A Candle in the Dark

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Archive for November 2008

When solitons collide

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Adapted from notes I copied during a course I attended

Consider a 1+1 dimensional scalar field \phi(x,t) where x corresponds to the space dimension, and t corresponds to the time dimension. The Lagrangian and the Hamiltonian energy density for the field are given by,

\mathcal{L}=\frac{1}{2} \partial_{\mu}\phi \partial^{\mu}\phi - U(\phi)

\mathcal{H}=\frac{1}{2}(\partial_0 \phi)^2 + \sum_i\frac{1}{2}(\partial_i \phi)^2 + U(\phi)

Here, the Einstein summation convention is used, where repeated indices are summed over. \partial_{\mu} is used to denote the four-vector (\frac{1}{c}\frac{\partial}{\partial t}, \nabla)

The evolution of this scalar field is governed by the Euler-Lagrange equation,

\frac{\partial \mathcal{L}}{\partial \phi} = \partial_{\mu}\left(\frac{\partial \mathcal{L}}{\partial \left( \partial_{\mu}\phi \right)}\right)

Plugging in Lagrangian for the scalar field I’ve written above, the E-L equation reduces to,

\Box \phi = -\frac{\partial U(\phi)}{\partial \phi}

where, \Box is the D’Alembertian. ie, \Box \phi = \phi_{tt}  - \phi_{xx}

A specific example, where the solutions of the E-L equation have interesting properties, is given by the potential,

U(\phi)=\frac{\lambda}{2} \left(\phi^2-a^2\right)^2

The lowest energy solutions which satisfy the E-L equations are known as the classical vaccum. In this specific example, they are given by

\phi(x)=a
\phi(x)=-a

You can see that these solutions correspond to the minimum of the potential. Additionally, the total energy is also minimized by these solutions, because the contribution of the space and time derivatives to the Kinetic Energy density term is zero.

Apart from these vacuum solutions, there exist finite energy solutions of the E-L equations for this potential. From the criteria that the energy must be finite when the Hamiltonian energy density is integrated over x, such a solution must tend to the classical vacuua solutions at the extremities.

Such a solution is known as the kink solition, and is given by,

\phi(x)=a \tanh{\mu x}

where, \mu^2=\lambda a^2

You can check that this solution satisfies the Euler-Lagrange equation by plugging it in. Note that the solution is time-independent. Furthermore, when x \rightarrow \pm \infty, \phi \rightarrow \pm a. Therefore, the solution interpolates between the classical vacuua solutions.

To understand why it’s called a kink soliton, let’s plot its functional form, and the corresponding Energy density. For the plots below, I’ve set a=1, \lambda=1

Plot of the kink soliton solution

Plot of the kink soliton solution

Hamiltonian density for the kink-soliton solution

Energy density for the kink-soliton solution

As you can see, the solution tends to the vaccum solution (here, a=1) at either end and the energy density appears as a stationary wave packet which is like a kink in a rope.

Now, there are certain global symmetries that Lagrangian has, which are absent in the kink soliton solution. For instance, if we set x \rightarrow -x, the Lagrangian in Eq. (1), and consequently the corresponding action, is invariant. However, the solution \phi(-x) \rightarrow -\phi(x), is a new solution and interpolates the other way (ie, \phi \rightarrow \mp a as x \rightarrow \pm \infty. This is called the anti-kink solition, and has the same energy as the kink soliton.

There are other symmetries where the Lagrangian is invariant which give rise to new solutions. For example, a spatial translation will give a solution \phi_{new}=\phi(x-x_0), which shifts the location of the center of the kink. A lorentz transformation x \rightarrow \gamma(x-vt) will give a new time dependent solution, \phi_{new}=\phi\left(\gamma(x-vt)\right), where \gamma=1/\sqrt{1-v^2/c^2}. This corresponds to a kink which starts out at the origin and moves to the right with a constant velocity.

Therefore, the most general form of the kink soliton can be written as,

\phi_K(x,t) = a \tanh{\mu(x-x_0-vt)}

This corresponds to a soliton which is initially centered at x_0 and moves with a velocity v to the right. The anti-kink soliton corresponding to this is simply -\phi_K(x,t)

Now, consider a box which has both a kink and an anti-kink solition placed away from each other and with velocities in opposite directions. What happens to them as they approach each other? They can’t pass each other, otherwise they’d violate the boundary condition (remember that the kink and anti-kink solitons tend to different vacuum solutions at either end). So you’d expect them to bounce off each other.

In fact, we can even numerically test this, because we know that they are governed by the Euler-Lagrange equation. If \phi(x,t) represents both the kink and anti-kink soliton, then initially, one has

\phi(x,t) = \left\{\begin{array}{c} a\tanh{\mu(x+x_0-vt)} \quad x \leq 0 \\ -a\tanh{\mu(x-x_0+vt)} \quad x \geq 0 \end{array}\right\}

then, the E-L equation can be rewritten as two first order differential equations

\left(\begin{array}{c} \partial_t \phi(x,t) \\ \partial_t \pi(x,t) \end{array}\right) = \left(\begin{array}{c} \pi(x,t) \\ \partial_x^2\phi(x,t) - \frac{\partial U(\phi)}{\partial \phi}\end{array}\right)

where, \pi(x,t) = \partial_t \phi(x,t) is the momentum density

To solve the system of equations above, I discretized the spatial derivative using the central difference method, and used a fourth-order Runge Kutta solver with corresponding intial and boundary condition discussed above. You can see how the kink and anti-kink soliton bounce off each other in the video below.

Kinky, ain’t it?

Wikipedia has a gallery of images on solitons for the Sine-Gordon model, where a Sine potential is used instead of the quartic potential. There are also a lot of nice websites on the fantastic properties that such non-linear PDE’s exhibit.

See Also
Sine-Gordon Solitons and Soliton Collisions:
Pendulum Model

Soliton-Lab Art Gallery

Written by parseval

November 30, 2008 at 6:20 am

Posted in physics, science, videos

Quick thoughts on the Mumbai tragedy

with 3 comments

Terrible events in Bombay today. My heart goes out to the families & people caught in this. A live update of the situation from BBC can be found here.

I am incredibly angry with the completely irresponsible TV coverage from channels like NDTV, CNN-IBN and Times Now. You DO NOT show live feed about positions, type or number of combat troops. You don’t tell that the special forces are in floor X and are about to launch an attack at location Y. All the terrorists need is a contact sitting outside watching the telecast with a cell phone. You’d expect better of a Padma Shri award winning journalist like Barkha Dutt.

We definitely need a proper crisis management and response team (as Ratan Tata rightly indicated), which would handle the media, bystanders, etc during such an event.

EDIT:
I’ve lost a lot of respect towards Barkha Dutt. What a colossally insensitive idiot.

Also, helplines and a list of injured and dead

Written by parseval

November 27, 2008 at 3:11 am

Posted in events, media, terrorism

Poll

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I see that wordpress has a new poll feature. Let me test it out with a completely unscientific poll.

Written by parseval

November 14, 2008 at 7:19 pm

Posted in fun

Dignity in Defeat

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That was pretty impressive. If he’d conducted his campaign with such poise and dignity, instead of making unsubstantiated claims against his opponent and pandering to the crazy evangelical part of his party, I think the election might have been much closer.

Oh well, the election sure was fun to watch while it lasted. Now, we’ll find out if Obama can live up to the hype.

Written by parseval

November 5, 2008 at 3:42 am

Posted in politics

How can we stop such horrendous monstrosity?

with 7 comments

Rape victim, 13, stoned to death

A 13-year-old girl who said she had been raped has been stoned to death in Somalia after being accused of adultery by Islamic militants.

Dozens of men stoned Aisha Ibrahim Duhulow to death on October 27 in a stadium packed with 1,000 spectators in the southern port city of Kismayo, Amnesty International and Somali media reported, citing witnesses.

The Islamic militia in charge of Kismayo had accused her of adultery after she reported that three men had raped her, the rights group said.

I really don’t know. Some distant day in the future, I hope that through education and rationality such horrifying acts can be eliminated. But helplessly reading about the terrible injustice committed by raving lunatics is enormously depressing. She was just a child. A thousand spectators in a stadium 😦 It makes you lose hope in humanity.

Written by parseval

November 2, 2008 at 5:00 pm

Posted in events, religion

I should fire this physio

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This is from the (otherwise) greatest game in the world.

From CM 01/02

From CM 01/02

Written by parseval

November 1, 2008 at 3:44 am

Posted in fun, video games