Ground states of the nonlinear Schrödinger equation on a fractured strip converge, as the strip narrows, to the ground state of the 1-D equation with a delta potential.
Periodic traveling waves for nonlinear Schr\"odinger equations with non-zero conditions at infinity in $ \R ^2 $
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abstract
We consider the nonlinear Schr\"odinger equation with nonzero conditions at infinity in $\R^2$. We investigate the existence of traveling waves that are periodic in the direction transverse to the direction of propagation and minimize the energy when the momentum is kept fixed. We show that for any given value of the momentum, there is a critical value of the period such that traveling waves with period smaller than the critical value are one-dimensional, and those with larger periods depend on two variables.
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Ground states on a fractured strip and one dimensional reduction
Ground states of the nonlinear Schrödinger equation on a fractured strip converge, as the strip narrows, to the ground state of the 1-D equation with a delta potential.