Given the nonlinear PDE $$ \partial^2\phi+\phi^3=0 $$ I consider the characteristics $\xi=\kappa\cdot x$. Then I look for a solution in the form $$ \phi(x)=a\cdot\chi(\xi). $$ Provided $\kappa^2=\frac{a}{2}$, I get the ODE defining of the Jacobi ${\rm sn}$ function and the final solution takes the form $$ \phi(x)=a\cdot{\rm sn}(a\xi/\sqrt{2}+b,-1). $$ This solution has two arbitrary integration constants $a$ and $b$ that yield a family of solutions of the PDE we started from.
My question is this: How general could be considered such a solution for a nonlinear PDE? Is it a fundamental set from which all others could be obtained?
That's not quite right. The general solution to the ODE $y'' + y^3 = 0$ is $y = c \; \text{sn}(c(t - t_0)/\sqrt{2}, i)$.
Since this is a nonlinear equation, you can't take these as a "fundamental set": other solutions will not be linear combinations of these.
EDIT: For example, there will be radially symmetric solutions $y = f(r)$ which (in 3D) are solutions of the ODE
$$ f''(r) + \dfrac{2}{r} f'(r) + f(r)^3 = 0 $$
but I don't think this has closed-form solutions (other than $0$).