I happen to know that the equation $$0=-\lambda \phi (t)^3+\mu ^2 \phi (t)+\phi ''(t)$$ has a simple solution: $$\phi(t) = \frac{\mu \tanh \left(\frac{\mu \left(t-t_0\right)}{\sqrt{2}}\right)}{\sqrt{\lambda }}$$ which is easily verified:

Block[
    {eqn, \[Phi]},
    \[Phi][t_] := \[Mu]/Sqrt[\[Lambda]] Tanh[(\[Mu] (t - t0))/Sqrt[2]];
    eqn = \[Phi]''[t] + \[Mu]^2 \[Phi][t] - \[Lambda] \[Phi][t]^3 == 0;
    FullSimplify[eqn, 
  Assumptions -> And[\[Mu] > 0, \[Lambda] > 0, t > t0]]
 ]
Out[137]=True

But Mathematica has a hard time providing a useful solution. It gives some very strange functions which I cannot reduce in any way to the known solution. In fact if you plot them they look like the given solution, rotated by 90 degrees. Is there a way to get it to simplify? Imagine that you did not know the solution above but had the intuition that the result could not be too complicated. How could you use Mathematica to find it? Here is the present solution using DSolve:

$$\left\{\left\{\phi (t)\to -\frac{2 i c_1 \sqrt{\lambda } \sqrt{-\frac{1}{\mu ^2-\sqrt{\mu ^4-2 c_1 \lambda }}} \text{sn}\left(\frac{\sqrt{\mu ^2 t^2+\sqrt{\mu ^4-2 \lambda c_1} t^2+2 \mu ^2 c_2 t+2 \sqrt{\mu ^4-2 \lambda c_1} c_2 t+\mu ^2 c_2{}^2+\sqrt{\mu ^4-2 \lambda c_1} c_2{}^2}}{\sqrt{2}}|\frac{\mu ^2-\sqrt{\mu ^4-2 \lambda c_1}}{\mu ^2+\sqrt{\mu ^4-2 \lambda c_1}}\right)}{\mu ^2+\sqrt{\mu ^4-2 c_1 \lambda }}\right\},\left\{\phi (t)\to \frac{2 i c_1 \sqrt{\lambda } \sqrt{-\frac{1}{\mu ^2-\sqrt{\mu ^4-2 c_1 \lambda }}} \text{sn}\left(\frac{\sqrt{\mu ^2 t^2+\sqrt{\mu ^4-2 \lambda c_1} t^2+2 \mu ^2 c_2 t+2 \sqrt{\mu ^4-2 \lambda c_1} c_2 t+\mu ^2 c_2{}^2+\sqrt{\mu ^4-2 \lambda c_1} c_2{}^2}}{\sqrt{2}}|\frac{\mu ^2-\sqrt{\mu ^4-2 \lambda c_1}}{\mu ^2+\sqrt{\mu ^4-2 \lambda c_1}}\right)}{\mu ^2+\sqrt{\mu ^4-2 c_1 \lambda }}\right\}\right\}$$

I happen to know that the equation $$0=-\lambda \phi (t)^3+\mu ^2 \phi (t)+\phi ''(t)$$ has a simple solution: $$\phi(t) = \frac{\mu \tanh \left(\frac{\mu \left(t-t_0\right)}{\sqrt{2}}\right)}{\sqrt{\lambda }}$$ which is easily verified:

Block[
    {eqn, \[Phi]},
    \[Phi][t_] := \[Mu]/Sqrt[\[Lambda]] Tanh[(\[Mu] (t - t0))/Sqrt[2]];
    eqn = \[Phi]''[t] + \[Mu]^2 \[Phi][t] - \[Lambda] \[Phi][t]^3 == 0;
    FullSimplify[eqn, 
  Assumptions -> And[\[Mu] > 0, \[Lambda] > 0, t > t0]]
 ]
Out[137]=True

But Mathematica has a hard time providing a useful solution. It gives some very strange functions which I cannot reduce in any way to the known solution. In fact if you plot them they look like the given solution, rotated by 90 degrees. Is there a way to get it to simplify? Do not suppose the form of the solution given above in your answer to the question. How could you use Mathematica to find it? Here is the present solution using DSolve:

$$\left\{\left\{\phi (t)\to -\frac{2 i c_1 \sqrt{\lambda } \sqrt{-\frac{1}{\mu ^2-\sqrt{\mu ^4-2 c_1 \lambda }}} \text{sn}\left(\frac{\sqrt{\mu ^2 t^2+\sqrt{\mu ^4-2 \lambda c_1} t^2+2 \mu ^2 c_2 t+2 \sqrt{\mu ^4-2 \lambda c_1} c_2 t+\mu ^2 c_2{}^2+\sqrt{\mu ^4-2 \lambda c_1} c_2{}^2}}{\sqrt{2}}|\frac{\mu ^2-\sqrt{\mu ^4-2 \lambda c_1}}{\mu ^2+\sqrt{\mu ^4-2 \lambda c_1}}\right)}{\mu ^2+\sqrt{\mu ^4-2 c_1 \lambda }}\right\},\left\{\phi (t)\to \frac{2 i c_1 \sqrt{\lambda } \sqrt{-\frac{1}{\mu ^2-\sqrt{\mu ^4-2 c_1 \lambda }}} \text{sn}\left(\frac{\sqrt{\mu ^2 t^2+\sqrt{\mu ^4-2 \lambda c_1} t^2+2 \mu ^2 c_2 t+2 \sqrt{\mu ^4-2 \lambda c_1} c_2 t+\mu ^2 c_2{}^2+\sqrt{\mu ^4-2 \lambda c_1} c_2{}^2}}{\sqrt{2}}|\frac{\mu ^2-\sqrt{\mu ^4-2 \lambda c_1}}{\mu ^2+\sqrt{\mu ^4-2 \lambda c_1}}\right)}{\mu ^2+\sqrt{\mu ^4-2 c_1 \lambda }}\right\}\right\}$$