Physica D 238 (2009) 1524–1549 Contents lists available at ScienceDirect Physica D journal homepage: www.elsevier.com/locate/physd Breakdown of weak-turbulence and nonlinear wave condensation Gustavo Düring a , Antonio Picozzi b , Sergio Rica a,c,∗ a Laboratoire de Physique Statistique, Ecole Normale Supérieure, UPMC Paris 06, Université Paris Diderot, CNRS, 24 rue Lhomond, 75005 Paris, France b Institut Carnot de Bourgogne, CNRS-UMR 5209, Université de Bourgogne, Dijon, France c Facultad de Ingeniería y Ciencias, Universidad Adolfo Ibáñez, Avda. Diagonal las Torres 2640, Peñalolén, Santiago, Chile article info Article history: Received 1 July 2008 Received in revised form 18 December 2008 Accepted 13 April 2009 Available online 3 May 2009 Communicated by A.C. Newell Keywords: Weak-turbulence theory Nonlinear Schrödinger equation Wave condensation abstract The formation of a large-scale coherent structure (a condensate) as a result of the long time evolution of the initial value problem of a classical partial differential nonlinear wave equation is considered. We consider the nonintegrable and unforced defocusing NonLinear Schrödinger (NLS) equation as a representative model. In spite of the formal reversibility of the NLS equation, the nonlinear wave exhibits an irreversible evolution towards a thermodynamic equilibrium state. The equilibrium state is characterized by a homogeneous solution (condensate), with small-scale fluctuations superposed (uncondensed particles), which store the information necessary for ‘‘time reversal’’. We analyze the evolution of the cumulants of the random wave as originally formulated by D.J. Benney and P.G. Saffman [D.J. Benney, P.G. Saffman, Proc. Roy. Soc. London A 289 (1966) 301] and A.C. Newell [A.C. Newell, Rev. Geophys. 6 (1968) 1]. This allows us to provide a self-consistent weak-turbulence theory of the condensation process, in which the nonequilibrium formation of the condensate is a natural consequence of the spontaneous regeneration of a non-vanishing first-order cumulant in the hierarchy of the cumulants’ equations. More precisely, we show that in the presence of a small condensate amplitude, all relevant statistical information is contained in the off-diagonal second order cumulant, as described by the usual weak-turbulence theory. Conversely, in the presence of a high-amplitude condensate, the diagonal second-order cumulants no longer vanish in the long time limit, which signals a breakdown of the weak-turbulence theory. However, we show that an asymptotic closure of the hierarchy of the cumulants’ equations is still possible provided one considers the Bogoliubov’s basis rather than the standard Fourier’s (free particle) basis. The nonequilibrium dynamics turns out to be governed by the Bogoliubov’s off-diagonal second order cumulant, while the corresponding diagonal cumulants, as well as the higher order cumulants, are shown to vanish asymptotically. The numerical discretization of the NLS equation implicitly introduces an ultraviolet frequency cut-off. The simulations are in quantitative agreement with the weak turbulence theory without adjustable parameters, despite the fact that the theory is expected to breakdown nearby the transition to condensation. The fraction of condensed particles vs energy is characterized by two distinct regimes: For small energies (H ≪ H c ) the Bogoliubov’s regime is established, whereas for H  H c the small-amplitude condensate regime is described by the weak-turbulence theory. In both regimes we derive coupled kinetic equations that describe the coupled evolution of the condensate amplitude and the incoherent field component. The influence of finite size effects and of the dimensionality of the system are also considered. It is shown that, beyond the thermodynamic limit, wave condensation is reestablished in two spatial dimensions, in complete analogy with uniform and ideal 2D Bose gases. © 2009 Elsevier B.V. All rights reserved. 1. Introduction In the present work we study the statistical initial value problem for the nonintegrable Hamiltonian defocusing NonLinear Schrödinger (NLS) equation or Gross–Pitaevskii equation in space dimensions higher or equal than 2. The random nonlinear wave is known to exhibit a thermalization process characterized by an irreversible evolution of the field towards an equilibrium state. In the defocusing regime of ∗ Corresponding author at: Laboratoire de Physique Statistique, Ecole Normale Supérieure, UPMC Paris 06, Université Paris Diderot, CNRS, 24 rue Lhomond, 75005 Paris, France. Tel.: +33 144323472; fax: +33 144323472. E-mail address: rica@lps.ens.fr (S. Rica). 0167-2789/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physd.2009.04.014