Numerical study of the logarithmic Schrodinger equation with repulsive harmonic potential
Résumé
We consider the Schrodinger equation with a
logarithmic nonlinearity and a repulsive harmonic potential. Depending
on the parameters of the equation, the solution may or may not be
dispersive. When dispersion occurs, it does with an exponential rate
in time. To control this, we change the unknown function through a
generalized lens transform. This approach neutralizes the possible
boundary effects, and could be used in the case of the Schrodinger
equation without potential. We then employ standard splitting methods on the new
equation via a nonuniform grid, after the logarithmic nonlinearity has been regularized. We
also discuss the case of a power nonlinearity and give some results concerning the error estimates of the first-order Lie-Trotter splitting method for both cases of nonlinearities. Finally extensive numerical experiments are reported to investigate the dynamics of the equations.
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