Slow Modulations of Periodic Waves in Hamiltonian PDEs, with Application to Capillary Fluids

Abstract

Since its elaboration by Whitham almost 50 years ago, modulation theory has been known to be closely related to the stability of periodic traveling waves. However, it is only recently that this relationship has been elucidated and that fully nonlinear results have been obtained. These only concern dissipative systems though: reaction–diffusion systems were first considered by Doelman et al. (Mem Am Math Soc 199(934):viii+105, 2009), and viscous systems of conservation laws have been addressed by Johnson et al. (Invent Math, 2013). Here, only nondissipative models are considered, and a most basic question is investigated, namely, the expected link between the hyperbolicity of modulated equations and the spectral stability of periodic traveling waves to sideband perturbations. This is done first in an abstract Hamiltonian framework, which encompasses a number of dispersive models, in particular the well-known (generalized) Korteweg–de Vries equation and the less known Euler–Korteweg system, in both Eulerian coordinates and Lagrangian coordinates. The latter is itself an abstract framework for several models arising in water wave theory, superfluidity, and quantum hydrodynamics. As regards its application to compressible capillary fluids, attention is paid here to untangle the interplay between traveling waves/modulation equations in Eulerian coordinates and those in Lagrangian coordinates. In the most general setting, it is proved that the hyperbolicity of modulated equations is indeed necessary for the spectral stability of periodic traveling waves. This extends earlier results by Serre (Commun Partial Differ Equ 30(1–3):259–282, 2005), Oh and Zumbrun (Arch Ration Mech Anal 166(2):99–166, 2003), and Johnson et al. (Phys D 239(23–24):2057–2065, 2010). In addition, reduced necessary conditions are obtained in the small-amplitude limit. Then numerical investigations are carried out for the modulated equations of the Euler–Korteweg system with two types of “pressure” laws, namely, the quadratic law of shallow-water equations and the nonmonotone van der Waals pressure law. Both the evolutionarity and the hyperbolicity of the modulated equations are tested, and regions of modulational instability are thus exhibited.

Appendices

Appendix 1: A Concrete Computation

We derive here the averaged equations associated with Benjamin’s impulses $\rho {u}$ and ${v}{u}$ for (1) and (2), respectively. This computation is given for concreteness, even though it is contained in the abstract computation made in Sect. 2. Taking the inner product of (44) and (45), with $({u}_0,\rho _0)$ and $({w}_0,{v}_0)$ respectively, then, averaging and integrating by parts in $\theta$, we obtain

$$\begin{aligned}&\begin{array}{l}-K\langle \partial _\theta (\rho _0 ({u}_0-\sigma )) {u}_1+ \partial _\theta (\tfrac{1}{2}({u}_0-\sigma )^2)\rho _1+A_0(\partial _\theta \rho _0)\rho _1\rangle \\ \quad +\,\partial _T\langle \rho _0{u}_0\rangle +\partial _X \langle \rho _0 {u}_0^2\rangle +\langle \rho _0(\partial _X{g}_0+K\partial _\theta B_0)\rangle =0,\end{array}\\&\begin{array}{l} k\langle \partial _\theta ({w}_0-j {v}_0){w}_1-(j\partial _\theta {w}_0){v}_1-a_0(\partial _\theta {v}_0){v}_1\rangle \\ \quad +\,\partial _S\langle {{v}}_0{w}_0\rangle -\partial _Y\langle \tfrac{1}{2} {{w}}_0^2\rangle +\langle {v}_0 (\partial _Yp_0+k\partial _\theta b_0)\rangle =0, \end{array} \end{aligned}$$

where all terms with index one cancel out because of the profile Eqs. (42) and (43), which imply indeed that

$$\begin{aligned}&\partial _\theta (\rho _0 ({u}_0-\sigma ))=0,\quad \partial _\theta (\tfrac{1}{2}({u}_0-\sigma )^2)+A_0(\partial _\theta \rho _0)=0,\\&\partial _\theta ({w}_0-j {v}_0)=0,\quad j\partial _\theta {w}_0+a_0(\partial _\theta {v}_0)=0. \end{aligned}$$

This is straightforward for the equations on the left, and for those on the right we observe that the second-order differential operators $A_0$ and $a_0$ have been defined in such a way that

$$\begin{aligned} A_0(\partial _\theta \rho _0)=\partial _\theta {g}_0,\quad a_0(\partial _\theta {v}_0)=\partial _\theta p_0. \end{aligned}$$

Finally, we recover the equations in (53) and (55) obtained by averaging the impulses’ conservation laws by checking that

$$\begin{aligned} \langle \rho _0(\partial _X{g}_0+K\partial _\theta B_0)\rangle&= \partial _X\left\langle \rho _0{g}_0 +K(\partial _\theta \rho _0)\frac{\partial \fancyscript{E}}{\partial \rho _x}(\rho _0,K\partial _\theta \rho _0)-\fancyscript{E}(\rho _0,K\partial _\theta \rho _0) \right\rangle \\ \langle {v}_0 (\partial _Yp_0+k\partial _\theta b_0)\rangle&= \partial _Y\left\langle {v}_0 p_0+{{e}}({v}_0,k\partial _\theta {v}_0) -k(\partial _\theta {v}_0)\frac{\partial {{e}}}{\partial {v}_y}({v}_0,k\partial _\theta {v}_0)\right\rangle . \end{aligned}$$

Let us check the first equality, the second one being identical through the symmetry . We have

$$\begin{aligned}&\partial _X\left\langle \rho _0{g}_0 +K(\partial _\theta \rho _0)\frac{\partial \fancyscript{E}}{\partial \rho _x}(\rho _0,K\partial _\theta \rho _0)-\fancyscript{E}(\rho _0,K\partial _\theta \rho _0) \right\rangle -\langle \rho _0(\partial _X{g}_0+K\partial _\theta B_0)\rangle \\&\quad =\left\langle (\partial _X\rho _0){g}_0 +\partial _X\left( K(\partial _\theta \rho _0)\frac{\partial \fancyscript{E}}{\partial \rho _x}(\rho _0,K\partial _\theta \rho _0)-\fancyscript{E}(\rho _0,K\partial _\theta \rho _0)\right) \right\rangle \!-\!K\langle \rho _0\partial _\theta B_0\rangle . \end{aligned}$$

Recalling the definition of ${g}_0$ and integrating once by parts (in $\theta$ of course), we obtain

$$\begin{aligned} \langle (\partial _X\rho _0) {g}_0\rangle&= \left\langle (\partial _X\rho _0)\frac{\partial \fancyscript{E}}{\partial \rho }(\rho _0,K\partial _\theta \rho _0)+ (\partial _\theta \partial _X\rho _0)\frac{\partial \fancyscript{E}}{\partial \rho _x}(\rho _0,K\partial _\theta \rho _0)\right\rangle \\&= \partial _X\langle \fancyscript{E}(\rho _0,K\partial _\theta \rho _0)\rangle - \left\langle (\partial _XK)(\partial _\theta \rho _0)\frac{\partial \fancyscript{E}}{\partial \rho _x}(\rho _0,K\partial _\theta \rho _0)\right\rangle \\&= \partial _X\left\langle \fancyscript{E}(\rho _0,K\partial _\theta \rho _0) -K(\partial _\theta \rho _0)\frac{\partial \fancyscript{E}}{\partial \rho _x}(\rho _0,K\partial _\theta \rho _0)\right\rangle \\&+K\partial _X\left\langle (\partial _\theta \rho _0)\frac{\partial \fancyscript{E}}{\partial \rho _x}(\rho _0,K\partial _\theta \rho _0) \right\rangle . \end{aligned}$$

So it remains to show simply that

$$\begin{aligned} \langle \rho _0\partial _\theta B_0\rangle =\partial _X\left\langle (\partial _\theta \rho _0)\frac{\partial \fancyscript{E}}{\partial \rho _x}(\rho _0,K\partial _\theta \rho _0) \right\rangle . \end{aligned}$$

Once again, this follows from integration by parts. Indeed,

$$\begin{aligned} -\langle (\partial _\theta \rho _0) B_0\rangle \begin{array}{l}=\left\langle - (\partial _\theta \rho _0) \dfrac{\partial ^2\fancyscript{E}}{\partial \rho \partial \rho _x}(\rho _0,K \partial _\theta \rho _0)(\partial _X\rho _0)+(\partial _\theta \rho _0)\partial _X\left( \dfrac{\partial \fancyscript{E}}{\partial \rho _x}(\rho _0,K \partial _\theta \rho _0)\right) \right. \\ \left. \;\;\;-K(\partial ^2_\theta \rho _0)\left( \dfrac{\partial ^2\fancyscript{E}}{\partial \rho _x^2}(\rho _0,K \partial _\theta \rho _0)\right) (\partial _X\rho _0)\right\rangle \\ = \left\langle -\partial _\theta \left( \dfrac{\partial \fancyscript{E}}{\partial \rho _x}(\rho _0,K \partial _\theta \rho _0)\right) (\partial _X\rho _0) +(\partial _\theta \rho _0)\partial _X\left( \dfrac{\partial \fancyscript{E}}{\partial \rho _x}(\rho _0,K \partial _\theta \rho _0)\right) \right\rangle \\ = \partial _X\left\langle (\partial _\theta \rho _0)\frac{\partial \fancyscript{E}}{\partial \rho _x}(\rho _0,K\partial _\theta \rho _0) \right\rangle . \end{array} \end{aligned}$$

Appendix 2: A Convenient Structural Assumption

Our purpose here is to check that, under a reasonable structure assumption on the Hamiltonian $\fancyscript{H}$, the operator ${\fancyscript{A}}^\nu$ behaves properly, and if the kernel of ${\fancyscript{A}}^{(0)}$ has the expected size, our parameterization hypothesis for periodic profiles is met.

Structure of Hamiltonian To go further into the analysis of the abstract Eq. (5), we need to be more specific about the form of the Hamiltonian $\fancyscript{H}$. Inspired by our examples (1)(3), (2)(4), and (9), we write the $\mathbf{U}$-space as ${\mathbb R}^N={\mathbb R}^n\times {\mathbb R}^{N-n}$ for some integer $n, 0\le n\le N$, require that

$$\begin{aligned} \mathbf{U}=\left( \begin{array}{c} \mathbf{v}\\ \mathbf{u}\end{array}\right) ,\quad \fancyscript{H}(\mathbf{U})=\fancyscript{I}(\mathbf{v},\mathbf{u})+\fancyscript{E}(\mathbf{v},\mathbf{v}_x), \end{aligned}$$

and assume that $\fancyscript{H}+c\fancyscript{Q}$ is uniformly strongly convex in both $\mathbf{v}_x$ and $\mathbf{u}$ on the range of $(\mathbf{U},\mathbf{v}_x)$-values and speeds $c$ under consideration. Note that a simple way to make this assumption independent of $c$ is to assume that $\mathbf{J}^{-1}$ has a block structure of the form

as is the case for the Euler–Korteweg system.

1.1 Appendix 2.1: Compactness of Resolvents

We briefly sketch here a proof of the fact that our structural assumption on $\fancyscript{H}$ ensures that ${\fancyscript{A}}^{(0)}$ with domain $H^3({\mathbb R}/{\mathbb Z};{\mathbb R}^n)\times H^1({\mathbb R}/{\mathbb Z};{\mathbb R}^{(N-n)})$ have a nonempty resolvent set and compact resolvents. In turn, this implies that, for all $\nu , {\fancyscript{A}}^\nu$ is a relatively compact perturbation of ${\fancyscript{A}}^{(0)}$.

Our structural assumptions readily yield

$$\begin{aligned} {\fancyscript{A}}^{(0)}=\mathbf{J}k\partial _\theta \mathbf{A}^{\!(0)},\qquad \mathbf{A}^{\!(0)}=\mathbf{\Sigma }_3+\mathbf{\Sigma }_2k\partial _\theta - k\partial _\theta \mathbf{\Sigma }_2^*-k\partial _\theta \mathbf{\Sigma }_1k\partial _\theta , \end{aligned}$$

with $\mathbf{\Sigma }_1=\mathbf{\Sigma }_1^*, \mathbf{\Sigma }_3=\mathbf{\Sigma }_3^*$,

with $\mathbf{\sigma }_1$ and $\mathbf{\sigma }_3$ being uniformly positive definite. Then, standard energy estimates enable us to show that

$$\begin{aligned} \begin{array}{rcl} \left| \langle \mathbf{U}, (z-{\fancyscript{A}}^{(0)})\mathbf{U}\rangle \right| &{}\ge &{}|\mathrm{Re}(z)|\Vert \mathbf{U}\Vert ^2-C\Vert \mathbf{v}\Vert _{H^2}^2-C\Vert \mathbf{u}\Vert _{H^1}^2,\\ \left| \left\langle \left( \begin{array}{c} (k\partial _\theta )^3\mathbf{v}\\ C_0k\partial _\theta \mathbf{u}\end{array}\right) , (z-{\fancyscript{A}}^{(0)})\mathbf{U}\right\rangle \right| &{}\ge &{} \frac{1}{2} \langle (k\partial _\theta )^3\mathbf{v},\mathbf{\sigma }_1(k\partial _\theta )^3\mathbf{v}\rangle \ +\ \frac{C}{2}\langle k\partial _\theta \mathbf{u},\mathbf{\sigma }_3k\partial _\theta \mathbf{u}\rangle \\ &{}&{} -C|\mathrm{Im}(z)|[\Vert \mathbf{v}\Vert _{H^{5/2}}^2+\Vert \mathbf{u}\Vert _{H^{1/2}}^2]-C\Vert \mathbf{U}\Vert ^2, \end{array} \end{aligned}$$

where $C$ is a positive constant independent of $z$. From this we obtain that there exist $\eta >0$ and $C'>0$ such that if $|\mathrm{Re}(z)|\ge \eta [1+|\mathrm{Im}(z)|^6]$, then

$$\begin{aligned} \Vert \mathbf{U}\Vert _{H^3\times H^1}\ \le \ C'\Vert (z-{\fancyscript{A}}^{(0)})\mathbf{U}\Vert . \end{aligned}$$

(76)

This already shows that, for such a $z, (z-{\fancyscript{A}}^{(0)})$ has a closed range and is one-to-one with a continuous inverse. To check that the previous range is dense, we only need to examine whether the formal adjoint is indeed one-to-one (on smooth functions). This amounts to showing that

$$\begin{aligned} (-\bar{z}-\mathbf{A}^{\!(0)}\mathbf{J}k\partial _\theta )\mathbf{V}\ =\ 0 \end{aligned}$$

(77)

has no nontrivial smooth solution $\mathbf{V}$. Applying the operator $\mathbf{J}k\partial _\theta$ to (77), we deduce from (76), applied to $\bar{z}$ and $\mathbf{U}=\mathbf{J}k \partial _\theta \mathbf{V}$, that $k\partial _\theta \mathbf{V}=0$, which in turn implies $\mathbf{V}=0$, because of (77) and the fact that $z$ is nonzero. This proves that for the preceding $\eta >0$, if $|\mathrm{Re}(z)|\ge \eta [1+|\mathrm{Im}(z)|^6]$, then $z$ lies in the resolvent set of ${\fancyscript{A}}^{(0)}$.

Since we did not use any Poincaré inequality in our previous arguments, they apply to ${\fancyscript{A}}$ with domain $H^3({\mathbb R};{\mathbb R}^n)\times H^1({\mathbb R};{\mathbb R}^{(N-n)})$ and show that it has a nonempty resolvent set.

1.2 Appendix 2.2: Parameterization of Periodic Orbits

Let us prove that, under our structural assumption on $\fancyscript{H}$, there is no restriction in assuming, as was done in Theorem 1, that Whitham’s parameterization by $(k,\mathbf{M},P)$ is admissible. More precisely, we will show that if nearby periodic traveling wave profiles form an $N+2$-dimensional manifold, and if the generalized kernel of ${\fancyscript{A}}^{(0)}$ is of dimension $N+2$, then those nearby profiles are parameterized by $(k,\mathbf{M},P)$. This extends to our Hamiltonian framework a result previously shown by Serre Serre (2005) for parabolic conservation laws.

To set things on a more formal ground, we define on some open neighborhood $\mathcal {U}$ of wave values $(\underline{\Xi },{\underline{c}},\underline{\mathbf{v}}(0),\underline{\mathbf{v}}_{x}(0),\underline{\varvec{\lambda }})$ the map

$$\begin{aligned} \mathcal {R}:\qquad \begin{array}{rcl}\mathcal {U}&{}\longrightarrow &{}{\mathbb R}^{2n},\\ (\Xi ,c,\mathbf{v}_{0},\mathbf{v}_{0,x},\varvec{\lambda })&{}\longmapsto &{}([\mathbf{v}]_0^{\Xi },[\mathbf{v}_x]_0^{\Xi }),\end{array} \end{aligned}$$

where $[\cdot ]_0^{\Xi }$ denotes the jump $[f]_0^{\Xi }=f(\Xi )-f(0)$, and $\mathbf{U}$ is the solution of

$$\begin{aligned} \mathsf{E}( \fancyscript{H}+c \fancyscript{Q})[\mathbf{U}] = \varvec{\lambda },\quad \mathbf{v}(0)=\mathbf{v}_{0},\quad \mathbf{v}_{x}(0)=\mathbf{v}_{0,x}. \end{aligned}$$

We identify in the usual way nearby periodic traveling wave profiles with elements of the zero set of $\mathcal {R}$.

Proposition 3

Assume that $\underline{\mathbf{U}}$ is nontrivial and that $\mathcal {R}$ has constant rank $2n-1$. Then the generalized kernel of ${\fancyscript{A}}^{(0)}$ is of dimension $N+2$ if and only if, up to translation, nearby periodic traveling wave profiles may be regularly parameterized by $(k,\mathbf{M},P)$.

Proof

Our proof is based upon the fact that the dimension of the generalized kernel of ${\fancyscript{A}}^{(0)}$ is the algebraic multiplicity of zero as a root of some Evans function ${D}(\cdot )$ (see Gardner (1993)). Indeed, viewing the spectral problem

$$\begin{aligned} z\mathbf{V}\ =\ {\fancyscript{A}}\mathbf{V}\end{aligned}$$

(78)

for $(z,\mathbf{V})=(z,(\mathbf{v},\mathbf{u})^T)$ as a system of coupled differential equations of third order in $\mathbf{v}$ and first order in $\mathbf{u}$, we may introduce its fundamental solution $R(z;\cdot )$ normalized by $R(z;0)=\mathrm{Id}_{{\mathbb R}^{3n}\times {\mathbb R}^{(N-n)}}$ and define

$$\begin{aligned} {D}(z)\ =\ \det ([R(z;\cdot )]_0^{\underline{\Xi }}). \end{aligned}$$

Then the condition on the dimension of the generalized kernel of ${\fancyscript{A}}^{(0)}$ reads

$$\begin{aligned} {D}(z)\ =\ az^{N+2}\ +\ \mathcal {O}(z^{N+3}) \end{aligned}$$

for some nonzero $a$Gardner (1993). We want to convert this into some information about profile parameterization.

Let us denote by $\mathbf{V}^j(z;\cdot )$ the solution to (78) corresponding to the $j$th column of the matrix $R(z;\cdot )$, that is, $\mathbf{V}^j(z;\cdot )$ solves (78) and $(\mathbf{v}^j(z;0),\mathbf{v}^j_x(z;0),\mathbf{v}^j_{xx}(z;0),\mathbf{u}^j(z;0))^T$ is the $j$th vector of the canonical basis of ${\mathbb R}^{3n}\times {\mathbb R}^{(N-n)}$. The Evans function is then written

$$\begin{aligned} {D}(z)\ =\ \left| \begin{array}{rrlcrl} &{}[\!\!\!\!&{}\mathbf{v}^1]&{} \cdots &{}[\!\!\!\!&{}\mathbf{v}^{N+2n}]\\ &{}[\!\!\!\!&{}\mathbf{v}^1_x]&{} \cdots &{}[\!\!\!\!&{}\mathbf{v}^{N+2n}_x]\\ &{}[\!\!\!\!&{}\mathbf{v}^1_{xx}]&{} \cdots &{}[\!\!\!\!&{}\mathbf{v}^{N+2n}_{xx}]\\ &{}[\!\!\!\!&{}\mathbf{u}^1]&{}\cdots &{}[\!\!\!\!&{}\mathbf{u}^{N+2n}]\\ \end{array}\right| , \end{aligned}$$

where we have dropped the marks $0$ and $\underline{\Xi }$ on jumps. To go further, using our structural assumptions, we write³

$$\begin{aligned} {\fancyscript{A}}=\mathbf{J}\partial _x\mathbf{A},\qquad \mathbf{A}=\mathbf{\Sigma }_3+\mathbf{\Sigma }_2\partial _x-\partial _x\mathbf{\Sigma }_2^*-\partial _x\mathbf{\Sigma }_1\partial _x \end{aligned}$$

with $\mathbf{\Sigma }_1=\mathbf{\Sigma }_1^*, \mathbf{\Sigma }_3=\mathbf{\Sigma }_3^*$,

$\mathbf{\sigma }_1$ and $\mathbf{\sigma }_3$ being uniformly positive definite. Integrating (78) from $0$ to $\underline{\Xi }$ yields

$$\begin{aligned} z\int \limits _0^{\underline{\Xi }} \mathbf{V}^j\ =\ \mathbf{J}\left( \begin{array}{c}\mathbf{\sigma }_1(0)[\mathbf{v}^j_{xx}]+\ *[\mathbf{v}^j_x]+*[\mathbf{v}^j]+*[\mathbf{u}^j]\\ \mathbf{\sigma }_3(0)[\mathbf{u}^j]+\ *[\mathbf{v}^j] \end{array}\right) . \end{aligned}$$

Therefore, up to a nonzero multiplicative constant, ${D}(z)$ is also written

$$\begin{aligned} \left| \begin{array}{r@{\quad }r@{\quad }l@{\quad }c@{\quad }r@{\quad }l} &{}[\!\!\!\!&{}\mathbf{v}^1]&{} \cdots &{}[\!\!\!\!&{}\mathbf{v}^{N+2n}]\\ &{}[\!\!\!\!&{}\mathbf{v}^1_x]&{} \cdots &{}[\!\!\!\!&{}\mathbf{v}^{N+2n}_x]\\ &{}z\!\!\!\!&{}\int \limits _0^{\underline{\Xi }} \mathbf{V}^1 &{}\cdots &{}z\!\!\!\!&{}\int \limits _0^{\underline{\Xi }} \mathbf{V}^{N+2n} \end{array}\right| . \end{aligned}$$

Now, corresponding to the impulse equation, we also have

$$\begin{aligned} \begin{array}{rcl} z\dfrac{\partial \fancyscript{Q}}{\partial U_{\alpha }}(\mathbf{V}^j_\alpha )&{}=&{} \partial _x\left( \underline{\mathbf{U}}\cdot \mathsf{Hess}(\fancyscript{H}+{\underline{c}}\fancyscript{Q})(\mathbf{V}^j)+\mathbf{V}^j\cdot \mathsf{E}(\fancyscript{H}+{\underline{c}}\fancyscript{Q})(\underline{\mathbf{U}})-\dfrac{\partial (\fancyscript{H}+{\underline{c}}\fancyscript{Q})}{\partial U_{\alpha }}(\mathbf{V}^j_\alpha )\right) \\ &{}&{}+\partial _x\left( \dfrac{\partial ^2(\fancyscript{H}+{\underline{c}}\fancyscript{Q})}{\partial U_{\alpha ,x}\partial U_{\beta }}(\underline{\mathbf{U}}^j_{\alpha ,x})(\mathbf{V}^j_{\beta }) +\dfrac{\partial ^2(\fancyscript{H}+{\underline{c}}\fancyscript{Q})}{\partial U_{\alpha ,x}\partial U_{\beta ,x}}(\underline{\mathbf{U}}^j_{\alpha ,x})(\mathbf{V}^j_{\beta ,x})\right) \\ &{}=&{}\partial _x\left( \underline{\mathbf{U}}\cdot \mathbf{A}\mathbf{V}^j+\mathbf{\Sigma }_1\underline{\mathbf{U}}_x\cdot \mathbf{V}^j_x+((\mathbf{\Sigma }_2-\mathbf{\Sigma }_2^*)\underline{\mathbf{U}}_x-\mathbf{\Sigma }_1\underline{\mathbf{U}}_{xx})\cdot \mathbf{V}^j\right) , \end{array} \end{aligned}$$

where the convention is, as previously, that linearization and derivatives are taken at $\underline{\mathbf{U}}$. By integrating the preceding relation, we obtain

$$\begin{aligned} z\int \limits _0^{\underline{\Xi }}\dfrac{\partial \fancyscript{Q}}{\partial U_{\alpha }}(\mathbf{V}^j_\alpha ) \ =\ (\mathbf{\sigma }_1\underline{\mathbf{v}}_x)(0)\cdot [\mathbf{v}^j_x]+\ *\ [\mathbf{v}^j]+\ *\ z\int \limits _0^{\underline{\Xi }} \mathbf{V}^j. \end{aligned}$$

We still need to check that it is not a trivial relation. But since $\underline{\mathbf{U}}$ is nontrivial, there is a point where $\underline{\mathbf{v}}_x$ is nonzero; otherwise, $\underline{\mathbf{v}}$ would be constant and, thus, $\underline{\mathbf{u}}$ and $\underline{\mathbf{U}}$ would also be constant, a contradiction. Then, since assumptions of the proposition and terms of the equivalence we are currently proving are invariant by translation, we may assume that $\underline{\mathbf{v}}_x(0)$ is nonzero. Now let us choose $\ell$ such that the $\ell$th component of $\mathbf{\sigma }_1(0)\underline{\mathbf{v}}_x(0)$ is nonzero, and, for any $\mathbf{V}=(\mathbf{v},\mathbf{u})^T\in {\mathbb R}^N={\mathbb R}^n\times {\mathbb R}^{(N-n)}$, let us denote by $\mathbf{v}_{*}$ the vector of ${\mathbb R}^{(n-1)}$ obtained from $\mathbf{v}$ by deleting the $\ell$th component. Then, up to a nonzero multiplicative constant, ${D}(z)$ is

$$\begin{aligned} z^{N+1}\ \left| \begin{array}{r@{\quad }r@{\quad }l@{\quad }c@{\quad }r@{\quad }l} &{}[\!\!\!\!&{}\mathbf{v}^1]&{} \cdots &{}[\!\!\!\!&{}\mathbf{v}^{N+2n}]\\ &{}[\!\!\!\!&{}(\mathbf{v}^1_{*})_x]&{} \cdots &{}[\!\!\!\!&{}(\mathbf{v}^{N+2n}_{*})_x]\\ &{}&{}\int \limits _0^{\underline{\Xi }} \frac{\partial \fancyscript{Q}}{\partial U_{\alpha }}(\mathbf{V}^1_\alpha )&{}\cdots &{}&{}\int \limits _0^{\underline{\Xi }} \frac{\partial \fancyscript{Q}}{\partial U_{\alpha }}(\mathbf{V}^{N+2n}_\alpha )\\ &{}&{}\int \limits _0^{\underline{\Xi }} \mathbf{V}^1&{}\cdots &{}&{}\int \limits _0^{\underline{\Xi }} \mathbf{V}^{N+2n} \end{array}\right| . \end{aligned}$$

Up to a change of basis we may assume that $\mathbf{V}^1(0;\cdot )=\underline{\mathbf{U}}_x, \mathbf{V}^j(0;\cdot )=\mathbf{U}_{\lambda _{j-2n}}$ for $2n+1\le j\le N+2n$ and $(\mathbf{V}^1(0;\cdot ),\cdots ,\mathbf{V}^{2n}(0;\cdot ))$ is a basis of the linear span of $\{\mathbf{U}_{(\mathbf{v}_{0})_1},\cdots ,\mathbf{U}_{(\mathbf{v}_{0})_{n}}\mathbf{U}_{(\mathbf{v}_{0,x})_1},\cdots ,\mathbf{U}_{(\mathbf{v}_{0,x})_{n}}\}$. With this choice, after setting $\widetilde{\mathbf{V}}^1=\mathbf{V}_z(0;\cdot ), {D}(z)$ is written up to a nonzero multiplicative constant

$$\begin{aligned} z^{N+2}\ \left| \begin{array}{r@{\quad }r@{\quad }l@{\quad }r@{\quad }l@{\quad }c@{\quad }r@{\quad }l} &{}[\!\!\!\!&{}\widetilde{\mathbf{v}}^1]&{}[\!\!\!\!&{}\mathbf{v}^2]&{} \cdots &{}[\!\!\!\!&{}\mathbf{v}^{N+2n}]\\ &{}[\!\!\!\!&{}(\widetilde{\mathbf{v}}_{*}^1)_x]&{}[\!\!\!\!&{}(\mathbf{v}^2_{*})_x]&{} \cdots &{}[\!\!\!\!&{}(\mathbf{v}^{N+2n}_{*})_x]\\ &{}&{}\int \limits _0^{\underline{\Xi }} \frac{\partial \fancyscript{Q}}{\partial U_{\alpha }}(\widetilde{\mathbf{V}}^1_\alpha )&{}&{}\int \limits _0^{\underline{\Xi }} \frac{\partial \fancyscript{Q}}{\partial U_{\alpha }}(\mathbf{V}^2_\alpha )&{}\cdots &{}&{}\int \limits _0^{\underline{\Xi }} \frac{\partial \fancyscript{Q}}{\partial U_{\alpha }}(\mathbf{V}^{N+2n}_\alpha )\\ &{}&{}\int \limits _0^{\underline{\Xi }} \widetilde{\mathbf{V}}^1&{}&{}\int \limits _0^{\underline{\Xi }} \mathbf{V}^2&{}\cdots &{}&{}\int \limits _0^{\underline{\Xi }} \mathbf{V}^{N+2n} \end{array}\right| \ +\ \mathcal {O}(z^{N+3}). \end{aligned}$$

Since $\widetilde{\mathbf{V}}^1$ satisfies $\underline{\mathbf{U}}_x\ =\ {\fancyscript{A}}\widetilde{\mathbf{V}}^1$, it differs from $\mathbf{U}_c$ by an element of the kernel of ${\fancyscript{A}}$ that is spanned by $(\mathbf{V}^1(0;\cdot ),\cdots ,\mathbf{V}^{N+2n}(0;\cdot ))$. This yields that, up to a multiplicative nonzero constant and an additive remainder $\mathcal {O}(z^{N+3}), {D}(z)$ reads

$$\begin{aligned} z^{N+2}\ \left| \begin{array}{r@{\quad }r@{\quad }l@{\quad }r@{\quad }l@{\quad }c@{\quad }r@{\quad }l} &{}[\!\!\!\!&{}\mathbf{v}_c]&{}[\!\!\!\!&{}\mathbf{v}^2]&{} \cdots &{}[\!\!\!\!&{}\mathbf{v}^{N+2n}]\\ &{}[\!\!\!\!&{}((\mathbf{v}_{*})_{c})_x]&{}[\!\!\!\!&{}(\mathbf{v}^2_{*})_x]&{} \cdots &{}[\!\!\!\!&{}(\mathbf{v}^{N+2n}_{*})_x]\\ &{}&{}\int \limits _0^{\underline{\Xi }} \frac{\partial \fancyscript{Q}}{\partial U_{\alpha }}((\mathbf{U}_c)_\alpha )&{}&{}\int \limits _0^{\underline{\Xi }} \frac{\partial \fancyscript{Q}}{\partial U_{\alpha }}(\mathbf{V}^2_\alpha )&{}\cdots &{}&{}\int \limits _0^{\underline{\Xi }} \frac{\partial \fancyscript{Q}}{\partial U_{\alpha }}(\mathbf{V}^{N+2n}_\alpha )\\ &{}&{}\int \limits _0^{\underline{\Xi }} \mathbf{U}_c&{}&{}\int \limits _0^{\underline{\Xi }} \mathbf{V}^2&{}\cdots &{}&{}\int \limits _0^{\underline{\Xi }} \mathbf{V}^{N+2n} \end{array}\right| \end{aligned}$$

or

$$\begin{aligned} z^{N+2}\ \left| \begin{array}{crlrlcrl} \underline{\mathbf{v}}_x(0)&{}[\!\!\!\!&{}\mathbf{v}_c]&{}[\!\!\!\!&{}\mathbf{V}^2_1(0;\cdot )]&{} \cdots &{}[\!\!\!\!&{}\mathbf{v}^{N+2n}(0;\cdot )]\\ (\underline{\mathbf{v}}_{*})_{xx}(0)&{}[\!\!\!\!&{}((\mathbf{v}_{*})_{c})_x]&{}[\!\!\!\!&{}(\mathbf{v}^2_{*})_x(0\cdot )]&{} \cdots &{}[\!\!\!\!&{}(\mathbf{v}^{N+2n}_{*})_x(0;\cdot )]\\ 1&{}&{}\quad 0&{}&{}\quad 0&{}\cdots &{}&{}\quad 0\\ \fancyscript{Q}(\underline{\mathbf{U}})(0)&{}&{}\int \limits _0^{\underline{\Xi }} \frac{\partial \fancyscript{Q}}{\partial U_{\alpha }}((\mathbf{U}_c)_\alpha )&{}&{}\int \limits _0^{\underline{\Xi }} \frac{\partial \fancyscript{Q}}{\partial U_{\alpha }}(\mathbf{V}^2_\alpha (0;\cdot ))&{}\cdots &{}&{}\int \limits _0^{\underline{\Xi }} \frac{\partial \fancyscript{Q}}{\partial U_{\alpha }}(\mathbf{V}^{N+2n}_\alpha (0;\cdot ))\\ \underline{\mathbf{U}}(0)&{}&{}\int \limits _0^{\underline{\Xi }} \mathbf{U}_c&{}&{}\int \limits _0^{\underline{\Xi }} \mathbf{V}^2(0;\cdot )&{}\cdots &{}&{}\int \limits _0^{\underline{\Xi }} \mathbf{V}^{N+2n}(0;\cdot ) \end{array}\right| . \end{aligned}$$

We are ready to complete the proof by observing the latter determinant. Indeed our assumption on $\mathcal {R}$ implies that the $(2n-1)$st rows of the preceding matrix are linearly independent. Furthermore, the kernel of the corresponding linear map is the tangent space at $\underline{\mathbf{U}}$ of the profile manifold (profiles being identified when equal up to translation). Thus the differential map of $\mathbf{U}\mapsto (\Xi ,\int _0^{\Xi }\fancyscript{Q}(\mathbf{U}),\int _0^{\Xi }\mathbf{U})$ is invertible on this tangent space if and only if the preceding determinant is nonzero. Consequently, this map is full-rank if and only if the generalized kernel of ${\fancyscript{A}}^{(0)}$ is of dimension $N+2$. $\square$

Note that, one may obtain an alternative proof of Theorem 1 by introducing Floquet exponents in the preceding arguments as in analogous computations in Serre (2005).

Appendix 3: Galilean Invariance

To check that the hyperbolicity of the Euler–Korteweg modulation Eqs. (56) does not depend on $\sigma$, it is convenient to rewrite those equations in a form that is similar to (69) for (58). Substituting $\varrho \sigma +j$ for the mean momentum $\langle \rho _0{u}_0\rangle$ in (56), and manipulating the remaining mean values as in the proof of Theorem 2, we obtain the system

$$\begin{aligned} \left\{ \begin{array}{l} \partial _T K +\partial _X (\sigma K)=0,\\ \partial _T\varrho +\partial _X (\varrho \sigma +j)=0,\\ \partial _T \left( \dfrac{\varrho \sigma +j+D}{\varrho }\right) + \partial _X\left( \dfrac{1}{2}\dfrac{(\varrho \sigma +j)(\varrho \sigma +j+2D)}{\varrho ^2} +\mathrm{g}\right) =0, \\ \partial _T(\varrho \sigma +j) +\partial _X\left( \dfrac{(\varrho \sigma +j)^2+2jD}{\varrho }+\varrho \mathrm{g}+K\Theta -\mathrm{E}\right) =0, \end{array}\right. \end{aligned}$$

(79)

together with the generalized Gibbs relation (65) $\mathrm{d}\mathrm{E}=\mathrm{g}\mathrm{d}\varrho +\Theta \mathrm{d}K+\dfrac{j}{\varrho }\mathrm{d}D$. Then we easily check that (79) is invariant by the Galilean transformation

$$\begin{aligned} (T,X,K,\varrho ,\sigma ,D)\mapsto (T,X-\underline{\sigma } T,K,\varrho ,\sigma -\underline{\sigma },D) \end{aligned}$$

for any $\underline{\sigma }$. Since this transformation leaves invariant all the “thermodynamic” variables $(K,\varrho ,D,\mathrm{g},\Theta ,j,\mathrm{E})$, it leaves (79) invariant simply because it does so for the reduced system

$$\begin{aligned} \left\{ \begin{array}{l} \partial _T K +\partial _X (\sigma K)=0,\\ \partial _T\varrho +\partial _X (\varrho \sigma )=0,\\ \partial _T \sigma + \partial _X\left( \dfrac{1}{2} \sigma ^2+\dfrac{j+D}{\varrho }\sigma \right) =0, \\ \partial _T(\varrho \sigma ) +\partial _X\left( \varrho \sigma ^2 + 2 j \sigma \right) =0.\end{array}\right. \end{aligned}$$