On a unit torus $T^n$ (or equivalently, on $\mathbb{R}^n$ with periodic boundary conditions), the linear Helmholtz equation:

$\nabla^2 \phi + k^2 \phi=0$

will have no non-trivial solutions for generic values of $k$, while for special values of $k$ it will have a finite-dimensional vector space of solutions, with a basis of functions:

$\phi_m(x)=\exp(2\pi i m_j x^j)$

for each $n$-tuple of integers $m=(m_1,m_2,...m_n)$ such that:

$m_1^2+m_2^2+...+m_n^2=\left(\frac{k}{2\pi}\right)^2$

Now, consider a non-linear version of this problem, such as:

$\nabla^2 \phi + k^2 \phi + \lambda \phi^3=0$

still on $T^n$. The solutions will no longer comprise a vector space, but rather a manifold.

My question is: how can I determine the cardinality of the dimension of that manifold? Will it again be zero-dimensional generically and finite for some special parameter values, or will the behaviour change?