### Abstract

Nonlinear wave systems integrable by Inverse Scattering Method (ISM) could demonstrate a complex behavior that demands the statistical description. The theory of this description composes a new chapter in the theory of wave turbulence - Turbulence in Integrable Systems. All systems integrable by ISM are separated in two classes: strongly and weakly integrable. Systems of both classes have infinite array of motion constants but only for strongly integrable systems this array is complete. As a result, the scattering is trivial in these systems. It means that all the collision terms in kinetic equations of arbitrary high order are identically zero. The examples of strongly integrable systems are: KdV, NLSE, and KP-2 equations. In strongly integrable systems one can choose as initial data a statistically homogenous random field with a given pair correlation function such that this function is invariant in time. The spatial spectrum of an equilibrium state can be chosen in an arbitrary way. In weakly integrable systems (KP-1, three-wave system, etc) the kinetic equations are nontrivial. They have infinite but incomplete set of motion constants. These kinetic equations have infinite amount of Rayley-Jeans-type stationary solutions, though their general stationary solutions are not explored yet.

Original language | English (US) |
---|---|

Pages (from-to) | 219-234 |

Number of pages | 16 |

Journal | Studies in Applied Mathematics |

Volume | 122 |

Issue number | 3 |

DOIs | |

State | Published - Apr 2009 |

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### ASJC Scopus subject areas

- Applied Mathematics

### Cite this

**Turbulence in integrable systems.** / Zakharov, Vladimir E.

Research output: Contribution to journal › Article

*Studies in Applied Mathematics*, vol. 122, no. 3, pp. 219-234. https://doi.org/10.1111/j.1467-9590.2009.00430.x

}

TY - JOUR

T1 - Turbulence in integrable systems

AU - Zakharov, Vladimir E

PY - 2009/4

Y1 - 2009/4

N2 - Nonlinear wave systems integrable by Inverse Scattering Method (ISM) could demonstrate a complex behavior that demands the statistical description. The theory of this description composes a new chapter in the theory of wave turbulence - Turbulence in Integrable Systems. All systems integrable by ISM are separated in two classes: strongly and weakly integrable. Systems of both classes have infinite array of motion constants but only for strongly integrable systems this array is complete. As a result, the scattering is trivial in these systems. It means that all the collision terms in kinetic equations of arbitrary high order are identically zero. The examples of strongly integrable systems are: KdV, NLSE, and KP-2 equations. In strongly integrable systems one can choose as initial data a statistically homogenous random field with a given pair correlation function such that this function is invariant in time. The spatial spectrum of an equilibrium state can be chosen in an arbitrary way. In weakly integrable systems (KP-1, three-wave system, etc) the kinetic equations are nontrivial. They have infinite but incomplete set of motion constants. These kinetic equations have infinite amount of Rayley-Jeans-type stationary solutions, though their general stationary solutions are not explored yet.

AB - Nonlinear wave systems integrable by Inverse Scattering Method (ISM) could demonstrate a complex behavior that demands the statistical description. The theory of this description composes a new chapter in the theory of wave turbulence - Turbulence in Integrable Systems. All systems integrable by ISM are separated in two classes: strongly and weakly integrable. Systems of both classes have infinite array of motion constants but only for strongly integrable systems this array is complete. As a result, the scattering is trivial in these systems. It means that all the collision terms in kinetic equations of arbitrary high order are identically zero. The examples of strongly integrable systems are: KdV, NLSE, and KP-2 equations. In strongly integrable systems one can choose as initial data a statistically homogenous random field with a given pair correlation function such that this function is invariant in time. The spatial spectrum of an equilibrium state can be chosen in an arbitrary way. In weakly integrable systems (KP-1, three-wave system, etc) the kinetic equations are nontrivial. They have infinite but incomplete set of motion constants. These kinetic equations have infinite amount of Rayley-Jeans-type stationary solutions, though their general stationary solutions are not explored yet.

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U2 - 10.1111/j.1467-9590.2009.00430.x

DO - 10.1111/j.1467-9590.2009.00430.x

M3 - Article

AN - SCOPUS:63849230580

VL - 122

SP - 219

EP - 234

JO - Studies in Applied Mathematics

JF - Studies in Applied Mathematics

SN - 0022-2526

IS - 3

ER -