On a kinetic equation arising in weak turbulence theory for the nonlinear Schrödinger equation

Abstract

We present a study of weak solutions to a kinetic equation of coagulation-fragmentation type. This quadratic equation (QWTE) is the leading order approximation for long times to the isotropic space-homogeneous weak turbulence equation for the nonlinear Schrödinger equation with defocussing cubic nonlinearity.<br /> We first recall the weak turbulence theory for that nonlinear Schrödinger equation, and we formally derive (QWTE). We then present the general theory of weak solutions to (QWTE), comprising among other things existence of solutions, conservation of mass and energy, and convergence to equilibria. A particularly interesting feature here is the instantaneous onset of a Dirac measure at zero for any nontrivial initial data.<br /> The better part of this work is concerned with solutions to (QWTE) that exhibit self-similar behaviour. Due to the two conservation laws it is necessary to introduce a modified notion of self-similarity for (QWTE). In that setting we prove existence of self-similar profiles with finite mass, and either finite or infinite energy. We further present several results on the qualitative behaviour of these profiles, and we pose two conjectures that are backed with consistency analysis and numerics.