Optical Solitons - General understanding and Project ideas

[Hard Maths]

Homework Statement

I have written a working program that can obtain a numerical solution to the nonlinear Schrödinger equation, given at initial pulse, using the split-step Fourier method.

I have performed the first task of writing a working programme and have shown that if the initial pulse is sech(x) then there is no change over time and if i try 3sech(x) then the pulse changes but returns to the initial profile at t=pi/2.

Below is the next stage of the project,

The Attempt at a Solution

I have tried first and 3rd point and obtained results, but nothing i can really write a project on.

I have a pile of books on solitons, i have spent days researching on the internet and i still have no good ideas that will make for a decent project, i really need some help on this.

 

[Manman]

If this is asking too much, maybe someone can help me with some simpler more focussed questions.

For something to be a soliton solution, does it have to be sech shape and does it have have to repeat?

I know that 3sech(x) has a period of pi/2 but 1.5sech(x) has no period that i can see, so this must mean that not all sech(x) pulses can be soliton solutions, only ones with a certain amplitude, is this right?

(i am meant to be able to show how the gaussian pulse evolves into a fundamental solition)
I have found that the Gaussian pulse becomes a sech shape at a time of 1.5 but after this point, no periodicity occurs, nor does it remain a sech shape, can someone shed some light on this, is this a correct result?

 

[Manman]

Please can someone help me out, I'm really struggling!