KdV approximation conjecture for generalized moving pulse solutions in the FPU model

Let $m\in\mathbb{N}$ and $\tilde{c}>0$. For sufficiently small $\varepsilon>0$, consider the FPU system and the speed $c=1-\varepsilon^2\tilde{c}$. A generalized moving pulse solution of permanent form is a solution $q_n(t)=v(n-ct)$ with a smooth profile $v\colon[-1/\varepsilon^m,1/\varepsilon^m]\to\mathbb{R}$. KdV approximation conjecture. For every $m\in\mathbb{N}$ and $\tilde{c}>0$, there exist $C,\varepsilon_0>0$ such that, for every $\varepsilon\in(0,\varepsilon_0)$, the FPU system possesses such a solution with

$$\sup_{|\xi|<1/\varepsilon^m}|v(\xi)-h(\xi)|\leq C\varepsilon^m,$$

where $h\colon\mathbb{R}\to\mathbb{R}$ is smooth,

$$\lim_{\xi\to\pm\infty}h(\xi)=0,$$

and

$$\sup_{\xi\in\mathbb{R}}\left|h(\xi)-\varepsilon^2A_{\mathrm{sol},\tilde{c}}(\varepsilon\xi)\right|\leq C\varepsilon^3.$$

Here $A_{\mathrm{sol},\tilde{c}}$ is the solitary wave of the associated KdV equation with speed $\tilde{c}>0$. This is the KdV solitary-wave prediction near the speed of sound; the supplied text gives the conjectural existence and approximation statement but no resolution status.