Nonlinear waves
Nonlinear waves¶
So far in this section we have focused on small-amplitude waves which can be well-described by linea partial differential equations in space and time. In thesese cases there is a fairly systematic method for solving the equations. We find a set of soluions which at fixed time provide a complete set of functions of space (such as sines and/or cosines) in which any initial condition can represented as a linear combination of these functions. The time evolution then follows by summing the time-evolved components. This is he essence of solutions to both Maxwell’s and Schrödinger’s equations as well as to the diffusion equation. The result is waves which may or may not disperse, with modes of different wavelength having distinct phase velocities.
When we pass to nonlinear equations, there is generally no such systematic method for solving them, except perhaps in the case that the nonlinearity is weak. Qualitatively different phenomena can emerge, such as shocks, singularities, and turbulence, in which through nonlinear effects, energy can be passed to shorter and shorter distance scales (or in the case of 2d turbulence, longer and longer scales).
To get a hint of how this might happen, let us consider he following nonlinear ODE:
where we take \(\epsilon = g/\Omega^2 \ll 1\). We can then expand
where \({\ddot x_0} + \Omega^2 x_0 = 0\). This is solved by \(x_0 = X \cos\Omega t\) (we have shifted an extra phase away by redefining the origin of \(t\).) Now at order \(\epsilon\) we have:
We will solve this with Green’s function techniques. That is, if we find \(G\) such that \(\partial_t^2 G(t,t') + \Omega^2 G(t,t') = \delta(t - t')\), then we merely need write
You can convince yourself that
satisfies this equation, and because of the Heaviside step function ensures that response follows stimulus. Returbing to the problem above, we find
so you can see that higher frequencies get excited at this leading nonlinear order.