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).

  • $\begingroup$ If you multiply the equation by $\dot x(t)$, you can integrate once and get a first order differential equation for $x(t)$ with one arbitrary constant in the equation. Did you try this? What do you get? (I am too lazy to try it myself, sorry.) $\endgroup$ – Harald Hanche-Olsen Apr 28 '14 at 19:20

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.

  • $\begingroup$ Thank you for your answer! Would you care to comment about the other, approximate solution? $\endgroup$ – PPR Apr 28 '14 at 19:45
  • 2
    $\begingroup$ @Psycho_pr I don't know how to qualitatively justify that the approximate solutions are ``good enough'', but their physical meaning is rather clear. The $\tanh$-solution is interpreted as a particle (a kind of domain wall) travelling along $x$. This is one of the classical solutions of the $\varphi^4$-theory. Now one would like to have solutions representing any number of such particles, to study, say, their scattering. If one considers sin-Gordon instead of $\varphi^4$, such solutions (of the corresponding PDE) can be found explicitly using a more complicated ansatz than a stationary wave. $\endgroup$ – Start wearing purple Apr 28 '14 at 19:58

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