Inmaculada Baldomá Barraca (DMAT, CRM)
In a wide range of physical, chemical, and biological systems modelling the interaction among different species, the dynamics of each species is usually governed by a diffusion mechanism together with a reaction term, which takes into account the interactions with the other species. For example, such systems arise when modelling chemical reactions processes as pattern formation mechanisms as aggregation of amoebae Dictyostelium discoideum, in certain ecological models [Mur01] or even to describe transformations of cardiac muscle cells [ES22].
A reaction–diffusion equation is a system of differential equations, with a diffusion term:
\begin{equation}
\partial_\tau U = D \Delta U + F(U,a) \;\;\;\;\;\;(1)
\end{equation}
where \(U = U(\tau,\mathbf{x}) \in \mathbb{R}^N\), \(\mathbf{x} \in\mathbb{R}^2\), \(D\) is the diffusion matrix, \(F\) is the (nonlinear) reaction term, \(\Delta\) is the Laplacian, and \(a\) is a parameter (for instance a catalytic concentration in a chemical reaction) or a set of parameters.
We consider the so-called oscillatory systems, which are a particular class of reaction–diffusion equations. These systems produce oscillations when \(\Delta U = 0\) (the so-called homogeneous situation, meaning that the medium in which the reaction takes place does not influence the system). They correspond to systems of the form (1) for which the dynamical system
$$\frac{dU}{d\tau} = F(U,a)$$
has an asymptotically stable periodic orbit. More precisely, we focus on dynamical systems exhibiting a supercritical Hopf bifurcation at \((U_0,a_0)\). In this setting, we look for solutions of the form
\begin{equation*}
U(\tau,\mathbf{x},a) = U_0
+ \varepsilon\big[ A(t,\mathbf{x}) e^{i\omega\tau} v
+ \overline{A}(t,\mathbf{x}) e^{-i\omega\tau} \bar v \big]
+ \mathcal{O}(\varepsilon^2),
\end{equation*}
where \(\overline{z}\) denotes the complex conjugate, \(\varepsilon^2 = a – a_0\), \(\omega\) is the frequency of the oscillation, and \(A(t,\mathbf{x})\in\mathbb{C}\) is the so-called amplitude.
Under generic conditions and neglecting terms of order higher than two, it can be shown (see [Kur03]) that the amplitude \(A\) satisfies the celebrated complex Ginzburg–Landau equation (CGL):
\begin{equation}
\label{eq:CGL}
\partial_t A = (1+i\alpha)\Delta A + A – (1+i\beta) A |A|^2 \;\;\;\;\;\;(2)
\end{equation}
with \(\alpha,\beta \in \mathbb{R}\) depending on \(F\) and \(D\).
The CGL is an ubicuous equation, widely studied in physics; for different regimes, it exhibits a pletora of patterns and it is far for been completely understood (see [AK02] and [SS23] for a extense exposition about open problems).
Following the classical literature, [KH81], [Hag82], [Gre80], [YK76], we consider solutions of the following type:
Definition 1. Let \(n\in \mathbb{N}\). In polar coordinates, \(A(t,r,\varphi)\) is a rigidly rotating Archimedean \(n\)-armed spiral wave if \(A\) is solution of (2) of the form
\begin{equation}
A(t,r,\varphi )= \mathbf{f}(r ) \exp (i(\Omega t+ n\varphi+ \Theta (r ) )) \;\;\;\;\;\;(3)
\end{equation}
where \(\mathbf{f},\Theta\) are \(\mathcal{C}^2\) when \(r\geq 0\). The boundary conditions are
\begin{equation*}
\mathbf{f}(0)=\Theta'(0)=0 ,\quad \lim_{r\to \infty} \mathbf{f}(r)=\sqrt{1-k_*^2}, \quad
\lim_{r\to \infty} \Theta'(r)=-k_*
\end{equation*}
with \(\Omega, k_*\) satisfying the dispersion relations:
\begin{equation}
C=\sqrt{1-k_*^2}, \qquad \Omega = \Omega (k_*) = – \beta +k_*^2 (\beta- \alpha). \;\;\;\;\;\;(4)
\end{equation}
For any \(\Omega t + n\varphi\) fixed, the spiral wave goes to
\begin{equation*}
A_*(\Omega t + n\varphi -k_*r+\theta(r)) : =C e^{i(\Omega t + n\varphi – k_*r+\theta(r))},
\end{equation*}
when \(r\to \infty\), with \(C,\Omega\in \mathbb{R}\) satisfying (4) an \(\theta'(r) \to 0\). The countour lines of \(A*\)
\[\mathrm{Re} \big (A_*(\Omega t + n\varphi-k_*r ) e^{-i\Omega t}\big )=\cos ( n\varphi -k_*r )=c
\]
are Archimedian spirals (see Figure 1).
Figure 1: Representation of two Archimedean \(n\)-armed spiral waves for winding numbers \(n=1,2,3,4\). These two spirals correspond to the contour lines \(\cos (-k_* r +n\varphi)=c\neq \pm 1\).
We introduce the twist parameter \(q\) as:
\begin{equation}
q=q(\alpha,\beta)=\frac{\beta-\alpha}{1+\alpha \beta} \;\;\;\;\;\;(5)
\end{equation}
which is well defined for \(\alpha,\beta\) satisfying \(|\alpha-\beta|\ll 1\). This singular pertorbation parameter plays an important role in the form of the spiral waves. Indeed, when \(q=0\), it is not difficult to check that the surface \((r\cos \varphi, r\sin \varphi, \mathrm{Re} ( e^{-i\Omega t}{A}(t,r,\varphi)))\), for \(r\gg 1\) looks like the one in the left hand side of Figure 2, that is there are not spirals waves when \(q=0\). However, when \(q\neq 0\) the level curves of \(\mathrm{Re} ( e^{-i\Omega t}{A}(t,r,\varphi)))\) starts to bend and the spirals arise as \(r\to \infty\).
Figure 2: The cuore corresponds to \(r=r_0 \gg 1\).
The result proven in [ABMS25] is the following:
Theorem 1. Let \(n\in \mathbb{N}\). There exist \(C_n\) and a unique odd function of the form
\begin{equation}
\kappa_*(q)= \frac{2}{q}e^{-\frac{C_n}{n^2} -\gamma } e^{-\frac{\pi}{2n|q|}} (1+\mathcal{O}(\big |\log |q|\big |^{-1})), \quad q \to 0 \;\;\;\;\;\;(6)
\end{equation}
with \(\gamma\) the Euler-Mascheroni constant, such that the CGL equation has an Archimedian spiral wave as in Definition 1 if and only if the asymptotic wave number \(k_*=\kappa_*(q)\) with \(q=q(\alpha,\beta)\) as in (5) and the frequency \(\Omega\) satisfies (4).
The existence of the function \(\kappa_*(q)\) was already known since the 80’s by the pionneer works by N. Kopell and L. Howard [KH81] and the formula (6) was conjectured by [Hag82] using formal matching techniques. This formula has been used widely in the literature, [CH93, AK02] but has not been proven until now. The novelty in our work, [AB11, ABMS16] and finally [ABMS25], is the use of a functional analysis approach that allow us, after 40 years, to design a beyond all orders selection mechanism of the asymptotic wavenumber.
References
[AB11] M Aguareles and I Baldomá, Structure and Gevrey asymptotic of solutions representing topological defects to some partial differential equations, Nonlinearity 24 (2011), no. 10, 2813–2847. MR 2842186 (2012g:34056)
[ABMS16] M Aguareles, I Baldomá, and T M-Seara, On the asymptotic wavenumber of spiral waves in λ − ω systems, Nonlinearity 30 (2016), no. 1, 90–114.
[ABMS25] M Aguareles, I Baldomá, and T M-Seara, A rigorous derivation of the asymptotic wavenumber of spiral waves in the complex ginzburglandau equation, Journal of European Mathematical Society https://ems.press/journals/jems/articles/14298785, published online first (2025).
[AK02] Igor S Aranson and Lorenz Kramer, The world of the complex ginzburglandau equation, Reviews of modern physics 74 (2002), no. 1, 99.
[CH93] Mark C Cross and Pierre C Hohenberg, Pattern formation outside of equilibrium, Reviews of modern physics 65 (1993), no. 3, 851.
[ES22] André H Erhardt and Susanne Solem, Bifurcation analysis of a modified cardiac cell model, SIAM Journal on Applied Dynamical Systems 21 (2022), no. 1, 231–247.
[Gre80] J. M. Greenberg, Spiral waves for λ − ω systems, SIAM J. Appl. Math. 39 (1980), no. 2, 301–309. MR 588502
[Hag82] Patrick S. Hagan, Spiral waves in reaction-diffusion equations, SIAM J. Appl. Math. 42 (1982), no. 4, 762–786. MR 84c:92069
[KH81] N Kopell and LN Howard, Target pattern and spiral solutions to reactiondiffusion equations with more than one space dimension, Add. Appl. Math. 2 (1981).
[Kur03] Yoshiki Kuramoto, Chemical oscillations, waves and turbulence. Mineola, 2003.
[Mur01] James D Murray, Mathematical biology II: spatial models and biomedical applications, vol. 3, Springer New York, 2001.
[SS23] Björn Sandstede and Arnd Scheel, Spiral waves: linear and nonlinear theory, Mem. Amer. Math. Soc. 285 (2023), no. 1413, v+126. MR 4580296
[YK76] Tomoji Yamada and Yoshiki Kuramoto, Spiral waves in a nonlinear dissipative system, Progress of Theoretical Physics 55 (1976), no. 6, 2035–2036.