Model equations for mode coupling saturation in unstable plasmas

Some simple one‐dimensional model equations for instability saturation by resonant mode coupling which are of the general form ${\mathrm{\partial u/\partial t;\hspace{0.17em}+\hspace{0.17em}\sigma }}_1\mathrm{\hspace{0.17em}u\hspace{0.17em}\partial u/\partial x\hspace{0.17em}+\hspace{0.17em}M[u]\hspace{0.17em}=\hspace{0.17em}O}$ , where $\text{M[u]\hspace{0.17em}=\hspace{0.17em}i/2π∫\hspace{0.17em}d\hspace{0.17em}k\hspace{0.17em}ω(k)∫\hspace{0.17em}d\hspace{0.17em}x\hspace{0.17em}′\hspace{0.17em}u(x\hspace{0.17em})\hspace{0.17em}}\exp \mathrm{\hspace{0.17em}[k(x\hspace{0.17em}-\hspace{0.17em}x\prime )]}$ are examined theoretically and numerically. For nondispersive $\mathrm{\omega (k)}$ , it is found the temporal evolution is characterized by the formation and merging of shocks. Also, the level of saturation can depend on the physical size of the unstable system. If disperisive effects are strong, the solution is characterized by the formation of solitons.