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S. Gilg, D.E. Pelinovsky and G. Schneider

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Validity of the NLS approximation for periodic quantum graphs

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Nonlinear Differential Equations and Applications 23, 63 (30 pages) (2016)

**Abstract:**

We consider a nonlinear Schrodinger (NLS) equation on a spatially extended
periodic quantum graph. With a multiple scaling expansion, an effective amplitude
equation can be derived in order to describe slow modulations in time and space
of an oscillating wave packet. Using Bloch wave analysis and Gronwall's inequality,
we estimate the distance between the macroscopic approximation which is obtained
via the amplitude equation and true solutions of the NLS equation on the periodic
quantum graph. Moreover, we prove an approximation result for the amplitude
equations which occur at the Dirac points of the system.

**Keywords**:

Nonlinear Schrodinger equation; periodic quantum graphs, justification of amplitude equations,
Bloch transform, Dirac points