Symmetry subspaces¶

Computing in an invariant subspace of a symmetry group: fewer unknowns and isolated solutions, at the price of hiding what breaks the symmetry.

When the equations are equivariant under a group (reflections, shifts, rotations by \(2\pi/m\)), the states with a given symmetry form an invariant subspace. Computing in it divides the size of the problem and, for continuous symmetries, removes the neutral directions that make solutions non-isolated.

Conatus uses symmetry subspaces in three places:

Invariant solutions. find_equilibrium works in the odd subspace \(u(-x) = -u(x)\) of a periodic problem, where translations are broken (Invariant solutions). Exact coherent states of 3D shear flows are searched in their shift–reflect and shift–rotate subspaces.
Bifurcations. A symmetry given as a matrix (branch_switch(..., symmetry=S), normal_form(..., symmetry=S)) classifies symmetry-breaking pitchforks (Continuation and bifurcations).
Eigenvalue problems. A linearized problem restricted to one parity class about a reflection, or to the azimuthal classes of an \(m\)-fold rotation in curvilinear geometry, has its own, smaller spectrum. Parities incompatible with the equations are refused (SymmetryBreakingError). These subspaces are declared on problems of the expert layer (tutorial 3. Symmetries and symmetry subspaces).

What a subspace hides¶

An instability that breaks the symmetry does not exist inside the subspace. A solution found there is not declared stable until its spectrum in the full space has been computed: equilibrium_eigenvalues and floquet_multipliers work in the full space, and symmetry_breaking_check reports the growth rates outside the subspace. explain() on a symmetry subspace states what it hides.