Solution of Euler-Lagrange equation

[parton]

I have the following Lagrangian:

$$\mathcal{L} = 1/2 \partial_{\mu} \varphi \partial^{\mu} \varphi - 1/2 b ( \varphi^{2} - a^{2} )^{2}$$, where $$a,b \in \mathbb{R}_{>0}$$ and $$\varphi$$ is a real (scalar) field and x are spacetime-coordinates.

I calculated the Euler-Lagrange eq. and get: $$\square \varphi + 2 b ( \varphi^{2} - a^{2} ) \varphi = 0$$

My problem is now to find constant solutions and static ones like $$\varphi(x) = f(x-x_{0})$$ where $$x_{0}$$ is constant. But I don't know how to solve the differential equation above. Does anyone have an idea?

 

[AiRAVATA]

Constant solutions are easy to calculate (there are 3 of them), as $\square \varphi$ is zero if $\varphi$ is constant.

Now, for the other question, I'm assuming that for static solution you mean independent of time. If so, $\square \varphi = \nabla^2 f$, which leads to the stationary NLS ecuation with constant potential, and there is plenty of literature on how to solve it.

 

[matematikawan]

Constant solutions are easy to calculate (there are 3 of them), as $\square \varphi$ is zero if $\varphi$ is constant.

I recognise the solution if $\varphi$ is constant.

$\varphi = a$ , $\varphi = -a$ and $\varphi = 0$.

 

[parton]

Thank you for your replies!

Remark: I think it should be: $$\square \varphi = - \nabla^{2} f$$

But isn't there any other (easier) approach to solve this differential equation? I thought of something like finding a first integral or something like that, but I can't. Or maybe there is a (simple) physical explanation which could lead to a solution?

 

[AiRAVATA]

There should be a soliton-like solution arising from from energy conservation. For example, let $x \in \mathbb{R}$, then

$$-f'' + 2b(f^2-a^2)f=0.$$

Multiplying by f' and integrating,

$$-\frac{f'^2}{2} + b(\frac{f^2}{2}-a^2)f^2 = C,$$

where C is a constant of integration determined by the boundary conditions (i.e. $f,f'\rightarrow 0$ as $x \rightarrow \infty$). The remainding equation is separable and solvable (in terms of elliptic functions maybe?), and if you study the face plane of the second order equation, you can prove that there is a soliton-like solution (depending of the existence of homoclinic or heteroclinic orbits).

In more variables, you should be able to build a Hamiltonian and see if that approach gives you some additional info.

 

[matematikawan]

$$-\frac{f'^2}{2} + b(\frac{f^2}{2}-a^2)f^2 = C,$$

Yeap, it is really difficult to expressed f in term of x. Soliton?
wave that maintains its shape while it travels at constant speed. http://en.wikipedia.org/wiki/Soliton" [Broken] without equation!

But what are face plane? homoclinic or heteroclinic orbits ?

 

[AiRAVATA]

If C=0 is not that hard to solve, the equation is separable and i think that the change of variable f = sech² x will do the trick. Even if the constant is different than zero, you can solve the integral in terms of elliptic functions (I believe).
Regarding the phase plane or portrait (my native language betrayed me there), you can read all about it in a basic DE book like the one of Braun or Boyce, homoclinic and heteroclinic orbits can be seen in more advanced books, like the second volume of Hubbard's dynamical systems book.