Annals of PDE

, 4:7 | Cite as

A Two-Soliton with Transient Turbulent Regime for the Cubic Half-Wave Equation on the Real Line

  • Patrick GérardEmail author
  • Enno Lenzmann
  • Oana Pocovnicu
  • Pierre Raphaël


We consider the focusing cubic half-wave equation on the real line
$$\begin{aligned} i \partial _t u + |D| u = |u|^2 u, \ \ \widehat{|D|u}(\xi )=|\xi |\hat{u}(\xi ), \ \ (t,x)\in {\mathbb {R}}_+\times {\mathbb {R}}. \end{aligned}$$
We construct an asymptotic global-in-time compact two-soliton solution with arbitrarily small \(L^2\)-norm which exhibits the following two regimes: (i) a transient turbulent regime characterized by a dramatic and explicit growth of its \(H^1\)-norm on a finite time interval, followed by (ii) a saturation regime in which the \(H^1\)-norm remains stationary large forever in time.


Multi-soliton modulation theory Wave turbulence Growth of Sobolev norms Half-wave equation Cubic Szegő equation 

Mathematics Subject Classification

35B40 35L05 35Q41 35Q51 37K40 



P.G. is supported by Grant ANAE of French ANR, and partially supported by the ERC-2014-CoG 646650 SingWave. E.L. is supported by the Swiss National Science Foundation (SNF) through Grant No. 200021-149233. O.P. was supported by the NSF grant under Agreement No. DMS-1128155 during the year 2013-2014 that she spent at the Institute for Advanced Study. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the NSF. P.R is supported by the ERC-2014-CoG 646650 SingWave and is a junior member of the Institut Universitaire de France. Part of this work was done while P.R was visiting the Mathematics Department at MIT, Boston, which he would like to thank for its kind hospitality. Another part was done while P.G., O.P., and P.R. were in residence at MSRI in Berkeley, California, during the Fall 2015 semester, and were supported by the NSF under Grant No. DMS-1440140.


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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques d’OrsayUniv. Paris-Sud, CNRS, Université Paris-SaclayOrsayFrance
  2. 2.Departement für Mathematik und InformatikUniversität BaselBaselSwitzerland
  3. 3.Department of MathematicsHeriot-Watt University and The Maxwell Institute for the Mathematical SciencesEdinburghUK
  4. 4.Laboratoire Jean-Alexandre DieudonnéUniversité Nice Sophia AntipolisNiceFrance

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