Translation invariants of Fourier-mode phases ($n\cdot \theta ,{\sum }_j{n}_j{q}_j=0$)

Let a translation symmetry with displacement $s\in {\mathbb{R}}^d$ act on nonzero complex Fourier modes byEach phase changes by $q_j\cdot s$. Therefore a real linear combination of locally unwrapped phases, ${\sum }_j{n}_j{\theta }_j$, is unchanged exactly when ${\sum }_j{n}_j{q}_j=0$. This is a kernel of a linear map calculation for the matrix whose columns are the wavevectors.

A globally defined expression on angular phases requires more care: $e^{i{\sum }_j{n}_j{\theta }_j}$ is independent of the representatives chosen modulo $2\pi$ precisely when every $n_j$ is an integer. The globally defined translation-invariant characters therefore satisfy both $n\in {\mathbb{Z}}^m$ and ${\sum }_j{n}_j{q}_j=0$. An arbitrary real vector in the matrix's kernel of a linear map gives a local unwrapped invariant, but need not give a single-valued function of the angular phases.

For example, take four planar wavevectors $q_1=(1,0)$, $q_2=(0,1)$, $q_3=(1,1)$ and $q_4=(1,-1)$. The two independent angular combinations are ${\theta }_3-{\theta }_1-{\theta }_2$ and ${\theta }_4-{\theta }_1+{\theta }_2$, each interpreted modulo $2\pi$. Translations can set the first two phases to zero; the other two then equal these combinations.

More generally, when four planar wavevectors span a common reciprocal lattice of rank two, their translation action has a closed two-dimensional torus as its image and the quotient of the four-phase torus is two-dimensional. Real rank two alone guarantees only two local independent combinations; without a commensurability condition, the translation image can be dense in a larger torus. At zero amplitude, a phase is undefined, so smooth centre manifold equations should first use real parts and imaginary parts of the complex amplitudes.

Consider complex Fourier modes transforming under a translation symmetry as $A_j\mapsto e^{i{k}_j\cdot s}{A}_j$. A monomialtransforms by multiplication with $e^{i[{\sum }_j(p_j-q_j)k_j]\cdot s}$. It can occur in the equation for ${\dot{A}}_{\ell }$ of an equivariant dynamical system only ifThis follows because ${\dot{A}}_{\ell }$ must transform with the same wavevector as $A_{\ell }$. The condition enforces translation equivariance; other symmetries can impose additional restrictions on the polynomial coefficients.

Writing $d=p-q$, a monomial with total polynomial degree $N$ and a prescribed integer vector $d$ exists exactly when $\Vert d{\Vert }_1\le N$ and $N-\Vert d{\Vert }_1$ is an even integer. To prove this, write $p_j=\max (d_j,0)+r_j$ and $q_j=\max (-d_j,0)+r_j$ with nonnegative integers $r_j$. Then $N=\Vert d{\Vert }_1+2{\sum }_j{r}_j$. Conversely, any nonnegative even surplus can be distributed among the $r_j$.

For one nonzero wavevector, a cubic term in its own equation must have $p-q=1$ and $p+q=3$, giving $A^2\overline{A}=A|A{|}^2$. With several Fourier modes, a relation $k_1+k_2=k_3$ instead permits the quadratic term $A_1{A}_2$ in the third equation. The wavevectors decide whether this resonant interaction is possible before any polynomial coefficient is calculated.