A.H.M. Kierkels On a kinetic equation in weak turbulence theory for the nonlinear Schrödinger equation 2016 http://arxiv.org/abs/1606.07290
2015
Arthur H. M. Kierkels and Juan J. L. Velázquez On the transfer of energy towards infinity in the theory of weak turbulence for the nonlinear Schrödinger equation J. Stat. Phys., 159(3): 668-712 2015 http://dx.doi.org/10.1007/s10955-015-1194-0
Abstract: 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.
Juan J. L. Velázquez and Arthur H. M. Kierkels On self-similar solutions to a kinetic equation arising in weak turbulence theory for the nonlinear Schrödinger equation 2015 http://arxiv.org/abs/1511.01292