| Abstract: |
| We consider the NLS equation with a cubic nonlinearity
posed on the half line. In the defocusing case, we prove that if
the Dirichlet data have a sufficient polynomial decay then the
Neumann data also have a proper decay so that the Fokas method is
applicable. Moreover, we prove that the solution in the $L^4$
space norm converges to $0$ as $t\rightarrow \infty$
establishing also the absence of solitons for this case. In the
focusing case, decay of the Neumann data is proven (for properly
decaying Dirichlet data), but only under the assumption of decay
of the solution as $t \rightarrow \infty$. A Crank-Nicolson finite
differences nonlinear scheme is introduced for the numerical
solution of the problem on the positive semi-axis with
experimental rates of convergence of order $2$. We present
numerical simulations for the soliton propagation with various
initial conditions.
The results are joint with Spyros Kamvissis. |
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