arXiv subjects
Publications and source records attributed to A. H. M. Kierkels.
The results from J. Stat. Phys. 159:668-712 & 163:1350-1393, on a quadratic kinetic equation in the analysis of the long time asymptotics of weak turbulence theory for the nonlinear Schrödinger equation, are summarized and placed in context. Additionally, two conjectures on self-similar solutions are presented, and backed with consistency analysis and numerics.
We construct a family of self-similar solutions with fat tails to a quadratic kinetic equation. This equation describes the long time behaviour of weak solutions with finite mass to the weak turbulence equation associated to the nonlinear Schrödinger equation. The solutions that we construct have finite mass, but infinite energy. In J. Stat. Phys. 159:668-712, self-similar solutions with finite mass and energy were constructed. Here we prove upper and lower exponential bounds on the tails of these solutions.
We study the mathematical properties of a kinetic equation which describes the long time behaviour of solutions to the weak turbulence equation associated to the cubic nonlinear Schrödinger equation. In particular, we give a precise definition of weak solutions and prove global existence of solutions for all initial data with finite mass. We also prove that any nontrivial initial datum yields the instantaneous onset of a condensate, i.e. a Dirac mass at the origin for any positive time. Furthermore we show that the only stationary solutions with finite total measure are Dirac masses at the origin. We finally construct solutions with finite energy, which is transferred to infinity in a self-similar manner.