Amplitude equations for weakly nonlinear surface waves in variational problems
Abstract
Among hyperbolic Initial Boundary Value Problems (IBVP), those coming from a variational principle ‘generically’ admit linear surface waves, as was shown by Serre [J. Funct. Anal. 2006]. At the weakly nonlinear level, the behavior of surface waves is expected to be governed by an amplitude equation that can be derived by means of a formal asymptotic expansion. Amplitude equations for weakly nonlinear surface waves were introduced by Lardner [Int. J. Engng Sci. 1983], Parker and coworkers [J. Elasticity 1985] in the framework of elasticity, and by Hunter [Contemp. Math. 1989] for abstract hyperbolic problems. They consist of nonlocal evolution equations involving a complicated, bilinear Fourier multiplier in the direction of propagation along the boundary. It was shown by the authors in an earlier work [Arch. Ration. Mech. Anal. 2012] that this multiplier, or kernel, inherits some algebraic properties from the original IBVP. These properties are crucial for the (local) wellposedness of the amplitude equation, as shown together with Tzvetkov [Adv. Math., 2011]. Properties of amplitude equations are revisited here in a somehow simpler way, for surface waves in a variational setting. Applications include various physical models, from elasticity of course to the directorfield system for liquid crystals introduced by Saxton [Contemp. Math. 1989] and studied by Austria and Hunter [Commun. Inf. Syst. 2013]. Similar properties are eventually shown for the amplitude equation associated with surface waves at reversible phase boundaries in compressible fluids, thus completing a work initiated by BenzoniGavage and Rosini [Comput. Math. Appl. 2009].
AMS subject classification: 35L53, 35L50, 74B20, 35L20.
Keywords: surface wave, weakly nonlinear expansion, amplitude equation, nonlocal Burgers equation, nonlocal Hamilton–Jacobi equation, Hamiltonian structure, Oseen–Frank energy, phase boundaries.
Contents
1 Introduction
In view of its topic and bibliography, this paper may look as though it were written in the honor of either John Hunter or Denis Serre. In fact, it is dedicated to a mathematician of the same generation, on the occasion of his 65th birthday, and this is not by chance. Guy Métivier has indeed been very influential in the work of both authors since the 1990s, and especially regarding two underlying topics in this paper, namely the stability of shocks and geometric optics.
Everything began with the discovery of surface waves^{1}^{1}1Emphasized words are explained in the bulk of the paper. associated with  somehow idealized  propagating phase boundaries [5], which thus departed from the case of classical shocks investigated earlier by Majda [17]. Surface waves are special instances of socalled neutral modes that cannot occur in connection with classical shocks, but they do occur for some undercompressive shocks such as reversible phase boundaries. This fact led to several developments that are out of purpose here. What we are concerned with now is to gain insight on the step beyond the localintime existence results ‘à la Majda’ for propagating discontinuities. One way is to consider weakly nonlinear asymptotics on longer time scales. Regarding surface waves associated with phase boundaries, this approach was started in [9]. Earlier studies were mostly concerning surface waves in elasticity [16, 19, 20]. Research on weakly nonlinear surface waves in more general hyperbolic boundary value problems was launched by Hunter [14]. A general feature of weakly nonlinear surface waves is that they are governed by a (very) complicated, nonlocal amplitude equation. More recently, the authors of the present paper investigated which properties of amplitude equations could be inferred from the fully nonlinear boundary value problem [7]. At about the same time, a then student of Métivier managed to rigorously justify, for dissipative boundary value problems, the asymptotic expansion in which the leading order term corresponds to weakly nonlinear surface waves [18].
Here we focus on the properties of amplitude equations for variational problems, first for abstract problems and then for phase boundaries. Roughly speaking, amplitude equations associated with surface waves in variational problems are found to be locally wellposed. The abstract part in § 2 provides in particular a way of revisiting the case of elasticity that is much simpler than in [7] and also applies to more general energies, such as the Oseen–Frank energy for liquid crystals considered by Austria and Hunter [3, 4]. The more specific part § 3 closes the loop about phase boundaries, which do not fit the abstract framework of § 2 and may nevertheless be viewed as a variational problem.
2 Amplitude equations in abstract variational problems
2.1 General framework
This paper is concerned with nonstationary models arising from a variational principle. The most basic ones are associated with spacetime Lagrangians of the form
where is a smooth, multidimensional domain, is a vector valued unknown, denotes its partial derivative with respect to , and denotes its spatial gradient. To be more specific about notations, if for , , we denote by the components of , and the entries of the matrix valued function are denoted by
Our first assumption on the spatial energy density is that it smoothly depends on its arguments, and satisfies the identities
The identities in (H1) and (H2) are satisfied in particular when depends quadratically on . We ask (H1) so as to ensure that all uniform, constant states are critical points of both the spacetime Lagrangian and the spatial energy defined by
in the sense that the variational gradients of and vanish at . Let us point out indeed that the variational gradient of is
with, using Einstein’s convention on summation over repeated indices,
Thanks to (H1) both and vanish when does not depend on . The reason for asking (H2) will be given afterwards.
The variational problem we are interested in concerns the more general critical points of that satisfy ‘natural’ boundary conditions associated with . This was precisely the kind of problem addressed by Austria [4] in his thesis. If we consider ‘test functions’ that vanish at times and , but not necessarily at the boundary of , we see that
where
and denotes the unit normal vector to that points inside^{2}^{2}2This unusual choice is made for convenience, so as to avoid too many minus signs in calculations. . Therefore, the directional derivative here above equals zero for all if and only if and . This is the motivation for considering the nonlinear boundary value problem
One may notice that the addition of a null Lagrangian, that is, a functional of identically zero variational derivative to leaves invariant the interior equations in (NLBVP) but changes the boundary conditions. This is what happens for instance with the Oseen–Frank energy
in which the last term corresponds to a null Lagrangian. Up to the addition of a Lagrange multiplier associated with the constraint to this energy, (NLBVP) then corresponds to a model introduced by Saxton [21] and Alì and Hunter [1] for nematic liquid crystals. This specific boundary value problem and a simplified version of it were studied by Austria [4, 3]. Otherwise, a most famous model that fits the abstract setting in (NLBVP) is given by the equations describing hyperelastic materials with traction free boundary condition, on which there is abundant literature. The main purpose of this work is to shed light on the weakly nonlinear surface waves associated with (NLBVP), under minimal assumptions on the energy . By staying at an abstract level we can indeed avoid many technical details, and find out which properties of the weakly nonlinear surface wave equations are inherited from the fully nonlinear boundary value problem. This was already our point of view in our earlier paper [7]. Even though variational problems may be viewed as special cases of the Hamiltonian problems considered in [7, §2], the present study is at the same time simpler and more general in terms of the assumptions on the energy  for instance the Oseen–Frank energy satisfies (H1) and (H2) but not the more stringent assumptions made in [7].
As already observed, (H1) ensures that uniform constant states automatically satisfy the interior equations in (NLBVP). This is also true for the boundary conditions when depends quadratically on , but for more general energies we can have .
2.2 Linear surface waves
From now on, we assume that is such that , so that solves (NLBVP). Then small perturbations about are expected to be governed by the linearized problem
where and is the vector valued operator whose components are defined by differentiating at , which gives
This is where the assumption (H2) comes in. Indeed, we are interested in boundary value problems that are scale invariant. More precisely, we would like (LBVP) to be invariant with respect to any rescaling of the type , . Of course, the first requirement is that the domain be scale invariant.
From now on, will implicitly be assumed to be a halfspace^{3}^{3}3The reader may think of as , so that , but we prefer keeping the notations for the components of in the calculations, for symmetry reasons.. Regarding the interior equations in (LBVP), (H1) and a weakened version of (H2) would be sufficient to ensure scale invariance. As a matter of fact, the general expression for the differential operator is given by
For the zeroth order terms in vanish because of (H1), while the first order ones cancel out as soon as we have the symmetry
We do need the stronger assumption that these derivatives are equal to zero for the boundary operator to be a homogeneous, first order operator. This is why we assume (H2). Introducing the convenient notations
we see that under (H1) and (H2) the operators and reduce to
where
Remarkably enough, (LBVP) then exactly falls within the framework considered by Serre in [23], up to introducing the reduced, quadratic energy density defined by
and assuming that it is strictly rankone convex. This is our next assumption, which ensures that the Cauchy problem for the system in is wellposed, whatever the chosen reference state .
About the Cauchy problem associated with (LBVP), one may summarize Serre’s findings as follows.
Theorem 1 (Serre [23]).
Under assumptions (H1)(H2)(H3), the Cauchy problem associated with (LBVP) is always strongly wellposed in one space dimension (), and in arbitrary space dimensions, it is strongly wellposed in if and only if the global energy
is convex and coercive on . If this is the case, then for all for all in an open subset of the cotangent space to , there exists , , such that (LBVP) admits nontrivial solutions of the form
The time frequency depends on the wave vector and solves the equation , where is the Lopatinskii determinant associated with (LBVP). In addition, if the space of surface waves associated with is onedimensional then is a simple root of , that is, . Finally, the surface wave profile solves an ODE , where the matrix is stable, in the sense that its eigenvalues are of negative real part.
The results stated in Theorem 1 follow from Theorems 3.1, 3.3, 3.5, and Proposition 4.1 in [23]. Roughly speaking, they mean that if (LBVP) does not admit any ‘exploding’ mode solution then it admits surface waves, which propagate with speed in ‘generic’ directions along the boundary , and decay to zero away from the boundary. They even decay exponentially fast, that is, the square integrable functions decay exponentially fast at infinity since they are of the form with a stable matrix, which amounts to the fact that the zeroes of lie in the socalled elliptic frequency domain.
2.3 Weakly nonlinear asymptotics
Once we have linear surface waves, it is natural to try and understand the influence of nonlinearities on their evolution. In this respect, we look for solutions of (NLBVP) admitting a (formal) weakly nonlinear expansion
where and are of course related by , and and are supposed to be bounded as well as their derivatives in the tangential variable and the slow time , and square integrable in the transverse variable . By plugging this expansion into (NLBVP) we see that for all the first order profile must be solution to
where the operators and are obtained from the operators and involved in (LBVP) merely by replacing each derivative by . Linear surface waves yield special solutions of (P1) of the form
More generally, we can find all the solutions of (P1) by Fourier transform in , under the following assumption.

The pair , with and cotangent to , is such that there are no normal mode solutions to of the form with , , and the space of solutions to (P1) of the form with is onedimensional.
In other words, (H4) asks that be associated with a line, and not a greater space, of surface waves.
Lemma 1.
Under assumptions (H1)(H2)(H4), the space of square integrable, realvalued solutions to (P1) is made of functions of the form , where is defined by its Fourier transform
for all , with such that is a fixed, nontrivial linear surface wave solution to (P1).
Proof.
By Fourier transform in , if we denote by the dual variable to , (P1) is equivalent to
where the operators and are obtained respectively from and by substituting for . More explicitly, they are defined by
Because of (H1)(H2), (LBVP) is invariant by the rescaling for all . Since (P1) is obtained from (LBVP) by setting , , this implies that is solution to (P1) if and only if is solution to
In particular, is solution to (P1) if and only if is solution to
Substituting the notation for , this is exactly at fixed . The latter thus has a onedimensional space of solutions, since this is the case for the solutions of the form of (P1), by (H4). To make this more precise, let us denote by a nontrivial linear surface wave solution to (P1), using temporarily the subscript to avoid confusion with other solutions to (P1). Then, for any solution to (P1), for all , there must exist a scalar such that . Furthermore, in order to be realvalued, we must have for all .
To conclude, we remove the subscript from , and define as claimed. By complex conjugation we see that for any solution to , satisfies
In particular, this implies that solves for all  and not only for . Then all square integrable, realvalued solutions to (P1) are such that , for all and all . We conclude by inverse Fourier transform. ∎
Note that for all , is exponentially decaying when , since this is the case for when , and that is as smooth in as in , except at . More importantly here, the fact that is solution to (Q1) is crucial for the symmetry properties of the amplitude equation studied below.
Recalling that the first order profile in the asymptotic expansion of must solve (P1) and is allowed to depend on the slow time , Lemma 1 shows that its general form is . Now, by plugging the expansion in (NLBVP) we find that the second order profile must solve
where the quadratic operators and are obtained by differentiating twice and respectively, which yields the operators and detailed below, and by replacing each derivative by . In order to write explicitly and in a rather short way, let us introduce a few more notations, for the third order derivatives of that do not automatically vanish under the assumptions (H1)(H2),
Then we have, under (H1)(H2),
(1) 
(We could of course notice that , but it is more convenient, for symmetry reasons, to keep these two sums.)
2.4 Derivation of amplitude equations
Theorem 2.
We assume that (H1)(H2)(H3)(H4) hold true, and introduce as in Lemma 1. For (P2) to have a square integrable solution the amplitude must solve the quadratic, nonlocal equation
with
for , where we have used the shortcuts
In particular, we have
and is symmetric  that is, is invariant under all permutations of . Furthermore, under the additional assumption that the matrix from Theorem 1 has no Jordan blocks, the part of is positively homogeneous degree one, while is positively homogeneous degree two.
Proof.
By Fourier transform in , (P2) is equivalent to
For this problem to have a square integrable solution , the righthand side must satisfy a Fredholmtype condition, and it turns out that this condition can be simply written in terms of . Indeed, an integration by parts shows the identity, for all and ,
which obviously reduces to
if solves (Q1). As already observed, this is the case for . We thus find that for to solve , we must have
The next important observation is that, since and are closely related to each other, the right handside here above can be ‘absorbed’ back into the integral. Indeed, recall that and are obtained from and  defined in (1)  by substituting for each derivative , so that we can write
Hence by integration by parts,
Therefore, the equation that must satisfy reads
Since , the first integral equivalently reads , and
hence the definition of
where denotes the sign of . Since and are all quadratic in , it just remains to read the contribution of the three other integrals to the amplitude equation by substituting for and by using repeatedly the formula . This yields the claimed, lengthy expression for the kernel
Both and turn out to be symmetric in their arguments thanks to the symmetries in the coefficients and . It is indeed clear from the symmetries of that each term
in the sum involved in is invariant under the transpositions and . The symmetry of is a little bit trickier to check. In fact, we can see by recalling the meaning of the notations
and by using that
that the twelve sums that are summed altogether to define are either invariant or pairwise exchanged by the transpositions and . This is shown on the pictures below.