Symmetric vibrations of higher dimensional nonlinear wave equations

Abstract

We prove a result characterizing conditions for the existence and uniqueness of solutions of a certain Diophantine equation, then using techniques from equivariant bifurcation theory, we apply the result to prove symmetric Hopf bifurcation type theorems for both dissipative and non-dissipative autonomous wave equations, for a large set of spatial dimensions. For the latter only the classical implicit function theorem is used. The set of admissible spatial dimensions is the union of the perfect squares together with finitely many non-perfect squares.

Appendix

In this section we provide more details of selected proofs.

1.1 Proof of Proposition 6.6

For $p\in G^s$ we see that

$$\begin{aligned} L_{\omega }L^{-1}_{\omega }p = p. \end{aligned}$$

(44)

so differentiating with respect to $\omega$ yields:

$$\begin{aligned} (\partial _{\omega } L^{-1}_{\omega })p = -L^{-1}_{\omega } (\partial _{\omega } L_{\omega })L^{-1}_{\omega }p\in G^{s} \end{aligned}$$

since $L^{-1}_{\omega }$ gives a derivative according to Proposition 6.5 while $\partial _{\omega } L_{\omega }$ takes two derivatives, the latter which follows from Eq. (14). Differentiating both sides of Eq. (44) a second time and counting derivatives shows that for $p\in G^s$, $(\partial ^2_{\omega } L^{-1}_{\omega })p\in G^{s-1}$. Now by Proposition 6.3 since F is smooth as a function in its arguments but “loses a derivative” as the codomain has one less derivative, it follows that the map

$$\begin{aligned} G^s\times B^s\times U_{\omega ^{+}} \times \mathbf {R}\ni (p,b,\omega , \rho )\mapsto L^{-1}_{\omega }\Pi _{G^{s-1}} F(w(\rho )+p+b,\omega )\in G^{s-2} \end{aligned}$$

is $C^2$, i.e. although the latter actually lies in $G^s \subset G^{s-2}$, it is $C^2$ when viewed as having the larger codomain. Hence the map ${\mathcal {G}}(p,b,\omega , \rho )$ inherits this regularity and by a Taylor expansion argument, it can be shown that the implicit function does too.

1.2 Proof of Theorem 2; Sect. 6.4

First we need to compute the larger kernel $K'$. As we are working in the space $H^s$, using the exponential basis, everything in this space has a Fourier expansion of the form

$$\begin{aligned} v = \sum _{\mathbf {m}\in \mathbf {Z}^d} f_{\mathbf {m}}e^{i \mathbf {m}\cdot x}. \end{aligned}$$

Setting $L_{\omega _{+}}v=0$ as in Sect. 6.1 this leads to $|\mathbf {m}|^2=N$, so the modes defining the kernel $K'$ are precisely those with $|\mathbf {m}|^2 = N$.

To see that $\Pi _{K'} {\hat{\varphi }}=\Pi _{K'} \varphi$, we recall from Sect. 6.3 that our solution $\varphi (\rho )$ is symmetric and has the expansion

$$\begin{aligned} \varphi (\rho )(x)=w(\rho )(x)+r(\rho )(x)\in \text{ Fix}_{S_d} Y^{s} \end{aligned}$$

where $\rho$ is a real amplitude parameter,

$$\begin{aligned} w(\rho )(x):=\rho \Psi _{\left[ \left[ {\mathbf {m}}^{+} \right] \right] }(x)=\rho \Big ( \sum _{{\mathbf {k}} \in \left[ \left[ {\mathbf {m}}^{+} \right] \right] } \cos ({\mathbf {k}}\cdot x) \Big ), \end{aligned}$$

and $r(\rho )(x) \in R^{s}$, the $\text{ Fix}_{S_d} Y^{s}$-$L^2$ orthogonal complement of $K:=\text {Span}\{ \Psi _{\left[ \left[ {\mathbf {m}}^{+} \right] \right] } \}$.

In particular, this may be written in the exponential basis as

$$\begin{aligned} \varphi (\rho )(x) = \frac{1}{2}\rho \Big ( \sum _{{\mathbf {k}}\in \mathbf {Z}^d, |{\mathbf {k}}|^2 = N} e^{i({\mathbf {k}}\cdot x)} + e^{-i({\mathbf {k}}\cdot x)} \Big )+\sum _{{\mathbf {k}}\in \mathbf {Z}^d, |{\mathbf {k}}|^2 \ne N} f_{{\mathbf {k}}}(\rho )e^{i {\mathbf {k}} \cdot x} \end{aligned}$$

for some $f_{{\mathbf {k}}}(\rho )$. Hence using the above expansion we see that

$$\begin{aligned} \Pi _{K'} {\hat{\varphi }} = \Pi _{K'} \varphi (\rho )\left( x+\frac{2\pi }{( m^{+}_1+\cdots +m^{+}_d)}{\mathbf {1}} \right) = \Pi _{K'}\varphi (\rho ). \end{aligned}$$

Finally, to check that the period vector $\frac{2\pi }{( m^{+}_1+\cdots +m^{+}_d)}{\mathbf {1}}$ is the minimum period vector of $\varphi (\rho )$, the latter Fourier expansion for $\varphi (\rho )$ in the exponential basis reveals that any period vector of the form $T {\mathbf {1}}$ where $T\in \mathbf {R}$ satisfies $T\in \frac{2\pi }{( m^{+}_1+\cdots +m^{+}_d)} \mathbf {Z}$ by substitution and comparison of coefficients with the nonzero coefficient coming from the mode $\mathbf {m}^{+}$.