A one-dimensional model for dispersive wave turbulence
A family of one-dimensional nonlinear dispersive wave equations is introduced as a model for assessing the validity of weak turbulence theory for random waves in an unambiguous and transparent fashion. Special form of solution (sum of dispersive waves plus one unknown function) is assumed, and the family of equations is transformed into a system of four nonlinear ordinary differential equations. These equations can be integrated explicitly in terms of elliptic functions. Instead of the analytical solution, the authors present a numerical one for several values of parameters. The equation considered in Fourier space is generalized to the case when forcing and dissipation are present. For this case, the direct and inverse turbulent cascades are obtained and reduced to the Kolmogorov solution. The numerical simulation based on this model is a subject for detailed discussion. To bring the obtained results nearer to the theoretical ones, a critical review of the assumption of weak turbulence theory is made. The authors discuss the possibilities for improvement to the theory.
system of four nonlinear ordinary differential equations
scientific article; zbMATH DE number 994021