Abstract
As discussed in Chapter 6, numerical simulations of the cubic NLS equation in two dimensions show that at critical dimension, the function a(t) defined in (6.1.3) does not converge to a finite limit as the singularity is approached, but rather continues to decrease slowly. This reflects the presence of a correction to the self-similar rate of blowup. Consequently, the rescaled solution u of (6.1.2) does not tend uniformly to the self-similar profile given up to a simple rescaling, by \( u\left( {\xi ,\tau } \right) = e^{i\tau } Q\left( {\xi ,a} \right) \) with a nonzero constant value of a.
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© 1999 Springer-Verlag New York, Inc.
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(1999). Critical Collapse. In: Sulem, C., Sulem, PL. (eds) The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse. Applied Mathematical Sciences, vol 139. Springer, New York, NY. https://doi.org/10.1007/978-0-387-22768-9_8
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DOI: https://doi.org/10.1007/978-0-387-22768-9_8
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-98611-1
Online ISBN: 978-0-387-22768-9
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