Nonlinear Schrodinger Equation Solving

[Ottimista]

Hi Guys,

I have to find equation and starting condition to solve nonlinear Schrodinger Equation with periodic edge condition.

This method should control the propagation of fiber optical signal. In details I need a case in which the energy conservation is discriminating.

Can you help me?

Edited December 1, 2015 by Chiara

[ajb]

The first thing I would do is see what methods have been already applied to similar situation in the literature and see if I can modify some of these.

Edited December 1, 2015 by ajb

[Ottimista]

Thanks for your reply.

 

Your suggestion is quite general & already known to me. To be more specific, would you be able to provide me with a definitive indication on how to resolve the "Nonlinear Schrodinger Equation" if I sent you " http://arxiv.org/pdf/1410.7009v2.pdf " where the methods are applied to the semilinear wave equation?

Otherwise would you be also able to find an example for the Nonlinear Schrodinger Equation, where the energy conservation is discriminating, based on section 7 (Numerical tests, equation (101) with the initial conditions (102) ) in the link I'm mentioning above?

Thanks very much.

[ajb]

To be more specific, would you be able to provide me with a definitive indication on how to resolve the "Nonlinear Schrodinger Equation" if I sent you " http://arxiv.org/pdf/1410.7009v2.pdf " where the methods are applied to the semilinear wave equation?

Such things are outside of my area of expertise. I am unlikely to be be able to offer much assistance. You maybe better off looking for forums dedicated to research level mathematics.

[Ottimista]

Can you indicate some forums dedicated to research level mathematics?

Thanks very much.

[ajb]

http://math.stackexchange.com/

 

http://mathoverflow.net/

 

The second one is more aimed at professional mathematicians.

[studiot]

Are you looking at envelope-soliton or perturbation solutions to the NLS?

 

googling the former even brings up MAtlab code

 

https://www.google.co.uk/search?hl=en-GB&source=hp&biw=&bih=&q=envelope-soliton+solution+of+NLS+equation&gbv=2&oq=envelope-soliton+solution+of+NLS+equation&gs_l=heirloom-hp.3...1281.21203.0.21547.41.16.0.25.2.0.187.2095.0j16.16.0....0...1ac.1.34.heirloom-hp..24.17.2048.vG4e-ILW9QU

[Ottimista]

Yes, I'm looking at "soliton-like solutions". I need to find a valid case as it was already found for the semilinear wave equation (see pages 26-27 in "http://arxiv.org/pdf/1410.7009v2.pdf "), where energy conservation is discriminating. If energy is conserved, the solution of the problem is correct, otherwise it's not.

[studiot]

I have just realised that your link allows access to the paper,

I do not normally have access to arxiv papers.

 

So I am still trying to understand what exactly you are looing for.

 

Have you looked in Drazin and Johnson

Solitons

Cambridge University Press?

 

Here are a couple of examples

 

 

 

As regards the Hamilton approach, have you looked in Sewell

 

Maximum and Minimum Principles

 

Both these books are from the Cambridge University series in advanced applied maths.

 

 

[imatfaal]

Just as an aside - cos the maths is way way over my head - Arvix papers are openly available to all

 

For instance

1. I went to - arxiv.org

2. Searched on a name - http://arxiv.org/find/all/1/all:+AND+bruce+AND+andrew+james/0/1/0/all/0/1

3. Scrolled past the three papers by collectives which amongst the hundreds of authors will contain almost every common forename

4. Clicked on the link to get the abstract - http://arxiv.org/abs/1507.05405

5. Clicked on the PDF in the box on the RHS to get the article in readable format - http://arxiv.org/pdf/1507.05405v1.pdf

 

And AJB - I don't understand any of your paper either but I am sure it is very impressive and good

[studiot]

Thanks, imatfaal, it's the internet stuff that's mostly way over my head.

 

If it's possible to b____r it up, netwise, I have found the way.

 

Three further references.

 

Firstly J E Marsden and J R Hughes

 

(Marsden seems to pop up a lot in higher level applied maths)

 

Mathematical Foundations of Elasticity

 

Has a good deal about non linear dynamics and you semi linear Schrondinger equation, in the chapter on functional analysis.

 

Avery Modern book

 

Material Inhomogeneities in Elasticity

 

Gerrard Maugin analysis various NLS equations quite deeply, including numerical emthods.

And boasts an extensive modern reference list.

 

Finally also look at

 

Philippe Boulanger and Michael Hayes

 

Bivectors and Waves in Mechanics and Optics.