# Klein-Gordon equation with position-dependent mass [closed]

Does there exist a general solution for a differential equation like: $$\ddot{\phi}(x,t) - \partial^2_x\phi(x,t) + \phi(x,t)m^2(x) = 0,$$ where $$m(x)$$ is a known function.

• Depends on the form of m(x) – 0x90 Oct 6 at 10:56
• It is smooth and has a compact support. I only have a numerical solution which states $m(x)$. – AbuuzArbuz Oct 6 at 11:07
• I think that under certain conditions on the function $m(x)$, a solution can be found by using Green's function. But to get a specific solution you should mention the function. – Manvendra Somvanshi Oct 6 at 11:38

The Poschl-Teller equation with $$m^2(x)= n(n+1) {\rm sech}^2 x$$, $$n\in {\mathbb Z}$$ is often found in soliton systems. It has closed-form solutions with bound states and scattering states.