Consider the following linear wave equation. $$ u_t+cu_x-\gamma u_{xx}+\delta u_{xxx}=0 $$
If we know the following initial data, $$ u(x,0)= 3\cos^2(x)+\sin(x) $$ how to get an explicit solution?
I know that the general solution is: $$ v(x,t)=A\exp( ik[x-(c-\delta k^2)t] )\exp(-\gamma k^2t) $$
Even if I compare $u(x,0)=v(x,0)$, I can not solve it.
Assume (for simplicity) that our function is defined on $\mathbb{R}$ with no specific boundary conditions. So there is no restriction on $k$. Hence $k\in \mathbb{R}$ and the general solution is $u(x,t)=\int_{\mathbb{R}}A(k)v_{k}(x,t)dk$ (superposition of modes numbered by $k$). So, $u(x,0)$ is given by $\int_{\mathbb{R}}A(k)e^{ikx}dk$. We can use Fourier transform to calculate $A(k)$ and use it to find solution of our equation with respect to given initial conditions.