Consider the reaction diffusion equation $u_t=u_{xx}+f(u)$ and a travelling wave solution $V(x,t)=\phi(\xi)$ with $\xi=x-ct$. Then $V$ is an equilibrium of $$ u_t=u_{\xi\xi}+cu_\xi+f(u). $$ Linearizing in $V$ gives the linear operator $$ L_c:=\partial_{\xi\xi}+c\partial_\xi+f'(V) $$ which is self-adjoint if and only if $c=0$.
Making the change of variables $u(\xi,t)=e^{c/2}v(\xi,t)$ gives the self-adjoint operator $$ \tilde{L}_c:=\partial_{\xi\xi}+f'(V)-\frac{c^2}{4}. $$
The kernel of $\tilde{L}_c$ is spanned by $e^{c/2}V'$.
I would like to prove that a solution $u(\xi,t)$ can be decomposed into the sum $$ u(\xi,t)=\underbrace{V(x-ct-q_1(t))}_{=:V_1(\xi,t)}+\underbrace{V(x-ct-q_2(t))}_{=:V_2(\xi,t)}+r(\xi,t) $$ where $r(\xi,t)$ suffices the orthogonality conditions $$ \int_\mathbb{R}e^{c\xi}r(\xi,t)V'(\xi-q_1(t))\, d\xi=\int_\mathbb{R}e^{c\xi}r(\xi,t)V'(\xi-q_2(t))\, d\xi=0. $$
The initial condition is $$ u(x,0)=V(x-q_1(0))+V(x-q_2(0)). $$
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The hint I was given was to use the Implicit Function Theorem for $(q_1,q_2)$. But I dont know exactly how to do so.
Hint: Start by defining the functionals $$ Y_1(q_1,q_2,u):=\int_\mathbb{R}e^{c\xi}(u(\xi,t)-V_1(\xi,t)-V_2(\xi,t))V'(\xi-q_1)d\xi, $$ $$ Y_2(q_1,q_2,u):=\int_\mathbb{R}e^{c\xi}(u(\xi,t)-V_1(\xi,t)-V_2(\xi,t))V'(\xi-q_2)d\xi. $$ Then you have to prove that $Y_1(0,0,V_1+V_2)=Y_2(0,0,V_1+V_2)=0$. Once you have all of this, you have to compute $$ \frac{\partial Y_1}{\partial q_1}, \quad \frac{\partial Y_1}{\partial q_2}, \quad \frac{\partial Y_2}{\partial q_1}, \quad \frac{\partial Y_2}{\partial q_2}. $$ Finally, prove that the matrix $$ \left(\dfrac{\partial Y_i}{\partial q_j}\right)(0,0,V_1+V_2),$$ is an invertible matrix*. The last step is just to correctly define the modulation-parameters considering the factor $-ct$. For this, just notice that on the previous computation you could add an arbitrary traslation parameter $q_3\in\mathbb{R}$ inside $V_1,V_2,V'$ on the definition of the functionals $Y_i$, which does not affect any of the previous computations.
Please don't hesitate to ask me if you have any doubt.
*Edit: Note that at this point you are already able to apply the Implicit Function Theorem. The last part is just to write rigorously that you can do this for every time $t\in\mathbb{R}$. Roughly, you are just modulating at each time.