Freezing problems, symmetries and waves
This section is dedicated to the study of the equation F(x,p)=0 where one wishes to freeze a continuous symmetry. When the equation $F(x, p) = 0$ has a continuous symmetry described by a Lie group $G$ and action $g\cdot x$ for $g\in G$, one can reduce the symmetry of F by considering the constrained problem[Beyn]:
\[\left\{ \begin{array}{l}\tag{W} F(x, p) - s\cdot T\cdot x=0 \\ \langle T\cdot x_{ref},x-x_{ref}\rangle=0 \end{array}\right.\]
where $T$ is a generator of the Lie algebra associated to $G$, $x_{ref}$ is a reference solution and $s$ is the speed. This is known as the freezing method.
Similarly, one can reduce several symmetries by considering
\[\left\{ \begin{array}{l} F(x, p) - \sum\limits_{i=1}^{N_g}\ s_i\cdot T_i\cdot x=0 \\ \langle T_i\cdot x_{ref},x-x_{ref}\rangle=0,\quad i=1,\cdots,N_g. \end{array}\right.\]
We call a solution
(x,p,s)of (W) a wave.
Wave stability
There are several ways to compute the stability of a wave $(x^w,p,s^w)$. From [Sandstede], this requires to compute the spectrum of
\[d_1F(x,p)- \sum\limits_{i=1}^{N_g}\ s_i\cdot T_i\tag{EV}.\]
However, there is (potentially) the zero eigenvalue associated to the eigenvectors $T_i\cdot x^w$. In practice, because the symmetry is discrete numerically, we find a small eigenvalue instead of 0.
Another way to compute the same spectrum is to proceed as follows. Using (W) as a definition of the functional $G((x,s),p)\in\mathbb R^{N+1}$, the eigenproblem for computing the stability of a wave $(x^w,s^w)$ is
\[A x = σ Bx\tag{GEV}\]
where $B = diag(1,\cdots,1,0)$ and $A:=dG$. An advantage of (GEV) over (EV) is that the trivial eigenvalues are removed but it comes at an increased cost. We can improved this situation as follows.
Case $N_g=1$
Let us have a look at (GEV) more closely. We need to solve for the eigenvalues $\sigma$ and the eigenvectors $(x_1,c_1)$ solutions of
\[\left\{ \begin{array}{l}\tag{W} J x_1+c_1A_{12} = \sigma x_1 \\ \langle A_{21},x_1\rangle + A_{22}c_1=0 \end{array}\right.\]
Case $A_{22}\neq 0$
If $A_{22}\neq 0$, the eigen problem is equivalent to
\[Jx_1 - c_1\frac{\langle A_{21},x_1\rangle}{A_{22}} A_{12}= \sigma x_1\]
Case $A_{22} = 0$
If $A_{22} = 0$, the eigen problem is equivalent to $x_1=α A_{21} + x_1^\bot$ with $\langle A_{21},x_1^\bot\rangle=0$. Hence, I find $\langle A_{21},Jx_1^\bot\rangle+c_1\langle A_{21},A_{12}\rangle=0$
\[Jx_1^\bot-\frac{\langle A_{21},Jx_1^\bot\rangle}{\langle A_{21},A_{12}\rangle}A_{21}=σ x_1^⊥\]
Encoding of the functional for the freezed problem
The freezing method is encoded in the composite type TWProblem which we loosely refer to as a Travelling Wave (TW) problem.
Computation with newton
We provide a simplified call to newton to locate the freezed solution
newton(prob::TWProblem, orbitguess, options::NewtonPar; kwargs...)Continuation
We also provide a simplified call to continuation to continue the freezed solution as function of a parameter:
continuation(prob::TWProblem, orbitguess, lens::Lens, contParams::ContinuationPar; jacobian = :MatrixFree, kwargs...)Note that in this case, the eigen solver passed in contParams is converted into an appropriate generalized eigensolver.