Suppose I have a nonlinear second order Cauchy PDE $\dfrac{\partial p(x,t)}{\partial t}=N(p(x,t))$, where $N:L^2(\mathbb{R})\rightarrow L^2(\mathbb{R})$, and a known fixed point $u(x)$. Mathematically, the only boundary condition known is that any solution $p$ decays to 0 as ${x\rightarrow\pm \infty}$.

From physical reasoning however, it is known that $u$ decays rapidly away from $x=0$, and hence, I am able to solve for $u$ numerically on a finite domain by prescribing zero Dirichlet boundary conditions on a finite domain around 0 (i.e, on $[-L,L]$, where $L$ is chosen judiciously).

Now, I want to linearize $N$ around $u$, and find eigenvalues of the resulting linear operator $L_{u}$. I have a dilemma as to what boundary conditions should I impose on the eigenfunctions to be able to solve this problem numerically.

Are there any standard ways to do this ? Is there a way to justify same boundary conditions as $u$.

Suppose I have a nonlinear second order Cauchy PDE $\dfrac{\partial p(x,t)}{\partial t}=N(p(x,t))$, where $N:L^2(\mathbb{R})\rightarrow L^2(\mathbb{R})$, and a known fixed point $u(x)$. Mathematically, the only boundary condition known is that any solution $p$ decays to 0 as ${x\rightarrow\pm \infty}$.

From physical reasoning however, it is known that $u$ decays rapidly away from $x=0$, and hence, I am able to solve for $u$ numerically on a finite domain by prescribing zero Dirichlet boundary conditions on a finite domain around 0 (i.e, on $[-L,L]$, where $L$ is chosen judiciously).

Now, I want to linearize $N$ around $u$, and find eigenvalues of the resulting linear operator $L_{u}$. I have a dilemma as to what boundary conditions should I impose on the eigenfunctions to be able to solve this problem numerically on a finite domain.

Are there any standard ways to do this ? Is there a way to justify same boundary conditions as $u$.