Nonlinear waves in rotating fluids
In this presentation the non-trivial dynamics of nonlinear dispersive waves affected by the Coriolis force is discussed. Applications include surface and internal waves in the ocean, magnetic sound in plasma, and other phenomena. The corresponding model equation (rKdV equation) derived by the author has the form (υt + c0υx + αυυx + βυxxx)x = γυ ,where c0 is the linear long wave velocity, α and β are, respectively, the nonlinearity and dispersion parameters, and γ is proportional to the Coriolis frequency. This equation is not known to be integrable (except for the limits of γ = 0 and β = 0; its solutions are defined by interplay of “two dispersions:” the KdV-type (β) and rotation-type (γ). Some specific features of this model found in different times are:
1. For the periodic and localized solutions, the mass integral is zero.
2. There are no solitary waves on a constant background at all (“antisoliton theorem”).
3. In the long-wave case (β = 0) there exists a family of stationary periodic waves with a limiting wave consisting of parabolic pieces.
4. An initial KdV soliton attenuates due to radiation and disappears as a whole entity in a finite time (“terminal damping”).
5. A long-time asymptotics of this solution can be a wave packet corresponding to the nonlinear Schrödinger equation.
6. A soliton can exist on a long-wave background which compensates radiation losses.
7. Two solitons on such background reveal rather complex dynamics.
Depending on time limits, some or all these processes will be discussed. Also some data of laboratory experiments and oceanic modeling will be shown.