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I've heard a couple of times that in the Biharmonic Nonlinear Schrodinger Equation,

$i\psi_z + \Delta ^2 \psi + |\psi | ^{2\sigma } \psi =0 $, $\psi (x, 0) = \psi _0 (x) \in H^2( \mathbb{R} ^d ) $

adding a phase term to initial solution makes for a blowup in finite time, e.g. :

$\psi _0 = R^ {(0)} _B (r) $, the solitonic solution, exsists globaly, but it is said that adding a phase term such as $\psi _0 = R^ {(0)} _B (r) e^ {i r^2} $, would suffice for a blowup at some $0 < Z_c < \infty $.

There is some motivation for this in the "Lense Transformation" of the Nonlinear Schrodinger (NLS), but I wasn't able to find any proof, numerics or further intuition for why it should work in the BNLS.

Any thoughts?

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  • $\begingroup$ What about the normal conditions (generally on the sign of the Hamiltonian, plus some constraints on the IC) that one has for the NLSE? This might give some insight. See Ch. 5 of Sulem and Sulem for details. $\endgroup$
    – Nick P
    Commented Sep 11, 2014 at 8:00
  • $\begingroup$ What's the name of the Sulem and Sulem book? I'll look into it $\endgroup$
    – Amir Sagiv
    Commented Sep 27, 2014 at 8:25
  • $\begingroup$ Sulem, C., & Sulem, P. L. (Eds.). (1999). The nonlinear Schrödinger equation: self-focusing and wave collapse (Vol. 139). Springer. $\endgroup$
    – Nick P
    Commented Sep 27, 2014 at 18:06
  • $\begingroup$ I took a look at it. In the 1 dimensional case, it might change the hamiltonian sign and thus admit a singular solution. Obviously, there is no proof of singularity for BNLS, but the direction of Hamiltonian sign doesn't look promising either, since BNLS Hamiltonian is always non-positive. @NickP $\endgroup$
    – Amir Sagiv
    Commented Nov 10, 2014 at 11:40
  • $\begingroup$ the other thing to look for then would be the pseudo conformal invariance of the associated action, corresponding to the so called lens transformation. i don't know much about the BNLS, but some googling yielded this paper which might be of assistance: math.tau.ac.il/~fibich/Manuscripts/BNLS_SIAP10.pdf $\endgroup$
    – Nick P
    Commented Nov 11, 2014 at 0:20

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