I want to to numerically solve the equation
i∂tU=−(∂2x+∂2y)U−n2IU+nmImU
for the field U(x,y,t), where I=UU∗ and n2,nm are constants, m>4.
There are two integrals of motion for this equation:
E=∫Idxdy
H=∫(∂μU∂μU−n2I2/2+nmIm/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?