# Appropriately handling boundary conditions in a PDE eigenvalue problem

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 $u$.

• Would the domain still be unbounded? – nicoguaro Oct 12 '17 at 3:18
• @nicoguaro mathematically yes it is still real line, however I need to solve on a finite domain numerically. – mystupid_acct Oct 12 '17 at 3:22
• That's not entirely true. You can choose a really big domain and impose Dirichlet boundary conditions. You could use an orthogonal basis such as Hermite polynomials. You could also use some kind of artificial boundary condition to impose the unbound character of your domain. – nicoguaro Oct 12 '17 at 3:25
• Well as I mentioned in the post, I am doing exactly as your first suggestion. My question is to find some justification for doing so, and any examples in literature where it is done for eigenvalue problem. – mystupid_acct Oct 12 '17 at 3:30
• This question is related to what you are looking for. – nicoguaro Oct 12 '17 at 3:50