In each of these cases, the dimension of the arrival manifold $\mathcal{N}$ is unchanged, and as a consequence, the difference $\dim(\mathcal{M}) - \text{codim}(\mathcal{W})$ is exactly decreased by 1. More precisely, since only bistable travelling fronts and standing pulses stable at infinity are considered, this difference is actually exactly equal to $-1$. Assumption 1 of Theorem 4.2 is therefore still satisfied.

Concerning assumption 2 of Theorem 4.2, it is also fulfilled in each of these cases, due to the key following observation: in the proof of each of the three lemmas proving that this assumption holds (Lemmas 5.7, 6.3 and 7.5), the freedom provided by the variables $b^u$ and $b^s$ is not used — only the freedom provided by the time variable $\xi$ and by the potential $V$ are. As a consequence, the fact that the unstable manifold of $E_-(V)$ is replaced by its strongly unstable manifold does not affect the validity of the conclusion of the lemma, and neither does the fact that the stable manifold of $E_+(V)$ is replaced by its strongly stable manifold. In other words, the key assumption 2 of Theorem 4.2 still holds.

In each case and for each of the two replacements 1 and 2, the conclusions of Theorem 4.2 thus still hold, and ensure that, locally generically with respect to $V$, the profiles of travelling fronts or of (a)symmetric standing pulses locally correspond to transverse intersections between the image of $m \mapsto \Phi(m, V)$ and $\mathcal{W}$ in $\mathcal{N}$. But the fact that $\dim(\mathcal{M}) - \text{codim}(\mathcal{W})$ is now equal to $-1$ actually precludes the very existence of such transverse intersections. In other words, locally generically with respect to $V$, profiles of bistable travelling fronts or of (a)symmetric pulses stable at infinity approaching their limit at $-\infty$ through its strongly stable manifold or their limit at $+\infty$ through its strongly stable manifold do simply (locally) not exist, which is the intended conclusion. The emptiness of such a transverse intersection due to the value $-1$ of the difference $\dim(\mathcal{M}) - \text{codim}(\mathcal{W})$ is illustrated by Figure 10.2.

<figure><img src="image_12.png" /><figcaption>Figure 10.2: Whereas the sum of the dimensions of $W^u(E_-)$ and $W^s(E_+)$ has the minimal value for a nonempty transverse intersection between these two manifolds to exist, for $W^{su}(E_-)$ and $W^s(E_+)$ (or for $W^u(E_-)$ and $W^{ss}(E_+)$) this sum is smaller, so that a transverse intersection between such manifolds must be empty. This figure actually depicts the intersection defining a transverse asymmetric bistable standing front, but the same principle applies for bistable travelling fronts, and elementary symmetric (or transverse asymmetric) standing pulses that are stable at infinity.</figcaption></figure>

The remaining arguments, ensuring the first extension to global statements for potentials quadratic past a certain radius (subsections 5.2, 6.1 and 7.1), and then the second extension to general potentials (subsections 9.1 and 9.2), are unchanged.
To complete these arguments, a few milestones of the proof for travelling fronts are detailed in subsection 10.5 below.

## 10.4 Proof of conclusion 1 of Theorem 1.8

Let $R$ denote a positive quantity, let us recall the notation $V_{quad-R}$ introduced in (4.2) and $V_{quad-R-Morse}$ introduced in (5.2), and let us consider the set

$$ V_{quad-R-Morse-ss-eig} = \{V \in V_{quad-R-Morse} : \text{ at every minimum point of } V, \\ \text{the smallest eigenvalue of } D^2V \text{ is simple}\} $$

(the subscript “ss-eig” stands for “simple smallest eigenvalue”).

**Proposition 10.2.** The set $V_{quad-R-Morse-ss-eig}$ is a dense open subset of $V_{quad-R-Morse}$ (and thus of $V_{quad-R}$).

*Proof.* Openness follows from the continuity of the roots (eigenvalues of $D^2V$ at a minimum point) of a polynomial with respect to its coefficients. To prove the density, let $V$ be in $V_{quad-R-Morse}$, and let us assume that there exists a minimum point $e$ of $V$ such that the smallest eigenvalue $\mu_1$ of $D^2V(e)$ is not simple. Let $\delta$ denote a positive quantity, small enough so that the closed ball $\bar{B}_{\mathbb{R}^d}(e, \delta)$ is included in $\bar{B}_{\mathbb{R}^d}(0_{\mathbb{R}^d}, R)$ and contains no critical point of $V$ but $e$. Let $\rho$ denote a smooth function $[0, +\infty) \to \mathbb{R}$ satisfying

$$ \rho(r) = 1 \quad \text{for } r \text{ in } [0, 1/2] \quad \text{and} \quad \rho(r) = 0 \quad \text{for } r \text{ in } [1, +\infty), $$

and let $\varepsilon$ denote a small positive quantity to be chosen below. Let $u_1$ denote an unit eigenvector of $D^2V(e)$ associated to $\mu_1$, and let us consider the perturbed potential $V_{pert}$ defined as:

$$ V_{pert}(u) = V(u) - \frac{\varepsilon}{2}((u - e) \cdot u_1)^2 \rho(|u - e|/\delta). $$

Then, $e$ is still a critical point of $V_{pert}$ and, for every $v$ in $\mathbb{R}^d$,

$$ D^2V_{pert}(e)(v, v) = D^2V(e)(v, v) - \varepsilon(v \cdot u_1)^2. $$

As a consequence, $u_1$ is still an eigenvector of $D^2V_{pert}(e)$, the corresponding eigenvalue $\mu_1 - \varepsilon$ is simple, and the other eigenvalues of $D^2V_{pert}(e)$ are the same as those of $D^2V(e)$ (the difference $D^2V_{pert}(e) - D^2V(e)$ vanishes on the orthogonal subspace to $u_1$ in $\mathbb{R}^d$), these other eigenvalues are therefore greater than $\mu_1 - \varepsilon$. In addition, if $\varepsilon$ is small enough, then $\mu_1 - \varepsilon$ is positive (so that $e$ is still a minimum point of $V_{pert}$) and the closed ball $\bar{B}_{\mathbb{R}^d}(e, \delta)$ contains no critical point of $V_{pert}$ but $e$. The same procedure, repeated for each minimum point of $V$ such that the smallest eigenvalue of $D^2V$ at this minimum point is not simple, provides an arbitrarily small perturbation of $V$ belonging to the set $V_{quad-R-Morse-ss-eig}$, and therefore proves the intended density. $\square$

Let $V_{Morse-ss-eig}$ denote the subset of $V_{full}$ containing Morse potentials $V$ such that, at every minimum point of $V$, the smallest eigenvalue of the Hessian $D^2V$ at this minimum point is simple. Proceeding as in subsection 9.3, the same arguments as in the proof of Proposition 10.2 above show that this set $V_{Morse-ss-eig}$ is a generic subset of $V_{full}$, which proves conclusion 1 of Theorem 1.8.
## 10.5 Proof of conclusion 2 of Theorem 1.8 for bistable travelling fronts

The aim of this subsection is to complete the idea of the proof of conclusion 2 of Theorem 1.8 provided in subsection 10.3 with a few milestones of this proof, in the case of travelling fronts (only).

As for conclusion 1 of Theorem 1.7, the first goal is to prove the intended conclusion among potentials that are quadratic past a certain (positive) radius $R$. This is stated by the following proposition, which is an extension of Proposition 5.1. It calls upon the notation $\mathcal{F}_V$ introduced in (5.1).

**Proposition 10.3.** There exists a generic subset of $\mathcal{V}_{\text{quad-R}}$, included in $\mathcal{V}_{\text{quad-R-Morse-ss-eig}}$ such that, for every potential $V$ in this subset, every travelling front $(c, u)$ in $\mathcal{F}_V$ is transverse, bistable, and its profile $u$ approaches its limit at $+\infty$ ($-\infty$) tangentially to the eigenspace corresponding to the smallest eigenvalue of $D^2V$ at this point.

### 10.5.1 Reduction to a local statement

Let $V_0$ denote a potential function in $\mathcal{V}_{\text{quad-R-Morse-ss-eig}}$, and let $e_{-,0}$ and $e_{+,0}$ denote non-degenerate minimum points of $V_0$. Let us consider the neighbourhood $\nu_{\text{robust}}(V_0, e_{-,0}, e_{+,0})$ of $V_0$ introduced in subsection 5.2, and let us denote by $\tilde{\nu}_{\text{robust}}(V_0, e_{-,0}, e_{+,0})$ the intersection $\nu_{\text{robust}}(V_0, e_{-,0}, e_{+,0}) \cap \mathcal{V}_{\text{quad-R-Morse-ss-eig}}$. The following proposition is a variant (extension in the case of bistable travelling fronts) of Proposition 5.2. The notation is similar, except for the “tilde” added to the symbols of the various sets, in order to differentiate them for the corresponding sets introduced in Proposition 5.2.

**Proposition 10.4.** For every positive speed $c_0$, there exist a neighbourhood $\tilde{\nu}_{V_0, e_{-,0}, e_{+,0}, c_0}$ of $V_0$ in $\mathcal{V}_{\text{quad-R}}$, included in $\tilde{\nu}_{\text{robust}}(V_0, e_{-,0}, e_{+,0})$, a neighbourhood $\tilde{C}_{V_0, e_{-,0}, e_{+,0}, c_0}$ of $c_0$ in $(0, +\infty)$, and a generic subset $\tilde{\nu}_{V_0, e_{-,0}, e_{+,0}, c_0, \text{gen}}$ of $\tilde{\nu}_{V_0, e_{-,0}, e_{+,0}, c_0}$ such that, for every $V$ in $\tilde{\nu}_{V_0, e_{-,0}, e_{+,0}, c_0, \text{gen}}$, every front travelling at a speed $c$ in $\tilde{C}_{V_0, e_{-,0}, e_{+,0}, c_0}$ and connecting $e_-(V)$ to $e_+(V)$, for the potential $V$, is transverse and its profile $u$ approaches its limit at $+\infty$ ($-\infty$) tangentially to the eigenspace corresponding to the smallest eigenvalue of $D^2V$ at this point.

*Proof that Proposition 10.4 yields Proposition 10.3.* Proposition 5.1 already ensures the existence of a generic subset $\mathcal{V}_{\text{quad-R-}\hat{\text{H}}-\mathcal{F}}$ of $\mathcal{V}_{\text{quad-R}}$ such that, for every potential function $V$ in this subset, every travelling front $(c, u)$ in $\mathcal{F}_V$ is transverse. According to the arguments of subsection 9.4, such a front is necessarily bistable. Thus, only the conclusion of Proposition 10.4 relative to the asymptotic behaviour of the profile remains to be proved.

To this end, the arguments are the same as in subsection 5.2. We may introduce the sets $\tilde{\nu}_{V_0, c_0}$ and $C_{V_0, c_0}$ and $\nu_{V_0, c_0, \text{gen}}$, defined exactly as the corresponding sets (without tilde) in (5.3) (with $\nu_{\text{robust}}(V_0, e_{-,0}, e_{+,0})$ replaced with $\tilde{\nu}_{\text{robust}}(V_0, e_{-,0}, e_{+,0})$), and the same remaining arguments (replacing $\mathcal{V}_{\text{quad-R-Morse}}$ with $\mathcal{V}_{\text{quad-R-Morse-ss-eig}}$) show the existence of a generic subset of $\mathcal{V}_{\text{quad-R}}$, included in $\mathcal{V}_{\text{quad-R-Morse-ss-eig}}$, such that, for every potential $V$ in this subset, every bistable travelling front $(c, u)$ in $\mathcal{F}_V$ is transverse and its profile $u$ approaches its limit at both ends of $\mathbb{R}$ according to the intended conclusion. Intersecting
this generic subset with the one provided by Proposition 5.1 provides a generic subset of $V_{\text{quad-R}}$ for which all conclusions of Proposition 10.3 hold. $\square$

### 10.5.2 Proof of the local statement

The proof of Proposition 10.4 may be derived from the proof of Proposition 5.2, up to a few changes and thanks to some key arguments, all of which are exposed in subsection 10.3 above.

### 10.5.3 Extension to all potentials

The extension to all potentials is obtained by applying the same strategy as in subsection 9.1. Let us recall the notation $\mathcal{F}_{V,R}$ introduced in (9.1). The same arguments as in subsection 9.1 show that the intended extension is a consequence of the following extension of Proposition 9.1.

**Proposition 10.5.** For every positive quantity $R$, there exists a generic subset

$V_{\text{full-}\hat{\eta}-\mathcal{F}\text{-min-rate-R}}$ of $V_{\text{full}}$, included in $V_{\text{Morse-ss-eig}}$, such that, for every potential $V$ in this subset, every travelling front $(c, u)$ in $\mathcal{F}_{V,R}$ is transverse, bistable, and approaches its limit at $+\infty$ ($-\infty$) tangentially to the eigenspace corresponding to the smallest eigenvalue of $D^2V$ at this point.

*Proof.* Proposition 9.1 already provides a generic subset $V_{\text{full-}\hat{\eta}-\mathcal{F}}$ of $V_{\text{full}}$ such that, for every potential $V$ in this subset, every travelling front $(c, u)$ in $\mathcal{F}_{V,R}$ is transverse, and therefore bistable (subsection 9.4). Therefore, only the conclusion relative to the asymptotic behaviour of the profiles remains to be proved.

The proof of this conclusion is a variation of the proof of Proposition 9.1 and follows the ideas exposed in subsection 10.3: for some potential $V_0$ in $V_{\text{quad-(}R+1)}$ and for some non-degenerate minimum points $e_{-,0}$ and $e_{+,0}$ of $V$, and for every nonnegative integer $N$, two variants of the set $M_N$ defined in (9.5) (and of the open subset $O_{V_0, e_{-,0}, e_{+,0}, c_0, N}$ defined in (9.6)) can be introduced: one where $B^u$ is replaced by $B^{\text{su}}$, and one where $B^s$ is replaced by $B^{\text{ss}}$. In each of these two cases, the condition “$\Phi(M_N, V)$ is transverse to $W$ in $N$” can be read as “the intersection between $\Phi(M_N, V)$ and $W$ is empty”, due to the missing dimension induced by the change in each of these variants. Then, replacing the open subset $O_{V_0, e_{-,0}, e_{+,0}, c_0, N}$ by the intersection of its two variants, the remaining arguments are exactly the same. This proves Proposition 10.5 (and therefore also completes the proof of conclusion 2 of Theorem 1.8 for bistable travelling fronts). $\square$

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### Romain JOLY

Université Grenoble Alpes, CNRS UMR 5582, Institut Fourier,
38000 Grenoble, France.

romain.joly@univ-grenoble-alpes.fr

### Emmanuel RISLER

Université de Lyon, INSA de Lyon, CNRS UMR 5208, Institut Camille Jordan,
F-69621 Villeurbanne, France.

emmanuel.risler@insa-lyon.fr