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Butterfly Catastrophe for Fronts in a Three-Component Reaction–Diffusion System

Abstract

We study the dynamics of front solutions in a three-component reaction–diffusion system via a combination of geometric singular perturbation theory, Evans function analysis, and center manifold reduction. The reduced system exhibits a surprisingly complicated bifurcation structure including a butterfly catastrophe. Our results shed light on numerically observed accelerations and oscillations and pave the way for the analysis of front interactions in a parameter regime where the essential spectrum of a single front approaches the imaginary axis asymptotically.

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Acknowledgments

The work of M.C.-B. is supported by the Deutsche Forschungsgemeinschaft (DFG) under the grant CH 957/1-1. J.R. is grateful for the support of the Dutch NWO cluster NDNS+ and his previous employer, Centrum Wiskunde & Informatica (CWI), Amsterdam. P.v.H. is grateful for the support of his previous employer, Boston University, Boston. The authors would like to thank Björn Sandstede for inspiring discussions.

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Correspondence to Martina Chirilus-Bruckner.

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Communicated by Arnd Scheel.

Appendices

Appendix 1: Higher Order Corrections of the Front Profile

The stability analysis for uniformly traveling fronts \(Z_\mathrm{tf} = (U_\mathrm{tf}, V_\mathrm{tf}, W_\mathrm{tf})\) as derived in Proposition 1 makes use of information about the higher order terms of the front profile (1). To ease notation, we introduce the abbreviation

$$\begin{aligned} \sqrt{2} (U^0_\mathrm{tf})_{\eta }=\mathrm{sech}^2{\left( \frac{\eta }{\sqrt{2}}\right) } =: \rho ({\eta }) \,. \end{aligned}$$
(45)

Upon working in a (fast) co-moving frame \({\eta }= \xi -\varepsilon c t\) and using a regular expansion for \(U_\mathrm{tf}({\eta })\), that is, \(U_\mathrm{tf}({\eta })= U_\mathrm{tf}^0({\eta }) + \varepsilon U_\mathrm{tf}^1({\eta }) + \varepsilon ^2 U_\mathrm{tf}^2({\eta }) + \fancyscript{O}(\varepsilon ^3)\), with \(U_\mathrm{tf}^0({\eta }) = \tanh {\left( \frac{\eta }{\sqrt{2}}\right) }\), we obtain the following result.

Lemma 12

Let the conditions in Proposition 1 be fulfilled.

  1. (i)

    The higher correction term \(U_\mathrm{tf}^1({\eta })\) is an even function in the fast field \(I_\mathrm{f}\) (see (11)).

  2. (ii)

    Its derivative obeys the relation

    $$\begin{aligned} {\fancyscript{L}} (U^1_\mathrm{tf})_{\eta }&:= ((U^1_\mathrm{tf})_{\eta })_{{\eta }\eta } + (U^1_\mathrm{tf})_{\eta } -3 (U^0_\mathrm{tf})^2(U^1_\mathrm{tf})_{\eta }&=-c (U^0_\mathrm{tf})_{{\eta }\eta } + 6 U^0_\mathrm{tf} (U^0_\mathrm{tf})_{\eta } U^1_\mathrm{tf}\\&= -\frac{c}{\sqrt{2}} \rho _{\eta } +3\sqrt{2} U^0_\mathrm{tf} U^1_\mathrm{tf} \rho \,. \end{aligned}$$
  3. (iii)

    The next order correction term \(U_\mathrm{tf}^2({\eta })\) obeys the integral relation

    $$\begin{aligned}&c \int (U^1_\mathrm{tf})_{{\eta }\eta } \rho d \eta - 6 \int U^0_\mathrm{tf} (U^0_\mathrm{tf})_{\eta } U^2_\mathrm{tf} \rho d {\eta }- 6 \int U^0_\mathrm{tf} U^1_\mathrm{tf} (U^1_\mathrm{tf})_{\eta } \rho d \eta \\&\quad - 3 \int (U^0_\mathrm{tf})_{\eta } (U^1_\mathrm{tf})^2\rho d {\eta } \\&= c \int (U^1_\mathrm{tf})_{{\eta }\eta } \rho d \eta - 3\sqrt{2} \int U^0_\mathrm{tf} U^2_\mathrm{tf} \rho ^2 d {\eta }- 6 \int U^0_\mathrm{tf} U^1_\mathrm{tf} (U^1_\mathrm{tf})_{\eta } \rho d \eta \\&\qquad \qquad \quad - \frac{3}{\sqrt{2}} \int (U^1_\mathrm{tf})^2\rho ^2 d {\eta }\\&= 4\sqrt{2}\left( \alpha \left( \frac{1}{\sqrt{c^2 {\hat{\tau }}^2 +4}}\right) + \frac{\beta }{D} \left( \frac{1}{\sqrt{\frac{c^2 {\hat{\theta }}^2}{D^2} +4}}\right) \right) \,. \end{aligned}$$

Proof

The proof of this lemma is completely analogues to the proofs in Section 2.2 of van Heijster et al. (2008) upon plugging in the correct expressions for the derivatives of the slow components in the fast field \(I_\mathrm{f}\) (from (11)). The parity of \(U_\mathrm{tf}^1({\eta })\) follows from expanding \(U_\mathrm{tf}({\eta })\) in a regular fashion and studying the \( \fancyscript{O}(\varepsilon ) \)-level of the \(U\)-equation of (22). After plugging in the derived existence condition (14), we obtain the following equation for \(U^1_\mathrm{tf}\)

$$\begin{aligned} {\fancyscript{L}} U^1_\mathrm{tf}= c\left( \frac{\sqrt{2}}{3}-(U_\mathrm{tf}^0)_{\eta } \right) = \sqrt{2} c\left( \frac{1}{3}-\frac{1}{2}\rho \right) \,. \end{aligned}$$

Since the operator \({\fancyscript{L}}\) conserves parity and the right hand side of the above equation is even, we obtain that \(U_\mathrm{tf}^1({\eta })\) is an even function. The relations for \((U_\mathrm{tf}^1)_{\eta }\) and \(U_\mathrm{tf}^2\) follow from studying the derivative of the \(U\)-equation of (22) in the fast field in the co-moving frame:

$$\begin{aligned} U_{{\eta }{\eta }\eta } = -U_{\eta } + 3U^2U_{\eta } + \varepsilon (\alpha V_{\eta } + \beta W_{\eta } - c U_{\eta \eta }) \,. \end{aligned}$$

After plugging in the regular expansion for \(U_\mathrm{tf}\) and noting that \(V_{\eta }\) and \(W_{\eta }\) are \(\fancyscript{O}(\varepsilon )\) and constant in the fast field, we obtain

$$\begin{aligned} \fancyscript{O}(1): {\fancyscript{L}} (U^0_\mathrm{tf})_{\eta } =&0\,,\\ \fancyscript{O}(\varepsilon ): {\fancyscript{L}} (U^1_\mathrm{tf})_{\eta } =&-c (U^0_\mathrm{tf})_{{\eta }\eta } + 6 U^0_\mathrm{tf} (U^0_\mathrm{tf})_{\eta } U^1_\mathrm{tf}\,,\\ \fancyscript{O}(\varepsilon ^2): {\fancyscript{L}} (U^2_\mathrm{tf})_{\eta } =\,&\alpha V_{\eta } + \beta W_{\eta } - c (U^1_\mathrm{tf})_{{\eta }\eta } + 6 U^0_\mathrm{tf} (U^0_\mathrm{tf})_{\eta } U^2_\mathrm{tf} \\&+ 6 U^0_\mathrm{tf} U^1_\mathrm{tf} (U^1_\mathrm{tf})_{\eta } + 3 (U^0_\mathrm{tf})_{\eta } (U^1_\mathrm{tf})^2\,, \end{aligned}$$

where we indicated the correct asymptotic scaling of the slow components by taking their derivatives with respect to the slow variable. To obtain the integral relation involving \(U^2_\mathrm{tf}\), we note that \( \rho \) is in the kernel of \({\fancyscript{L}}\) and we apply a solvability condition on the \(\fancyscript{O}(\varepsilon ^2)\)-equation. This gives

$$\begin{aligned} 2\sqrt{2}(\alpha V_{\eta } + \beta W_{\eta }) =\,&c \int (U^1_\mathrm{tf})_{{\eta }\eta } \rho d \eta - 6 \int U^0_\mathrm{tf} (U^0_\mathrm{tf})_{\eta } U^2_\mathrm{tf} \rho d {\eta }\\&- 6 \int U^0_\mathrm{tf} U^1_\mathrm{tf} (U^1_\mathrm{tf})_{\eta } \rho d \eta - 3 \int (U^0_\mathrm{tf})_{\eta } (U^1_\mathrm{tf})^2\rho d \eta \,, \end{aligned}$$

where we used that \(\int \rho ({\eta }) dt = 2\sqrt{2}\,.\) Finally, from (12) we get that

$$\begin{aligned} V_{\eta }&= \lambda _v^+(v_*+1) = \frac{1}{2}\left( -\frac{c^2 {\hat{\tau }}^2}{\sqrt{c^2 {\hat{\tau }}^2 +4}}+ \sqrt{c^2 {\hat{\tau }}^2 +4}\right) =\frac{2}{\sqrt{c^2 {\hat{\tau }}^2 +4}}\,,\\ W_{\eta }&= \lambda _w^+(w_*+1) = \frac{1}{2}\frac{1}{D}\left( -\frac{\frac{c^2 {\hat{\theta }}^2}{D^2}}{\sqrt{\frac{c^2 {\hat{\theta }}^2}{D^2} +4}}+ \sqrt{\frac{c^2 {\hat{\theta }}^2}{D^2} +4}\right) = \frac{1}{D}\left( \frac{2}{\sqrt{\frac{c^2 {\hat{\theta }}^2}{D^2} +4}}\right) \,. \end{aligned}$$

This completes the proof. \(\square \)

Appendix 2: Proof of Lemma 6 and Theorem 1 via Evans Function Analysis

As alluded to, we will make use of the Evans function to compute the point spectrum of our operator. For slow-fast systems, as analyzed in this paper, it has been shown in Alexander et al. (1990) that the Evans function \(\fancyscript{D}(\lambda )\) can be split into an analytic fast part \(\fancyscript{D}_\mathrm{f}(\lambda )\) and a meromorphic slow part \(\fancyscript{D}_\mathrm{s}(\lambda )\) and a nonzero function \(d(\lambda )\). In Doelman et al. (1998, 2001, 2002), the NLEP-method was developed in context of two-component singularly perturbed reaction–diffusion equations by which these two parts of the Evans function can be explicitly computed. In van Heijster et al. (2008), this method was extended to \(N\)-component singularly perturbed reaction–diffusion equations with one fast and \((N-1)\)-slow components. In more detail, for \( N = 3 \) we can write the Evans function \(\fancyscript{D}(\lambda )\) as

$$\begin{aligned} \fancyscript{D}(\lambda ) = d(\lambda ) \fancyscript{D}_\mathrm{f}(\lambda ) \fancyscript{D}_\mathrm{s}(\lambda ) = d(\lambda ) t_1^+(\lambda )\left( t_{22}^+(\lambda )t_{33}^+(\lambda )-t_{23}^+(\lambda )t_{32}^+(\lambda )\right) \,, \end{aligned}$$

where \(t_1^+(\lambda )\) is an analytic transmission function corresponding to \(\fancyscript{D}_\mathrm{f}(\lambda )\) and \(t_{ij}^+(\lambda )\,, i=2,3\) are four slow-fast transmission functions corresponding to \(\fancyscript{D}_\mathrm{s}(\lambda )\). A transmission function measures how much information is transferred to \(\infty \) through the potential given by the square of the front for a function which we fix at \(-\infty \). See van Heijster et al. (2008) for more details and note that the term transmission function comes from scattering theory. Also observe that we use a slightly different notation then van Heijster et al. (2008), instead of \(t_{i}, t_{2i}\), and \(t_{3i}, i = 1, \ldots , 6\), we use \(t_{i}^{\pm }, t_{2i}^{\pm }\), and \(t_{3i}^{\pm }, i = 1, \ldots , 3\). We first determine the fast transmission function \(t_1^+(\lambda )\).

Lemma 13

(Fast transmission function) The fast–fast transmission function \(t_1^+(\lambda )\) is of the form

$$\begin{aligned} t_1^+(\lambda ) = \lambda - \varepsilon ^2 \hat{\lambda }_\mathrm{fast} \,, \end{aligned}$$
(46)

where, to leading order,

$$\begin{aligned} \hat{\lambda }_\mathrm{fast} =\frac{6}{\sqrt{2}}\left( \frac{\alpha }{\sqrt{c^2 {\hat{\tau }}^2 +4}} +\frac{\beta }{D\sqrt{\frac{c^2 {\hat{\theta }}^2}{D^2} +4}}\right) \, . \end{aligned}$$

Proof

From Section 4.1 in van Heijster et al. (2008), modified for uniformly traveling fronts, we know that the fast transmission function can be obtained by studying

$$\begin{aligned} \fancyscript{L} u = \lambda u - \varepsilon c u_{\eta }\,, \end{aligned}$$

where \(\fancyscript{L}\) is defined in Lemma 12. Moreover, from this work it follows that this problem possibly has positive eigenvalues near zero. Therefore, we write

$$\begin{aligned} \lambda = \varepsilon \lambda ^0 + \varepsilon ^2 {\hat{\lambda }}\,. \end{aligned}$$

Upon using a regular expansion for \(u({\eta })\), that is, \(u=u^0+ \varepsilon u^1 + \varepsilon ^2 u^2+ \fancyscript{O}(\varepsilon ^3)\), we obtain

$$\begin{aligned} \fancyscript{O}(1):&\quad \fancyscript{L} u^0 = 0\,,\\ \fancyscript{O}(\varepsilon ):&\quad \fancyscript{L} u^1 = \lambda ^0 u^0 - c u_{\eta }^0 + 6 U_\mathrm{tf}^0 U_\mathrm{tf}^1 u^0\,,\\ \fancyscript{O}(\varepsilon ^2):&\quad \fancyscript{L} u^2 = \lambda ^0 u^1 + {\hat{\lambda }}u^0 - c u_{\eta }^1 + 6 U_\mathrm{tf}^0 U_\mathrm{tf}^2 u^0 + 6 U_\mathrm{tf}^0 U_\mathrm{tf}^1 u^1 + 3 (U_\mathrm{tf}^1)^2 u^0\,. \end{aligned}$$

The leading order equation yields \(u^0({\eta }) = C \rho ({\eta })\,,\) with \(C \in \mathbb {R}\) and \(\rho \) defined in (45). Applying a solvability condition to the \(\fancyscript{O}(\varepsilon )\)-equation yields

$$\begin{aligned} 0 = \lambda ^0 C \int \rho ^2 d {\eta }- c C \int \rho \rho _{\eta } d {\eta }+ 6 C \int U_\mathrm{tf}^0 U_\mathrm{tf}^1 \rho ^2 d \eta \,. \end{aligned}$$

The last two integrals vanish since they are odd, see Lemma 12, and since the first integral is unequal to zero, this yields \(\lambda ^0=0\), i.e. \(\lambda \) is of \(\fancyscript{O}(\varepsilon ^2)\). Thus, the equation for \(u^1\) becomes

$$\begin{aligned} \fancyscript{L} u^1 = - c C \rho _{\eta } + 6 C U_\mathrm{tf}^0 U_\mathrm{tf}^1 \rho \,. \end{aligned}$$

By Lemma 12, we get that \(u^1 = \sqrt{2} C (U_\mathrm{tf}^1)_{\eta }\). Finally, using all this information, the \(\fancyscript{O}(\varepsilon ^2)\)-equation reduces to

$$\begin{aligned} \fancyscript{L} u^2 = C {\hat{\lambda }}\rho - \sqrt{2} c C (U_\mathrm{tf}^1)_{{\eta }\eta } + 6 C U_\mathrm{tf}^0 U_\mathrm{tf}^2 \rho + 6 \sqrt{2} C U_\mathrm{tf}^0 U_\mathrm{tf}^1 (U_\mathrm{tf}^1)_{\eta } + 3 C (U_\mathrm{tf}^1)^2 \rho \,\,, \end{aligned}$$

and a solvability condition gives

$$\begin{aligned} 0=\,&{\hat{\lambda }}\int \rho ^2 d {\eta }- \sqrt{2} c\int (U_\mathrm{tf}^1)_{{\eta }\eta } \rho d {\eta }+ 6 \int U_\mathrm{tf}^0 U_\mathrm{tf}^2 \rho ^2 d {\eta }\\&+ 6 \sqrt{2} \int U_\mathrm{tf}^0 U_\mathrm{tf}^1 (U_\mathrm{tf}^1)_{\eta } d {\eta }+ 3\int (U_\mathrm{tf}^1)^2 \rho ^2 d {\eta }\,\,. \end{aligned}$$

The last four integral terms are equal up to factor \(\sqrt{2}\) to the integral condition in Lemma 12, and since \(\int \rho ^2 d {\eta }= \frac{4 \sqrt{2}}{3}\), we get

$$\begin{aligned} {\hat{\lambda }}=\frac{6}{\sqrt{2}}\left( \frac{\alpha }{\sqrt{c^2 {\hat{\tau }}^2 +4}}+\frac{\beta }{D\sqrt{\frac{c^2 {\hat{\theta }}^2}{D^2} +4}}\right) \,. \end{aligned}$$

This completes the proof of Lemma 13. \(\square \)

Lemma 14

The slow transmission functions \(t_{22}^+, t_{23}^+, t_{32}^+, t_{33}^+\) are given by

$$\begin{aligned} t_{22}^+({\hat{\lambda }}) = 1- \frac{2\sqrt{2}}{\sqrt{G_v}} C_2\,,&\quad \quad t_{23}^+({\hat{\lambda }}) = - \frac{2\sqrt{2}}{D \sqrt{G_w}} C_2\,, \\ t_{32}^+({\hat{\lambda }}) = - \frac{2\sqrt{2}}{\sqrt{G_v}} C_3\,,&\quad \quad t_{33}^+({\hat{\lambda }}) = 1- \frac{2\sqrt{2}}{D \sqrt{G_w}} C_3\,,\nonumber \end{aligned}$$
(47)

with

$$\begin{aligned} G_v = c^2 {\hat{\tau }}^2 + 4({\hat{\lambda }}{\hat{\tau }}+1), \quad G_w = c^2 \frac{{\hat{\theta }}^2}{D^2} + 4({\hat{\lambda }}{\hat{\theta }}+1) \,, \end{aligned}$$

and \(C_2\) and \(C_3\) are two different integration constants for two different slow basis functions \(\Phi _{2,3}\).

Proof

The proof of this lemma is very similar to the derivation of the slow transmission functions in van Heijster et al. (2008), see especially Sections 4.2 and 5.3. Note that it is even less complicated here since we only make one excursion through a fast field and we do not have to introduce the intermediate transmission functions \(s_{ij}\). Therefore, we omit the proof of this lemma and only state that we need the same asymptotic scalings as in Section 5.3 of van Heijster et al. (2008), get the same eigenvalues and eigenvectors, and in the end need to match the slow components and their derivatives over their jump through the fast field \(I_\mathrm{f}\). \(\square \)

From Lemma 14, it follows that

$$\begin{aligned} t_{22}^+t_{33}^+-t_{23}^+t_{32}^+= 1-\frac{2\sqrt{2}}{\sqrt{G_v}} C_2 - \frac{2\sqrt{2} }{D\sqrt{G_w}}C_3\,, \end{aligned}$$

and the Evans function thus reads

$$\begin{aligned} \fancyscript{D}({\hat{\lambda }}) = d({\hat{\lambda }}) \left( t_1^+({\hat{\lambda }})-\frac{2\sqrt{2}}{\sqrt{G_v}} C_2t_1^+({\hat{\lambda }}) - \frac{2\sqrt{2} }{D\sqrt{G_w}}C_3t_1^+({\hat{\lambda }})\right) \,, \end{aligned}$$
(48)

where \(d({\hat{\lambda }})\) is a nonzero function and note that we suppressed the explicit dependence on \({\hat{\lambda }}\) in \(G_{v,w}\). In order to finish the proof of Theorem 1, we need to determine the constants \(C_{2}\) and \(C_3\). Therefore, we look at the higher order corrections of the fast component of the eigenvalue problem (26) in the fast field \(I_\mathrm{f}\). From the proof of Lemma 14 it followed that we also need to rescale the slow variables

$$\begin{aligned} (v,w)({\eta }) = \varepsilon ({\tilde{v}},{\tilde{w}})({\eta }) \,. \end{aligned}$$

(Note that the proof of this last lemma is actually not in the present work, but can be found in §5.3 of van Heijster et al. (2008).) With the above rescalings, the fast \(u\)-component of (26) becomes

$$\begin{aligned} u_{{\eta }\eta }+(1-3(U_\mathrm{tf})^2)u = -\varepsilon c u_{\eta } + \varepsilon ^2 (\alpha {\tilde{v}} + \beta {\tilde{w}} + {\hat{\lambda }}u) \,. \end{aligned}$$

Using the regular expansions \(u({\eta })= u^0({\eta }) + \varepsilon u^1({\eta }) + \varepsilon ^2 u^2({\eta }) + \fancyscript{O}(\varepsilon ^3)\) and \(U_\mathrm{tf}({\eta }) = U_\mathrm{tf}^0({\eta }) + \varepsilon U_\mathrm{tf}^1({\eta }) + \varepsilon ^2 U_\mathrm{tf}^2({\eta }) + \fancyscript{O}(\varepsilon ^3)\), we get

$$\begin{aligned} \fancyscript{O}(1):&\quad \fancyscript{L} u^0 = 0 \,,\\ \fancyscript{O}(\varepsilon ):&\quad \fancyscript{L} u^1 = -c (u^0)_{\eta } + 6 U_\mathrm{tf}^0U_\mathrm{tf}^1 u^0 \,,\\ \fancyscript{O}(\varepsilon ^2):&\quad \fancyscript{L} u^2 \!=\! -\!c (u^1)_{\eta } \!+\! \alpha {\tilde{v}} \!+\! \beta {\tilde{w}} + {\hat{\lambda }}u^0 + 6 U_\mathrm{tf}^0U_\mathrm{tf}^1 u^1 +6 U_\mathrm{tf}^0U_\mathrm{tf}^2 u^0 +3 (U_\mathrm{tf}^1)^2u^0 \,, \end{aligned}$$

where \(\fancyscript{L}\) is defined in Lemma 12. The leading order equation yields that

$$\begin{aligned} u^0({\eta }) = C_{2,3} \rho ({\eta }), \end{aligned}$$

see (45), where we use two different constants for the two different slow basis functions \(\Phi _{2,3}\). Comparing the \(\fancyscript{O}(\varepsilon )\)-equation with Lemma 12, yields

$$\begin{aligned} u^1({\eta }) = \sqrt{2} C_{2,3} (U_\mathrm{tf}^1({\eta }))_{\eta } \,. \end{aligned}$$

Finally, integrating the \(\fancyscript{O}(\varepsilon ^2)\)-equation against \(\rho ({\eta })\), and implementing the above expressions for \(u^0({\eta })\) and \(u^1({\eta })\) gives

$$\begin{aligned} 0=&-\sqrt{2} c C_{2,3} \int (U_\mathrm{tf}^1)_{{\eta }\eta } \rho d \eta + \alpha {\tilde{v}} \int \rho d \eta + \beta {\tilde{w}} \int \rho d {\eta }+ {\hat{\lambda }}C_{2,3} \int \rho ^2 d {\eta }\\&+ 6 \sqrt{2} C_{2,3} \int U_\mathrm{tf}^0U_\mathrm{tf}^1 (U_\mathrm{tf}^1)_{\eta } \rho d {\eta }\!+\! 6 C_{2,3}\!\int U_\mathrm{tf}^0U_\mathrm{tf}^2 \rho ^2 d \eta +3 C_{2,3}\!\int (U_\mathrm{tf}^1)^2u^0 \rho ^2 d \eta \\ =&\,2\sqrt{2} \alpha {\tilde{v}} + 2\sqrt{2} \beta {\tilde{w}}+ 4\frac{\sqrt{2}}{3}{\hat{\lambda }}C_{2,3}-8 C_{2,3}\left( \frac{\alpha }{\sqrt{c^2 {\hat{\tau }}^2 +4}}+\frac{\beta }{D \sqrt{\frac{c^2 {\hat{\theta }}^2}{D^2} +4}}\right) \,, \end{aligned}$$

where we used the integral condition of Lemma 12 in the last step and exploited that \({\tilde{v}}\) and \({\tilde{w}}\) are the constant values of the slow components in the fast field. When we closely examine the above equation, we recognize the fast transmission function \(t_1^+({\hat{\lambda }})\) given by (46). More precisely, the above equality reduces to

$$\begin{aligned} 0=&\sqrt{2} \alpha {\tilde{v}} + \sqrt{2} \beta {\tilde{w}}+ C_{2,3}\frac{2\sqrt{2}}{3} t_1^+({\hat{\lambda }}) \,. \end{aligned}$$

Now, we have constructed our slow basis functions in such a fashion that \({\tilde{v}}=1\) and \({\tilde{w}}=0\) for \(\Phi _2\) and vice versa for \(\Phi _3\). Therefore, \(C_{2}\) and \(C_3\) are given by

$$\begin{aligned} C_{2} = -\frac{3}{2} \frac{\alpha }{t_1^+({\hat{\lambda }})}\,,\quad \quad C_{3}= -\frac{3}{2} \frac{\beta }{t_1^+({\hat{\lambda }})}\,. \end{aligned}$$

Implementing this in the Evans function representation of (48) gives

$$\begin{aligned} \fancyscript{D}({\hat{\lambda }}) =\,&d({\hat{\lambda }}) \left( t_1^+({\hat{\lambda }})+\frac{3\sqrt{2}}{\sqrt{G_v}} \alpha + \frac{3\sqrt{2} }{\sqrt{G_w}}\frac{\beta }{D}\right) \\ =\,&d({\hat{\lambda }}) \left( {\hat{\lambda }}+ 3\sqrt{2}\alpha \left( \frac{1}{\sqrt{c^2 {\hat{\tau }}^2 + 4({\hat{\lambda }}{\hat{\tau }}+1)}} - \frac{1}{\sqrt{c^2 {\hat{\tau }}^2 +4}}\right) \right. \\&\left. + 3\sqrt{2}\frac{\beta }{D}\left( \frac{1}{\sqrt{c^2 \frac{{\hat{\theta }}^2}{D^2} + 4({\hat{\lambda }}{\hat{\theta }}+1)}}-\frac{1}{\sqrt{\frac{c^2 {\hat{\theta }}^2}{D^2} +4}}\right) \right) \,, \end{aligned}$$

which completes the proof of Theorem 1.

Appendix 3: Number of Small Eigenvalues (Proof of Lemma 7)

We examine the roots of the Evans function (27), that is, of

$$\begin{aligned} E({\hat{\lambda }}) = {\hat{\lambda }}+ 3\sqrt{2}\left( F_0 + F({\hat{\lambda }})\right) , \end{aligned}$$

where \(F({\hat{\lambda }}) = \alpha \sqrt{F_1} + \frac{\beta }{D}\sqrt{F_2}\) and

$$\begin{aligned} F_0&= -\frac{\alpha }{\sqrt{c^2{\hat{\tau }}^2 + 4}} - \frac{\beta }{\sqrt{c^2{\hat{\theta }}^2 + 4D^2}}\,, \quad F_1({\hat{\lambda }}) = \frac{1}{c^2{\hat{\tau }}^2 + 4 + 4{\hat{\tau }}{\hat{\lambda }}}\,, \\ F_2({\hat{\lambda }})&= \frac{1}{c^2{\hat{\theta }}^2/D^2 + 4 + 4{\hat{\theta }}{\hat{\lambda }}}. \end{aligned}$$

We shall prove in this section that \(E\) possesses at most three (complex) roots. Since one of these is always zero, in case of two roots the other one must be real, and if there are three, either all are real or the nonzero ones form a complex conjugate pair. In fact, we shall prove that for \(\alpha , \beta > 0\) there are precisely two real eigenvalues, for \(\alpha , \beta \le 0\) there is one, and in case \(\alpha \beta <0\) we give conditions for when the number is one or three.

Let us briefly relate this to the spectrum of the linearization \(\mathbb {L}_c\) in a front, where changes in the number of roots of \(E\) are changes in the number of eigenvalues. In the analysis below, we find that eigenvalues are lost to or born from a branch cut \(B_* \subset \mathbb {R}-\setminus \{0\}\) for \(E\). This relates to the spectral theory of \(\mathbb {L}_c\) via the so-called absolute spectrum as defined in Sandstede and Scheel (2000), which is the branch cut in our case and more generally the boundary for analytic continuations of an Evans function (Sandstede and Scheel 2004).

Before entering into the proof, we make some preliminary observations on the domain and holomorphic (analytic) nature of \(E\). Clearly, \(E({\hat{\lambda }}) = {\hat{\lambda }}\) for \(\alpha =\beta =0\), which is holomorphic on \(\mathbb {C}\) and has zero as its only root. Concerning singularities and the occurrence of square root terms \(\sqrt{F_j}\) in \(E\), these can cancel each other in the sense that \(F\equiv 0\) for \(\alpha \beta \ne 0\) if and only if \(\alpha \beta <0\) and the two square root terms merge into a single one. The latter means that the possible singularities in \({\hat{\lambda }}\), given by

$$\begin{aligned} {\hat{\lambda }}_{{\hat{\tau }}}=-\frac{1}{{\hat{\tau }}-\frac{c^2{\hat{\tau }}}{4}}\,,\qquad {\hat{\lambda }}_{{\hat{\theta }}}=-\frac{1}{{\hat{\theta }}-\frac{c^2{\hat{\theta }}}{4D^2}} \end{aligned}$$

are equal, which is equivalent to

$$\begin{aligned} \frac{c^2}{4}\left( {\hat{\tau }}- \frac{{\hat{\theta }}}{D^2}\right) + \frac{1}{{\hat{\tau }}} - \frac{1}{{\hat{\theta }}} = 0. \end{aligned}$$
(49)

If in addition \(\Sigma =0\), where

$$\begin{aligned} \Sigma := \frac{\alpha }{\sqrt{{\hat{\tau }}}} + \frac{\beta }{D\sqrt{{\hat{\theta }}}}, \end{aligned}$$
(50)

then both \(F(\lambda )\) and \(F_0\) vanish. In case (49) and (50) hold, we have \(E({\hat{\lambda }}) = {\hat{\lambda }}\), as for \(\alpha =\beta =0\). Simple examples for (49) are \({\hat{\tau }}={\hat{\theta }}\), and \(c=0\) or \(D=1\). (Recall that we have stipulated that \(D>1\). However, this is not necessary for this proof.) If, in addition, \(\alpha =-\beta /D\) then also (50) is fulfilled.

Notably, condition (49) is independent of \(\alpha , \beta \), which is a splitting of parameter space we shall exploit later.

In the generic case, where (50) fails, the nature of \(E\) is different. We understand square roots as the principal branch with branch cut the negative real axis, that is, \({\hat{\lambda }}\in \mathbb {C}\) has arguments in \((-\pi ,\pi ]\), and the image of the square roots has non-negative real parts. Hence, the images of \(\sqrt{F_j}\) have non-negative real parts and \(E\) is holomorphic only on \(\mathbb {C}\setminus B_*\) with branch cut for \(E\) being the half line \(B_*:=(-\infty ,\max ({\hat{\lambda }}_{{\hat{\tau }}},{\hat{\lambda }}_{{\hat{\theta }}})] \subset \mathbb {R}_-\setminus \{0\}\), where \({\hat{\lambda }}_{{\hat{\tau }}}\) and \({\hat{\lambda }}_{{\hat{\theta }}}\) defined above are the branch points and also singularities of \(E\).

The proof that there are at most three roots goes by a homotopy argument to a case of merged singularities (49). Before entering into the homotopy, we proceed with following preliminary observations, where the number of roots is understood to be counted with multiplicity.

Lemma 15

Assume \((\alpha ,\beta )\ne (0,0)\) and \(D,{\hat{\tau }},{\hat{\theta }}>0\).

  1. (i)

    For \(\alpha \beta \ne 0\) there are no roots of \(E\) in the closed interval between the singularities \({\hat{\lambda }}_{{\hat{\tau }}}\) and \({\hat{\lambda }}_{{\hat{\theta }}}\).

  2. (ii)

    Suppose \(\alpha =0\) or \(\beta =0\), or (49) holds, and define \(\Sigma \) as in (50). Then \(E\) has precisely two roots if \(\Sigma >0\) and one, if \(\Sigma \le 0\). None of these roots lies in \(B_*\).

  3. (iii)

    For \(\alpha \beta \ne 0\), \(E\) possesses a root \({\hat{\lambda }}_B\in B:=(-\infty ,\min ({\hat{\lambda }}_{{\hat{\tau }}},{\hat{\lambda }}_{{\hat{\theta }}})]\subset B_*\) if and only if \(\alpha \beta <0\), \({\hat{\lambda }}_B = \frac{3\sqrt{2}\alpha }{\sqrt{c^2 {\hat{\tau }}^2 +4}} + \frac{3\sqrt{2}\beta }{\sqrt{c^2 {\hat{\theta }}^2 + 4D^2}} \in B\), and \(\alpha \), \(\beta \) lie on the curves of the hyperbolic conic section \(\fancyscript{C}\) in the \((\alpha ,\beta )\)-plane given by \(A\alpha ^2 + a\alpha = B\beta ^2 + b\beta \) with strictly positive

    $$\begin{aligned} A&= \frac{3\sqrt{2}}{{\hat{\tau }}}, \; B= A\frac{{\hat{\tau }}}{D^2{\hat{\theta }}}, \; a = \sqrt{c^2{\hat{\tau }}^2+4}\frac{c^2{\hat{\theta }}^2+4D^2}{4{\hat{\tau }}{\hat{\theta }}D^2}, \; \\ b&= \sqrt{c^2{\hat{\theta }}^2+4D^2}\frac{c^2{\hat{\tau }}^2+4}{4{\hat{\tau }}{\hat{\theta }}D^2}. \end{aligned}$$

    Such a root of \(E\) is simple (on the principal part of the square roots).

  4. (iv)

    If \(E(\lambda )=0\) then \(p(z)=0\), where \(z=\sqrt{c^2{\hat{\tau }}^2+4({\hat{\tau }}\lambda +1)}\) and \(p(z)\) is a polynomial of degree at most 8 with real coefficients. In particular, \(E\) has at most 8 roots and for any bounded parameter set there is an \(R>0\) such that there is no root of \(E\) with modulus larger than or equal to \(R\) for parameters chosen from this set.

Proof

  1. (i)

    For \({\hat{\lambda }}\) between the singularities, exactly one of the square root terms involving \({\hat{\lambda }}\) is purely imaginary while all other terms are real. Hence, the imaginary part of \(E\) is given by the reciprocal of a square root, and as such never vanishes on its domain.

  2. (ii)

    In all cases only one square root term in \(E\) involving \({\hat{\lambda }}\) remains, and \(E=0\) can be arranged to read \({\hat{\lambda }}- \Sigma C_1/{\sqrt{C_2}} = -\Sigma C_1/{\sqrt{C_2+{\hat{\lambda }}}}\), where \(C_1,C_2>0\). Upon rescaling \(\tilde{\lambda } = {\hat{\lambda }}/C_2\) and setting \(C= C_1/C_2^{3/2}\) this becomes

    $$\begin{aligned} \tilde{\lambda }- \Sigma C = -\Sigma C/\sqrt{1+\tilde{\lambda }}. \end{aligned}$$
    (51)

    Let us consider \(\tilde{\lambda }\in \mathbb {R}\) first and note that \(C>0\). Then the left hand side of (51) is always real, but the right hand side is not for \(\tilde{\lambda }<-1\), that is, \({\hat{\lambda }}\) is in the interior of \(B\). Hence, there are no roots on \(B\). For \(\tilde{\lambda }\ge -1\), the graph of the left hand side of (51) is always a line with positive slope, which meets the graph of the right hand side at \(\tilde{\lambda }=0\). The latter is concave and unbounded for \(\Sigma >0\), which generates two intersections counted with multiplicity. In case \(\Sigma \le 0\) the only intersection point is \(\tilde{\lambda }=0\). Upon squaring both sides in the above equation and multiplying by the denominator, we find a polynomial of degree three, which means there are at most three complex roots. Dividing by \(\tilde{\lambda }\) this polynomial reduces to

    $$\begin{aligned} \tilde{\lambda }^2+ (1-2\Sigma C)\tilde{\lambda }+ (\Sigma C-2)\Sigma C=0. \end{aligned}$$

    Since roots come in complex conjugate pairs, the fact that for \(\Sigma >0\) there are two real roots already proves that a third non-real solution is not possible and so there are precisely two real ones. In the case \(\Sigma <0\) we can rule out a complex conjugate pair of roots as follows. From the quadratic equation we infer that the real part of such roots would be \(\Sigma C-1/2<0\) so that the left hand side of (51) in this root has negative real part. However, the right hand side has positive coefficient in this case and the principal branch of the square root has non-negative real parts. Hence, also the reciprocal has positive real part and so for \(\Sigma <0\) the only root is \({\hat{\lambda }}=0\).

  3. (iii)

    For \({\hat{\lambda }}\in B\) both square root terms in \(E\) involving \({\hat{\lambda }}\) are purely imaginary, and the real part of \(E\) reads

    $$\begin{aligned} {\hat{\lambda }}- \frac{3\sqrt{2}\alpha }{\sqrt{c^2 {\hat{\tau }}^2 +4}}-\frac{3\sqrt{2}\beta }{\sqrt{c^2 {\hat{\theta }}^2 + 4D^2}}, \end{aligned}$$

    which gives the claimed location of \({\hat{\lambda }}\), and since \(\mathrm {Re}(\partial _{{\hat{\lambda }}} E({\hat{\lambda }}_B)) = 1\) it is a simple root. As the imaginary part is a sum of square root reciprocals, it can only vanish if these have opposing sign, which means \(\alpha \beta <0\). Substituting the location of \({\hat{\lambda }}\) and dividing out the trivial root \({\hat{\lambda }}=0\), a straightforward calculation yields the claimed conic section.

  4. (iv)

    Set \(z^2=c^2{\hat{\tau }}^2+4({\hat{\tau }}{\hat{\lambda }}+1)\), that is, \({\hat{\lambda }}= (z^2-c^2{\hat{\tau }}^2-4)/(4{\hat{\tau }})\) and substitute this into \(E({\hat{\lambda }})\). Then, using \(z=\sqrt{z^2}\), at most one square root term involving \(z^2\) remains and \(E=0\) can be arranged so that the right hand side is a pure square root. Upon squaring, the left hand side is the sum of a rational function with numerator of degree at most 4 and denominator of degree at most 2. The right hand side is the reciprocal of an at most quadratic polynomial so that the equation can be rearranged as \(p(z)=0\) with suitable polynomial \(p\) of degree at most 8 with real coefficients. Compare (53) below. Its at most 8 complex roots are confined to a bounded region and continuous dependence of roots on the parameters gives \(R\). \(\square \)

We are now ready to prove Lemma 7. Figure 12 illustrates the setting.

Fig. 12
figure 12

Schematic illustrations in case \({\hat{\lambda }}_{{\hat{\tau }}}> {\hat{\lambda }}_{{\hat{\theta }}}\). a the conic section, \(\fancyscript{C}\), at the beginning of the homotopy (bold) and at the end (dashed), and the region where \({\hat{\lambda }}_B\not \in B\) (shaded), as well as the signs of \(\Sigma \) in the various regions. b The relevant part of \(\fancyscript{C}\), \(\fancyscript{C}^*\), at the beginning of the homotopy to (49) (bold) and at the end (dashed). The numbers of roots, \(n\), in the various regions inside the homotopy. The shaded regions have \(\alpha \beta >0\)

For parameters as in item (ii) of Lemma 15, the claim holds so that we next assume parameters with \(\alpha \beta \ne 0\) and where (49) fails, i.e., \({\hat{\lambda }}_{{\hat{\tau }}} \ne {\hat{\lambda }}_{{\hat{\theta }}}\).

The case \(\alpha \beta > 0\) . For \(\alpha \beta > 0\), take any homotopy in \((c,D,{\hat{\tau }},{\hat{\theta }})\) from the given parameters to a point for which (49) holds, for instance to \(D=1, {\hat{\tau }}={\hat{\theta }}\). Clearly, during the homotopy there is no sign change of \(\alpha \) and \(\beta \). Lemma 15, (iv), provides an \(R\) so that there is no root on \(\{|z|=R\}\) during the homotopy and Lemma 15, (i), (iii) for \(\alpha \beta >0\), imply that there are no roots on the branch cut \(B_*\) during this homotopy. Since \(\alpha \beta >0\) we readily estimate that the (finitely many) roots stay uniformly away from the singularities and hence from \(B_*\). This provides an open neighborhood \(U\) of \(B_*\) without roots during the homotopy. Now \(E\) is holomorphic in the closed simply connected set \(G = \{z\in \mathbb {C} : |z|\le R\}\setminus U\) and there are no roots on \(\partial G\) during the homotopy. Hence, the number of roots of \(E\) in \(G\) remains constant during the entire homotopy. Therefore, Lemma 15, (ii), implies that \(E\) possesses up to two roots in \(\mathbb {C}\setminus U\) also for the initially given parameters, and since there are none in \(U\) this also holds in \(\mathbb {C}\). More precisely, \(\alpha ,\beta >0\) implies \(\Sigma >0\) so that by Lemma 15, (ii), there are two roots, and \(\alpha ,\beta <0\) implies \(\Sigma <0\) so that there is one root in this case.

The case \(\alpha \beta < 0\) . If we still can choose a homotopy to a point where (49) holds such that no root approaches \(B\) during the homotopy, the argument for \(\alpha \beta > 0\) applies and it follows that there are one or two roots depending on the sign of \(\Sigma \). However, this is not possible in general. The two main points to consider are the possible gain or loss of roots when crossing \(\fancyscript{C}\) and at the endpoint of the homotopy. Let \(n\) denote the parameter dependent number of roots. The strategy in the following to show that a homotopy in \((c,D,{\hat{\tau }},{\hat{\theta }}, \alpha ,\beta )\) can be chosen so that

(a) :

\(n\) changes at most by two when crossing \(\fancyscript{C}\),

(b) :

at most one (and usually no) such crossing occurs during the homotopy,

(c) :

at the homotopy endpoint, \(n\) changes in such a way that \(1\le n\le 3\) throughout the homotopy. In fact, the precise number of roots in different regions in parameter space will be derived.

(a) Crossing \(\fancyscript{C}\) . Assume parameters lie on \(\fancyscript{C}\) so that there exists a simple and unique root \(\lambda _B\in B\) according to Lemma 15 (iii). To see all Riemann surface branches, consider the multivalued images of the square root terms and record all \(\lambda \) on the principal branch with argument in \((-\pi ,\pi ]\) upon perturbing parameters. The four sign distributions on the square roots in \(F\) yield all images \(E_\sigma ^s(\lambda ) = \lambda + F_0 +\sigma \sqrt{F_1(\lambda )} + s \sqrt{F_2(\lambda )}\), \(\sigma ,s\in \{-1,+1\}\), where we drop the \(1\)’s to ease notation so that \(E=E_+^+\) is the image on the principal branch. For \(\lambda \in B\) the square root term images in \(F\) form two complex conjugate pairs on the imaginary axis, while \({\hat{\lambda }}+F_0\) is real. Thus, \(\mathrm {Im}({E_+^+(\lambda _B)})=0\) implies \(F(\lambda _B)=0\) and hence \(E_-^-(\lambda _B)=0\), while \(E_-^+(\lambda ), E_+^-(\lambda )\ne 0\) since (49) fails. The analytic continuation of \(E_\sigma ^s\) yields continuous curves of roots (and typically a branch switching), so that the only possible roots on the principal branch for perturbations of parameters away from \(\fancyscript{C}\) are continuations of \(\lambda \) as roots of the analytic continuations of \(E_+^+\) or \(E_-^-\). But this means that typically \(n\) changes by two if the root crosses \(B\) as parameters cross \(\fancyscript{C}\). Indeed, this always occurs as implicit differentiation shows: let us write \(E\) in the compact form

$$\begin{aligned} E({\hat{\lambda }}) = {\hat{\lambda }}+ F_0 + \frac{c_1}{\sqrt{c_2+{\hat{\lambda }}}} + \frac{c_3}{\sqrt{c_4+{\hat{\lambda }}}}, \end{aligned}$$
(52)

with \(c_1=\beta /(2D\sqrt{\theta })\), \(c_2=-{\hat{\lambda }}_{{\hat{\theta }}}\), \(c_3=\alpha /(2\sqrt{{\hat{\tau }}})\), \(c_4=-{\hat{\lambda }}_{{\hat{\tau }}}\). Since \({\hat{\lambda }}_B\) is a simple root the implicit function theorem applies to its parameter dependence. Since \(c_1\ne 0\) (\(\alpha \beta <0)\) let us consider the \(c_1\)-dependence of this root; dependencies on other parameters driving off \(\fancyscript{C}\) are analogous. From

$$\begin{aligned} \mathrm {Im}(\partial _{c_1} E({\hat{\lambda }}_B)) = \frac{1}{\sqrt{-c_2-{\hat{\lambda }}_B}} \ne 0 \end{aligned}$$

it follows that roots cross the branch cut \(B\) when parameters cross \(\fancyscript{C}\), and hence move to another branch of the Riemann surface, and \(n\) changes by two.

In particular, when the initial parameters lie on \(\fancyscript{C}\) and \({\hat{\lambda }}_B\in B\), the number of roots lies between the numbers of roots near \(\fancyscript{C}\).

(b) Number of crossings of \(\fancyscript{C}\) . In order to choose a homotopy that avoids unnecessary crossings of \(\fancyscript{C}\), we normalize \(\fancyscript{C}\) by rescaling \(\hat{\beta } = \frac{b}{B}\beta \), and then \(\hat{\alpha }=\frac{b}{\sqrt{AB}}\alpha \), which changes \(\fancyscript{C}\) to \(\hat{\fancyscript{C}}\) given by \(\hat{\alpha }^2 + \hat{a} \hat{\alpha } = \hat{\beta }^2 + \hat{\beta }\) with

$$\begin{aligned} \hat{a} = \frac{a}{b}\sqrt{\frac{B}{A}} = \sqrt{\frac{c^2{\hat{\theta }}^2+4D^2}{c^2{\hat{\tau }}+4}\frac{{\hat{\tau }}}{D^2{\hat{\theta }}}} = \sqrt{\frac{c^2/D^2+4/{\hat{\theta }}}{c^2+4/{\hat{\tau }}}} = \sqrt{\frac{{\hat{\lambda }}_{{\hat{\theta }}}}{{\hat{\lambda }}_{{\hat{\tau }}}}}. \end{aligned}$$

Note that a homotopy to a point that satisfies (49), i.e., \({\hat{\lambda }}_{{\hat{\theta }}} = {\hat{\lambda }}_{{\hat{\tau }}}\), is a homotopy of \(\hat{a}\) to \(1\). The formula for \(\hat{a}\) yields a homotopy so that \(\hat{a}\) monotonically grows or decays to \(1\), e.g., changing \({\hat{\tau }}\) only. This homotopy yields a homotopy in \(\{\alpha \beta \ne 0\}\) such that \((\hat{\alpha },\hat{\beta })\) remain unchanged during the homotopy that changes \(\hat{a}\). Since a fixed \((\hat{\alpha },\hat{\beta })\) lies on \(\hat{\fancyscript{C}}\) at a unique value of \(\hat{a}\), it follows that there is at most one crossing of \(\hat{\fancyscript{C}}\) during the combined homotopy, which will be the homotopy in the following.

In fact, only the branch of \(\fancyscript{C}\) contained in \(\{\Sigma <0\}\) is relevant at all, and there is no crossing for \(\Sigma >0\) as shown next.

The relevant component of \(\fancyscript{C}: \fancyscript{C}^*\) . Let us check the condition \({\hat{\lambda }}_B\in B\), that is, \({\hat{\lambda }}_B<\min \{{\hat{\lambda }}_{{\hat{\tau }}},{\hat{\lambda }}_{{\hat{\theta }}}\}\) from Lemma 15 (iii) on \(\fancyscript{C}\). Substituting the formulas for these quantities, we readily compute that the \((\hat{\alpha }, \hat{\beta })\)-value for \({\hat{\lambda }}_B=\min \{{\hat{\lambda }}_{{\hat{\tau }}},{\hat{\lambda }}_{{\hat{\theta }}}\}\) lies on one branch of \(\fancyscript{C}\). More precisely, for \({\hat{\lambda }}_{{\hat{\tau }}}<{\hat{\lambda }}_{{\hat{\theta }}}\) this point lies at \(\hat{\alpha }=0\), \(\hat{\beta }=-1\) and for \({\hat{\lambda }}_{{\hat{\tau }}}>{\hat{\lambda }}_{{\hat{\theta }}}\) at \(\hat{\alpha }=-\hat{a}\), \(\hat{\beta }=0\). Since \({\hat{\lambda }}_B\) cannot cross a singularity as \((\hat{\alpha },\hat{\beta })\) move on a curve of \(\fancyscript{C}\), the line \(\{{\hat{\lambda }}_B={\hat{\lambda }}_{{\hat{\tau }}}\}\) is tangent to \(\fancyscript{C}\) at the intersection point; compare Fig. 12(a). Due to the conic section geometry, the line does not cross the other branch of \(\fancyscript{C}\). Since increasing \(\hat{\alpha }\) means increasing \({\hat{\lambda }}_B\), the other branch of \(\fancyscript{C}\) has \({\hat{\lambda }}_B\not \in B\). Therefore, always only the branch of \(\fancyscript{C}\) in the region \(\{\Sigma < 0\}\) is relevant; denote this by \(\fancyscript{C}^*\).

The location of \(\fancyscript{C^*}, \mathrm {sgn}(\Sigma )\) and the set \(\Sigma _0\) . To show \(\fancyscript{C}^*\) is contained in \(\{\Sigma <0\}\), consider the sign of \(\Sigma \) during the homotopy. It follows from (50) that

$$\begin{aligned} \mathrm {sgn}(\Sigma ) = \mathrm {sgn}\left( \sqrt{A}\alpha + \sqrt{B}\beta \right) = \mathrm {sgn}(\hat{\Sigma }),\, \hat{\Sigma }:=\hat{\alpha }+ \hat{\beta }. \end{aligned}$$

Together with the fact that \(\hat{\fancyscript{C}}\) can be written as

$$\begin{aligned} \hat{\alpha }= -\frac{\hat{a}}{2} \pm \sqrt{\frac{\hat{a}^2}{4} + \hat{\beta }^2 + \hat{\beta }}, \end{aligned}$$

we readily infer that \(\Sigma =0\), i.e., \(\hat{\alpha }= -\hat{\beta }\), requires \(\hat{a} = -1\), which is outside the homotopy range. Therefore, \(\{\Sigma =0\}\) is disjoint from \(\fancyscript{C}\) during the homotopy (recall that \(\fancyscript{C}\) and also \(\hat{\fancyscript{C}}\) change during the homotopy). Hence, since either \((\hat{\alpha },\hat{\beta }) = (-\hat{a}, 0)\) or \((0,-1)\) lies on \(\hat{\fancyscript{C}}^*\) and has \(\Sigma <0\), it holds that \(\Sigma <0\) on the entire \(\fancyscript{C}^*\) in \(\{\hat{\alpha }\hat{\beta }<0\}\) and during the homotopy. In conclusion, crossing \(\hat{\fancyscript{C}}^*\) in case \(\alpha \beta <0\) requires \(\Sigma <0\). Moreover, since the homotopy makes a locally bounded monotone change in \(\hat{\fancyscript{C}}\), there is an open region within \(\{\hat{\Sigma }<0\}\) that is disjoint from all \(\hat{\fancyscript{C}}\) during the homotopy. Let \(\Sigma _0\) denote this set; compare Fig. 12(b).

(c) Change of \(n\) at the homotopy endpoint. Next to crossing \(\fancyscript{C}^*\), the only possible change in the number of roots are bifurcations from the branch point at the end of the homotopy. (Due to Lemma 15, this cannot happen in the interior of \(B_*\), and roots remain away from the disjoint singularities inside the homotopy.) Recall that the singularities, \({\hat{\lambda }}_{{\hat{\tau }}}, {\hat{\lambda }}_{{\hat{\theta }}}\), merge into the branch point at the end of the homotopy, where (49) holds. Consider the form (52) of \(E\) with \(c_4=c_2+\delta \ge c_2\) so that in case \({\hat{\lambda }}_{{\hat{\tau }}}< {\hat{\lambda }}_{{\hat{\theta }}}\) we have \(c_1=\beta /(2D\sqrt{\theta })\), \(c_3=\alpha /(2\sqrt{{\hat{\tau }}})\) and \(c_2={\hat{\lambda }}_{{\hat{\theta }}}\), \(\delta ={\hat{\lambda }}_{{\hat{\theta }}}- {\hat{\lambda }}_{{\hat{\tau }}}\). Otherwise \(c_1\) and \(c_3\) are interchanged and \(c_2={\hat{\lambda }}_{{\hat{\theta }}}\). Note that always \(c_1+c_3=\Sigma \). Since we are concerned only with roots approaching \(-c_2\), we write \({\hat{\lambda }}= w^2-c_2\), \(w\in \mathbb {C}\), which gives

$$\begin{aligned} F(w^2-c_2) = \left( \frac{c_1}{w} + \frac{c_3}{\sqrt{w^2+\delta }}\right) . \end{aligned}$$

The full Evans function in this form, denoted by \(E_\delta \), reads \(E_\delta (w^2)=w^2-c_2+F_0 + F\) and roots satisfy

$$\begin{aligned} w^2-c_2+F_0 +\frac{c_1}{w} = - \frac{c_3}{\sqrt{w^2+\delta }}. \end{aligned}$$

Upon squaring and multiplication by denominators this yields the polynomial equation

$$\begin{aligned} P_\delta (w):=(w^3+(F_0-c_2)w+c_1)^2(w^2+\delta )-c_3^2 w^2=0. \end{aligned}$$
(53)

We readily compute that \(P_0(0)=0\), \(\partial _w P_0(0) = 0\), and \(\partial _w^2 P_0(0) = 2(c_1^2-c_3^2)\). Hence, in \(\{\alpha \beta <0\}\) we have \(\partial _w^2 P_0(0)=0\) if and only if \(\Sigma =0\), which means that in \(\{\alpha \beta <0, \Sigma \ne 0\}\) precisely two roots of \(P_\delta \) bifurcate from \(w=0\) at \(\delta =0\). (Here we suppressed the dependence of \(c_j\) on \(\delta \), which is irrelevant for the argument.)

However, these bifurcating roots are not necessarily roots of \(E_\delta \). Indeed, consider the bifurcation of real roots, which, away from \(\fancyscript{C}^*\), is necessarily for \(z=w^2>0\). Here the graph of \(E_\delta \) converges locally uniformly to that of \(E_0\) as \(\delta \rightarrow 0\). See Fig. 13(a). We readily check that \(\partial _z^2 E_\delta \) has a unique positive root for \(\delta >0\), so that the only option for a change in the number of roots is a change by one through a sign change of the asymptotics as \(z\rightarrow 0\). If \(\delta >0\) we have \(\mathrm {sgn}(F) = \mathrm {sgn}(c_1)\), while \(\mathrm {sgn}(F) = \mathrm {sgn}(c_1+c_3)\) for \(\delta =0\). Hence, if \(\mathrm {sgn}(c_1) = \mathrm {sgn}(c_1+c_3)\) no real roots bifurcate, while for \(\mathrm {sgn}(c_1) \ne \mathrm {sgn}(c_1+c_3)\) one real root is lost as \(\delta \rightarrow 0\).

Which case occurs in terms of the original parameters depends on the ordering of singularities. For \({\hat{\lambda }}_{{\hat{\tau }}}< {\hat{\lambda }}_{{\hat{\theta }}}\) (\({\hat{\lambda }}_{{\hat{\tau }}}> {\hat{\lambda }}_{{\hat{\theta }}}\)) a change of \(n\) by one occurs when \(\mathrm {sgn}(\alpha ) \ne \mathrm {sgn}(\Sigma )\) (\(\mathrm {sgn}(\beta ) \ne \mathrm {sgn}(\Sigma )\)) and otherwise \(n\) remains constant or changes by two. Therefore, \(n\) can change by two at the end of the homotopy only in the component of \(\{\alpha \beta <0, \Sigma \ne 0\}\) that contains \(\fancyscript{C}^*\). Compare Fig. 12(b).

Counting roots. Let us now gather what we learned about the number of roots \(n\): in the quadrant of \(\{\alpha \beta <0\}\) containing the irrelevant branch of the conic section, \(\fancyscript{C}\setminus \fancyscript{C}^*\), \(n\) can only change at the end of the homotopy, where \(n=2\) for \(\Sigma > 0\) and \(n=1\) for \(\Sigma <0\). Since such a change is at most by one, it follows that inside the homotopy (i.e. without the endpoint) \(n=2\) in this quadrant of \(\{\alpha \beta <0\}\).

Within the quadrant of \(\{\alpha \beta <0\}\) containing \(\fancyscript{C}^*\), let \(\Sigma ^-\) denote the subset where \(\Sigma <0\) and \(\Sigma ^+\) where \(\Sigma >0\). In \(\Sigma ^+\), \(n\) changes at the end of the homotopy, where it drops by one, so that inside the homotopy \(n=3\), and this extends to \(\{\Sigma =0\}\). In \(\Sigma ^-\) we have \(n=1\) at the end of the homotopy from a possible drop by one or two, so that \(1\le n \le 3\) inside the homotopy.

In conclusion, \(1\le n \le 3\) for all parameter values. More precisely, the global distribution of \(n\) in parameter space is as follows; see Fig. 13(b): \(n\) must be constant (inside the homotopy) in the connected components of \(\{\hat{\alpha }\hat{\beta }<0\}\setminus \fancyscript{C}^*\). Since crossing \(\{\Sigma =0\}\) does not change the number of roots we have \(n=3\) in the component containing \(\Sigma ^+\). Since crossing \(\fancyscript{C}^*\) changes \(n\) by two it follows that \(n=1\) in the component containing \(\Sigma _0\).

Fig. 13
figure 13

In both panels \(n\) in brackets is the value at the end of the homotopy, if different from inside the homotopy in the given region. a Sketch of the graphs of \(E_\delta \) for \(\delta >0\) (solid) and \(\delta =0\) (dashed) illustrating the loss of one real root into the branch point \(z=0\), which occurs for \(\mathrm {sgn}(\beta ) \ne \mathrm {sgn}(\Sigma )\) if \({\hat{\lambda }}_{{\hat{\tau }}}> {\hat{\lambda }}_{{\hat{\theta }}}\), and for \(\mathrm {sgn}(\alpha ) \ne \mathrm {sgn}(\Sigma )\) if \({\hat{\lambda }}_{{\hat{\tau }}}< {\hat{\lambda }}_{{\hat{\theta }}}\). b The distribution of \(n\) in case \({\hat{\lambda }}_{{\hat{\tau }}}> {\hat{\lambda }}_{{\hat{\theta }}}\) with curves and regions are as in Fig. 12(b)

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Chirilus-Bruckner, M., Doelman, A., van Heijster, P. et al. Butterfly Catastrophe for Fronts in a Three-Component Reaction–Diffusion System. J Nonlinear Sci 25, 87–129 (2015). https://doi.org/10.1007/s00332-014-9222-9

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Keywords

  • Three-component reaction–diffusion system
  • Front solution
  • Geometric singular perturbation theory
  • Evans function
  • Center manifold reduction

Mathematics Subject Classification

  • 35K55
  • 35B25
  • 35B35
  • 35B32