Let $\left\{a,\lambda\right\}\subset\mathbb{R}$.
Let the following differential equation for a function $x\left(t\right)\in\mathbb{R}^{\mathbb{R}}$ be given: $$ \boxed {\ddot{x}\left(t\right)=4\lambda\left(x\left(t\right)^{2}-a^{2}\right)x\left(t\right) } $$
I am trying to find all solutions to this equation which obey the following boundary condition: $x\left(-\infty\right)=-a$ and $x\left(\infty\right)=a$.
I have found one such family of solutions, indexed by $\tau\in\mathbb{R}$: $$\boxed{x\left(t\right)=a\,\tanh\left[\frac{\omega}{2}\left(t-\tau\right)\right]}$$ It is easy to verify this is indeed a solution.
My question is:
- Is it the only set of solutions for these boundary conditions? If yes, how to prove no others exist? If not, what are all the other solutions?
- I suspect that there is another set of (at least approximate) solutions: $$ \boxed{x\left(t\right) = a\prod_{j=1}^{n} \tanh\left[\frac{\omega}{2}\left(t-\tau_j\right)\right] }$$ where $n\in2\mathbb{N}+1$, and $\left\{\tau_j\right\}_{j=1}^n\subset\mathbb{R}$ are such that $\tau_j < \tau_{j+1} \forall j\in\left\{1,\dots,n-1\right\}$. If these are solutions, how do you prove that? (I tried induction and failed) If they are not solutions, in what way are they approximate solutions? (what is the margin of error?)
This problem comes from trying to find instanton solutions to the double well potential in quantum mechanics (see Coleman "Aspects of Symmetry" page 272).
These are indeed all solutions. Multiplying the equation by $2\dot{x}$, one easily finds a first integral (the energy): $$\dot{x}^2=2\lambda \left(x^2-a^2\right)^2+E,\tag{1}$$ and then boundary conditions imply $E=0$.
Now write $x=a\tanh u$, then (1) with $E=0$ gives the equation $\dot{u}^2=2\lambda a^2$, with the only solutions given by linear functions.