TY - JOUR

T1 - One-dimensional wave turbulence

AU - Zakharov, Vladimir

AU - Dias, Frédéric

AU - Pushkarev, Andrei

N1 - Funding Information:
V. Zakharov and A. Pushkarev acknowledge support of the US Army, under the grant DACA 42-00-C-0044, of NSF, under the grant DMS 0072803, and of INTAS, under the grant 00-292. V. Zakharov and F. Dias acknowledge support of NATO, under the linkage grant EST.CLG.978941.

PY - 2004/8

Y1 - 2004/8

N2 - The problem of turbulence is one of the central problems in theoretical physics. While the theory of fully developed turbulence has been widely studied, the theory of wave turbulence has been less studied, partly because it developed later. Wave turbulence takes place in physical systems of nonlinear dispersive waves. In most applications nonlinearity is small and dispersive wave interactions are weak. The weak turbulence theory is a method for a statistical description of weakly nonlinear interacting waves with random phases. It is not surprising that the theory of weak wave turbulence began to develop in connection with some problems of plasma physics as well as of wind waves. The present review is restricted to one-dimensional wave turbulence, essentially because finer computational grids can be used in numerical computations. Most of the review is devoted to wave turbulence in various wave equations, and in particular in a simple one-dimensional model of wave turbulence introduced by Majda, McLaughlin and Tabak in 1997. All the considered equations are model equations, but consequences on physical systems such as ocean waves are discussed as well. The main conclusion is that the range in which the theory of pure weak turbulence is valid is narrow. In general, wave turbulence is not completely weak. Together with the weak turbulence component, it can include coherent structures, such as solitons, quasisolitons, collapses or broad collapses. As a result, weak and strong turbulence coexist. In situations where coherent structures cannot develop, weak turbulence dominates. Even though this is primarily a review paper, new results are presented as well, especially on self-organized criticality and on quasisolitonic turbulence.

AB - The problem of turbulence is one of the central problems in theoretical physics. While the theory of fully developed turbulence has been widely studied, the theory of wave turbulence has been less studied, partly because it developed later. Wave turbulence takes place in physical systems of nonlinear dispersive waves. In most applications nonlinearity is small and dispersive wave interactions are weak. The weak turbulence theory is a method for a statistical description of weakly nonlinear interacting waves with random phases. It is not surprising that the theory of weak wave turbulence began to develop in connection with some problems of plasma physics as well as of wind waves. The present review is restricted to one-dimensional wave turbulence, essentially because finer computational grids can be used in numerical computations. Most of the review is devoted to wave turbulence in various wave equations, and in particular in a simple one-dimensional model of wave turbulence introduced by Majda, McLaughlin and Tabak in 1997. All the considered equations are model equations, but consequences on physical systems such as ocean waves are discussed as well. The main conclusion is that the range in which the theory of pure weak turbulence is valid is narrow. In general, wave turbulence is not completely weak. Together with the weak turbulence component, it can include coherent structures, such as solitons, quasisolitons, collapses or broad collapses. As a result, weak and strong turbulence coexist. In situations where coherent structures cannot develop, weak turbulence dominates. Even though this is primarily a review paper, new results are presented as well, especially on self-organized criticality and on quasisolitonic turbulence.

KW - Dispersive waves

KW - Hamiltonian systems

KW - Quasisolitons

KW - Solitons

KW - Wave collapse

KW - Weak turbulence

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U2 - 10.1016/j.physrep.2004.04.002

DO - 10.1016/j.physrep.2004.04.002

M3 - Review article

AN - SCOPUS:3042652794

VL - 398

SP - 1

EP - 65

JO - Physics Reports

JF - Physics Reports

SN - 0370-1573

IS - 1

ER -