= Translation-mode solvability for a bistable front
{title2=$c_1=-\int U_0\prime h(U_0)/\int(U_0\prime)^2$}
Suppose $u_t=Du_{xx}+f_0(u)+\varepsilon h(u)$ has a balanced stationary front $U_0(z)$, where $z=(x-q)/\sqrt D$ and $U_0''+f_0(U_0)=0$. Take $\dot q/\sqrt D=\varepsilon c_1+\cdots$ and $u=U_0+\varepsilon U_1+\cdots$. At first order, $LU_1=-c_1U_0'-h(U_0)$ with $L=d^2/dz^2+f_0'(U_0)$. Differentiating the front equation proves $LU_0'=0$. Integration by parts against the decaying <translation> mode gives
$$
c_1=-\frac{\int_{-\infty}^\infty U_0'h(U_0)\,dz}{\int_{-\infty}^\infty(U_0')^2\,dz}.
$$
This is the <Fredholm solvability condition> determining the velocity, derived from the <self-adjoint differential operator> rather than assumed. An orthogonality condition or a fixed midpoint removes the freedom to add a multiple of $U_0'$ to $U_1$. For the <bistable cubic reaction-diffusion equation>, $h=-aU_0(1-U_0)$ gives $c_1=-\sqrt2a$, and its first-order forcing vanishes pointwise, so the phase-fixed shape correction is zero.
Back to article page
