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Blow-up of solutions of nonlinear Schrödinger equations with oscillating nonlinearities

The author is supported by Izmir Institute of Technology's BAP Grant 2015IYTE43.
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  • The finite time blow-up of solutions for 1-D NLS with oscillating nonlinearities is shown in two domains: (1) the whole real line where the nonlinear source is acting in the interior of the domain and (2) the right half-line where the nonlinear source is placed at the boundary point. The distinctive feature of this work is that the initial energy is allowed to be non-negative and the momentum is allowed to be infinite in contrast to the previous literature on the blow-up of solutions with time dependent nonlinearities. The common finite momentum assumption is removed by using a compactly supported or rapidly decaying weight function in virial identities - an idea borrowed from [18]. At the end of the paper, a numerical example satisfying the theory is provided.

    Mathematics Subject Classification: Primary: 35B44, 35Q41, 35Q55; Secondary: 35B45.


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  • Figure 1.  The oscillating coefficient $A_\Omega$ over an extended interval

    Figure 2.  Mollifiers $l$ and $r$

    Figure 3.  Weight function $\varphi$

    Figure 4.  The graph of $m(x)$

    Figure 5.  The graph of $m'(x)$

    Figure 6.  The graph of initial datum $\text{Re}[u_0(x)]$ ($\rho = 0.1$) and $m(x)$

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