We consider the global regularity problem for defocusing nonlinear Schrödinger systems
on Galilean spacetime , where the field is vector-valued, is a smooth potential which is positive, phase-rotation-invariant, and homogeneous of order outside of the unit ball for some exponent , and is a smooth, compactly supported forcing term. This generalises the scalar defocusing nonlinear Schrödinger (NLS) equation, in which and . It is well known that in the energy-subcritical and energy-critical cases when or and , one has global existence of smooth solutions from arbitrary smooth compactly supported initial data and forcing term , at least in low dimensions. In this paper we study the supercritical case where and . We show that in this case, there exists a smooth potential for some sufficiently large , positive and homogeneous of order outside of the unit ball, and a smooth compactly supported choice of initial data and forcing term for which the solution develops a finite time singularity. In fact the solution is locally discretely self-similar with respect to parabolic rescaling of spacetime. This demonstrates that one cannot hope to establish a global regularity result for the scalar defocusing NLS unless one uses some special property of that equation that is not shared by these defocusing nonlinear Schrödinger systems.
As in a previous paper of the author (Anal. PDE 9:8 (2016), 1999–2030) considering the analogous problem for the nonlinear wave equation, the basic strategy is to first select the mass, momentum, and energy densities of , then itself, and then finally design the potential in order to solve the required equation.
"Finite time blowup for a supercritical defocusing nonlinear Schrödinger system." Anal. PDE 11 (2) 383 - 438, 2018. https://doi.org/10.2140/apde.2018.11.383