Quick Integral Stuck, context: bounded Energy, solitons

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Homework Statement



Integrating ##^{\frac{\pi}{2}}\int_{\frac{-\pi}{2}}(1-u^{2})^\frac{1}{2}u_{x}dx##, and using the result : ##\int(1-u^{2})^{\frac{1}{2}}=\frac{1}{2}u(1-u^{2})^{\frac{1}{2}}+\frac{1}{2}arcsin(u)##

Homework Equations



I'm pretty sure it is just the integral itself were I am going to wrong. But to provide context, just in case, it is a bounding the energy question using the Bogomolnyi arguement from a soliton course. Here's the two lines, I am just stuck on how we get from the top line to the last, so the 2nd term in both lines.

##E \geq \frac{1}{2}(u_{x}\pm(1-u^{2})^{\frac{1}{2}})^{2}dx \mp (1-u^{2})^\frac{1}{2} u_{x} dx ##
## = \frac{1}{2}(u_{x}\pm(1-u^{2})^{\frac{1}{2}})^{2}dx \mp [\frac{1}{2}u(1-u^{2})^{\frac{1}{2}}+\frac{1}{2}arcsin(u)]^{\frac{\pi}{2}}_{\frac{-\pi}{2}} ##


The Attempt at a Solution



Using ##dx=dx/du * du ## I see how we can replace ##u_{x} dx## with ##du## and so this allows us to use the result provided, BUT surely then the limits too need changing - to ##\pm 1 ##????

Thanks in advance.
 
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Answers and Replies

  • #2
haruspex
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It would help to know what x and u refer to physically here.
In general, bounds on an integral should specify the variable. They don't have to be values of the variable of integration, but if the variable is omitted then it would be usual that the variable of integration is the implied one. If the author intended that the bounds still refer to x after the substitution then that should be made clear. So it might not be wrong, just sloppy. It depends what happens later.
 
  • #3
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It would help to know what x and u refer to physically here.
In general, bounds on an integral should specify the variable. They don't have to be values of the variable of integration, but if the variable is omitted then it would be usual that the variable of integration is the implied one. If the author intended that the bounds still refer to x after the substitution then that should be made clear. So it might not be wrong, just sloppy. It depends what happens later.
u is the height of the soliton wave. u=u(x,t).
 
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haruspex
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u is the height of the soliton wave. u=u(x,t).
OK, so clearly the limits shown refer to x values. What is the next step in the text? Are the right bounds used for u?
 
  • #5
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OK, so clearly the limits shown refer to x values. What is the next step in the text? Are the right bounds used for u?
The next step is getting the answer to be 1, for this part of the integral I am talking about. Whereas I'm thinkin it's arcsin(pi/2) - arcsin(-pi/2) not arcsin(1)-arcsin(-1)...(all divided by 2). cheers.
 
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haruspex
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The next step is getting the answer to be 1, for this part of the integral I am talking about. Whereas I'm thinkin it's arcsin(pi/2) - arcsin(-pi/2) not arcsin(1)-arcsin(-1)...(all divided by 2). cheers.
Is u a known function of x? I assumed not. So the correct expression should be ##\arcsin(u(\frac{\pi}{2},t)) - \arcsin(etc.)##, no? But without knowing how u at those points we can go no further.
 
  • #7
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Is u a known function of x? I assumed not. So the correct expression should be ##\arcsin(u(\frac{\pi}{2},t)) - \arcsin(etc.)##, no? But without knowing how u at those points we can go no further.
Yes, that's what I thought, so you get arcsin(1)- arcsin(-1) not arcsin(pi/2)-arcsin(-pi/2) which gives the right answer.
 
  • #8
haruspex
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Yes, that's what I thought, so you get arcsin(1)- arcsin(-1) not arcsin(pi/2)-arcsin(-pi/2) which gives the right answer.
I'm not sure which you are saying is the right answer, and I'm not saying it gives either of those two answers.
To me,it gives ##\arcsin(u(\frac{\pi}2),t))## etc. If you know by some means that ##u(\frac{\pi}2,t)## = 1 then you get arcsin(1), i.e. pi/2.

arcsin(pi/2) looks most unlikely.
 

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