Freezing problems, symmetries and waves

This section is dedicated to the study of an equation (in x) 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 the problem 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

\[\left\{ \begin{array}{l} F(x, p) - \sum\limits_{i=1}^{N_g}\ s_i\cdot T_i\cdot x=0 \\ \langle T_i\cdot x_{ref},x-x_{ref}\rangle=0,\quad i=1,\cdots,N_g. \end{array}\right.\]

Wave stability

There are several ways to compute the stability of a wave $(x^w,s^w)$. 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.

Another way to compute the same spectrum is to proceed as follows. Using (W) as a definition for 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 find

\[\left\{ \begin{array}{l}\tag{W} J x_1+c_1A_{12} = \sigma x_1 \\ \langle A_{21},x_1\rangle + A_{22}c_1=0 \end{array}\right.\]

If $A_{22}\neq 0$, the eigen problem is equivalent to

\[Jx_1 - c_1\frac{\langle A_{21},x_1\rangle}{A_{22}} A_{12}= \sigma x_1\]

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.

References

  • Beyn

    Beyn and Thümmler, Phase Conditions, Symmetries and PDE Continuation.

  • Sandstede

    Sandstede, Björn. “Stability of Travelling Waves.” In Handbook of Dynamical Systems, 2:983–1055. Elsevier, 2002. https://doi.org/10.1016/S1874-575X(02)80039-X.