15 Wave-wave nonlinearities
15.1 Introduction
Wave non-linearity is a vast subject that is not particularly specific to plasma physics be-
cause similar non-nonlinear behavior can occur in any medium that supports wavepropa-
gation. However, because of the enormous variety of plasma waves, there is usually at least
one plasma context where any given type of wave nonlinearity is an important issue.
Wave nonlinearity occurs when non-linear terms become sufficiently largethat they
cannot be ignored. Three general types of non-linear wave behavior will be discussed:
mode-mode coupling instabilities, self-modulation, and solitons. Before discussing these
various behaviors in detail, a qualitative overview will first be presented showing how cer-
tain wave nonlinearities typically present themselves.
Mode-mode coupling instabilities
Suppose a linear wave is made to propagate in a plasma by inserting in the plasmaan
antenna connected to a sine wave generator tuned to an appropriate frequency.As a spe-
cific example, suppose the plasma frequency isωpe/ 2 π=100 MHz and the generator is
tuned toω/ 2 π=500 MHz so as to excite an electromagnetic plasma wave with dispersion
relationω^2 =ω^2 pe+k^2 c^2. The wave propagates through the plasma and is detected by
a distant receiving probe. Instead of connecting the receiving probe to a radioreceiver or
an oscilloscope, the receiving probe is connected to a spectrum analyzer, adevice which
provides a graphic display of signal amplitude versus frequency. The signal showsup on
this display as a sharp peak at 500 MHz as sketched in Fig.15.1(a). If the signal gener-
ator amplitude is increased or decreased, the spectrum analyzer peak goes up or down in
proportion, thereby indicating the signal strength.
An odd behavior is observed when the generator amplitude is increased above some
critical threshold. At generator amplitudes below this threshold, the spectrum analyzer
displays just the single peak at 500 MHz, but above this threshold, two additional peaks
abruptly appear at two different frequencies and these new peaks have amplitudes consider-
ably lower than the 500 MHz peak. These two additional peaks are typically at frequencies
which sum to 500 MHz and the frequencies of the two additional peaks are typically very
different from each other. For example, one of the two additional peaks might be at 1 MHz
and the other at 499 MHz. These two peaks are called daughter waves and their spectrum
is usually broader and their amplitude “springier” than the 500 MHz “pump” wave. If the