Solitons - Static Kinks

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Thread starter 4 years ago
#1
Hi guys, I'm working through a sheet to help me learn about Solitons. I'm on section 3 now, which is about static kinks, and I can't really seem to start this exercise. Can someone give me a hint, show me what I'd have to do generally?

http://imgur.com/ywh8nQq

Thanks guys
0
4 years ago
#2
(Original post by 0range)
Hi guys, I'm working through a sheet to help me learn about Solitons. I'm on section 3 now, which is about static kinks, and I can't really seem to start this exercise. Can someone give me a hint, show me what I'd have to do generally?

http://imgur.com/ywh8nQq

Thanks guys
Isn't this just an application of the chain rule? Differentiating equation (11) w.r.t. x is the same as differentiating w.r.t. u and multiplying by du/dx. The square roots cancel leaving you with the V'(u) term.
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Thread starter 4 years ago
#3
(Original post by Gregorius)
Isn't this just an application of the chain rule? Differentiating equation (11) w.r.t. x is the same as differentiating w.r.t. u and multiplying by du/dx. The square roots cancel leaving you with the V'(u) term.

and then

So then you're left with

Would that be right? I think I've done something wrong
0
4 years ago
#4
(Original post by 0range)

and then

So then you're left with

Would that be right? I think I've done something wrong
That's right, you've got the "if" part. Now see if you can do the "only if" part. Take that equation

and write it as

Can you see where to go now?
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Thread starter 4 years ago
#5
(Original post by Gregorius)
That's right, you've got the "if" part. Now see if you can do the "only if" part. Take that equation

and write it as

Can you see where to go now?
Hmm I may need a tiny bit more help. What I was thinking was that I'd need to integrate from here somehow. Like move the

and integrate both sides so I'd have a

on the left after integration, but I'm not sure what the other side would be..
0
4 years ago
#6
(Original post by 0range)
Hmm I may need a tiny bit more help. What I was thinking was that I'd need to integrate from here somehow. Like move the

and integrate both sides so I'd have a

on the left after integration, but I'm not sure what the other side would be..
We've got to

Now ponder what you get from

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Thread starter 4 years ago
#7
(Original post by Gregorius)
We've got to

Now ponder what you get from

I really still can't see it I was thinking of multiplying both sides with d/du and then integrating both sides ?

Would the bit to ponder not just turn into 0??
0
4 years ago
#8
(Original post by 0range)
I really still can't see it I was thinking of multiplying both sides with d/du and then integrating both sides ?

Would the bit to ponder not just turn into 0??
That expression evaluates to

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Thread starter 4 years ago
#9
(Original post by Gregorius)
That expression evaluates to

Right I can see how you got that (I'm confusing myself a lot here)
To be able to do that, would you not have to differentiate the rhs as well? (so the v'(u))
and would that not be going in the wrong direction?
0
4 years ago
#10
(Original post by 0range)
Right I can see how you got that (I'm confusing myself a lot here)
To be able to do that, would you not have to differentiate the rhs as well? (so the v'(u))
and would that not be going in the wrong direction?
We've simply re-written the left hand side of one of our equations...So now we can say

is the same as the equation

which you can integrate straight away. The end result will be the required equation for
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