Communications in Mathematical Physics

, Volume 369, Issue 2, pp 433–456 | Cite as

Solvable Cubic Resonant Systems

  • Anxo BiasiEmail author
  • Piotr Bizoń
  • Oleg Evnin


Weakly nonlinear analysis of resonant PDEs in recent literature has generated a number of resonant systems for slow evolution of the normal mode amplitudes that possess remarkable properties. Despite being infinite-dimensional Hamiltonian systems with cubic nonlinearities in the equations of motion, these resonant systems admit special analytic solutions, which furthermore display periodic perfect energy returns to the initial configurations. Here, we construct a very large class of resonant systems that shares these properties that have so far been seen in specific examples emerging from a few standard equations of mathematical physics (the Gross–Pitaevskii equation, nonlinear wave equations in Anti-de Sitter spacetime). Our analysis provides an additional conserved quantity for all of these systems, which has been previously known for the resonant system of the two-dimensional Gross–Pitaevskii equation, but not for any other cases.


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We thank Ben Craps, Javier Mas, and Alexandre Serantes for discussions. This research has been supported by FPA2014-52218-P from Ministerio de Economia y Competitividad, by Xunta de Galicia ED431C 2017/07, by European Regional Development Fund (FEDER), by Grant María de Maetzu Unit of Excellence MDM-2016-0692, by Polish National Science Centre Grant Number 2017/26/A/ST2/00530 and by CUniverse research promotion project by Chulalongkorn University (Grant CUAASC). A.B. thanks the Spanish program “ayudas para contratos predoctorales para la formación de doctores 2015” and its mobility program for his stay at Jagiellonian University, where part of this project was developed.


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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Departamento de Física de PartículasUniversidade de Santiago de Compostela and Instituto Galego de Física de Altas Enerxías (IGFAE)Santiago de CompostelaSpain
  2. 2.Institute of PhysicsJagiellonian UniversityKrakówPoland
  3. 3.Department of Physics, Faculty of ScienceChulalongkorn UniversityBangkokThailand
  4. 4.Theoretische NatuurkundeVrije Universiteit Brussel and The International Solvay InstitutesBrusselsBelgium

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