Ebert-Van Saarloos [@EVS:2000], the large-$t$ corrections to this shape are of the form $1/t$ (there is no term of order $1/\sqrt{t}$). All these features should not depend on whether we address Eq. (\[eq:FKPPphi\]) or Eq. (\[eq:tipcutoffsmooth\]) above, except for the signs of $\alpha$ and $\xi$.

We write (t,x)=e\^[--z]{}g(t,z) , where $z=\frac{\xi^2}{4t}$, and the following ansatz are taken:

&=x-2t+32t++\
g(t,z)&= g\_[-12]{}(z)+g\_0(z)+

\[eq:ansatzg\] The variable $\xi$ may be positive in the linear domain (it is the case for the usual Ebert-Van Saarloos solution) or negative: Therefore, we write $\xi=\varepsilon\sqrt{4t\,z}$, where $\varepsilon=\pm 1$. The ansatz for the front position contained in $\xi$ already incorporates the two known [@B:1983] dominant terms at large $t$, namely $2t-\frac32\ln t$. The $-\frac{2c}{\sqrt{t}}$ term was new in Ref. [@EVS:2000].

Thanks to these ansatz, the original equation splits into a hierarchy of equations for the functions $g$. The first two equations of this set read

&z g\_[-12]{}\^+(12-z) g\_[-12]{}\^+12 g\_[-12]{}=0 ,\
&z g\_[0]{}\^+(12-z) g\_[0]{}\^+g\_[0]{}= c g\_[-12]{}-32 (g\_[-12]{}-g\^\_[-12]{}) .

\[eq:hierarchy\] The first equation of the hierarchy is the Kummer equation z+(b-z)-aw=0 with $w=g_{-\frac12}$, $a=-\frac12$ and $b=\frac12$. Two independent solutions are, for example, the two Kummer functions (or ${}_1F_1$ hypergeometric functions) M(a,b,z)  z\^[1-b]{}M(a-b+1,2-b,z) namely, in our case, M(-12,12,z)     M(0,32,z). The latter is 