the variations are taken over η and φ (a modified version of the classical variational principle introduced by Luke [25]); this observation is confirmed by the calculation where the formal first variations of η and φ are denoted by respectively η and φ and we have used integration by parts and Green's integral formula.We proceed by using the change of variable (1.5), which transforms (1.11) into the new variational principle this variational principle recovers the transformed equations (1.6)-(1.9).The next step is to perform a (formal) Legendre transform to obtain a formulation of the hydrodynamic problem as a spatial Hamiltonian system (in which the variable x plays the role of 'time').The presence of second-order derivatives in the Lagrangian however necessitates the use of a higher-order Legendre transform (see Lanczos [24, Appendix I]), by means of which obtain the Hamiltonian system with variables η, Φ and these equations are accompanied by the boundary conditions which emerge when computing the variational derivatives.Equations (1.12), (1.13) are reversible, that is invariant under the transformation (η, ω, ρ, ξ, Φ, Ψ)(x) → (η, −ω, −ρ, ξ, −Φ, Ψ)(−x); this symmetry is inherited from (1.6)-(1.9),which are invariant under (η(x), Φ(x, y)) → (η(−x), Φ(−x, y)).They are also invariant under the transformation Φ → Φ + c for any constant c.To eliminate this symmetry one replaces (Φ, Ψ) with new variables ( Φ, Φ 0 , Ψ, Ψ 0 ), where and additional constraints The formulation of the hydrodynamic problem as a spatial Hamiltonian system is discussed rigorously in Section 2, where a precise definition of a Hamiltonian system is given and Hamilton's equations are derived.Full details of the changes of variable, which are performed explicitly, are also given; the result is a quasilinear evolution equation of the form for the variable u = (η, ρ, ω, ξ, Γ, Ψ) in the phase space where the overline denotes the subspace of functions with zero mean value; the domain of the linear operator L is and the nonlinear term on the right-hand side of (1.14), which satisfies N ε (u) = O( (ε, u) u ), maps a neighbourhood of the origin in R 2 × D(L) analytically into X.Here we have written α = α 0 + ε 1 and β = β 0 + ε 2 , where α 0 and β 0 are fixed, and the superscript ε denotes the dependence upon this parameter.
In Section 3 we show that the spectrum of L is discrete.By reducing the spectral problem to a non self-adjoint Sturm-Liouville problem, we show that a complex number λ is an eigenvalue of L if and only if and deduce that σ(L) consists of (a) a countably infinite family {λ k } k∈Z\{0} of simple real eigenvalues, where {λ k } ∞ k=1 are the positive real solutions of equation (1.15), so that λ k ∈ (kπ, (k + 1)π) for k = 1, 2, . . .and for large k, and four additional eigenvalues (counted according to multiplicity) which are shown in Figure 4.Note in particular that a Hamiltonian-Hopf bifurcation occurs at each point (β 0 (s), α 0 (s)) of the curve C: two pairs of purely imaginary eigenvalues become complex by colliding at the points ±is on the imaginary axis.
Remarkably, we can treat (1.14) as a dynamical system with countably infinitely many coordinates by showing that L is a Riesz spectral operator, that is its generalised eigenvectors form a Riesz basis for X (a Schauder basis obtained by an isomorphism from an othonormal basis).In particular, at a point (β 0 (s), α 0 (s)) of the curve C (a 'Hamiltonian-Hopf point') we can write where e, f and e λ k are suitably normalised generalised eigenvectors with (L − isI)e = 0, (L − isI)f = e and (L − λ k I)e λ k = 0.In the above notation Homoclinic solutions of (1.12) are of particular interest since they correspond to solitary waves.We detect them using centre-manifold reduction (see Mielke [28,29] for the version of the reduction theorem used here).Denoting the central and hyperbolic subspaces of X at a Hamiltonian-Hopf point by one finds that all small, globally bounded solutions to (1.14) lie on a centre manifold of the form {u 2 = r(u 1 ; ε)}, where the reduction function r : The flow on the centre manifold is governed by the reduced system The shaded region indicates the parameter regime in which homoclinic bifurcation is detected; dots and crosses denote respectively simple and algebraically double, geometrically simple eigenvalues.
which is itself a reversible Hamiltonian system (with two degrees of freedom).One of the key requirements in Mielke's theorem is that the operator L 2 = L| X 2 has L p -maximal regularity in the sense that the differential equation In fact L p -maximal regularity for some p > 1 implies L p -maximal regularity for all p > 1 (see Mielke [27]), and an operator has L In Section 4 we however demonstrate directly that a Riesz spectral operator with no imaginary eigenvalues has L 2 -maximal regularity, and stipulate L 2 -maximal regularity as a hypothesis in Mielke's theorem.This approach is more direct than that taken in the above references to the Kirchgässner reduction, in which central and hyperbolic subspaces of a suitable phase space are defined by Dunford integrals and the bisectorality condition is verified by a priori estimates.Writing (ε 1 , ε 2 ) = (µ, 0), so that positive values of µ correspond to points on the 'complex' side of C (the shaded region in Figure 4), one finds after a Darboux and normal-form transformation that the reduced equation (1.16) can be formulated as the Hamiltonian system ) ) is a real polynomial function of its arguments which satisfies it contains the terms of order 3, ..., n 0 + 1 in the Taylor expansion of Hµ (A, B, Ã, B).Equations (1.17) inherit the reversibility of (1.12): they are invariant under the transformation (A, B)(x) → ( Ā, − B)(−x).Neglecting the remainder term in the Hamiltonian and introducing the scaled variables where δ = µ 2 , confirms that the system is at leading order equivalent to the nonlinear Schrödinger equation where c 1 and d 1 are the coefficients of respectively µ|A| 2 and |A| 4 in the Taylor expansion of Hµ NF .We compute these coefficients explicitly in Appendix B and find that where b 1 , b 2 are the coefficients in equation (1.10) and τ 1 > 0 is defined in equation (4.4).
A rigorous analysis of (1.17) is given in Section 5. Returning to real coordinates q, p ∈ R 2 given by A = 1 , eliminating p and introducing the scaled variables where δ 2 = −c 1 µ and R θ is the matrix representing a rotation through the angle θ, transforms (1.17) into where C = −d 1 /c 1 and ) and in the limit δ = 0 has the explicit solution which is nondegenerate in the class of symmetric functions (see Section 5 for a precise statement of this result).This fact allows one to prove the following theorem with an implicit-function theorem argument.
Theorem 1.2.For each ν ∈ (0, 1) and each sufficiently small value of δ > 0 equation (1.18) has two homoclinic solutions Q δ± which are symmetric, that is invariant under the transformation , and satisfy the estimate for all X ∈ R.
Finally, let us briefly mention some related work in the literature.Buffoni & Groves [4] show that (1.17) has an infinite number of geometrically distinct homoclinic solutions which generically resemble multiple copies of one of the 'primary' homoclinic solutions found here.In the present context this result yields the existence of an infinite family of 'multi-pulse' hydroelastic solitary waves.A variational existence theory for hydroelastic solitary waves in the present parameter regime has been given by Groves, Hewer & Wahlén [10], while the Kirchgässner reduction (without the Hamiltonian framework) has also been applied to alternative models in which the ice sheet is modelled as a thin Euler-Bernoulli elastic plate (Parau & Dias [32]) and a Kirchhoff-Love elastic plate with nonzero thickness and inertial effects (Ilichev [17] and Ilichev & Tomashpolskii [18]).There are also several numerical studies of hydroelastic solitary waves in deep water (Milewski, Vanden-Broeck & Wang [30], Guyenne & Parau [16], Gao, Wang & Vanden-Broeck [8]), and an alternative approach to centre-manifold reduction has been given by Chen, Walsh & Wheeler [5].

Formulation as a spatial Hamiltonian system
In this section we formulate the hydrodynamic problem as a spatial Hamiltonian system.Starting with a variational principle for the 'flattened' hydrodynamic problem (1.6)-(1.9),we perform a formal Legendre transform to detect its spatial Hamiltonian structure, the correctness of which is confirmed a posteriori.
for all tangent vectors v ∈ T M | m .Hamilton's equations for (M, Ω, H) are the differential equations where the overline denotes the subspace of functions with zero mean value, and define the manifold The 2-form Ω on M defined by upon which it is given by the right-hand sides of equations (2.7)-(2.12). Proof.

Spectral analysis
In this section we examine the spectrum of the linear operator L : D(L) ⊆ X → X in detail.Our first result is obtained by a straightforward calculation.Proposition 3.1.A complex number λ is an eigenvalue of L if and only if its eigenspace is one-dimensional and spanned by respectively for λ = 0 and λ = 0 (which arises only for α 0 = 1).All eigenvalues are also algebraically simple, with the exception of the zero eigenvalue at α 0 = 1 and the purely imaginary eigenvalues ±is at the point (β 0 (s), α 0 (s)) of the curve in the parameter plane which are algebraically double.
Proof.Observe that λ solves (3.1) if and only if ν = λ 2 is an eigenvalue of the non-selfadjoint Sturm-Liouville problem This problem has a countable number of (not necessarily real) eigenvalues {ν n } n∈N 0 , which repeated according to algebraic multiplicity and listed according in increasing absolute value, are given asymptotically for large n by (2) there are precisely two additional eigenvalues (counted according to algebraic multiplicity) in the form of either (a) a complex-conjugate pair (with non-vanishing imaginary part) whose absolute value is less than ν D 0 (Figure 5  The solutions λ of (3.1) are recovered from the above analysis by the formula ν = λ 2 , so that in particular they occur in plus-minus pairs.Clearly (3.1) has a real solution in each interval ), n ∈ N 0 (see point (1) above), and it follows from point (2) that there are four additional solutions (counted according to multiplicity).The results in Proposition 3.1 and the fact that B(0) = 1 show that these four solutions are described by precisely one of the statements (a)-(e) (according to which of the scenarios in Figure 5 occurs).
The asymptotic formula for λ k follows by writing k = n + 1.
According to this lemma the purely imaginary eigenvalues of L appear in pairs ±is satisfying the dispersion relation α 0 + s 4 β 0 = s coth(s). (3.7) Figure 4 shows the dependence of these eigenvalues upon β 0 and α 0 .At each point of {α 0 = 1} two real eigenvalues become purely imaginary by colliding at the origin, while at each point of the curve C two pairs of purely imaginary eigenvalues become complex by colliding at non-zero points ±is on the imaginary axis.For later reference we record the formulae for an eigenvector e and generalised eigenvector f with eigenvalue is when (β 0 , α 0 ) ∈ C (the corresponding formulae for the zero eigenvalue at α 0 = 1 are e 0 = (1, 0, 0, 0, 0, 0) T , f 0 = (0, 1, − 1 3 , 0, 0, 0) ).Lemma 3.3.The operator L is regular, that is its spectrum consists entirely of isolated eigenvalues of finite algebraic multiplicity.
Proof.Since D(L) is compactly embedded in X it suffices to show that ρ(L) is non-empty, so that L has compact resolvent (Kato [20,Theorem III.6.29]).In the case α 0 = 1 a direct calculation shows that L is invertible with To deal with the case α 0 = 1 note that L| α 0 =1 is a compact perturbation of L| α 0 = 1 2 , so that the essential spectrum of these two operators (the set of λ for which (λI − L) is not Fredholm with index zero) is identical (see Schechter [34]).It follows that the spectrum of L| α 0 =1 consists of the solution set of (3.1); in particular its resolvent set is non-empty.
Finally, we show that the set of generalised eigenvectors of L form a Schauder basis for X, which is henceforth replaced by its complexification.In particular, we show that this set is a Riesz basis, that is a basis obtained from an orthonormal basis by an isomorphism (see Gohberg & Krein [9, §VI.2]); note that we use the Dirichlet norm for the space H1 (0, 1). in H1 (0, 1) × L2 (0, 1) that A is complete, and it is evidently orthonormal.
Corollary 3.5.Let P be the spectral projection onto the four-dimensional subspace of X corresponding to the eigenvalues shown in Figure 4, and let {e 1 , e 2 , e 3 , e 4 } be a basis for P [X] consisting of generalised eigenvectors of L. The set , is a Riesz basis for X.
Proof.We first note that the set {e 1 , e 2 , e 3 , e 4 } ∪ {e λ k } k∈Z\{0} is ω-linearly independent since it is the union of bases for the generalised eigenspaces of a regular operator (see Gohberg & Krein [9, p. 329]).
Lemma 3.7.The operator (where w k = w, e λ k , h k = h, e λ k ), we find that which is solved by Note that for k > 0 with a similar calculation for k < 0. It follows that and similarly ) and solves the given differential equation.
The uniqueness of the solution follows by noting that equation (3.8) has no nontrivial solution in L 2 (R, R) when h k = 0.

Centre-manifold reduction
Our strategy in finding solutions to Hamilton's equations (2.19)-(2.24)for (M, Υ, H ε ) consists in applying a reduction principle which asserts that it is locally equivalent to a finite-dimensional Hamiltonian system.The key result is the following theorem due to Mielke [28,29].
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 l is a parameter and L : D(L) ⊂ X → X is a densely defined, closed linear operator.Regarding D(L) as a Hilbert space equipped with the graph norm, suppose that 0 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 of a finite number of eigenvalues of finite multiplicity and is separated from the rest of σ(L) in the sense of Kato, so that X admits the decomposition X = X 1 ⊕ X 2 , where X 1 = P(X ), X 2 = (I − P)(X ) are the centre and hyperbolic subspaces of L defined by the spectral projection P corresponding the purely imaginary part of σ(L).
(H2) The operator L 2 = L| X 2 has L 2 -maximal regularity in the sense that the differential equa- (H3) There exist a natural number k and neighbourhoods Λ ⊂ R l of 0 and U ⊂ D(L) of 0 such that N is (k + 1) times continuously differentiable on U × Λ, its derivatives are bounded and uniformly continuous on U × Λ and N (0, 0) = 0, d 1 N [0, 0] = 0.
(iv) M λ is a symplectic submanifold of M and the flow determined by the Hamiltonian system ( M λ , Ωλ , Hλ ), 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 represents Hamilton's equations for ( M λ , Ωλ , Hλ ).
(ii) Substituting u = u 1 + r(u 1 ; λ) into (4.1) and eliminating u1 using (4.2) leads to the equation which can be used to recursively determine the terms in the Taylor series of r(u 1 ; λ) and N λ (u 1 ).
We proceed by choosing (β 0 (s), α 0 (s)) ∈ C, setting (ε 1 , ε 2 ) = (µ, 0), and applying Theorem 4.1 to (M, Υ, H ε ).Hypothesis (H3) is clearly satisfied for any natural number k, and we henceforth refer to functions which are continuously differentiable an arbitrary, but fixed number of times as 'smooth'.The spectral theory in Section 3 shows that (H1), (H2) are also satisfied; indeed, the (complexified) four-dimensional centre subspace of L is spanned by the generalised eigenvectors where ) so that the centre and hyperbolic subspaces of L are respectively The vectors are normalised such that (L − isI)E = 0, (L − isI)F = E with SE = Ē, SF = − F , and and the symplectic product of any other combination of the vectors E, F, Ē, F is zero (so that {E, F, Ē, F } is a symplectic basis for the centre subspace of L).Writing we therefore find that A, B are canonical coordinates for the reduced Hamiltonian system (see Remark 4.2(i)), which can therefore be written as (with a slight abuse of notation we abbreviate Hε | (ε 1 ,ε 2 )=(µ,0) to Hµ ); this system is reversible with reverser S : (A, B) → ( Ā, − B).Note that the quadratic, parameter-independent part of the Hamiltonian is The next step is to use a normal-form transform to simplify the Hamiltonian.For this purpose we use the following result due to Elphick [7].
where the complexification of H µ NF lies in ker L L * , and for M ∈ C 4×4 , where the coefficients of the polynomials in the complex polynomial rings depend upon µ and the gradient is taken with respect to Z = (A, B, Ā, B).
We proceed by characterising ker L L * using the following lemma, the statements in which are obtained from results by Murdock [31,Lemma 3.4 Writing the transformed reduced system as where we can compute the Taylor series of r(u 1 ; µ) and N µ (u 1 ), and hence H µ (A, B, Ā, B), recursively using the equation where µ j Hj k (A, B, Ā, B) denotes the part of the Taylor expansion of Hµ (A, B, Ā, B) which is homogeneous of order j in µ and k in (A, B, Ā, B).The coefficients c 1 and d 1 , whose values are required in Section 5 below, are computed in Appendix B; we find that c 1 < 0 and there exists a critical value s of s such that d 1 > 0 for s < s , which we now assume.

Homoclinic solutions
In this section we examine the reduced Hamiltonian system where the underscore indicates that the order-of-magnitude estimate remains valid when formally differentiated with respect to (A, B).The truncated system without the remainder terms was examined in detail by Iooss & Pérouème [19], who also studied the 'persistence' of certain solutions as solutions to the full system.Here we present an alternative, functional-analytic proof of the existence of two reversible homoclinic solutions to (5.1), (5.2).We begin by returning to real coordinates q = (q 1 , q 2 ) T , p = (p 1 , p 2 ) T given by and hence obtaining the real Hamiltonian system +R µ 2 (q, p), (5.4) in which so that P µ 1 (q, p), P µ 2 (q, p) are polynomials in µ, q and p and R µ 1 (q, p), R µ 2 (q, p) = O(|(q, p)|(µ, q, p)| n 0 ).

Figure 1 :
Figure 1: An ice sheet on the free surface of a two-dimensional perfect fluid.

Figure 2 :
Figure 2: The linear dispersion relation for a fixed β 0 .

Definition 2 . 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 2-form) and the Hamiltonian H : M → R is a smooth function.Its Hamiltonian vector field v H with domain D(v H ) ⊆ M is defined as follows.The point m ∈ M belongs to D(v H ) with v H | m := w ∈ T M | m if and only if s) − 1) ds dy, which is a smooth diffeomorphism X → X and M → M with inverse Φ = Γ + ρ y 0 s(Ψ(s) − 1) ds − ρ s) − 1) ds dy.

Lemma 3 . 2 .
Choose (β 0 , α 0 ) ∈ C. The point spectrum of L consists of a countably infinite family {λ k } k∈Z\{0} of simple real eigenvalues, where {λ k } ∞ k=1 are the positive real solutions of equation (3.1) and λ −k = −λ k for k = 1, 2, . . .together with (a) two plus-minus pairs of simple purely imaginary eigenvalues if α 0 > 1 and (β 0 , α 0 ) lies to the left of the curve C in the parameter plane, (b) a plus-minus pair of algebraically double purely imaginary eigenvalues ±is if (β 0 , α 0 ) is the point with parameter value s on the curve C, (c) a plus-minus quartet of genuinely complex eigenvalues if α 0 > 1 and (β 0 , α 0 ) lies to the right of the curve C in the parameter plane, (d) a plus-minus pair of simple purely imaginary eigenvalues and an algebraically double zero eigenvalue if α 0 = 1, (e) an additional plus-minus pair of simple real eigenvalues and a plus-minus pair of simple purely imaginary eigenvalues if α 0 < 1.

Figure 5 :
Figure 5: Geometric characterisation of the eigenvalues ν n as the points of intersection of the curve s = B(ν) with the parabola s = α 0 + β 0 ν 2 ; one real eigenvalue lies in each interval (ν D n , ν D n+1 ), n ∈ N 0 .(a) Two additional complex eigenvalues; (b) one additional algebraically double negative eigenvalue; (c)-(e) two additional real eigenvalues.

Lemma 4 . 3 .
Let n 0 ≥ 2. There exists a near-identity, canonical change of variables which transforms the Hamiltonian to is

Lemma 4 . 4 .Corollary 4 . 5 .
Let S = diag(is, is, −is, −is) and N = L − S. (i) The kernel of L L * : C[Z] → C[Z] is given by ker L L * = ker L N * ∩ ker L S * .(ii) The kernel of L N * is given by ker L N * = C[A, Ā, A B − ĀB].(iii)The kernel of L S * is given by ker L S * = C[A Ā, A B, B Ā, B B].The kernel of L L
closed (since it is constant) and weakly nondegenerate at each point of M .The triple (M, Ω, H ε ) is therefore a Hamiltonian system in the sense of Definition 2.1.Theorem 2.2.Consider the Hamiltonian system (M, Ω, H ε ).The domain of the corresponding Hamiltonian vector field v H ε is