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I am trying to understand the resolution of the following KdV equation. I try to demonstrate it by myself.

The solitary wave solution is :

At first, I created new variable as follows so I could transform the PDE in to an ODE.

A = A(p)

p = g(x,t)

g(x,t) = x - ct

I succeeded to transform the PDE to ODE by the chain rule. My problem is when I arrive at that integral :

I read a lot of article and I have found that that integral needs to be solve by hyperbolic trigonometric substitution :

I have found that this is the substitution, but I have found anywhere why it needs that specific one. It might be because I just don't see it as the last time I hade to integrate by trigonometric substitution is a few years ago.

Is there someone on PF that knows why it needs that specific substitution?

Thank you!!!

(I apologize for my bad english as the language I use everyday is french)

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# Hyperbolic substitution in KdV equation resolution

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