Conservative numerical method for the solution of the nonlinear Schrödinger equation

Question

I want to to numerically solve the equation

\(i\partial_tU=-(\partial_x^2+\partial_y^2)U-n_2IU+n_mI^mU\)

for the field \(U(x,y,t)\), where \(I=UU^*\) and \(n_2,n_m\) are constants, m>4.

There are two integrals of motion for this equation:

\(E=\int{I}dxdy\)

\(H=\int{(\partial_\mu U \partial^\mu U-n_2I^2/2+n_mI^m/m)}dxdy\)

But the standard numerical scheme of splitting into physical factors (used for example in Spin-Glass Model Governs Laser Multiple Filamentation by Wahb Ettoumi, Jérôme Kasparian, Jean-Pierre Wolf) actually does not preserve the second integral of motion.

Where can I find in the literature a discussion of conservation properties of numerical schemes for the nonlinear Schrödinger equation? Has someone already solved the problem of preserving both conserved integrals of motion?

Comments

So what is your question? Also, the arXiv reference seems to be wrong.

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I am interested numerical conservative and economical schemes by means of which I can solve this equation.

For an example I have given one of many articles:

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Spin-Glass Model Governs Laser Multiple Filamentation

Wahb Ettoumi, Jérôme Kasparian, Jean-Pierre Wolf

arXiv:1505.07041 [cond-mat.stat-mech]   (or arXiv:1505.07041v1 [cond-mat.stat-mech] for this version)

The vast majority of works use this scheme (splittings on physical factors) the numerical solution of the nonlinear Schrödinger equation.   

    I use this scheme long ago.   

    But recently decided to verify conservation laws and found out that everything is bad

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There is still no question. You just told us about your interests.

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Where in the literature can be found opianie conservative numerical schemes for SNLE?

Can someone has already solved this problem?@ak