Your query is probably meant to perturbative approach through which people obtain KdV equations having exact soliton solutions.

The stretched transformations and complete perturbative approach is to look in to the physics in a simplified manner by separating out the various orders of field variables.

for example in dusty plasma ---

First order expansions contribute for linear physics, inclusion of next higher order will give weakly nonlinear system being govern by Kdv equations and even higher terms give the information regarding phase shift of propagating solutions.

Separation at different orders not only helps getting exact solutions of some nonlinear problems but also helps understanding the contribution of higher order non linearity in system.

Next query is probably how we decide those powers of epsilons in stretched variables as well as in expansion. It is not done by some written rule but comparing

terms we think are of same importance we can decide those orders.

Just for example-- take momentum equation and linearize it. Now compare any two terms with time derivative and space derivative will give an intuition to coose those orders of epsilon.

Hope this all may be of some use. You may refer to my PRE paper and its references.

Stretching the coordinates helps see in detail what happens on different scales of distance or time. For example, a boundary layer is usually thin, but a lot happens there. By stretching the coordinates in the vicinity of the boundary layer, one can see in detail what goes on there.

Different stretching "recipes" are required for different physical problems because of the different small domains, over which the interesting physics takes place.

Thanks Sir

@ Sanat Tiwari ...is the power of epsilon gives the order of perturbation? and also my question is that not only the different transformation have different power of epsilon but also there is difference that some transformation have mach number in some transformation lambda is used..

Indeed, there is no one recipe for coordinate stretching. It's an "art" that requires experinene...

Thanks sir

@ Sanat Tiwari..... While deriving the expression for solitons, we jump into the soliton pulse frame and we assume that it is the plasma which is moving towards the soliton pulse such that the soliton pulse, 'phi(x)' is constant in time in its own frame of reference. 

Is it the reason to get back in lab coordinate by the transformation (x-v.t), v being the phase velocity of the pulse?

The answers out here are very helpful for a nascent like me. I thank you all for your explanation. The illustrations being a little more comprehensive and incorporating some links of papers comprising stretch coords would indeed be a little more helpful for our juniors.