I am studying field theory and am struggling to solve this $\text{PDE}.$ I know the solution must be rather simple but I don't manage to find it.
$$\frac{\partial^2 \phi(x,t)}{\partial t^2} + \frac{\partial^2 \phi(x,t)}{\partial x^2} + \lambda \phi(x,t)\left[\phi^2(x,t) -v^2 \right]=0.$$
With $\lambda$ and $v$ constants and boundary conditions:
$\phi(x,t)\rightarrow -v$ for $x\rightarrow - \infty$
$\ $
$\phi(x,t)\rightarrow +v$ for $x\rightarrow +\infty$
Any help is welcomed!
I'm only getting started with a course in PDEs but I threw the equation into Mathematica and it found $$\phi(x,t)=\pm\frac{v\sqrt{2c_4}\operatorname{sn}\left(c_3+c_2t-\frac{x\sqrt{-v^2\lambda-{c_2}^2-c_4{c_2}^2}}{\sqrt{1+c_4}};c_4\right) }{\sqrt{1+c_4}}$$ With $\operatorname{sn}(x,k)$ being the Jacobi s-noidal function, one of the many Jacobi elliptic functions. It has many definitions, but perhaps one of the more relevant ones to this example is as the solution to the differential equation $$\frac{\mathrm{d}^2y}{\mathrm{d}x^2}=2k^2y^3-(1+k^2)y$$ Which sort of resembles the structure of the original PDE, with the $y^3$ term appearing.