Bifurcation of Symmetric Domain Walls for the Bénard–Rayleigh Convection Problem

Abstract

We prove the existence of domain walls for the Bénard–Rayleigh convection problem. Our approach relies upon a spatial dynamics formulation of the hydrodynamic problem, a center manifold reduction, and a normal forms analysis of an eight-dimensional reduced system. Domain walls are constructed as heteroclinic solutions connecting suitably chosen periodic solutions of this reduced system.

Appendices

Some Properties of Linear Operators

1.1 Adjoint Operator

The explicit, but not so obvious, expression of the adjoint of operator ${\mathcal {L}}_{\mu }$ given below is necessary for computing the algebraic multiplicities of eigenvalues and the coefficients of the normal form.

Denote by $\langle \cdot ,\cdot \rangle$ the scalar product in $(L_{per}^{2}(\Omega ))^{8}$ and consider the closed subspace

$$\begin{aligned}&{\mathcal {H}}_0 = \big \{{\mathbf {U}}=(V_x,V_\bot ,W_x,W_\bot ,\theta ,\phi )\in (L_{per}^{2}(\Omega ))^{8} \;;\\&\int _{\Omega _{per}}V_{x}\,dy\,dz=0\big \} \subset (L_{per}^{2}(\Omega ))^{8}, \end{aligned}$$

which is the closure in $(L_{per}^{2}(\Omega ))^{8}$ of both ${\mathcal {X}}$ and the domain of definition ${\mathcal {Z}}$ of the operator ${\mathcal {L}}_\mu$ . We compute the adjoint ${\mathcal {L}} _{\mu }^{*}$ of ${\mathcal {L}}_{\mu }$ from the scalar product $\langle {\mathcal {L}}_{\mu }\mathbf {U,U}^{\prime }\rangle$, for ${\mathbf {U}}\in {\mathcal {Z}}$, and choose ${\mathbf {U}}^{\prime }\in {\mathcal {H}}_0$ such that ${\mathbf {U}}\mapsto \langle {\mathcal {L}}_{\mu } \mathbf {U,U}^{\prime }\rangle$ is a linear continuous form on ${\mathcal {H}} _0$. We obtain the linear operator

$$\begin{aligned} {\mathcal {L}}_{\mu }^{*}\mathbf {U=}\left( \begin{array}{c} -\mu ^{-1}\left( \Delta _{\bot }W_{x} -\langle \Delta _{\bot }W_{x}\rangle \right) \\ \nabla _{\bot }V_{x}-\mu ^{-1}\Delta _{\bot }W_{\bot }-\mu ^{-1}\nabla _{\bot }(\nabla _{\bot }\cdot W_{\bot })-\mu \phi {\mathbf {e}}_{z} \\ \nabla _{\bot }\cdot W_{\bot } \\ \mu V_{\bot } \\ -W_{z}-\Delta _{\bot }\phi \\ \theta \end{array} \right) , \end{aligned}$$

where

$$\begin{aligned} \langle \Delta _{\bot }W_{x}\rangle =\int _{\Omega _{per}}\Delta _{\bot }W_{x}(y,z)\,dy\,dz. \end{aligned}$$

The operator ${\mathcal {L}}_\mu ^*$ is closed in the space ${\mathcal {X}}^{*}$ defined by

$$\begin{aligned} {\mathcal {X}}^{*}=\big \{{\mathbf {U}}\in (L_{per}^{2}(\Omega ))^{3}\times (H_{per}^{1}(\Omega ))^{3}\times L_{per}^{2}(\Omega )\times H_{per}^{1}(\Omega )\;;\;&\\ W_{x}=W_{\bot }=\phi =0\text { on }z=0,1,\text { and } \int _{\Omega _{per}}V_{x}\,dy\,dz=0\big \},&\end{aligned}$$

with domain

$$\begin{aligned}&{\mathcal {Z}}^{*}=\big \{{\mathbf {U}}\in {\mathcal {X}}^{*}\cap (H_{per}^{1}(\Omega ))^{3}\times (H_{per}^{2}(\Omega ))^{3}\times H_{per}^{1}(\Omega )\times H_{per}^{2}(\Omega )\;;\; \\&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~V_{\bot }=\nabla _{\bot }\cdot W_{\bot }=\theta =0\text { on }z=0,1\big \}. \end{aligned}$$

The adjoint operator ${\mathcal {L}}_{\mu }^{*}$ has the same center spectrum as the operator ${\mathcal {L}}_{\mu }$. For our purposes we need to compute its kernel, an eigenvector associated with the eigenvalue $-ik$ of ${\mathcal {L}}_{\mu _0(k)}^*$, and one of the eigenvectors associated with the eigenvalue $-ik_{x}$ of ${\mathcal {L}} _{\mu _{c}}^{*}$. The kernel of ${\mathcal {L}}_{\mu }^{*}$ is easily computed by solving the equation ${\mathcal {L}}_\mu ^*{\mathbf {U}}=0$, and we find that it is spanned by the vector

$$\begin{aligned} \varvec{\varphi }_{0}^*=\left( 0,0,0, z(1-z),0,0,0,0,\right) ^{t}. \end{aligned}$$

We use this vector in the computation of the coefficients of the cubic normal form in “Appendix B.2”.

Next, for $\mu =\mu _0(k)$, the operator ${\mathcal {L}}_{\mu _0(k)}^*$ has the geometrically simple eigenvalues $\pm ik$, just as the operator ${\mathcal {L}} _{\mu _0(k)}$. In “Appendix A.2” we need the expression of an eigenvector ${{\varvec{\Psi }}}_{k,0}^{*}$ associated with the eigenvalue $-ik$. A direct calculation gives

$$\begin{aligned} {{\varvec{\Psi }}}_{k,0}^{*}(y,z)=\widehat{{\varvec{\Psi }}}_{k,0}^{*}(z), \quad \widehat{{\varvec{\Psi }}}_{k,0}^{*}(z)=\left( \begin{array}{c} -\frac{1}{\mu _{0}(k)k^{2}}\left( D^{3}V_{k}-\langle D^{3}V_{k}\rangle \right) \\ 0 \\ \frac{ik}{\mu _{0}(k)}V_{k} \\ -\frac{i}{k}DV_{k} \\ 0 \\ -V_{k} \\ -ik\phi _{k} \\ \phi _{k} \end{array} \right) , \end{aligned}$$

(A.1)

where

$$\begin{aligned} \langle D^3V_k\rangle =\int _{\Omega _{per}}D^3V_k(z)\,dy\,dz, \end{aligned}$$

$V_{k}$ is the solution of the boundary value problem (4.10), and $\phi _k$ is the unique solution of the boundary value problem

$$\begin{aligned} (D^{2}-k^{2})\phi _k=V_k ,\quad \phi _k |_{z=0,1}=0. \end{aligned}$$

Notice that the function $\phi _k$ is related to the function $\theta$ in the boundary value problem (2.4)–(2.5) through the equality $\theta =-\mu _0(k)\phi _k$.

Finally, in the computations in “Appendix B.2” we also need an eigenvector associated with the eigenvalue $-ik_{x}$ of ${\mathcal {L}} _{\mu _{c}}^{*}$ which is of the form

$$\begin{aligned} {\varvec{\Psi }}_{+}^{*}(y,z)=\widehat{{\varvec{\Psi }}}_{+}^{*}(z)e^{ik_{y}y}. \end{aligned}$$

We obtain that

$$\begin{aligned} \widehat{{\varvec{\Psi }}}_{+}^{*}(z)=\left( \begin{array}{c} -\frac{1}{\mu _{c}k_{c}^{2}} (D^{2}-k_{c}^{2}\cos ^{2}\alpha )DV \\ -\frac{\sin \alpha \cos \alpha }{\mu _{c}}DV \\ \frac{ik_{c}\sin \alpha }{\mu _{c}}V \\ -\frac{i\sin \alpha }{k_{c}}DV \\ -\frac{i\cos \alpha }{k_{c}}DV \\ -V \\ -ik_{c}(\sin \alpha )\phi \\ \phi \end{array} \right) , \end{aligned}$$

where V is the solution of the boundary value problem (4.15), and $\phi$ is the unique solution of the boundary value problem

$$\begin{aligned} (D^{2}-k_{c}^{2})\phi =V ,\quad \phi |_{z=0,1}=0. \end{aligned}$$

(A.2)

1.2 Algebraic Multiplicities of $\varvec{\pm ik}$ and $\varvec{\pm i\omega _1(k)}$

Consider the geometrically simple eigenvalues $\pm ik$ and the geometrically double eigenvalues $\pm i\omega _1(k)$ of the operator ${\mathcal {L}}_{\mu _{0}(k)}$ given in Lemma 4.1. We assume that $\mu _0'(k)\not =0$, and show that the algebraic multiplicities of these eigenvalues are equal to their geometric multiplicities. We prove the result for the eigenvalue ik, the arguments being the same for the eigenvalue $i\omega _{1}(k)$.

Assuming that the algebraic multiplicity of the eigenvalue ik is larger than its geometric multiplicity, there exists a vector ${{\varvec{\Psi }}}_{k,0}$ such that

$$\begin{aligned} ({\mathcal {L}}_{\mu _{0}(k)}-ik){{\varvec{\Psi }}}_{k,0}= {\mathbf {U}}_{k,0}. \end{aligned}$$

(A.3)

Differentiating the eigenvalue problem

$$\begin{aligned} {\mathcal {L}}_{\mu _{0}(k)}{{\mathbf {U}}}_{k,0}=ik{{\mathbf {U}}}_{k,0} \end{aligned}$$

with respect to k leads to the equality

$$\begin{aligned} ({\mathcal {L}}_{\mu _{0}(k)}-ik)\left( \frac{d}{dk}{\mathbf {U}} _{k,0}\right) =\left( i-\mu _{0}^{\prime }(k)\,\frac{\partial }{\partial \mu }{{\mathcal {L}} _{\mu }\big |_{\mu =\mu _{0}(k)}}\right) {{\mathbf {U}}} _{k,0}. \end{aligned}$$

Since $\mu _{0}^{\prime }(k)\ne 0$, this identity and the equality (A.3) imply that there is a solution ${\mathbf { \Phi }}_{k,0}$ of the linear equation

$$\begin{aligned} ({\mathcal {L}}_{\mu _{0}(k)}-ik){\varvec{\Phi }}_{k,0}=\frac{ \partial }{ \partial \mu }{{\mathcal {L}}_{\mu }\big |_{\mu =\mu _{0}(k)}} {{\mathbf {U}}} _{k,0}. \end{aligned}$$

(A.4)

As a consequence, the vector in the right hand side of the above equation is orthogonal to the kernel of the adjoint operator $({\mathcal {L}}_{\mu _{0}(k)}^{*}+ik)$, and in particular to the eigenvector ${{\varvec{\Psi }}}_{k,0}^{*}$ given by (A.1). A direct computation shows that their scalar product is equal to the positive number

$$\begin{aligned} \frac{1}{\mu _{0}^{2}(k)k^{2}}\left( \Vert D^{2}V_{k}\Vert ^{2}+2k^{2}\Vert DV_{k}\Vert ^{2}+k^{4}\Vert V_{k}\Vert ^{2}\right) +\Vert D\phi _{k}\Vert ^{2}+k^{2}\Vert \phi _{k}\Vert ^{2}>0. \end{aligned}$$

This contradicts the orthogonality condition, and proves that the algebraic multiplicity of the eigenvalue ik is equal to its geometric multiplicity.

Cubic Normal Form

1.1 Proof of Lemma 6.1

Proof

The existence of the polynomial ${\varvec{P}}_{\varepsilon }$ and the first two properties in Lemma 6.1 follow from the general normal form theorems in [8, Sections 3.2.1, 3.3.1, and 3.3.2]. In addition, $N(\cdot ,\cdot ,\varepsilon )$ is an odd polynomial of degree 3 such that $N(0,0,\varepsilon )=0$ and the identity

$$\begin{aligned} D_{Z}N(Z,{\overline{Z}},\varepsilon )L_{0}^{*}Z+D_{{\overline{Z}}}N(Z, {\overline{Z}},\varepsilon )\overline{L_{0}^{*}}{\overline{Z}}=L_{0}^{*}N(Z,{\overline{Z}},\varepsilon ), \end{aligned}$$

(B.1)

in which $L_{0}^{*}$ is the adjoint of $L_{0}$, holds for any $Z\in {\mathbb {C}}^{4}$ and $\varepsilon \in {\mathcal {V}}_{2}$. We write

$$\begin{aligned} N(Z,{\overline{Z}},\varepsilon )=N_{1}(Z,{\overline{Z}})\varepsilon +N_{3}(Z, {\overline{Z}}), \end{aligned}$$

where $N_{1}$ and $N_{3}$ denote the linear and cubic terms, respectively, of N. It is now straightforward to check that the linear part $N_{1}$ has the form in Lemma 6.1 (iii), and it remains to check the cubic terms $N_{3}$.

We set $N_{3}=({{\widetilde{N}}}_{+},{{\widetilde{M}}}_{+},\widetilde{N}_{-},{{\widetilde{M}}}_{-})$. Then the identity (B.1) becomes

$$\begin{aligned}&({\mathcal {D}}^{*}+ik_{x}){{\widetilde{N}}}_{+}=0,\quad ({\mathcal {D}}^{*}+ik_{x}){{\widetilde{M}}}_{+}={{\widetilde{N}}}_{+}, \\&({\mathcal {D}}^{*}+ik_{x}){{\widetilde{N}}}_{-}=0,\quad ({\mathcal {D}}^{*}+ik_{x}){{\widetilde{M}}}_{-}={{\widetilde{N}}}_{-}, \end{aligned}$$

in which

$$\begin{aligned} {\mathcal {D}}^{*}= & {} -ik_{x}A_{+}\frac{\partial }{\partial A_{+}} +(A_{+}-ik_{x}B_{+})\frac{\partial }{\partial B_{+}}-ik_{x}A_{-}\frac{ \partial }{\partial A_{-}}+(A_{-}-ik_{x}B_{-})\frac{\partial }{\partial B_{-} } \\&+ik_{x}\overline{A_{+}}\frac{\partial }{\partial \overline{A_{+}}}+( \overline{A_{+}}+ik_{x}\overline{B_{+}})\frac{\partial }{\partial \overline{ B_{+}}}+ik_{x}\overline{A_{-}}\frac{\partial }{\partial \overline{A_{-}}}+( \overline{A_{-}}+ik_{x}\overline{B_{-}})\frac{\partial }{\partial \overline{ B_{-}}}. \end{aligned}$$

Due to the equivariance of the normal form under the action of the symmetry ${\mathbf {S}}_{2}$, it is enough to determine $(\widetilde{N}_{+},{{\widetilde{M}}}_{+})$, the components $(\widetilde{N}_{-},{{\widetilde{M}}}_{-})$ being obtained by switching the indices $+$ and − in the expressions of $({{\widetilde{N}}}_{+},{{\widetilde{M}}}_{+})$.

Cubic monomials are of the form

$$\begin{aligned} A_{+}^{p_{+}}\overline{A_{+}}^{q_{+}}B_{+}^{r_{+}}\overline{B_{+}}^{s_{+}} A_{-}^{p_{-}}\overline{A_{-}}^{q_{-}}B_{-}^{r_{-}}\overline{B_{-}}^{s_-}, \end{aligned}$$

with nonnegative exponents such that

$$\begin{aligned} p_{+}+q_{+}+r_{+}+s_{+}+p_{-}+q_{-}+r_{-}+s_{-}=3. \end{aligned}$$

(B.2)

We claim that the cubic monomials in ${{\widetilde{N}}}_+$ and ${{\widetilde{M}}}_+$ also satisfy

$$\begin{aligned} S_{\pm }=p_{+}-q_{+}+r_{+}-s_{+}+p_{-}-q_{-}+r_{-}-s_{-}=1. \end{aligned}$$

(B.3)

Indeed, for any monomial as above, we have

$$\begin{aligned}&{\mathcal {D}}^{*}\left( A_{+}^{p_{+}}\overline{A_{+}} ^{q_{+}}B_{+}^{r_{+}}\overline{B_{+}}^{s_{+}}A_{-}^{p_{-}}\overline{A_{-}} ^{q_{-}}B_{-}^{r_{-}}\overline{B_{-}^{s_-}}\right) = \\&-ik_{x} S_{\pm } A_{+}^{p_{+}}\overline{A_{+}}^{q_{+}}B_{+}^{r_{+}}\overline{B_{+}} ^{s_{+}}A_{-}^{p_{-}}\overline{A_{-}}^{q_{-}}B_{-}^{r_{-}}\overline{B_{-}} ^{s_-} \\&+ r_+ A_{+}^{p_{+}+1}\overline{A_{+}}^{q_{+}}B_{+}^{r_{+}-1}\overline{ B_{+}} ^{s_{+}}A_{-}^{p_{-}}\overline{A_{-}}^{q_{-}}B_{-}^{r_{-}}\overline{ B_{-}}^{s_-} \\&+s_+ A_{+}^{p_{+}}\overline{A_{+}}^{q_{+}+1}B_{+}^{r_{+}}\overline{B_{+}} ^{s_{+}-1}A_{-}^{p_{-}}\overline{A_{-}}^{q_{-}}B_{-}^{r_{-}}\overline{B_{-}} ^{s_-} \\&+ r_- A_{+}^{p_{+}}\overline{A_{+}}^{q_{+}}B_{+}^{r_{+}}\overline{B_{+}} ^{s_{+}}A_{-}^{p_{-}+1}\overline{A_{-}}^{q_{-}}B_{-}^{r_{-}-1}\overline{B_{-} }^{s_-} \\&+s_- A_{+}^{p_{+}}\overline{A_{+}}^{q_{+}}B_{+}^{r_{+}}\overline{B_{+}} ^{s_{+}}A_{-}^{p_{-}}\overline{A_{-}}^{q_{-}+1}B_{-}^{r_{-}}\overline{B_{-}} ^{s_--1}, \end{aligned}$$

implying that the subspace of monomials for which the sum in the left hand side of (B.3) is constant is invariant under the action of ${\mathcal {D}}^*$. Ordering the monomials by decreasing exponents $p_+$, $q_+$, $r_+$, $s_+$, $p_-$, $q_-$, $r_-$, and $s_-$, this action is represented by a lower triangular matrix with equal elements on the diagonal given by $-ik_xS_{\pm }.$

Consequently, the polynomials ${{\widetilde{N}}}_+$ and ${{\widetilde{M}}}_+$, which belong to the kernel and generalized kernel of ${\mathcal {D}}_*+ik_x$, respectively, belong to the subspace for which (B.3) holds. This proves the claim. Furthermore, the commutativity of $N_{3}$ and $\varvec{ \tau }_{a}$, implies that monomials in $({{\widetilde{N}}}_{+},{{\widetilde{M}}}_{+})$ also satisfy

$$\begin{aligned} p_{+}-q_{+}+r_{+}-s_{+}-p_{-}+q_{-}-r_{-}+s_{-}=1. \end{aligned}$$

(B.4)

Collecting all possible monomials in $({\widetilde{N}}_{+},{\widetilde{M}}_{+})$ for which the conditions (B.2)–(B.4) hold, we compute

$$\begin{aligned}&({\mathcal {D}}^{*}+ik_{x})(A_{+}^{2}\overline{A_{+}})=0, \\&({\mathcal {D}}^{*}+ik_{x})(A_{+}^{2}\overline{B_{+}})=({\mathcal {D}}^{*}+ik_{x})(A_{+}\overline{A_{+}}B_{+})=A_{+}^{2}\overline{A_{+}}, \\&({\mathcal {D}}^{*}+ik_{x})(A_{+}B_{+}\overline{B_{+}})=A_{+}^{2} \overline{B_{+}}+A_{+}\overline{A_{+}}B_{+},\\&({\mathcal {D}}^{*}+ik_{x})(\overline{A_{+}}B_{+}^{2})=2A_{+}\overline{A_{+}}B_{+}, \\&({\mathcal {D}}^{*}+ik_{x})(B_{+}^{2}\overline{B_{+}})=2A_{+}B_{+} \overline{B_{+}}+\overline{A_{+}}B_{+}^{2}, \end{aligned}$$

and

$$\begin{aligned}&({\mathcal {D}}^{*}+ik_{x})(A_{+}A_{-}\overline{A_{-}})=0, \\&({\mathcal {D}}^{*}+ik_{x})(A_{+}A_{-}\overline{B_{-}})=({\mathcal {D}} ^{*}+ik_{x})(A_{+}\overline{A_{-}}B_{-})\\&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~=({\mathcal {D}}^{*}+ik_{x})(B_{+}A_{-}\overline{A_{-}})=A_{+}A_{-}\overline{A_{-}} \\&({\mathcal {D}}^{*}+ik_{x})(A_{+}B_{-}\overline{B_{-}})=A_{+}A_{-} \overline{B_{-}}+A_{+}\overline{A_{-}}B_{-}, \\&({\mathcal {D}}^{*}+ik_{x})(B_{+}A_{-}\overline{B_{-}})=A_{+}A_{-} \overline{B_{-}}+B_{+}A_{-}\overline{A_{-}}, \\&({\mathcal {D}}^{*}+ik_{x})(B_{+}\overline{A_{-}}B_{-})=A_{+}\overline{ A_{-}}B_{-}+B_{+}A_{-}\overline{A_{-}}, \\&({\mathcal {D}}^{*}+ik_{x})(B_{+}B_{-}\overline{B_{-}})=A_{+}B_{-} \overline{B_{-}}+B_{+}A_{-}\overline{B_{-}}+B_{+}\overline{A_{-}}B_{-}. \end{aligned}$$

Since ${\widetilde{N}}_{+}$ and ${\widetilde{M}}_{+}$ are necessarily linear combinations of these 14 monomials, the equalities above imply that they are of the form

$$\begin{aligned} {\widetilde{N}}_{+}= & {} A_{+}{\widetilde{P}}_{+}(u_{1},u_{2},u_{3},u_{4})+A_{-} {\widetilde{R}}_{+}(u_{5}), \\ {\widetilde{M}}_{+}= & {} B_{+}{\widetilde{P}}_{+}(u_{1},u_{2},u_{3},u_{4})+B_{-} {\widetilde{R}}_{+}(u_{5})\\&+A_{+}{\widetilde{Q}} _{+}(u_{1},u_{2},u_{3},u_{4})+A_{-}{\widetilde{S}}_{+}(u_{5}), \end{aligned}$$

with ${\widetilde{P}}_{+},{\widetilde{R}}_{+},{\widetilde{Q}}_{+},{\widetilde{S}} _{+}$ linear in their arguments, which are the quadratic expressions

$$\begin{aligned}&u_{1}=A_{+}\overline{A_{+}},\quad u_{2}=i(A_{+}\overline{B_{+}}-\overline{ A_{+}}B_{+}),\quad u_{3}=A_{-}\overline{A_{-}}, \\&u_{4}=i(A_{-}\overline{B_{-}}-\overline{A_{-}}B_{-}),\quad u_{5}=(A_{+} \overline{B_{-}}-\overline{A_{-}}B_{+}). \end{aligned}$$

This proves the expressions of the cubic terms of $N_{+}$ and $M_{+}$ in (iii). Finally, taking into account the action of the reversibility ${\mathbf {S}}_{1}$, it is straightforward to check that the coefficients $\beta _{j}$, $b_{j}$, $\gamma _{5}$, and $c_{5}$ are real. $\quad \square$

1.2 Computation of the Quotient $\varvec{g=b_3/b_1}$

For the computation of the coefficients $b_1$ and $b_3$, we follow the method in [8, Section 3.4.1]. We restrict to the 8-dimensional center manifold

$$\begin{aligned} {\mathcal {M}}_{\pm }(\varepsilon )=\{{\mathbf {U}}_{c}+\varvec{\Phi }({\mathbf {U}}_{c},\varepsilon )\;;\;{\mathbf {U}}_{c}\in E_\pm \} . \end{aligned}$$

Recall that solutions on this submanifold are invariant under the action of ${\mathbf {S}}_3\varvec{\tau }_\pi$. Combining the transformations from the center manifold reduction in Sect. 5.1 and the normal form in Lemma 6.1, we write

$$\begin{aligned} {\mathbf {U}}= & {} A_{+}\varvec{\zeta }_{+}+B_{+}{\varvec{\Psi }}_{+}+A_{-} \varvec{\zeta }_{-}+B_{-}{\varvec{\Psi }}_{-} +\,\overline{A_{+} \varvec{\zeta }_{+}}+\overline{B_{+}{\varvec{\Psi }}_{+}}+\overline{A_{-} \varvec{\zeta }_{-}}+\overline{B_{-}{\varvec{\Psi }}_{-}} \\&+\,\widetilde{\varvec{\Phi }}(A_{+}, B_{+},A_{-}, B_{-}, \overline{ A_{+} }, \overline{B_{+}},\overline{A_{-}}, \overline{B_{-}},\varepsilon ), \end{aligned}$$

in which $Z=(A_{+}, B_{+},A_{-}, B_{-})$ satisfies the normal form (6.4). Substituting ${\mathbf {U}}$ given by this formula in the dynamical system (3.3), and using the expressions of the derivatives of $A_{+}$, $B_{+}$, $A_{-}$, $B_{-}$ given by the normal form in Lemma 6.1, we obtain an equality for the variables $A_{+}$, $B_{+}$, $A_{-}$, $B_{-}$ and their complex conjugates. We find the coefficients of the normal form, and in particular $b_1$ and $b_3$, by identifying the coefficients of suitably chosen monomials in this equality.

We denote by $\varvec{\Phi }_{rstu}$ the coefficient of the monomial $A_{+}^{r}\overline{A_{+}}^{s}A_{-}^{t}\overline{A_{-}}^{u}$ in the expansion of $\widetilde{\varvec{\Phi }}$. Identifying successively the coefficients of the monomials $A_{+}^{2}\overline{A_{+}}$, $A_{+}A_{-} \overline{A_{-}}$, and then $A_+^2$, $A_+\overline{A_+}$, $A_+A_-$, $A_+ \overline{A_-}$, $A_-\overline{A_-}$, we find the equalities

$$\begin{aligned}&i\beta _{1}\varvec{\zeta }_{+}+b_1{\varvec{\Psi }}_{+} =({\mathcal {L}} _{\mu _{c}}-ik_x )\varvec{\Phi }_{2100}+2{\mathcal {B}}_{\mu _{c}}(\varvec{\Phi }_{2000},\overline{\varvec{\zeta }_{+}})+2{\mathcal {B}}_{\mu _{c}}({{\varvec{\Phi }} }_{1100},\varvec{\zeta }_{+}), \\&i\beta _{3}\varvec{\zeta }_{+}+b_3{\varvec{\Psi }}_{+} =({\mathcal {L}} _{\mu _{c}}-ik_x )\varvec{\Phi }_{1011}+2{\mathcal {B}}_{\mu _{c}}(\varvec{\Phi }_{1010},\overline{\varvec{\zeta }_{-}})\\&~~~~~~~~~~~~~~~~~~~~~~~~~~\quad +2{\mathcal {B}}_{\mu _{c}}(\mathbf { \ \Phi }_{1001},\varvec{\zeta }_{-})+ 2{\mathcal {B}}_{\mu _{c}}(\mathbf { \Phi }_{0011},\varvec{\zeta }_{+}), \end{aligned}$$

and

$$\begin{aligned}&({\mathcal {L}}_{\mu _{c}}-2ik_x )\varvec{\Phi }_{2000} =-{\mathcal {B}} _{\mu _{c}}(\varvec{\zeta }_{+},\varvec{\zeta }_{+}), \end{aligned}$$

(B.5)

$$\begin{aligned}&{\mathcal {L}}_{\mu _{c}}\varvec{\Phi }_{1100} =-2{\mathcal {B}}_{\mu _{c}}( \varvec{\zeta }_{+},\overline{\varvec{\zeta }_{+}}), \end{aligned}$$

(B.6)

$$\begin{aligned}&({\mathcal {L}}_{\mu _{c}}-2ik_x )\varvec{\Phi }_{1010} =-2{\mathcal {B}}_{\mu _{c}}(\varvec{\zeta }_{+},\varvec{\zeta }_{-}) , \end{aligned}$$

(B.7)

$$\begin{aligned}&{\mathcal {L}}_{\mu _{c}}\varvec{\Phi }_{1001} =-2{\mathcal {B}}_{\mu _{c}}( \varvec{\zeta }_{+},\overline{\varvec{\zeta }_{-}}), \end{aligned}$$

(B.8)

$$\begin{aligned}&{\mathcal {L}}_{\mu _{c}}\varvec{\Phi }_{0011} =-2{\mathcal {B}}_{\mu _{c}}( \varvec{\zeta }_{-},\overline{\varvec{\zeta }_{-}}). \end{aligned}$$

(B.9)

We determine the coefficients $b_1$ and $b_3$ by taking the scalar product of the first two equalities above with the vector ${\varvec{\Psi }}_{+}^{* }$ in the kernel of the adjoint operator $({\mathcal {L}}_{\mu _{c}}-ik_x )^*$ computed in “Appendix A.1”,

$$\begin{aligned} b_1\langle {\varvec{\Psi }}_{+},{\varvec{\Psi }}_{+}^{* }\rangle= & {} \langle 2 {\mathcal {B}}_{\mu _{c}}(\varvec{\Phi }_{2000},\overline{\varvec{\zeta } _{+}})+2{\mathcal {B}}_{\mu _{c}}(\varvec{\Phi }_{1100},\varvec{\zeta } _{+}),{\varvec{\Psi }}_{+}^{* }\rangle , \end{aligned}$$

(B.10)

$$\begin{aligned} b_3\langle {\varvec{\Psi }}_{+},{\varvec{\Psi }}_{+}^{* }\rangle= & {} \langle 2 {\mathcal {B}}_{\mu _{c}}(\varvec{\Phi }_{1010},\overline{\varvec{\zeta } _{-}})+2{\mathcal {B}}_{\mu _{c}}(\varvec{\Phi }_{1001},\varvec{\zeta } _{-})\nonumber \\&+2{\mathcal {B}}_{\mu _{c}}(\varvec{\Phi }_{0011},\varvec{\zeta } _{+}), {\varvec{\Psi }}_{+}^{* }\rangle , \end{aligned}$$

(B.11)

where $\varvec{\Phi }_{2000}$, $\varvec{\Phi }_{1100}$, $\varvec{\Phi }_{1010}$ , $\varvec{\Phi }_{1001}$, and $\varvec{\Phi }_{0011}$ are solutions of the linear equations (B.5)–(B.9).

In the equations (B.5) and (B.7), the linear operator $( {\mathcal {L}}_{\mu _c}-2ik_x)$ is invertible, except in the case $\alpha =\pi /6$ when $2k_x=k_c$. Nevertheless, we only have to solve the equations in the subspace of vectors which are invariant under the action of ${\mathbf {S}}_3 \varvec{\tau }_\pi$ and the restriction of $({\mathcal {L}}_{\mu _c}-ik_c)$ to this subspace is invertible, since its two-dimensional kernel is spanned by $\varvec{\zeta }_0$ and $\overline{\varvec{\zeta }_0}$ which do not belong to this subspace. Consequently, $\varvec{\Phi }_{2000}$ and $\mathbf { \Phi }_{1010}$ are uniquely determined. In the equations (B.6), (B.8) and (B.9), the linear operator ${\mathcal {L}}_{\mu _c}$ has a one-dimensional kernel spanned by the vector $\varvec{\varphi }_0$ in Lemma 4.2 (i), and the kernel of its adjoint is spanned by the vector $\varvec{\varphi }_0^*$ in “Appendix A.1”. The solvability condition is easily checked in all cases, so that we can solve these equations up to an element in the kernel of ${\mathcal {L}}_\mu$. The choice of this element in the kernel does not influence the result from (B.10)–(B.11), since ${\mathcal {B}}_\mu$ is invariant upon adding a multiple of $\varvec{\varphi }_0$.

After long and intricate computations we obtain that

$$\begin{aligned} g=\frac{b_3}{b_1} = \frac{b_{31}(\sin ^{2}\alpha )+b_{31}(\cos ^{2}\alpha )+b_{31}(0)}{\frac{1}{2}b_{31}(1)+b_{31}(0)}, \end{aligned}$$

(B.12)

in which

$$\begin{aligned} b_{31}(\Theta ) = A_{31}(\Theta ) + B_{31}(\Theta ){\mathcal {P}}^{-1} + C_{31}(\Theta ){\mathcal {P}}^{-2}, \end{aligned}$$

with

$$\begin{aligned} A_{31}(\Theta )= & {} 2\mu _c^3 \langle (D^2-4k_c^2\Theta )^2 V_1,R_1 \rangle , \\ B_{31}(\Theta )= & {} 4\mu _c^3\Theta \left( \langle V_1,R_2 \rangle + \langle V_2,R_1 \rangle \right) , \\ C_{31}(\Theta )= & {} -\frac{2\mu _c\Theta }{k_c^2} \langle (D^2-4k_c^2\Theta ) V_2,R_2 \rangle , \end{aligned}$$

where

$$\begin{aligned} R_1= & {} VD\phi +(1-2\Theta )\phi DV,\\ R_2= & {} \left( D^2-4k_c^2(1-\Theta )\right) \left( VDV\right) -4\Theta (DV)(D^2V), \end{aligned}$$

and $V_1$, $V_2$ are the unique solutions of the boundary value problems

$$\begin{aligned} \begin{array}{l} (D^2-4k_c^2\Theta )^3 V_1 + 4k_c^2 \mu _c^2\Theta \, V_1=R_1, \\ V_1=DV_1=(D^{2}-4k_{c}^{2}\Theta )^{2}V_1=0\text { in }z=0,1, \end{array} \end{aligned}$$

and

$$\begin{aligned} \begin{array}{l} (D^2-4k_c^2\Theta )^3 V_2 + 4k_c^2 \mu _c^2\Theta \, V_2=R_2, \\ V_2=(D^{2}-4k_{c}^{2}\Theta )V_2=(D^{2}-4k_{c}^{2}\Theta )DV_2=0\text { in } z=0,1, \end{array} \end{aligned}$$

respectively. Recall that V and $\phi$ are the unique symmetric solutions of the boundary value problems (4.15) and (A.2), respectively. Notice that $g\rightarrow 2$, as $\alpha \rightarrow 0$, which was the value of g in the case of the Swift-Hohenberg equation in [10].

Remark B.1

In this way we can also compute the coefficient $b_{0}$. By identifying the coefficients of the terms $\varepsilon A_+$, and then taking the scalar product with ${\varvec{\Psi }}_{+}^{* }$ we obtain

$$\begin{aligned} b_{0}\langle {\varvec{\Psi }}_{+},{\varvec{\Psi }}_{+}^{* }\rangle =\langle {\mathcal {L}}^{(1)}\varvec{\zeta }_{+},{\varvec{\Psi }}_{+}^{* }\rangle , \end{aligned}$$

in which ${\mathcal {L}}^{(1)}$ is the derivative with respect to $\mu$ of the operator ${\mathcal {L}}_\mu$ in (A.4) taken at $\mu =\mu _{c}$. A direct computation gives

$$\begin{aligned} b_{0}\langle {\varvec{\Psi }}_{+},{\varvec{\Psi }}_{+}^{* }\rangle= & {} \frac{1}{\mu _{c}^{2}k_{c}^{2}}\left( \Vert D^{2}V\Vert ^{2}+2k_{c}^{2}\Vert DV\Vert ^{2}+k_{c}^{4}\Vert V \Vert ^{2}\right) \nonumber \\&+ \Vert D\phi \Vert ^{2}+k_{c}^{2}\Vert \phi \Vert ^{2}>0 , \end{aligned}$$

(B.13)

and implies that $\langle {\varvec{\Psi }}_{+},{\varvec{\Psi }}_{+}^{* }\rangle <0$, since $b_0<0$. We point out that it is not obvious to determine the sign of this scalar product directly from the explicit formulas of $\mathbf { \ \Psi }_{+}$ and ${\varvec{\Psi }}_{+}^{* }$.